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Pricing OTC energy derivatives: credit, debit, funding value adjustment, and wrong way risk

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Pricing OTC energy derivatives: credit, debit, funding value adjustment, and wrong way risk

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Content Pricing OTC Energy Derivatives Credit, Debit, Funding Value Adjustment, and Wrong Way Risk Naemeh N. Esfahani December 2015 A Dissertation Presented to the Faculty of the Graduate School UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirement for the Degree of DOCTOR OF PHILOSOPHY in Industrial and Systems Engineering Committee: __________________________________ Professor Sheldon M. Ross __________________________________ Professor Fernando Zapatero __________________________________ Professor Maged Dessouki 3 I would like to dedicate this thesis to Amin who supported me throughout this journey with his love and compassion. Com- pletion of this work would not have been possible without him. 4 Contents Abstract 1 1 Introduction 3 1.1 Risk Management and Pricing . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Credit Value Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Debit Value Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Derivative Fair Valuation under Default Risk . . . . . . . . . . . . . . . 13 1.5 Wrong and Right Way Risk . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.6 Funding Value Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.7 Discounting Under Default Risk . . . . . . . . . . . . . . . . . . . . . . . 17 1.8 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Credit Value Adjustment for Energy Derivatives 21 2.1 Modeling Exposure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Commodity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.1 Payoff and Valuation . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.2 Two-Factor Commodity Spot Model . . . . . . . . . . . . . . . . . 28 2.2.3 Standard Forward Swap Rate Model . . . . . . . . . . . . . . . . 29 2.2.4 Swap Rate Volatility . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3 Modeling Probability of Default . . . . . . . . . . . . . . . . . . . . . . . 33 2.3.1 Reduced Form Models . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3.2 Structural Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.4 Modeling Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.4.1 Alpha Multiplier . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.4.2 Copula Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4.3 Deterministic Hazard Rate Model . . . . . . . . . . . . . . . . . . 42 2.4.4 Stochastic Hazard Rate Model . . . . . . . . . . . . . . . . . . . . 43 2.5 Linking Survival Probabilities to Market Data . . . . . . . . . . . . . . . 44 2.5.1 Market Implied Survival Probabilities . . . . . . . . . . . . . . . . 45 i Contents Contents 2.5.2 Hazard Rate Curve . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.6 Swap CVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3 Discounting under Default Risk 51 3.1 Equivalent Market Measures and Change of Numeraire . . . . . . . . . . 53 3.2 Risky Zero Bond Equivalent Martingale Measure . . . . . . . . . . . . . 56 3.3 Double Curve Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.4 Modified Swap Valuation Under Risky Curve . . . . . . . . . . . . . . . . 63 4 Defaultable Energy Derivatives Valuation 67 4.1 Forward Contract Valuation . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2 Swap Contract Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.3 Risky Swap Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.4 Risky Swap as a Basket of Forwards . . . . . . . . . . . . . . . . . . . . . 79 5 PDE Representation of OTC and Exchange Traded Derivatives 83 5.1 Cost of Credit vs. Cost of Capital . . . . . . . . . . . . . . . . . . . . . . 85 5.1.1 Margining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.1.2 Collateralization . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.2 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.3 Cost of Capital in Collateralized Derivative Valuation . . . . . . . . . . 90 5.3.1 Derivative price is always positive . . . . . . . . . . . . . . . . . . 94 5.3.2 Derivative price is always negative . . . . . . . . . . . . . . . . . . 94 5.4 Cost of Credit in Uncollateralized Derivative Valuation . . . . . . . . . . 95 5.4.1 Derivative price is always positive . . . . . . . . . . . . . . . . . . 98 5.4.2 Derivative price is always negative . . . . . . . . . . . . . . . . . . 99 6 Numerical Results 101 6.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Appendix I 119 Acknowledgments 121 Bibliography 123 ii Abstract In this work we develop a framework for pricing bilateral contracts subject to both coun- terparty and dealer default risk, with the main focus on forward and swap contracts predominantly traded in energy markets. Unlike futures contracts that are traded on an Exchange and settled daily through clear-housing process, forward contracts can bear full risk of default by both parties. Market practice to account for associated credit risk has been to add adjustments to risk-neutral value of the contract, based on expected loss due to possibility of default. Calculation of expected loss relies on deriving incre- mental default probabilities based on assumptions on hazard rate models, and become computationally intensive. In our approach, pricing formula is based on valuation of future payoffs discounted under discount factors reflective of dealer and counterparty credit spreads. Default risk is essentially accounted for through discounting in our framework. Technical derivations of the valuation formulas in our methodology is based on change of numeraire and using a market measure suitable for the specific pricing problem under consideration. The new measure accounts for wrong/right way risk through correlation of payoff process with the probability measure. This methodology integrates Credit and Debit Value Adjustments into a single pricing formula, accounting for cost of credit in pricing equation, where credit risk is warehoused internally. In contrast with a bilateral setting that in extreme case of zero collateralization results 1 Contents Contents in bearing full counterparty default risk, exchanges have traditionally relied on full col- lateralization by all participants in order to virtually eliminate default risk. Margining mechanism assures that adequate amount of cash is deposited in margin accounts on a daily basis by all counterparties, equivalent to the value of the contract. Posting col- lateral in the form of cash requires participants to access funds available to them only at unsecured funding rate, which is higher than risk free rate, while the margin account accrues interest close to risk-free rate. This asymmetry results in cost of capital to the market participant, and has historically not been addressed in pricing of exchange traded derivatives that are traditionally assumed to be risk-free. In the second part of this work we derive partial differential equations for the two scenarios where cost of credit and cost of capital are considered in the pricing equation. The results show that under certain generic assumptions on funding rate, both over-the- counter and exchange trading result in identical pricing PDEs. Our main finding here is that margining is essentially a mechanism by which credit risk is transferred back to the counterparties, while the inherent nature of risk is preserved. In the final chapter of this work, we illustrate numerical results with case studies based on crude oil forward and swap contracts and examine the impact of counterparty and dealer credit spread and right/wrong way risk on contract value. 2 1 Introduction Black and Scholes (1973) and Merton (1973) European stock option model (BSM) and its extensions applied to different underlying asset classes and derivatives payoffs revo- lutionized modern finance theory. Among other assumptions, BSM relies on a market model where contractual obligations are met by market participants, and therefore pos- sibility of loss due to default or counterparty risk is not considered in pricing equation. Exchange Traded Derivatives (ETD) such as futures and futures options are essentially fully collateralized plain vanilla financial instruments that carry negligible counterparty risk. Traditionally exchanges have mitigated counterparty risk through daily margining and settlement of the contracts, which have been in place to assure that all members of the exchange post adequate collateral in variation margin accounts to offset their re- ported daily losses. In order to be able to efficiently complete a trading day process which includes daily valuation and settlement of derivatives, exchanges only allow trad- ing of highly standardized derivatives that are significantly limited in specs and scope. Limited choices of ETD, along with credit and margin requirements of the exchanges have resulted in the development of a secondary market where counterparties trade bi- laterally, Over-The-Counter (OTC). OTC derivatives demonstrate a far more complex variety both in terms of the underlying instrument and derivative’s payoff, and unlike ETD bear full counterparty credit risk in the absence of collateral agreement. The financial crisis of 2008 has shifted the focus of international regulators and fi- 3 Chapter 1 Introduction nancial institutions to more effective management of counterparty risk through stringent capital requirements for bilaterally traded contracts. The increase in regulatory capital requirements in the past few years has incented the market to instead recourse to trading through Central Counterparties (CCPs). CCPs act as an intermediary in bilateral con- tracts, and closely follow the exchange model in that they mitigate counterparty credit risk through daily margining operations. In addition the market is increasingly trading under the International Swaps and Derivatives Association (ISDA) master agreement that require bilateral collateralization mechanisms under their Credit Support Annex (CSA). Another development in financial markets has been to develop hedging programs that aims to transfer this type of risk through active trading of insurance contracts such as Credit Default Swaps (CDS). Consequently, the cost associated with OTC derivatives trading has increased substantially both due to capital requirements imposed by the reg- ulators and cashflow risk associated with trading with risky counterparties. These have been the key drivers towards accurate pricing of credit risk and efforts on transferring it back to either the counterparties or market participants better suited to carry credit risk on their books. Pricing OTC derivatives commonly consists of pricing the derivative assuming absence of default risk and subsequently adding/subtracting adjustment factors associated with certain types of risks that are not captured in the risk-free valuation. We refer to this pricing method as Adjustment Based Pricing throughout this work. Credit Value Ad- justment (CVA), Debit/Debt Value Adjustment (DVA) and more recently Funding Value Adjustment (FVA) among others are all terms that have attracted significant attention in the past few years among practitioners. While CVA is a well established concept and has been around for a considerable period of time, there are many controversial issues around pricing, hedging and fair value accounting of the rest of these adjustments, particularly in the case of DVA and its relation with FVA. 4 Introduction CVAistheexpectedlossonaportfolioofderivatives, arisingfromcounterpartydefault, and DVA is the expected gain that will be experienced by the bank in the event of own default 1 . FVA reflects the costs incurred by the bank to hedge or fund a derivatives portfolio 2 . FVAisdirectlyrelatedtothecostofborrowingofthebankandisfirmspecific. [28] summarizes an account of issues that arise with asset-liability asymmetry, double counting of DVA, and breakdown of theory of capital structure irrelevance by Modigliani- Miller, in addition to seemingly generating potential arbitrage opportunities when FVA is accounted for in the pricing equation. Funding cost and credit risk are two sides of the same coin in our view. If a derivatives contract is fully collateralized, counterparty credit risk can be theoretically eliminated, while the dealer will incur the cost of posting collateral. FullcollateralizationcanbeeitherdirectunderaCSAarrangement,orthrough an offsetting hedge on the exchange market where collateral posting is required. An example is hedging an OTC swap with Futures contracts that are cleared on an exchange. In this work we do not differentiate between the funding required for upfront costs and the payments made throughout time in the form of margin/collateral requirement. We assume that the bank incurs an average cost to access capital which then uses to fund financial activities both in the form of upfront payment or throughout the lifetime of the transaction. We emphasize that part of the disagreement lies in the distinction between two differ- ent methods for pricing; namely entry and exit pricing. Accounting regulations require pricing derivatives based on their “market value”. Market assumes the value of a port- folio if it were liquidated on the day (exit price). This approach does not account for the costs associated with entering the deal by the counterparts and assumes one market 1 Bank’s secured creditors will realize DVA gains after default. We will discuss this in further details later in this chapter. 2 Funding is required when an upfront payment is to be made for example in the form of an option premium. 5 Chapter 1 Introduction (risk-neutral) measure which prices all trades. Entry price, on the other hand, is the quoted price of a bank that provides enough incentive for the bank to enter the deal, in addition to paying for all the costs associated with maintaining the contract throughout its lifetime, inclusive of funding costs. Later, we will discuss the mechanics of realizing loss and gain in the event of default to shed some light on the root cause of disagreement by practitioners. 1.1 Risk Management and Pricing Riskmanagementhasdevelopedintoanessentialfunctionofcommercialoperationsofthe financial institutions, in the last few decades. Collapse of large financial institutions such as Long Term Capital Management and Enron, and more recently Lehman Brothers, all played a significant role in bringing risk management to spotlights. Quantitative approaches and probabilistic models are commonly adopted by financial institutions to account for various types of risks they face. Financial risk is commonly managed under three main umbrellas; market, credit and operational risk. Market risk is a type of risk that arises from movement of underlying asset prices in a portfolio of exposures. The exposures can be both linear and non-linear in terms of the underlying asset price 3 . It is also a common practice for market risk functions to be involved in measuring and managing position limits, liquidity risk and concentration risk. Operational risk is a domain that deals with losses due to system and human error, and process inefficiencies. This type of risk is rarely quantified and is managed heuristically. Credit risk is the third type of risk that arises due to losses resulting from the failure of a counterparty to fulfill contractual obligations. Counterparty exposure at default, counterparty default 3 These linear instruments are essentially what referred to as basic instruments, or linear portfolios thereof. On the other hand, non-linear instruments are contingent claims whose values depend on the price process of the basic securities. 6 1.1 Risk Management and Pricing probability, as well as recovery rate are the main factors that determine severity of losses in an event of default. Traditional ways of mitigating counterparty credit risk include efforts to limit future losses through netting and collateral agreements. In the event of default, netting agree- ment helps reduce total exposure, if there are offsetting contracts with the same counter- party. Total exposure in these cases is the sum of the positive and negative exposure, and can potentially be significantly smaller than the exposure generated by individual con- tracts. Collateralization can also reduce counterparty risk through reduction of effective exposure. When collateral agreement is in place, each sides of a bilateral contract can ask for collateral to be posted in the form of cash, cash equivalent, or assets, should exposure level surpass a pre-agreed level. This reduces effective exposure to a level initially agreed to by the counterparts. In economic theory, investors should demand compensation for the risks they expose their capital to, in the form of risk premium, and credit risk is no exception. Pricing default risk has been a common practice in industries such as insurance, and has found its way to the financial sector. A now common method of mitigating risk is to introduce a charge in the pricing of contracts to account for losses due to specific risks. The inclusion of counterparty risk in pricing of commodities derivatives traded in the OTC market has gained significant amount of attention in the past few years and is readily applicable to valuationofforwardvsfuturescontracts. Futurescontractsaretradedinanexchangeand are marked to market daily. Exchanges mitigate counterparty risk through clearhousing mechanism and margin accounts. Therefore, counterparty risk is considered negligible when trading Futures. On the other hand, forward and swap contracts are traded in a bi-lateral setting in the OTC market, and outside clearing houses. These contracts can span over very long periods of time, therefore the possibility of losses due to counterparty default can not be neglected in their valuation. 7 Chapter 1 Introduction In this work, we mainly focus on bi-lateral forward and swap contract valuation, while the discussion can be generalized to broader financial markets. In particular interest rate contracts such as forward rate agreements and Interest Rate swaps can be valued through similar methodologies. We propose moving away from adjustment based pricing and account for default risk through appropriate discounting of future payoffs. This methodology does not require model based derivation of piecewise default probabilities and relies solely on market observed discount factors, which can greatly simplify pricing problem. In the next section we first introduce the building blocks of adjustment based pricing including CVA, DVA, and FVA, and later discuss discount based approach as an alternative for default pricing. 1.2 Credit Value Adjustment Exchange traded derivatives are not exposed to counterparty risk, because the exchange guarantees the cashflows promised by the derivative to the counterparties. However, the contracts privately negotiated between counterparties are subject to counterparty risk. Counterparty risk is the risk that either sides in a derivatives transaction default prior to expiration of a trade and do not meet contractual obligations. For years, the standard practice in the industry was to mark derivatives portfolios to market without taking the counterparty credit quality into account. As outlined earlier, netting and collateral agreements were introduced as the next step to effectively reduce counterparty risk through reduction of exposure at every point in time in the future. However, all cashflows were still discounted by the LIBOR curve, representing risk-free rate, and the resulting values were often referred to as risk-free values. In recent years with the development of Credit Default Swap (CDS) market, dealers were able to hedge counterparty risk through purchase of default insurances in the form 8 1.2 Credit Value Adjustment of CDS. The largest most significant contribution of CDS market was that now dealers could focus on specific types of risk and exposure to certain market variables were their expertise lied, while offsetting the unwanted default risk to third parties best suited to warehouse this type of risk. The details regarding each of these approaches are discussed in great details in [21]. The true portfolio value must incorporate the possibility of losses due to counterparty default. The price of counterparty risk when charged to counterparties, will generate funds that can be used to absorb potential losses in an event of default. It can also be used to fund hedging costs should the company choose to offset credit risk through purchase of credit derivatives. CVA is by definition the difference between the risk-free portfolio value and the true portfolio value that takes into account the possibility of a counterparty’s default. In other words, CVA is the market price of counterparty credit risk. In practice, CVA is calculated as follows: Let us assume that a bank 4 has a portfolio of derivatives contracts with a counterparty. Credit exposure, counterparty exposure, or exposure, are all terms that refer to the dollar amount that would be lost if the counterpartyweretodefault. Let Π + (t)denotethebank’sexposuretothecounterpartyat any future timet. This exposure takes into account all netting and collateral agreements between the bank and the counterparty. If the counterparty defaults, the bank will be able to recover a fraction of exposure that is usually assumed to be constant. Denoting this fraction by recovery rate, R c , and the time of counterparty default by τ c , we can write the present value of future loss as 5 [45]: L (s) =1 {s≤τc≤s+δs} (1−R c )P (t,s) Π + (s) (1.1) 4 We frequently use the terms bank or dealer to refer to the valuing party. 5 In our notation, subscript c and d stand for counterparty and dealer respectively. 9 Chapter 1 Introduction where P (u,v) is the stochastic discount factor process at time u with maturity v, and 1 {} is the indicator function that takes value of one if the argument inside the bracket is true, and zero otherwise. TheunilateralCVAisgivenbytherisk-neutralexpectationofEquation 1.1,conditional on information set available up to current time. CVA =E Q t [L] = (1−R c ) T t E Q t h P (t,s) Π + (s)|τ c =s i dF c (s) (1.2) whereT isthematurityofthelongesttransactionintheportfolio, andwehaveusedE Q t [·] as a shorthand for the conditional expectation E Q [·|F t ] under risk neutral probability measureQ. F c (s) denotes the risk neutral probability of counterparty default in a small interval s. These probabilities can be obtained from the term structure of CDS spreads, or credit spread of fixed income instruments such as defaultable zero-coupon bonds. In chapter 2 we introduce reduced form models of default time and models forF c (s). When instantaneous default probability is assumed to be constant, Equation 1.2 can be shown to equal to the expectation of exposure up to a random time which has exponential distribution (see Appendix I). The two approaches for deriving default probabilities from market observed instru- ments, namely CDS spreads and fixed income spreads, generally result in different as- sessmentofdefaultprobabilities. Itisalsoimportanttonotethedifferencebetweenactual (historical) and market implied default probabilities. Historical default probabilities from actual defaults are used in actuarial type applications, whereas default probabilities im- plied by the market (CDS or bond market) should be used for derivatives pricing. This particularly becomes relevant when such instruments are used to transfer counterparty risk. We will discuss this topic in further details in the following chapters. It should be noted that in Equation 1.2, the expectation of discounted exposure at time 10 1.2 Credit Value Adjustment t is conditional on counterparty default occurring at time t. When there is a significant dependence between exposure and counterparty credit quality or probability of default, this conditioning becomes material. The dependence between exposure and credit quality is referred to as Wrong/Right Way Risk. The risk is wrong way if exposure tends to increase when counterparty credit quality worsens or probability of default increases. Typical examples include a bank entering a swap contract with an oil producer where the bank receives fixed rate and pays floating rate tied to crude oil spot price. In this circumstance, lower oil prices simultaneously worsen credit quality of the oil producer and increase the value of the swap to the bank by raising the overall exposure to the oil producer. Right Way Risk on the other hand refers to the tendency of decreasing exposure with worsening counterparty credit quality. A reverse situation where the bank is the fix payer is an example of RWR 6 . In simplified approaches, exposure is assumed to be independent of counterparty de- fault, and Equation 1.2 simplifies to CVA = (1−R c ) T t E Q t h P (t,s) Π + (s) i dF c (s) (1.3) The term Expected Positive Exposure (EPE) in the literature refers to the risk neutral discounted expected positive part of the portfolio value EPE (t) =E Q t h P (t,s) Π + (s) i (1.4) which is now independent of counterparty default event. EPE can be computed ana- lytically only at the contract level for simplified cases such as plain vanilla options. In 6 Thepredominantsettingintheenergysectoristhatabankandacommodityproducerenterafix-payer swap contract where the bank is the fix payer. The producer secures stable cashflows throughout the lifetime of the contract, while the bank enjoys the upside in future commodity prices. The bank under this circumstances is exposed to the better of the two risks, namely RWR. 11 Chapter 1 Introduction more general cases calculating this value requires simulating future commodity prices and valuing the contracts to arrive at aggregate portfolio value when netting and collateral conditions are assumed. These simulations can be performed according to the exposure models that each bank maintains for individual counterparties. 1.3 Debit Value Adjustment Large banks were treated as risk-free entities by the market prior to the crisis of 2008. This meant that counterparties were charged unilateral CVA, while not receiving any benefits for exposing themselves to the bank’s default. After the events of 2008 however, the perception of major bank’s default changed, as reflected in their CDS spreads. It has increasingly become the consensus that in addition to counterparty default risk, dealer’s default risk should also be considered for fair valuation purposes. This extends unilateral CVA framework into a bilateral CVA where dealer’s credit risk is also considered in valuation. DVA is defined similar to CVA where the role of the counterparty and dealer is reversed. DVA is given by expected future gains due to bank’s own default as DVA = (1−R d ) T t E Q t h P (t,s) Π − (s) i dF d (s) (1.5) where Π − (s) is the negative part of portfolio value at any future time s, F d denotes cumulative default probability of the dealer, and R d is the recovery rate of the dealer. From the point of view of the dealer, negative exposure to the counterparty is positive exposure to the dealer, and this results in cashflows from the dealer to the counterparty. The counterparty will incur losses should the dealer default in this case. 12 1.4 Derivative Fair Valuation under Default Risk 1.4 Derivative Fair Valuation under Default Risk Including both CVA and DVA in the valuation of a derivative between a dealer and a counterparty results in the following valuation formula Π D (t) = Π (t)−CVA +DVA (1.6) = Π (t) − (1−R c ) T t E Q t h P (t,s) Π + (s) i dF c (s) + (1−R d ) T t E Q t h P (t,s) Π − (s) i dF d (s) where Π D denotes default risky derivative value, and Π denotes risk-free value of the derivative 7 . 1.5 Wrong and Right Way Risk As highlighted above, simplified calculations of CVA and DVA assume independence betweenexposureanddefaultprobabilities. Thisassumptionisunrealisticwhencontracts are between parties with cashflows strictly tied to the value of the underlying asset in a derivative’s contract. To illustrate, assume a swap contract between a bank and a producerofcrudeoilwheretheproduceragreestoreceiveafixedpriceforspecificvolumes of future crude oil production. In general, there is a strong linkage between the price of crude oil and the cashflows a typical oil producer generates. Lower future spot prices of crude oil will adversely affect the cashflows and ultimately raise default probability 8 . 7 This formula essentially excludes the possibility of simultaneous default by both parties. Essentially once either of the two parties default the position is liquidated and settlement of the contract takes place. 8 A recent evidence from the market showed that falling crude oil prices increased CDS spreads for energy sector almost instantly. 13 Chapter 1 Introduction In the case of the swap contract, it is reasonable to assume that the states of economy when the producer is liable for payments (price of crude oil is above the agreed fixed price) coincides with improved credit quality of the producer. Right-way risk is said to be present in this situation. On the other hand, wrong-way risk is present when higher exposures are correlated with worst credit quality of the counterparty. A clear understanding of the magnitude of this type of risk involves having accurate information on the nature of the business, and risks faced by the counterparty. An objective assessment of the amount of wrong-way or right-way risk in transactions also requires a good knowledge of the transactions the counterparty has entered into with other dealers. Consider an insurance firm that sells insurance contracts with similar terms. Under such circumstances where the company enters many similar trades with one or more dealers is likely to lead to wrong way risk for the dealers. This is because the company’s financial position and therefore its probability of default is likely to be affected adversely if the trades move against the company. The insurance firm has tried to diversify idiosyncratic risk through a diversified client base, however similar nature of the contracts leaves the firm exposed to systemic risk. Exposure should be measured with the assumption that counterparty default will oc- cur. Counterparty exposure in general is a stochastic amount and the simplest approach is to measure exposure at default as the expected exposure taken over a large number of simulated market conditions. In simulation terminology, the correlation between po- tential exposure and probability of default should be reflected in the simulated future exposure. The correlation term will result in the scenarios leading to default of the coun- terparty coincide with scenarios leading to above average exposure, to reflect WWR and vice versa for RWR. These usually computationally intensive procedures require long processing time and an IT infrastructure that most average firms lack. Other simplified methodologies are proposed to account for correlation while avoiding 14 1.6 Funding Value Adjustment the computational intensity. For example, a crude methodology would be to scale the re- ported exposure by a certain factor (say 150%) to account for WWR, or use a percentage of notional amount (say 80%) for RWR. This approach has the advantage of simplicity but lacks sufficient sensitivity in modeling different levels of correlation across time and exposure levels. In the next chapter we discuss methodologies implemented by practi- tioners and review academic publications on the approaches for modeling correlation in CVA calculations. 1.6 Funding Value Adjustment OTC derivatives pricing necessitates accounting for, or at the least addressing additional costs that are incurred due to departure from the conditions of market completeness. One major departure is the asymmetry that exists for borrowing and lending rates by the bank. While a bank can access capital at a rate higher than risk free rate to fund purchasing and hedging activities related to OTC derivatives, the proceeds from these activities can not earn funding rate 9 . Fair value pricing is based on the exit price in a complete market with no arbitrage. In a Black-Scholes-Merton world, the market is frictionless and the cost of borrowing and lending for replicating portfolio equals risk free rate. In the aftermath of financial crisis, funding costs increased substantially due to margin requirements on the hedges that banks held against OTC derivatives. Among other reason, this triggered efforts to look for ways to offset these costs to counterparties through charging FVA. Figure 1.1 describes wealth and risk transfer mechanism in a bilateral setting. At the inception of a contract, counterparty default risk is essentially transferred from the se- cured creditors of the counterparty to the shareholders of the bank. In a counterparty 9 This is mainly due to the fact that a bank can not actively trade in its own bond under Basel III regulation. A detailed account of this topic can be found in [20] 15 Chapter 1 Introduction default scenario, losses realized by the bank shareholders are gains realized by counter- party’s secure creditors. The bank therefore charges the counterparty a CVA amount which corresponds to expected losses under the terms of the contract. DVA and CVA are symmetric in the sense that the risk and wealth transfer act similarly when bank and counterparty roles are reversed. On the other hand, the bank’s own default prompts an internal wealth transfer from the shareholders to secure creditors. Secured creditors are essentially charging a spread to provide funds to a non-secured entity, namely the shareholders, for their risky activ- ities. The opponents of FVA accounting in fair value pricing [23, 1] argue that unlike CVA, funding costs are essentially wealth transfer within a firm, from shareholders to more secured creditors. On the other hand, some market practitioners [3, 2], insist that replicating portfolio value must be self-financing to bank’s shareholders and not the bank as a whole. This essentially means that they ignore the benefits that the bondholders of the bank will gain post bankruptcy and quote the prices based on incurred costs to the shareholders. Without proceeding into details of FVA in pricing, we consider a closely related situa- tion where a contract can be traded both over-the-counter without collateralization, or in an exchange-like setting where its value is fully collateralized through daily margining. In the former setting, CVA/DVA charges apply in pricing, while in the latter, neither of the counterparts are exposed to default risk. In this case while CVA/DVA charges are not relevant, the counterparty needs to fulfill margin requirements by posting collateral that can be accessed at an average cost of funding above risk-free rate. Our pricing partial differential equations that govern the two scenarios result in the same valuation for the contracts. We conclude that the two scenarios are essentially related to the same type of risk, where in the former, the bank warehouses credit risk, and in the latter the credit risk is transferred back to the counterparty. In both scenarios the counterparties incur 16 1.7 Discounting Under Default Risk costs, one in the form of CVA/DVA and the other in the form of cost of funding. Figure 1.1: Wealth transfer under counterparty default and funding fees 1.7 Discounting Under Default Risk A different but related recent development in the area of derivatives pricing involves di- vergence between London Interbank Offered Rates (LIBOR) rates and Overnight Indexed Swap (OIS) rates, and revisiting risk-free discounting . Traditionally LIBOR rates were used as proxies for risk-free implied forward rates for valuing derivatives. After the credit crunch of 2008 these rates diverged substantially from OIS rates. LIBOR rates are being increasingly viewed as loans to default-risky banks with a AA rating, whereas OIS rates are continually refreshed one-day loans, and are considered closer proxies for risk-free 17 Chapter 1 Introduction rates 10 . Valuation of interest rate derivatives with future cashflows tied to LIBOR rates is increasingly involving calculating payoffs based on a credit risky curve (LIBOR based) and discounting the expected cashflows using risk free (OIS) curves. [4] discusses pricing and discounting through a double-curve framework for interest rate derivatives, forward rate agreements (FRAs) , and swaps. Change of numeraire is the main technique used to translate cashflow dynamics tied to LIBOR rates to risk-neutral measure associated with OIS numeraire. Other works in this area include [43]. Dual curve pricing approach initiated in interest rate markets integrates CVA and DVA pricing in calculation of expected cashflows through discounting, and is increas- ingly becoming market consensus in this market. LIBOR rates are only a single example of expected rates of return that embed both risk free rate and the premium built in due to default risk. In general, one can think of defaultable corporate bond as a default-risky rate equivalent to LIBOR, with embedded credit risk premium equivalent to that of the specific counterparty. These rates introduce a new market measure for pricing deriva- tives. Valuation under equivalent martingale measure is a well-studied topic and has been applied to various pricing problems (see [18],[32], [9]). In this work we approach the problem of pricing defaultable energy derivatives through the same angle. We depart from adjustment based pricing through CVA/DVA and propose discounting future cash- flows through selecting a discount factors that accounts for default risk though higher yield. Application of discounting approach to energy derivatives is presented in details in chapter 4. 10 Another major reason for moving from LIBOR discounting to OIS discounting relates to recent LIBOR fixing scandal. In 2012 multiple criminal settlements revealed that a series of fraudulent actions by LIBOR members resulted in false inflation or deflation of reported LIBOR rates. We refer the interested reader to http://www.cftc.gov/PressRoom/PressReleases/pr6289-12. 18 1.8 Thesis Outline 1.8 Thesis Outline Thisworkmainlyfocusesonintroducingsomenewresultsandtechniquestopricedefault- risky forward and swap contracts, predominantly used in energy markets. The approach hereistodepartfromtheideaofadjustingrisk-freevalue,anduseappropriatediscounting for the future cashflows. The proposed methodology not only greatly simplifies pricing problem through reliance on existing infrastructure in contrast with building a new CVA engine, it also integrates RWR and WWR in the pricing equation through change of numeraire process. Finally, this work intends to contrast the cost associated with credit risk (cost of credit) with the cost to access capital (cost of capital), and arrives at the conclusion that the two are two sides of the same coin. In chapter 2 we introduce the general concepts and building blocks associated with calculating CVA, and describe the results applied to generic forward and swap contract 11 . In chapter 3 the idea of double curve discounting is introduced, where cashflows linked to a particular yield curve are discounted at a different discount rate. The central idea here is to translate the dynamics of cashflows expressed from one probability space to another. In this chapter we also describe change of numeraire technique for pricing, a well known technique (see [31]) whose application in option pricing dates back to the work of [18]. The main contributions of this work are presented in chapter 4 through chapter 6. In chapter 4 we attempt to extend the idea of discounting to future cashflows associated with energy forward and swap contracts, where the positive and negative parts of the cashflow are discounted at appropriate discount rates. chapter 5 elaborates on the types of risks associated with pricing OTC derivatives and contrasts that with margining mech- 11 This chapter is intended to aquatint the reader with both CVA and DVA, since DVA is essentially identical to CVA with the role of counterparty and dealer reversed. 19 Chapter 1 Introduction anism of the exchange markets which essentially attempt to eliminate the same type of risk that is otherwise present in a bilateral setting. The pricing PDEs for both OTC and exchange scenarios are presented here result in the same pricing equation. Finally chapter 6 provides numerical results. 20 2 Credit Value Adjustment for Energy Derivatives The industry approaches pricing default risk in two main ways: top-down vs. bottom-up. A top-down pricing is where exposures are aggregated by counterparty and credit costs associated with each portfolio is calculated at the portfolio level. Conversely, a bottom- up approach is when each contract is priced as an independent transaction, accounting for credit charge. First approach requires netting agreements in places that allows for payments to and from the counterparty to offset, therefore reducing total exposure. Di- versification plays a major role in reducing aggregate exposure at counterparty level. Second approach is mainly applied when a large structured transaction is priced with a distant maturity date, and requires significant resources and capital allocation to manage the position throughout its life. In this chapter we introduce the main building blocks that constitute a credit pricing engine. Our approach, however, is bottom-up where we discuss pricing forward and swap contracts. As we previously discussed, common practice in computing CVA is through full Monte Carlo simulation of the future exposures. As a first step, a realization of counterparty level exposure at all simulation horizons and interim dates are computed. These values are then adjusted to a higher or lower level depending on the structure of correlation betweenexposureandcounterpartydefaultprobabilities. Amodelofdefaultprobabilities 21 Chapter 2 Credit Value Adjustment for Energy Derivatives at different time horizons is also required to weigh the set of realized exposures. Final step involves numerical integration of the discounted expected expected loss, which is defined as the expectation of positive part of discounted exposures. The rest of this chapter discusses each element in a CVA pricing engine in further details. 2.1 Modeling Exposure There are three main steps in calculating the distribution of counterparty-level credit exposure [46]: • Scenario Generation: Future market scenarios are simulated for a fixed set of sim- ulation dates using evolution models of the risk factors. Risk factors are a set of fundamental macroeconomic factors such as exchange rate, interest rate, commod- ity spot prices, etc. that impact portfolio value. • Instrument Valuation: For each simulation date and for each realization of the underlying market risk factors, each contract is valued. This procedure takes into account both non-linear and linear instrument valuation, and associated expiry dates. • Portfolio Aggregation: For each simulation date and for each realization of the underlying market risk factors, counterparty-level exposure is obtained by applying necessary netting rules, and finally aggregating the values o single instruments to arrive at total portfolio value at each time increment. These steps result in a set of realizations of counterparty-level exposure, with each real- ization corresponding to one market scenario at each simulation date, as schematically illustrated in Figure 2.1 [14]. 22 2.1 Modeling Exposure Figure 2.1: Simulation framework for credit exposure When pricing a general credit derivative includes CVA, full Monte Carlo usually means simulating not only market evolution (continuously changing market observable risk fac- tors such as interest rates, FX rates, commodity prices, credit spreads), but also the full sequence of credit events (defaults and rating transitions for one or multiple reference names). CVA is then obtained as discounted expectation of the portfolio level expo- sure averaged over the simulation paths which include both market evolution and credit events. In practice, there are major challenges for estimating the underlying joint market and credit risk factor processes driving counterparty loses, and in particular for modeling ac- curately the co-dependence between future exposures and defaults. The implementation of integrated market and credit risk simulation models is conceptually straightforward, but proves to be computationally intensive. This is due to the fact that low default probability results in only a small fraction of the paths that include a credit event. The total number of paths must be very high in order to get a stable average. Given the computational complexity, stress-testing of the results and estimation of sensitivity to the model parameters, such as market-credit correlations, are rarely performed. 23 Chapter 2 Credit Value Adjustment for Energy Derivatives This approach is frequently referred to as brute-force Monte Carlo, to distinguish it from the methods which use importance sampling or numerical integration that can en- hancetheoriginalMonteCarlosimulationengineandavoidtherarecrediteventproblem. Because the simulation model of the payoff function of a credit derivative typically in- volves a limited number of variables, they are very fast to evaluate, making brute-force Monte Carlo feasible and widely used for credit derivative pricing [13]. However, in CVA valuation of other financial instruments including energy related financial assets, because both market and credit events need to be simulated, brute force Monte Carlo becomes increasingly unfeasible and costly to implement. 2.2 Commodity Models In the context of fair valuation of a single derivative under credit risk described in Equation 1.6, the general approach is first to derive the value of the contract under no credit risk. Specific to energy markets, forwards, swaps and swaptions are most com- monly traded instruments in OTC setting. The valuation of these contracts relies on a spot price model for underlying commodity (natural gas, crude oil, etc.). In the following sections, we describe the payoff function for each contract and present the closed form solutions for CVA based on two commonly used commodity spot models. 2.2.1 Payoff and Valuation 2.2.1.1 Forward Contract A forward contract is an agreement between a buyer to buy from a seller a commodity at a price K fixed today, for delivery on a future date. This is expressed by saying that the buyer has entered a payer forward rate agreement (FRA) or a forward contract. The 24 2.2 Commodity Models counterparty has entered a receiver forward contract. The payoff of this contract to the buyer at maturity will be 1 χ fwd (T ) =S (T )−K or the difference between spot price of the commodity at maturity,S (T ) and the agreed- upon fixed price K, with the appropriate sign for the buyer. The value of this contract from the point of view of the fix rate payer at time t, denoted by Π (t,χ fwd ), is given by the discounted expected payoff Π (t,χ fwd ) =E Q t [P (t,T ) (S (T )−K)] =P (t,T ) h E Q T t (S (T ))−K i =P (t,T ) [F (t,T )−K] (2.1) where the expectation is taken under risk neutral measure 2 , and F (t,T ) denotes the forward price of the contract at time t for expiry at time T. The fixed price K that sets the value of the contract at the time of inception equal to zero is given by K =F (t,T ). Analytical forms for forward contract depends on the assumptions on evolutions of spot price S T . 2.2.1.2 Swap Contract A swap contract is an obligation to the buyer to buy from the seller a commodity at the price K fixed today, at future times T i ∈T ={T 1 ,T 2 ,...,T n }. In a physical swap con- 1 Throughout this work we denote payoff function at maturity by χ x , where x stands for the specific contract. 2 Throughout this work we Q denotes the risk neutral measure associated with money market account, and Q T the risk neutral measure associated with risk-free zero coupon bond numeraire. We will discuss different risk neutral measures and numeraire processes in great details in chapter 3. 25 Chapter 2 Credit Value Adjustment for Energy Derivatives tract, the buyer of swap agrees to pay fixed priceK and receive the physical commodity, and hence enters a payer swap. The seller of the swap receives fixed price K and sells the physical commodity, hence entering a receiver swap. In the case of a financial swap contract, actual physical commodity does not change hands, and the difference between the current value of the commodity as specified in the contract and fixed price K is set- tled financially. A swap can be viewed as a portfolio of forward contracts with a single strike price K. The value of a payer commodity swap contract at time t is therefore Π (t,χ swp ) =E Q T t " n X i=1 P (t,T i ) (S (T i )−K) # = n X i=1 P (t,T i ) E Q T t (S (T i ))−K = n X i=1 Π (t,χ fwd i ) (2.2) where χ fwd i denote the payoff of forward contracts with maturity T i , i∈{1,...,n}. The industry approaches swap and its derivatives such as swaptions through modeling swap rate or forward swap rate as the underlying variable with its dynamics assumed through GBM process or its variations. Forward swap rate is defined as the weighted average of the sum of forward rates as follows: S n (t) = n X i=1 w i F (t,T i ) (2.3) where T≤T 1 <T 2 <...<T n , and the weights are associated with the discount factors as shown below. w i (t) = P (t,T i ) P n j=1 P (t,T j ) (2.4) 26 2.2 Commodity Models We differentiate commodity spot rate and commodity forward swap rate with the sub- script n for the latter, which indicates n payments in the future. A swap contract can be viewed as a basket of underlying forward rates, with its dynamics fully determined once the volatility and correlation structure of the underlying forward rates are specified. Swap contract payoff can be written in terms of forward swap rate as Π (t,χ swp ) = n X j=1 P (t,T j ) [S n (t)−K] (2.5) and the swap price K that sets the value of the contract at inception equal to zero is given by P n i=1 P (t,T i )F (t,T i ) P n j=1 P (t,T j ) (2.6) 2.2.1.3 Swaption Contract One main reason for relying on a model of forward swap rate vs. underlying forward rates has historically been related to pricing of swap option or swaption contracts. A European swaption or a fix-payer swaption gives the holder the right to enter into a swap with pre-determined strike price K, at a certain time in the future (exercise date). A company may use the fix-payer swaption to limit the price it would pay for a certain commodity swap at a future time T. Suppose that the n-payment swap rate starting at time T proves to be S n (T ). By comparing the cashflows on a swap where the fixed rate isS n (T ) to the cash flows on a swap where the fixed rate isK, the payoff of the swaption is given by(S n (T )−K) + . The cashflows are received n periods in the future from the start of the swap life. Suppose that the swap payment dates are T i in{T 1 ,T 2 ,...,T n }, where T ≤ T 1 < T 2 < ... < T n . Each cashflow is the payoff from a call option on the swap rate with strike K. In other words, a swaption is a single option on the swap rate 27 Chapter 2 Credit Value Adjustment for Energy Derivatives with repeated payoffs. Π (t,χ p−swptn ) =E Q T t   n X j=1 P (t,T j ) [S n (T )−K] +   (2.7) Analytical forms for swaption contract value depends on the assumptions on evolutions of forward swap rate evolution S n (t). In the following section we will present two popular closed form solutions for forward and swap contracts based on commodity spot and forward swap rate models. 2.2.2 Two-Factor Commodity Spot Model A two factor model for crude oil or natural gas spot price is presented in [10]. This model is initially discussed in a more classical convenience-yield model setting with stochastic volatility. Two factor commodity spot price at time t denoted by S (t) follows the dy- namics shown in lnS (t) =m (t) +M (t) +ϕ (t) (2.8) The processm (t) represents the short-term deviations whereasM (t) represents the long term equilibrium price level. ϕ (t) is a deterministic function of time inserted for calibra- tion purposes. The short and long term prices are assumed to follow the dynamics in the following dm (t) =−μ m (t)m (t)dt +σ m (t)dX (t) dM (t) =μ M (t)dt +σ M (t)dY (t) (2.9) 28 2.2 Commodity Models where X and Y are correlated Brownian motions with the instantaneous correlation factor defined by dX (t)·dY (t) =ρ mM (t)dt (2.10) Under these assumptions, forward contract value is given by the following closed form solutions: Π (t,χ fwd ) =P (t,T ) [F (t,T )−K] =P (t,T ) [exp (M (t,T ))−K] (2.11) where M (t,T ) is M (t,T ) =ϕ (T ) +m (t)e −μm(T−t) +M (t) +μ M (T−t) + 1 2 σ 2 (t,T ) (2.12) and σ 2 (t,T ) is the variance of spot price defined by σ 2 (t,T ) = σ 2 x 2μ m ( 1−e −2μm(T−t) +σ 2 M (T−t) + 2 ρ mM σ m σ M μ m 1−e −μm(T−t) ) (2.13) Swap price can be derived using the definition given in Equation 2.2. 2.2.3 Standard Forward Swap Rate Model This is the standard market model discussed in the literature for both commodity spot priceandforwardswaprateevolution. Intherestofthissectionwelimitthediscussionto 29 Chapter 2 Credit Value Adjustment for Energy Derivatives standard forward swap rate model, while emphasis that the model is readily applicable to commodity spot price. Suppose that in the risk-neutral world, forward swap rate development follows Black’s model with constant volatility σ n > 0 (see [22]), i.e. dS n (t) =σ n S n (t)dX (t) S n (T ) =S n (t)exp n −σ 2 n (T−t)/2 +σ n √ T−tX (t) o (2.14) where X is a standard normal random variable. In this case, the close form solution for swap contract value at time t is given by Equation 2.5, and payer swaption contract χ p−swptn , which entitles the holder the right to payK and enter in a swap contract at an expiry time T prior to T 1 is valued as Π (t,χ p−swptn ) = n X j=1 P (t,T j ) (S n (t) Φ (d 1 )−KΦ (d 2 )) (2.15) where t≤T≤T 1 ≤...≤T n , and d 1 = ln (S n (t)/K) +σ 2 n (T−t)/2 σ n √ T−t d 2 =d 1 −σ n √ T−t (2.16) S n (t) is the swap rate at initial time t, and σ n is the volatility of the swap rate. This formula is the natural extension of Black’s model for option valuation when the risk free discount factor is replaced by the term A (t) = P n j=1 P (t,T j ) called annuity term. Annuity corresponds to the value of a portfolio of zero coupon bonds, hence the price of a tradable asset. 30 2.2 Commodity Models 2.2.4 Swap Rate Volatility Swap contract definition given in Equation 2.2 shows that swap rate is essentially a linear portfolioorbasketoftheforwardrates, anditsdynamicsisfullydefinedoncethevolatility and correlation of the original forward rates are assumed. It follows that if commodity spot rate or consequently forward rates are assumed to follow a lognormal process, swap rate in general will not have a lognormal distribution. Only numerical methods such as MonteCarlocansuitablydeterminethefuturedistributionofthepayoffofaderivativeon the underlying swap rate. Alternative valuation methods however exist which attempts to approximate the dynamics of the portfolio of underlying forward contracts with a lognormally distributed swap rate dynamics. In the following we briefly describe several techniques to ascertain swap rate dynamics given a particular choice of forward rate model, while we refer the interested reader to [35, 30]. Levy’s lognormal moment matching method is based on Levy’s idea to approximate the distribution of a portfolio by a lognormal distribution and match the first two moments of the two processes. Gentle proposes approximating the arithmetic average by a geometric average for basket dynamics. We briefly describe Levy’s approach in the rest of this section. Let the basket of forward rates be defined by Equation 2.3, with weights specified by Equation 2.4. In addition, let the dynamics of the forward contracts be governed by a standard Black’s model dF (t,T i ) =σ i F (t,T i )dX i (t) F (T,T i ) =F (t,T i ) exp n −σ 2 i (T−t)/2 +σ i √ T−tX i (t) o (2.17) where the valuation of the swap rate is for a terminal timeT,t≤T≤T i . Letρ ij denote 31 Chapter 2 Credit Value Adjustment for Energy Derivatives the correlation between the random shocks X i , andX j . Levy’s idea is to match the first and second moments of the swap rate process S n which is lognormally distributed, with those of the basket. Denote basket’s first and second moments bym 1 andm 2 respectively. m 1 = n X i=1 w i F (t,T i ) m 2 = n X j=1 n X k=1 w j w k F (t,T j )F (t,T k ) exp (σ j σ k ρ jk T ) (2.18) These should match the first and second moments of the swap rate process defined by Equation 2.14. M 1 =S n (t) M 2 =S 2 n (t)exp n σ 2 n T o (2.19) Letting m 1 =M 1 , and m 2 =M 2 results in following swap rate variance σ 2 n T = log P n j=1 P n k=1 w j w k F (t,T j )F (t,T k ) exp (σ j σ k ρ jk T ) { P n i=1 w i F (t,T i )} 2 ! (2.20) Finally, the price of a European call option on this swap rate with strike K at expiry T is given by n X j=1 P (t,T j ){M 1 Φ (d 1 )−KΦ (d 2 )} (2.21) where d 1 = log (M 1 /K) + 1 2 σ 2 n T σ n √ T d 2 =d 1 −σ n √ T (2.22) 32 2.3 Modeling Probability of Default This concludes the introduction to commodity models and contract valuation. In the next section we describe the second building block of CVA engine, namely modeling corporate default and term structure of default probabilities. 2.3 Modeling Probability of Default As evident in the form of CVA in Equation 1.2, next component in computing CVA is modeling probability of counterparty default at time t which we denote by F τc (t). There are mainly two approaches to modeling probability of default, namely reduced form ap- proach and structural approach. In reduced form approach, the information regarding the structure of the counterparty is not determined through the model, instead, all coun- terparties are assumed to default at some point in the future. The model introduces time to default of the counterparty as a random process, usually characterized by hazard rate function, and uses information available through the market such as credit spread to numerically deduce the value of hazard rate function. Structuralmodelsontheotherhand,assumecompleteknowledgeofcounterpartyfirm’s value, defining this value through a stochastic process called the value process. Through these models, time to default of the counterparty is defined as the stopping time at which the value of the firm hits a pre-determined barrier. In the rest of this chapter we briefly explain the concepts in both approaches and highlight major application areas. 2.3.1 Reduced Form Models Hazard rate function is most frequently used by statisticians for specifying the distribu- tion of time to default. Denote time to default of the counterparty C byτ c , and letF c (t) 33 Chapter 2 Credit Value Adjustment for Energy Derivatives denote the distribution function of τ c , F c (t) =P (τ c ≤t), t≥ 0 (2.23) Hazard rate denoted by λ c , is defined as the instantaneous default probability for a company that has attained age t„ λ c (t) =P (t<τ c ≤t +4t|τ c >t) = F c (t + Δt)−F c (t) 1−F c (t) ≈ f c (t) Δt 1−F c (t) (2.24) has a conditional probability density interpretation: it gives the value of the conditional probability density function ofτ c at exact aget, given survival up to that time. Survival function denoted byS c is defined as the unconditional probability that the counterparty survives until a certain age t: S c (t) = 1−F c (t) =P (τ c >t), t≥ 0 (2.25) The relationship of the hazard rate function with the distribution function and survival function is as follows: λ c (t) = f c (t) 1−F c (t) =− ˙ S c (t) S c (t) (2.26) The survival function can be expressed in terms of the hazard rate function as S c (t) =e − t 0 λc(s)ds (2.27) 34 2.3 Modeling Probability of Default In addition, F c (t) = 1−S c (t) = 1−e − t 0 λc(s)ds (2.28) and f c (t) =S c (t)λ c (t) (2.29) In order to model default rate or survival probability, deterministic and stochastic models of survival rates are used. A typical assumption is that the hazard rate is constant, in which case default probability density function is given by: f c (t) =λ c e −λct (2.30) It follows that survival time is exponentially distributed with parameter λ c . Under this assumption, survival probability up to time t is given by: S c (t) =e −λct (2.31) As described above, modeling a default process is equivalent to modeling a hazard rate function. There are a number of reasons why modeling the hazard rate function may be a good idea. First, it provides us information on the immediate default risk of each entity known to be alive at exact age t . Second, the comparisons of groups of companies are easily made via hazard rate function. Third, hazard rate function based model can be easily adapted to more complicated situations, such as when there are several types of default or when stochastic default fluctuations should be considered. Fourth, there are manysimilaritiesbetweenhazardratefunctionandshortrate. Manymodelingtechniques 35 Chapter 2 Credit Value Adjustment for Energy Derivatives for the short rate processes can be readily borrowed to model hazard rate. 2.3.2 Structural Models Structural models are largely categorized into econometric and Merton-based. Econo- metric models attempt to model default rate as a regression on macroeconomic variables with random innovations. In Merton-based models first introduced by Merton [44], a default occurs if the value of the assets of the company is below the face value of the debt at a particular future time. Black and Cox [7] provide an important extension of Merton’s model. Their model has a first passage time structure where default takes place whenever the value of the assets of a company drops below a barrier level. Other ex- tensions of the Merton’s model are provided by Geske [19], Leland [37], Longstaff and Schwartz [40], Leland and Toft [38], and Zhou [52]. All of these papers look at the default probability of only one issuer. In the context of contracts with several counterparties, correlation between defaults of different counter- parties are considered by Zhou [51]and Hull e. al. [25]. They were the first to incorporate default correlation between different issuers into the Black and Cox first passage time structural model. Zhou [51] finds a closed form formula for the joint default probability of two issuers, but his results cannot easily be extended to more than two issuers. Hull et. al. [25] show how many issuers can be handled, but their correlation model requires computationally time-consuming numerical procedures. Hull, Predescu and White [24] extend the model to a large number of different obligors, and introduce a one factor model to model correlation between default times. We proceed with an introduction to a general framework for structural models. In general, if a variable x follows an arithmetic Brownian motion: dx =μ x dt +σ x dZ (2.32) 36 2.3 Modeling Probability of Default where μ x and σ x are constants, dZ is a Wiener process and x is an amount K above a barrier M at time zero, the probability of hitting the barrier M by time T is given by Φ −K−μ x T σ x √ T ! +exp −2Kμ x σ 2 x ! Φ −K +μ x T σ x √ T ! (2.33) where Φ is the cumulative normal distribution function. The probability density function of default time τ c at which the barrier is hit is: 1 √ 2πστ 3 /2 c exp − (K +μ x τ c ) 2 2σ 2 x τ c ! (2.34) Next, denote by V (t) the value of the assets of company at time t. Structural models assume that V follows a risk-neutral diffusion process described as: dV =V (μ V dt +σ V dZ) (2.35) or, d lnV = μ V −σ 2 V /2 dt +σ V dZ (2.36) A default occurs when lnV = lnH, whereH is the barrier value. Assuming thatV (0)> H the probability of a default between time 0 andT is given by substituting the following K = lnV (0)− lnH μ x =μ V −σ 2 V /2 σ x =σ V (2.37) 37 Chapter 2 Credit Value Adjustment for Energy Derivatives into Equation 2.33. Φ ln (H/V (0))− (μ V −σ 2 V /2) σ V √ T ! + exp 2 ln (H/V (0)) (μ V −σ 2 V /2) σ 2 V ! Φ ln (H/V (0)) + (μ V −σ 2 V /2) σ V √ T ! (2.38) This result can be used to find the probability of first hitting the barrier between timesT 1 andT 2 in terms of the value of the company at time 0. The probability density function for the time to default τ c is given by: K √ 2πσ V τ 3 /2 c exp − (ln (V (0)/H) + (μ V −σ 2 V /2)τ c ) 2 2σ 2 V τ c ! (2.39) 2.4 Modeling Correlation Last piece to the puzzle of modeling CVA is to model exposure at default. As explained earlier, practitionersusethesimplifiedCVAformulainEquation 1.3whenthereisenough evidence that counterparty default is an independent event from the value of exposure to the counterparty, eliminating the need to account for correlation. However, as it has become more evident in the past few years, it appears that credit worthiness of firms and the level of exposure share some of the underlying macroeconomic activities, resulting in various levels of correlation between the two. This correlation depends heavily on the nature of firm’s business model. Typical examples of WWR include a bank entering a swap with an oil producer where the bank receives fixed and pays floating crude oil price. In this circumstance, lower oil prices simultaneously worsens credit quality of the oil producer and increases the value of the swap to the bank by affecting the overall exposure to the oil producer. In the rest of this chapter we introduce some literature on factoring correlation into simulation of future exposure. 38 2.4 Modeling Correlation 2.4.1 Alpha Multiplier Basel II Accord 3 (BCBS (2006)) allows banks to use an internal ratings based approach to compute minimum capital requirements for counterparty credit risk of derivatives portfolio. In their framework, they introduce a multiplier on the calculated exposure level, which acts as a crude method to raise or reduce the calculated exposure at default based on assumed correlation between default event and exposure. The capital charges are determined through a simplified CVA formula, which uses the following quantitative inputs provided by the bank: • Probability of Default • Exposure at Default • Loss Given Default • Maturity Basel II defines a supervisory alpha multiplier of 1.4, but gives banks the option to estimate their own alpha multiplier based on their internal models. The definition of a portfolio’s alpha is given below 4 [17]: α = EC Total EC EPE (2.40) whereEC Total denotes the economic capital for counterparty credit risk from a joint sim- ulation of market and credit risk factors, and EC EPE is the economic capital when coun- 3 Basel II is the second of the Basel Accords, (now extended and effectively superseded by Basel III), which are recommendations on banking laws and regulations issued by the Basel Committee on Bank- ing Supervision. Basel II, initially published in June 2004, was intended to create an international standard for banking regulators to control how much capital banks need to put aside to guard against the types of financial and operational risks they face. 4 The original publication by the Bank for International Settlements can be found here: http://www.bis.org/publ/bcbs116.pdf . 39 Chapter 2 Credit Value Adjustment for Energy Derivatives terparty exposures are deterministic and equal to Expected Positive Exposure (EPE) 5 . AlphaprovidesameanstoconditionEPEestimatesonthe“badstates”oftheeconomy. Industry numerical exercises suggest that alpha may range from 1.1 for large global dealer portfoliostoover 1.5fornewuserswithconcentratedexposuresandlittlecurrentexposure [13]. 2.4.2 Copula Function The structure of one-factor or Gaussian copula model was suggested by Vasicek [49] and later applied to credit derivatives by Li[39], and Gregory and Laurent[36]. A copula func- tion is a function that links univariate marginal to their full multivariate distribution. In other words, it is essentially a method to build a multivariate distribution that preserves the marginal univariate distributions. If the univariate distribution of exposure and time to default is known, copula function can construct a multivariate random variable while maintaining marginal distributions as assumed. Let U 1 , ..., U n denote n uniform random variables. The joint distribution function C, or copula function, is defined as C (u 1 ,...,u n ,ρ) =P (U 1 ≤u 1 ,...,U n ≤u n ) (2.41) where ρ is the correlation structure between U 1 , ..., U n . For given univariate marginal distribution functions F 1 (x 1 ),...,F n (x n )the function F (x 1 ,...,x n ) =C (F 1 (x 1 ),...,F n (x n )) (2.42) results in a multivariate distribution function with univariate marginal distributions as 5 Economic capital refers to the capital set aside by a bank in support of its activities, as derived through an internal process that assesses the risk and return of those activities. 40 2.4 Modeling Correlation specified by F 1 (x 1 ),...,F n (x n ). A commonly used copula functions is bi-variate normal or Gaussian copula with the following form C (u,v) = Φ Φ −1 (u), Φ −1 (v),ρ ,−1≤ρ≤ 1 (2.43) The essence of the copula model is that correlation structure is not defined between the variables of interest directly. These variables of interest are mapped to other more man- ageable variables and correlation structure between the secondary variables determine the correlation structure between the initial variables. To elaborate further on this point, we consider the problem of modeling correlation in a swap CVA problem as laid out in [16]. Suppose that swap rate dynamic is given by Equation 2.14, and let X (t) =ρU (t) + q 1−ρ 2 V (t) (2.44) whereU, andY are independent random variables, and−1≤ρ≤ 1. Define counterparty time to default as τ c (t) =S −1 c (1− Φ (Y (t))) (2.45) where S c (t) is the survival probability function as defined in Equation 2.25, and Y is a standard normal random variable with it’s correlation structure defined as Y (t) =ρU (t) + q 1−ρ 2 W (t) (2.46) U,V,andW areallindependentnormalrandomvariables. TherelationshipinEquation 2.45 essentially maps the distribution of Y on τ c . The 5% point on the Y distribution is 41 Chapter 2 Credit Value Adjustment for Energy Derivatives mapped to the 5% point on the τ c distribution, the 10% point on the Y distribution is mapped to the 10% point on the τ c distribution, and so on. In other words, a normal distribution is mapped to an exponential distribution on a percentile to percentile basis. X and Y have two stochastic components each, the first element U is the same while the second elements V, and W are different and reflect the idiosyncratic component of the original stochastic processes. it follows that the correlation between X, and Y, as well as the correlation between S n , and τ c is given by the parameter ρ. From Equation 2.44, conditional onU,X, andY are normally distributed independent random variables P (X <x|U) = Φ x−ρU √ 1−ρ 2 ! P (Y <y|U) = Φ y−ρU √ 1−ρ 2 ! (2.47) 2.4.3 Deterministic Hazard Rate Model As a way to introduce correlation between exposure and counterparty time to default, Hull and White [26] define the hazard rate of the counterparty as a deterministic function of exposure. Denote by λ c (t) the counterparty hazard rate at time t, and EPE (t), the expected positive exposure to the counterparty at time t. The models proposed for wrong-way/rightway modeling are λ c (t) =e α(t)+β(t)EPE(t) (2.48) λ c (t) = ln n 1 +e α(t)+β(t)EPE(t) o (2.49) Each of these models has the property that λ c is a monotonic function of EPE, with λ c (t)> 0. The parameterβ measures the sensitivity ofλ c toEPE (t). The functionα is 42 2.4 Modeling Correlation determined using an iterative search procedure so that the average survival probability, calculated across all simulations, up to any time T matches that calculated from credit spreads. Whenβ > 0,λ c is an increasing function ofEPE, which corresponds to WWR, and when β < 0, λ c is a decreasing function of EPE, corresponding to RWR. 2.4.4 Stochastic Hazard Rate Model Brigo et. al. [8] introduce CIR++; a reduced form model that is stochastic in the coun- terparty hazard rate. They correlate the credit spread of this model with the exposure model. Again, denote counterparty hazard rate byλ c , and denote the cumulative hazard rate by Λ (t) = t 0 λ (s)ds. In this context, time to default τ c is given by τ c =F −1 (ξ) (2.50) whereξ is a standard exponential random variable. CIR++ model assumes the following form for hazard rate: λ c (t) =y (t) +ψ (t), t≥ 0 (2.51) where ψ is a deterministic function inserted for calibration purpose, and y is given by a Cox-Ingersoll-Ross process as: dy (t) =κ (μ−y (t))dt +ν q y (t)dZ (t) (2.52) where the parametersκ,μ,ν,y 0 are positive deterministic constants, andZ is a standard Brownian motion process under the risk neutral measures, representing the stochastic shock in y (t) dynamics. 43 Chapter 2 Credit Value Adjustment for Energy Derivatives As a model for exposure, they adopt a two factor model introduced in Equation 2.16. The correlation between asset price and default time is then introduced by correlating the shocks Z in the default intensity to the shocks X, and Y in the spot price. Assuming dZ (t)dX (t) =ρ ZX dt, dZ (t)dY (t) =ρ ZY dt It follows that the instantaneous correlation of interest is given by corr (dλ c (t),dS (t)) = σ m ρ ZX +σ M ρ ZY q σ 2 m +σ 2 M + 2ρ mM σ m σ M (2.53) 2.5 Linking Survival Probabilities to Market Data As outlined in the previous chapter, a key input to CVA pricing is counterparty survival probability, which can be inferred from different instruments traded on the counterparty name. Market practice is to use CDS spreads and build a hazard rate curve based on JP Morgan bootstrapping method. A second option is to read the survival probabilities from risky discount curves or zero coupon bonds. The implied survival probabilities are usuallydifferentfromeachoftheseinstrumentsduetoliquiditypremiumembeddedinthe quoted spreads. Without elaborating on the details of modeling liquidity in the spreads, weassumethatspreadsreflecttruedefaultprobabilities. Itisalsoworthnotingthatmany of the CDS spreads used for this purpose are quoted CDS or bond spreads by the dealers and the bid-ask spreads can be significant in some cases. CDS instruments trade only on limited single names and for limited terms. In addition, liquidity drops dramatically once CDS expiry exceeds 5 years. These all introduce empirical considerations when handling the actual data. 44 2.5 Linking Survival Probabilities to Market Data The main objective of building a term structure for hazard rate (or survival probabil- ities) is to determine the remaining hazard rates through interpolating or extrapolating between the observed market data. Bootstrapping consists mainly of two steps: the first involves deriving survival probabilities for different time horizons based on the market data of choice, and the second step is to deduce hazard rates from survival probabilities based on an assumed model that links the two. We briefly discuss these methodologies and refer the interested reader to [15] for more detailed discussion. 2.5.1 Market Implied Survival Probabilities A risky zero-coupon bond with zero recovery denoted byP c (t,T ) is modeled through the following risk-neutral expectation: P c (t,T ) =E Q t " exp − T t r (u)du ! 1 {τc>T} # =P (t,T )S c (T ) (2.54) whereP (t,T ) denotes the price of a risk-free zero-coupon bond, andS c (T ) is the survival probability untilT. The equation for a risky zero-coupon bond with a constant recovery R c becomes ˜ P c (t,T ) = (1−R c )P c (t,T ) +R c P (t,T ) (2.55) These equations can be used to read the survival probabilities from the bond market. Essentially this is what is known as bootstrapping of the term structure. Another ap- proach is to fit the parameters of a model such as those defined in subsection 2.4.3 or subsection 2.4.4 to market observed data, known as curve fitting. Curve fitting imposes artificial shape on the final term structure as assumed in the model. It is critical to 45 Chapter 2 Credit Value Adjustment for Energy Derivatives choose a good model that can represent market data as closely as possible. This is achieved through adding degrees of freedom (parameters) to the model, which in turn exposes the problem to over-fitting. A CDS contract for a reference entity is an insurance policy that gives the right to the policy holder to be compensated for losses given default of the reference entity. CDS premium is quoted in the form of spread on a notional that is specified in the terms of the contract. CDS contract has a floating and a fixed leg; the fixed leg pays fixed premium at a certain frequency until the expiry of the insurance. The floating leg makes a lump sum payment equal to the amount of losses after recovered portion is accounted. letT 1 ,...,T n denoten future times that fixed payments will occur. In addition, assume thatn observed CDS spreadss 1 ,...,s n are traded for payment at these times (i.e. s 1 is a CDS spread with one fixed payment at time T 1 , s 2 is a CDS spread with two payments atT 1 , andT 2 , ...). JP Morgan CDS model assumes that present value of the floating leg PV float (T j ) is given by PV float (t,T j ) = (1−R c ) j X i=1 P (t,T i ){S c (T i−1 )−S c (T i )} (2.56) and similarly, the present value of the fixed leg is given by PV fixed (t,T j ) =s j j X i=1 δ i P (t,T i ) S c (T i−1 ) +S c (T i ) 2 (2.57) whereδ i =T i −T i−1 . The following is the value of a CDS contract from the point of view of fixed payer. The fair spread sets contract price at inception equal to zero. V CDS (t,T j ) =PV float (t,T j )−s j PV fixed (t,T j ) (2.58) 46 2.5 Linking Survival Probabilities to Market Data 2.5.2 Hazard Rate Curve Hazard rate correspondence to survival probability is given by Equation 2.27. In practice a model of hazard rate is required to deduce the function from discrete market observed survival probabilities. A simplifying assumption that is commonly used by practitioners is piece-wise constant property of hazard rate. That is, when the observed survival probabilities are for n future dates T 1 ≤...≤T n , hazard rate is assumed to be constant in between the intervals: λ c (t) =λ i c ,t∈ (T i−1 ,T i ] (2.59) Therefore, the survival probability for a certain expiry is given by S c (T j ) = exp   − j X i=1 λ i c δ i   (2.60) BootstrappingprocedurestartswithsubstitutingEquation 2.60inEquation 2.58starting with V CDS (t,T 1 ), and solving for the hazard rates λ 1 c . The rest of the hazard rates are solved through recursively solving for λ j c from V CDS (t,T j ). Details can be found in [15]. Bootstrapping results in a term structure for hazard rate, given the assumption of constant piecewise hazard rate. As outlined above, deriving default probabilities from market data is model dependent and relies on several key assumptions. Recovery rates are also assumes to be fixed and are derived based on some historical default data. This shows that valuation can differ significantlybasedontheseelementsandbecomesspecifictothevaluingentity. Moreover, the infrastructure and intensity of the calculations become material once more elaborate models are considered. These are all issues that arise when market observed discount factors or spreads are attempted to fit a certain model of hazard rate. 47 Chapter 2 Credit Value Adjustment for Energy Derivatives 2.6 Swap CVA Following introduction of CVA building blocks, in this section we introduce commodity forward and swap CVA calculation. CVA of a forward contract with a single payoff in the future is given by the positive part of that payoff times credit spread This essentially results in price of a cap contract weighed by credit spread. Figure 2.2: Generic swap contract CVA Figure 2.2 presents CVA calculation steps for a generic swap contract with payments at nfuturetimesT ={T 1 ,...,T n }(see[48], and[21]). V (t,T i ,T n )denotesthepresentvalue of a swaption contract on a swap rate with payments at future dates T ={T i ,...,T n }, and{S (T i )−S (T i+1 )} denotes the probability of default during time interval [T i ,T i+1 ). Future expected loss in discrete form is therefore expressed as a portfolio of swaption contracts, each weighted by probability of default and LGD during each interval. Note that swaption contracts are written on different underlying swap rates, with each swap rate representing a basket of forward contracts with expiry in the future. [50] takes a Gaussian copula approach with constant correlation factor to derive a semi-analytical 48 2.6 Swap CVA result for swap contract CVA with WWR, and contrasts that with complex simulation results. 49 3 Discounting under Default Risk In the aftermath of the financial crisis events starting in 2007 the concept of risk free rate went under scrutiny by market participants. Traditionally London Inter Bank Overnight Rate (LIBOR) and LIBOR-swap rates were proxied for risk free rate for discounting futurecashflowsinderivativesvaluation. LIBORratesalsoreflectedthecostofborrowing for major financial institutions, therefore cashflows were discounted at LIBOR rate to implicitly account for the cost of funding. Overnight Indexed Swap (OIS) rates were also used in occasions. OIS and LIBOR rates historically followed each other very closely with spreads less than 50 basis points in normal market conditions. After the events of 2007, the spread peaked at 450 basis points, and has since been neither negligible nor deterministic. LIBORratesareshowntocarrythesameriskasaseriesofshort-termloanstofinancial institutionswithAAratings. Thereforetheseratesareconsideredaslow-riskbutnomore default-free. OIS rates on the other hand are interest rate swaps in which a fixed rate of interest is exchanged for a floating rate that is the geometric mean of a daily overnight rate. The calculation of payment on the floating side is designed to replicate the interest that is earned from rolling over a sequence of daily loans at the overnight rates. These are relatively short-lived rates that would be paid on continually refreshed overnight loans to borrowers in the overnight market. These rates are considered to more accurately reflect the risk free rates in the market. 51 Chapter 3 Discounting under Default Risk Replacing LIBOR discount curve with OIS introduces another issue in derivative pric- ing. A large number of interest rate derivatives have future payoff structure that are tied to LIBOR curve. Discounting under OIS implies that the underlying dynamics of LIBOR rates need to be transformed to allow for taking expectation under OIS numer- ator.In most of this chapter we introduce a new development in interest rate markets, namely double curve pricing that is mostly an effort to address this issue. Double curve pricing [4] or multi-curve pricing [43] views default risk differently, in that default pric- ing is incorporated in the payoff function and discounting, as opposed to adjustments to risk-neutral valuation. It was mainly developed to address the credit risk in LIBOR rates, through change of numeraire, and pricing under an equivalent martingale measure defined by OIS discount rates. Motivated by the idea of double curve pricing, in chapter 4 we approach pricing credit risk in energy OTC derivatives in a similar fashion with the following key differences: At a high level, double curve pricing deals with correct valuation of derivative payoff when expectation is taken under risk neutral measure associated with OIS curve numeraire, and payoffs dynamic is described based on LIBOR rates. In our approach however, the payoff is a function of two market observed components: underlying commodity prices of commodity futures contracts traded on an exchange, that assume no default risk, and risky zero coupon bonds that embed default risk. In addition, in the context of bilateral derivative pricing, our approach accounts for pricing the positive and negative parts of the exposure at correct discount curve, while this is not a concern in double curve pricing. While the setup of the problems are different between double curve discounting and OTC derivative valuation, the methodologies resemble in that change of measure technique is used and in this process the correlation between the payoff function and discount curve is reflected in a drift term in the adjusted dynamics of the underlying instrument. In order to clearly elaborate the concepts presented in double curve pricing, we take a 52 3.1 Equivalent Market Measures and Change of Numeraire step back and review the theory of equivalent martingale measures. Of these equivalent martingale measures, the ones associated with risk free, and risky zero coupon bonds are of central role in the developments in the next chapter. After introducing these fundamental concepts we highlight some of the key results of double curve pricing for interest rate swap at the end of this chapter. 3.1 Equivalent Market Measures and Change of Numeraire In a complete market, pricing contingent claims are based on their martingale property under risk neutral measure. Absence of arbitrage in a complete market implies the existence of a risk-adjusted probability measure Q such that the current price of any security is equal to the Q- expectation of it’s discounted future cashflow. A measure is a martingale measure only relative to some chosen numeraire asset 1 . Consider an economy where uncertainty is represented by a probability space (Ω,F,P), with two classes of securities: basic assets and derivatives which are contingent claims. Basic asset prices are market observed and are positive almost surely. Denote security prices by S 0 ,S 1 ,...S n and assume their dynamics are given by dS i S i =α i (t)dt +σ i (t)dW (t) (3.1) The coefficients are adapted to the filtrationF t generated by the Wiener process W. Under these assumptions, the market model is arbitrage-free, if and only if there exists 1 A detailed explanation can be found in [6] 53 Chapter 3 Discounting under Default Risk a martingale measureQ on this filtered probability space 2 , such that values of the form S i (t) S 0 (t) ,i = 0,...,n (3.2) are martingales underQ. Here we assume S 0 is a basic security. In other words, for this economy to be free of arbitrage, the price of anyF-measurable T-claim χ is such that χ S 0 (T) is Q-martingale, and its price at current time is given by the discounted expected payoff underQ-measure. Π (t,χ) =S 0 (t)E Q " χ (T ) S 0 (T ) |F t # (3.3) Historically, this risk-neutral probability measure was associated with the money market account as the numeraire. Money market account is essentially a riskless asset that con- tinuously re-invests initial dollar that it holds at initial time, and pays off instantaneous interest rate r (t). Let B (t) denote the value of the funds in the money market account at time t. Risk neutral martingale measure makes all processes of the form S (t)/B (t) - where S (t) is the arbitrage free price process of any non-dividend paying tradable asset - a martingale. Denote this risk neutral martingale measure by Q B , and the conditional expectation taken under this measure by E Q B [·|F t ]. The arbitrage-free price of any contingent claim χ, with a stochastic short rate r at time t is given by Π (t,χ) =E Q B e − T t r(s)ds ·χ (T )|F t (3.4) Using change of numeraire technique, [18] among others, show that many other risk neutral probability measures equivalent to risk-neutral measure Q B exist, and prove ex- 2 We often denoteQ∼P to show that the martingale measureQ is on probability space with associated real measure P. 54 3.1 Equivalent Market Measures and Change of Numeraire tremely useful in various asset pricing problems. More specifically, the price process of any non-dividend paying basic security in the relevant time period can act as the nu- meraire process for an equivalent risk-neutral martingale measure that makes the relative price of any security, a martingale. In other words, the fair price of a contingent claim does not depend on the choice of the risk-neutral probability measure. Depending on the structure of payoff function for any contingent claim, there exists best numeraire process that substantially simplifies the pricing problem. A very useful numeraire process is risk-free zero-coupon bond, when the pricing problem is related to an asset giving right to a single cashflow at a particular future date, T. A forward measure is defined through replacing the money market account numeraire with risk free zero-coupon bond as the numeraire process. For a fixed T, the T-forward measure denoted byQ T is defined as the martingale measure associated with the risk free zero-coupon bond numeraire process denoted by P (t,T ). Since it is a tradable asset, its price is aQ B −martingale, with payoff of 1 at time T, the price of risk free zero-coupon bond from Equation 3.4 is given by P (t,T ) =E Q B e − T t r(s)ds |F t (3.5) In addition, under forward probability measure, the arbitrage free price of a contingent claim χ with payoff at T is given by Π (t,χ) =P (t,T )E Q T " χ (T ) P (T,T ) |F t # =P (t,T )E Q T [χ (T )|F t ] (3.6) 55 Chapter 3 Discounting under Default Risk 3.2 Risky Zero Bond Equivalent Martingale Measure The selection of a specific variation of the numeraire processS 0 (t) depends on the payoff function of the asset under valuation and is ideally determined such that the ratio χ(T) S 0 (T) is simplified and the expectation can be calculated analytically. Again, letQ B denote the risk-neutral probability measure associated with money market account B. In general, any claim β that is positive almost surely (β t > 0,∀t) defines a numeraire process with the associated measureQ β , defined by Radon-Nikodym derivative of the form L β/B (t) := B (t)β (T ) B (T )β (t) (3.7) The price of a general T-claim χ underQ β is given by Π (t,χ) =β (t)E Q β t " χ (T ) β (T ) # =E Q B t " β (t) β (T ) L β/B χ (T ) # (3.8) In particular, assumeQ B -dynamics of theT-claimχ is given by a prototypical lognormal process with drift μ χ , and volatility σ χ dχ (t) χ (t) =μ χ dt +σ χ (t)dX (t) (3.9) where X is aQ B -Wiener process. TheQ B -dynamics of the likelihood process l β/B (t) is known to be given by a driftless lognormal process (see for example [6]) dl β/B (t) l β/B (t) =σ l (t)dY (t) (3.10) where Y is aQ B -Wiener process. 56 3.2 Risky Zero Bond Equivalent Martingale Measure The correlation structure between X and Y is defined by dX (t)·dY (t) =ρ XY (t)dt (3.11) The transformedQ β -dynamics of χ is given by applying Girsanov ’s theorem dχ (t) χ (t) ={μ χ −σ χ (t)σ l (t)ρ XY (t)}dt +σ χ (t)dZ (t) (3.12) where now Z is aQ β -Wiener process 3 . In the context of defaultable claim pricing, counterparty default time τ c is modeled as anF t -stopping time, or{τ c >t}∈F t , whereF t is the filtration on the probability space described above. It is convenient to divide the original filtration space into the subfil- trationF t =F τ t ∨H t whereF τ t =σ ({τ c >u},u≤t) is the subfiltration generated by default timeτ c andH t contains all information except default itself. It is further assumed that the conditional survival probability Q (τ c >t|H t ) =E h 1 {τc>t} |H t i is strictly pos- itive. Under this subfiltration structure, pricing formula of a general defaultable T-claim can be defined in terms of survival probability Q (τ c >t|H t ) which can be assumed to be positive in all states of the world. Given the subfiltration setting defined above, following equality holds for a general F-measurable T-claim χ (see [34]). E Q T h 1 {τc>T} χ (T )|F t i = 1 {τc>t} Q (τ c >t|H t ) E Q T h 1 {τc>T} χ (T )|H t i (3.13) 3 A well known property of change of measure is the following: Let β be a numeraire with the associated measure Q β , and B be the numeraire with associated measureQ B . For anyF-measurable claim C , we have E Q B C B T |F t =E Q β C β T |F t E Q B β B T |F t 57 Chapter 3 Discounting under Default Risk let theF-measurable T-claim ψ be a defaultable zero-coupon bond with a constant recovery rate R c , 0≤ R c ≤ 1. The payoff at maturity of this claim is given by (see for example [33]) P c (T,T ) =R c P (t,T ) + (1−R c )P (t,T )1 {τc>T} (3.14) where P (t,T ) is risk-free zero-coupon bond. The value of defaultable bond prior to default from Equation 3.13 is given by P c (t,T ) = E Q T [P c (T,T )|F t ] = 1 {τc>t} Q (τ c >t|H t ) E Q T h (1−R c )1 {τc>T} +R c P (t,T )|H t i (3.15) Notice that Equation 3.15 yields Π (t,ψ) = 0 after default, therefore, it does not give the right value of a defaultable bond after default with non-zero recovery. However, the relationship above is valid for our practice which focuses on pricing defaultable claims pre-default (see [34] for further details). Therefore, the value of defaultable bond prior to default coincides with P c (t,T ) = E Q T h (1−R c )1 {τc>T} +R c P (t,T )|H t i Q (τ c >t|H t ) (3.16) P c (t,T ) are market observable and take positive values almost surely 4 . When recovery rate is zero, Equation 3.16 reduces to [9] P c (t,T ) = E Q T h 1 {τc>T} P (t,T )|H t i Q (τ c >t|H t ) (3.17) We define a new probability measure Q T c associated with variations of the defaultable 4 This property is shown in [9] 58 3.3 Double Curve Pricing bond process P c (t,T ) as the numeraire. From Equation 3.8 It follows that the value of a generalF-measurable T-claim χ prior to default is Π (t,χ) = B (t)E Q B " χ (T ) B (T ) |F t # = P (t,T )E Q T " χ (T ) P (T,T ) |F t # = 1 {τc>t} P c (t,T )E Q T c " χ (T ) P c (T,T ) |H t # (3.18) where the expectation is under the measure associated with risky zero-bond numeraire. 3.3 Double Curve Pricing We now revisit double curve pricing. Valuation of interest rate derivatives increasingly involves calculating forward rates based on a credit risky curve (LIBOR) and discounting the expected cashflows using OIS curves 5 . In double curve pricing framework, a generic interbank rate (LIBOR for example) is assumed to be subject to counterparty default 5 The no-arbitrage relation in each (risky and risk-free) market is given by P x (t,T 2 ) =P x (t,T 1 )×L x (t;T 1 ,T 2 ) (3.19) where t≤ T 1 < T 2 and P x (t;T,T 2 ) denotes the forward discount factor from time T 2 to time T 1 prevailing at time t, and x{·,c} . Denote by L x (t;T 1 ,T 2 ) the simple compounded forward rate associated to P x (T 1 ,T 2 ), resetting at time T 1 and covering the time interval [T 1 ,T 2 ]. L x (t,T 1 ,T 2 ) = 1 T 2 −T 1 1 P x (T 1 ,T 2 ) − 1 LIBOR and riskless zero-bonds relationship in Equation 3.20 defines a new forward rate which are essentially simple interest earned by the deposit P c (T 1 ,T 2 ) L (t;T 1 ,T 2 ) = 1 T 2 −T 1 1 P c (T 1 ,T 2 ) − 1 = 1 T 2 −T 1 1 P (T 1 ,T 2 ){R + (1−R)Q (T 1 ,T 2 )} − 1 59 Chapter 3 Discounting under Default Risk risk. Following simple models can elaborate on the relation between credit risky rates (in this case derived from LIBOR) and risk free rates (OIS). Similar to prior notation, let P (t,T ), and P c (t,T ), t≤ T denote the price at time t of risk free and defaultable zero-coupon bonds with maturity T. The continuous term structure of these discount factors describes risk free and risky yield curves. In addition, let Ω,F,Q T , and Ω,F c ,Q T c denote the probability spaces with the filtrationF, and F c associatedwithnumeraireP (t,T ),andP c (t,T ), respectively. Denotebyτ c thedefault time of the generic interbank counterparty. Assuming independence between default and interest rate and denoting byR c the constant recovery rate, the value at timet of a risky deposit that starts at t and matures at T is P c (t,T ) =E Q B e − T t r(u)du R c + (1−R c ) 1 {τc>T} |F t =P (t,T ){R c + (1−R c )Q (t,T )} (3.20) whereF t is the information set at time t, and Q (t,T ) :=E Q B h 1 {τc>T} |F t i (3.21) Equation 3.20 is equivalent to Equation 3.15 when the expectation is conditional on the subfiltration excluding default information. Assume a derivative is written on a single underlying interest rate with n coupons payoffsχ ={χ 1 ,...,χ n } at future dates T ={T 1 ,...,T n } 6 . As in traditional valuation method, present value of the cashflows are calculated as the expected values of the i-th 6 Coupon payments take non-negative values. 60 3.3 Double Curve Pricing coupon payoff with respect to the risk neutral T i -forward measureQ T i Π (t,χ i ) =P (t,T i )E Q T i t [χ i ] (3.22) The price at time t of the derivative is the expected discounted cashflows using the risk free discount factor. Π (t,χ) = n X i=1 Π (t,χ i ) = n X i=1 P (t,T i )E Q T i t [χ i ] (3.23) While the cashflows are calculated under the market defined through risky curves, with their dynamics expressed in risky probability space, based on LIBOR curve, the expec- tation is taken under the martingale measure associated with risk-free yield curve. The computation of the expectation under one measure, and payoffs under a different mea- sure involves quanto adjustment commonly used for pricing cross-currency derivatives, described in the rest of this chapter. As outlined above, the cashflows in interest rate market are tied to LIBOR based forward rates, which in turn are assumed to have lognormal martingale dynamics of the form dL c (t,T 1 ,T 2 ) L c (t,T 1 ,T 2 ) =σ L (t)dX (t) (3.24) whereL c (t;T 1 ,T 2 )denotesthesimplecompoundedforwardrateassociatedwithP c (T 1 ,T 2 ), resetting at time T 1 and covering the time interval [T 1 ,T 2 ]. σ L (t) is the volatility of the forward rate process under the probability space (Ω,F c ,P c ), with the filtrationF c gen- erated by Brownian motion X under the risky T 2 -forward measureQ T 2 c , associated with 61 Chapter 3 Discounting under Default Risk numeraire P c (t,T 2 ). We emphasize that forward rate dynamics are expressed in Q T 2 c forward measure by adding the subscript. In order to convert this process to a process under the original probability space Q T 2 , with associated numeraire process P (t,T 2 ), a conversion process or numeraire ratio of the form l (t,T 2 ) = P c (t,T 2 ) P (t,T 2 ) (3.25) is defined. This process is the ratio of two tradable assets and evolves as a martingale under the associated risk-free T 2 -forward measureQ T 2 , dl (t,T 2 ) l (t,T 2 ) =σ l (t)dY (t) (3.26) where σ l (t) is the volatility of the process and Y is a Brownian motion under Q T 2 such that dX·dY =ρ Ll (t)dt (3.27) A forward process that has zero drift under Q T 2 c measure, after transformation, has the following dynamics with non-zero drift underQ T 2 measure dL (t;T 1 ,T 2 ) L (t;T 1 ,T 2 ) =−σ L (t)σ l (t)ρ Ll (t) +σ L (t)dZ (t) (3.28) where Z is a Brownian motion under the risk-free T 2 -forward measure Q T 2 , associated with numeraireP (t,T 2 ). This new forward rate is lognormally distributed with following 62 3.4 Modified Swap Valuation Under Risky Curve mean and variance E Q T 2 t " ln L (T 1 ;T 1 ,T 2 ) L (t;T 1 ,T 2 ) # = T 1 t −σ L (u)σ l (u)ρ Ll (u)− 1 2 σ 2 L (u) du Var Q T 2 t " ln L (T 1 ;T 1 ,T 2 ) L (t;T 1 ,T 2 ) # = T 1 t σ 2 L (u)du (3.29) or E Q T 2 t [L (T 1 ;T 1 ,T 2 )] =L (t;T 1 ,T 2 ) exp ( − T 1 t σ L (u)σ l (u)ρ Ll (u)du ) (3.30) 3.4 Modified Swap Valuation Under Risky Curve AssumeaninterestrateswapexchangesfloatingrateforfixedrateatdatesT ={T 1 ,...,T n }. The swap rate - and the corresponding fixed rate that sets the price of a swap contract at inception equal to zero - is defined as S n (t) = P n i=1 P c (t,T i )δ (T i−1 ,T i )L c (t;T i−1 ,T i ) P n i=1 P c (t,T i )δ (T i−1 ,T i ) (3.31) where by the subscript inS n (t) we emphasize that the swap payments occur at n future times. δ (T i−1 ,T i ) denotes the time interval fromT i−1 toT i 7 . The denominator is referred to as the annuity termA c (t) on the risky yield curve. Assuming the annuity term as the numeraire, theswaprateistheratiobetweentwotradableassets, thusamartingaleunder the associated Q T c measure with its dynamics given as a driftless geometric Brownian motion dS n (t) S n (t) =σ n (t)dU (t) (3.32) 7 This term is specific to interest rate swaps, and will be omitted in commodity swaps. 63 Chapter 3 Discounting under Default Risk where σ n (t) is the volatility of the process and U (t) is a Brownian motion under Q T c measure. Defineaswapconversionrate ˜ l astheratiobetweenQ T c andQ T -measureannuity terms respectively ˜ l (t) : = A c (t) A (t) = P n i=1 P c (t,T i )δ (T i−1 ,T i ) P n i=1 P (t,T i )δ (T i−1 ,T i ) = P n i=1 P (t,T i )δ (T i−1 ,T i )l (t,T i ) P n i=1 P (t,T i )δ (T i−1 ,T i ) (3.33) Since the swap conversion rate is the ratio between two tradable assets, its evolution is also given through a martingale process as d ˜ l (t) ˜ l (t) =σ ˜ l (t)dV (t) (3.34) Similarly, the volatility term σ ˜ l (t,T) is the volatility of the process and V (t) is a Brow- nian motion under Q T measure. Repeating the same argument above, the dynamics of swap rate under risk free swap measure Q T is given by dS (t) S (t) =−σ n (t)σ ˜ l (t,T)ρ Sn ˜ l (t) +σ n (t)dW (t) (3.35) where ρ Sn ˜ l is the correlation between the two Brownian motions dU (t)·dV (t) =ρ Sn ˜ l (t)dt (3.36) The mean and variance of log of swap rate is given by E Q T 2 t " ln S n (T 1 ) S n (t) # = T 1 t −σ n (u)σ ˜ l (u)ρ Sn ˜ l (u)− 1 2 σ S (u) du Var Q T 2 t " ln S n (T 1 ) S n (t) # = T 1 t σ 2 S (u)du (3.37) 64 3.4 Modified Swap Valuation Under Risky Curve and the expectation under risk neutral measure of swap process is obtained through the following expression E Q T 2 t [S n (T 1 )] =S n (t) exp ( − T 1 t σ s (u)σ ˜ l (u)ρ Sn ˜ l (u)du ) (3.38) Dual curve pricing approach integrates CVA and DVA pricing in calculation of expected cashflowsandisincreasinglybecomingmarketconsensusininterestrateproductsmarket. LIBOR is only a single example of expected return rates that embed both risk free rate and the premium built in due to default risk. In general, one can think of any bank or counterparty loan as a defaultable rate equivalent to LIBOR. In the next chapter we tackle the problem of counterparty credit risk pricing through the same concepts and extend credit risky derivative valuation through discounting with appropriate risky curves, and attempt at arriving at a single pricing formula that integrates CVA, DVA and WWR. 65 4 Defaultable Energy Derivatives Valuation Previous chapter outlined practitioners approach to pricing derivatives payoff contingent onunderlyingdynamicsthatreflectdefaultrisk. Underlyingmarketobservedinterestrate instruments in the previous scenario are LIBOR rates, which in turn define derivative’s payoff function. In the context of OTC energy derivatives pricing, our starting point is market observed commodity prices 1 . An ETD is a contingent claim on these underlying commodity prices. However in the case of OTC derivatives, the payoff function should accountfordefaultprobabilitiesthroughcreditrelatedmarketobservedinstruments. The payoff function of an OTC derivative is therefore a function of both commodity prices and default related instruments. In addition, default probabilities of both parties involved in an OTC derivative are relevant in pricing, and the choice of relevant default risk depends on whether the contract is in the money or out of the money to each party. Counterparty default probability is only relevant if possibility of losses exist, in other words, the contract is in the money from the point of view of the dealer. For instance, a sold option does not expose the seller to counterparty default. However, from the point of view of the counterparty, dealer default possibility and failure to fulfill future obligation 1 To be precise, spot and forward prices of commodities observed through futures contracts are market observed variables 67 Chapter 4 Defaultable Energy Derivatives Valuation remains, should the option go in the money. In this chapter we focus our attention on pricing future positive and negative cashflows at discount rates that reflect the relevant default probabilities. In particular, we derive the price of OTC forward and swap contract based on their payoff which assumes default risk. Thepriceisgivenbytheexpectationofthefuturepayoffunderriskneutralmeasure. Using change of measure, the T-forward expectation of the defaultable claim is translated to one which is numerated by defaultable zero-coupon bond. In this transformation processweaccountforwrong/rightwayriskthroughanaddeddrifttermintheunderlying spot dynamics. Finally we derive new swap rate based on the derivations above. Starting from the adjustment based pricing 2 for a general derivative in Equation 1.6, the defaultable value Π D is given in terms of risk-free value Π by Π D (t) = Π (t) − (1−R c ) T t E Q T t h P (0,s) Π + (s) i dF c (s) + (1−R d ) T t E Q T t h P (0,s) Π − (s) i dF d (s) (4.1) dF c (s) and dF d (s) are counterparty and dealer default probabilities in time period ds respectively,R c andR d arethecounterpartyanddealerexpectedrecoveryratesatdefault, and the expectations are taken under risk neutral probability measureQ T conditional on information available up to time t 3 . [27] discusses discounting with OIS (risk-free) vs. LIBOR (risky) curves and highlight three cases where discounting under both LIBOR and OIS result in the same valuation when Equation 1.6 is used for pricing. Essentially, modification in discount rate can 2 In the rest of this work we refer to pricing through CVA/DVA adjustment as adjustment based pricing in contrast with our proposed pricing based on discounting. 3 Throughout this chapter we will use the shorthand symbol E Q T t [·] for conditional expectation E Q T [·|F t ] under T-forward risk-neutral probability measure Q T . 68 Defaultable Energy Derivatives Valuation adjust for pricing default risk: • The portfolio promises a single positive payoff to the dealer (and negative payoff to the counterparty) at expiry time T. In this case, Π + (t) = Π (t) , Π − (t) = 0. Denote bys c (t), the adjusted credit spread for a zero coupon bond valued att with maturity T, issued by the counterparty. The risky value of the portfolio is given by: Π D (t) = Π (t)− (1−R c ) T t E Q T t [P (t,s) Π (s)]dF c (s) = Π (t) exp (−s c (T ) ΔT ) (4.2) • The portfolio promises a single negative payoff to the dealer (or positive payoff to the counterparty) at expiry time T. In this case, Π + (t) = 0 , Π − (t) = Π (t). Similar to previous case, let s d (t,T ) denote the adjusted credit spread for a zero coupon bond valued at t with maturity T issued by the dealer. The risky value of the portfolio is given by: Π D (t) = Π (t)− (1−R d ) T t E Q T [P (t,s) Π (s)]dF d (s) = Π (t) exp (−s d (t,T ) ΔT ) (4.3) • The derivatives portfolio promises a single payoff, which can be positive or negative at time T, and the two sides of loss are identical for dealer and counterparty, or (1−R c )dF c (t) = (1−R d )dF d (t) for all t. In this case the risky portfolio value is given by either of Equation 4.2 or Equation 4.3. These results show that the derivative can be valued under a discount rate which is the sum of risk free rate and counterparty or dealer credit spread for maturity T. This 69 Chapter 4 Defaultable Energy Derivatives Valuation essentially reflects discounting at defaultable zero-coupon bond yield. Any contract can be viewed as two contracts which promise payoff at expiry of either the positive or the negative part of the original contract, whereby appropriate discount factors can be used for default pricing as outlined above. In this chapter we mostly limit our discussion to forward and swap contracts most commonly traded in OTC energy markets. In addition, weextendtheconceptstoaccountforcorrelationbetweencreditriskandmarketexposure or wrong/right way risk. Through this extension we can stress swap values using different correlation scenarios between credit spread and exposure. 4.1 Forward Contract Valuation Let us revisit a commodity forward contract. These contracts are used both for hedging the risk of future price movements and for speculation on future commodity prices. An oil producer who has a planned volume of future production may be concerned about possible future movement in the spot prices. To hedge this price risk, the producer enters a forward contract with a bank, where the producer receives a fixed priceK, agreed upon today, for delivery of specific volumes of crude oil in the future. In this scenario, the bank will profit when the future spot price of the commodity is above the agreed price, in exchange for providing a guaranteed cashflow for future production volumes of the producer. A prototypical forward contract is an agreement to buy a commodity from the seller at a future date T, at a price K which is fixed today. The payoff of the forward contract at maturity T to the bank not assuming credit risk is given by χ fwd (T ) =S (T )−K (4.4) 70 4.1 Forward Contract Valuation where S (T ) denotes the spot price of commodity at time T. Under default risk, the payoff is given by χ D fwd (T ) = R c + (1−R c )1 {τc>T} (S (T )−K) + − R d + (1−R d )1 {τ d >T} (S (T )−K) − (4.5) whereχ D fwd (T ) denotes payoff at timeT of the forward contract subject to both counter- party and dealer default risk 4 . Hereτ x ,x{c,d} denote counterparty and dealer default times, R x , x{c,d} are counterparty and dealer recovery rates, and 1 {} is the indicator function 5 . In the above derivations we have essentially used the put call parity relationship for a forward contract (see[22]): Π fwd (t) = Π cplt (t)− Π fllt (t) (4.6) where Π cplt and Π fllt denotepricesofcapletandfloorletcontracts,definedby (S (T )−K) + and (S (T )−K) − respectively. Thebankcanbeviewedtohavesimultaneouslypurchased a caplet from and sold a floorlet to the counterparty. Counterparty risk is only relevant on the purchased caplet, while bank’s own default is a concern related to sold floorlet. Equation 4.5 accounts for default possibility of the relevant party on the relevant instru- ment, i.e. counterparty default for the caplet, and dealer default for the floorlet. The arbitrage-freepriceofthedefault-riskyforwardcontractattimet, denotedby Π t,χ D fwd , 4 In this chapter we refer to commodity producer as the counterparty and to the bank as the dealer. The bank pays fixed price to purchase the commodity at a future date. 5 Note that the above equation does not account for first to default time, rather it assumes that two default times are independent and there are two states of the world, where either dealer defaults and counterparty survives, or the counterparty survives and the dealer defaults. This payoff is consistent with unilateral CVA and DVA definition. In [21] Gregory discusses bilateral CVA which accounts for first to default time adjustment. 71 Chapter 4 Defaultable Energy Derivatives Valuation is given by the discounted expected payoff in Equation 4.5 under risk neutral measure 6 Π t,χ D fwd =E Q T [A (T )|F t ]−E Q T [B (T )|F t ] (4.7) where expectations are taken under T-forward risk neutral measureQ T , and A (T ) = P (t,T ) R c + (1−R c )1 {τc>T} (S (T )−K) + B (T ) = P (t,T ) R d + (1−R d )1 {τ d >T} (S (T )−K) − (4.8) P (t,T ) is risk-free zero-coupon bond price process for maturity T. From Equation 3.15 Π t,χ D fwd = 1 {τc>t} P (τ c >t|H t ) E Q T [A (T )|H t ] − 1 {τc>t} P (τ d >t|H t ) E Q T [B (T )|H t ] (4.9) The two terms on the R.H.S are survival claims. Define two probability measuresQ T x ,x∈ {c,d} associated with defaultable zero-coupon bond processes of the form P x (t,T ) =E Q T h P (t,T ) R x + (1−R x )1 {τx>T} |F t i x∈{c,d} (4.10) Usingfootnote 3,andapplyingthechangeofnumerairetechniquethroughRadon-Nikodym derivative given by Equation 3.7, we can rewrite Equation 4.9 in terms of expectations taken underQ T x ,x∈{c,d} measures Π t,χ D fwd = 1 {τc>t} P c (t,T )E Q T c h (S (T )−K) + |H t i − 1 {τ d >t} P d (t,T )E Q T d h (S (T )−K) − |H t i (4.11) 6 We frequently use Π t,χ D or Π D (t) to refer to the risky value of the contract which is otherwise valued Π (t) when default is not considered. 72 4.1 Forward Contract Valuation SinceH t includes all information inF t except default itself, all information related to commodity spot dynamics is included in the information set. Therefore, prior to default, the value of a forward contract is given by the difference between a caplet valued under the equivalent martingale measure Q T c defined by defaultable zero-bond issued by the counterparty, with recovery rate R c , and a floorlet valued under the equivalent measure Q T d defined by dealer defaultable zero-bond with recovery rate R d Π t,χ D fwd = Π t,χ D cplt − Π t,χ D fllt (4.12) In order to explicitly derive the price of a defaultable forward contract, we proceed with applying the Radon-Nikodym derivative. Assume commodity spot price has a general lognormal martingaleQ T -dynamics of the form: 7 dS (t) S (t) =μ S (t)dt +σ S (t)dZ (t) (4.13) where μ S (t) and σ S (t) are time-dependent drift and volatility parameters of the spot process, and Z is a Q T -Wiener process. Let l x , x∈{c,d} denote the two likelihood processes defined by Equation 3.7. TheQ T -dynamics of l x ,x∈{c,d} evolve as driftless processes of the forms dl c (t) l c (t) = σ lc (t)dX (t) , dl d (t) l d (t) =σ l d (t)dY (t) (4.14) where σ lx (t) , x∈{c,d} are the volatilities of the two processes, and X and Y are two Q T -Wiener processes such that dZ (t)·dX (t) = ρ Slc (t)dt, dZ (t)·dY (t) =ρ Sl d (t)dt (4.15) 7 The dynamics of spot price can be generalized to that of [8] for example. 73 Chapter 4 Defaultable Energy Derivatives Valuation The spot price process that hasμ S (t) drift underQ T measure, after transformation, has the followingQ T x -dynamics with a new drift process dS x (t) S x (t) = (μ S (t) +μ Slx (t))dt +σ S (t)dW x (t) (4.16) where W x ,x ={c,d} areQ T x -Wiener processes, and μ Slx (t) = −σ S (t)σ lx (t)ρ Slx (t) (4.17) The volatility terms remain invariant under change of measure. The new spot rates are lognormally distributed with following mean and variance E Q T x t " ln S x (T ) S x (t) # = T t μ S (u) +μ Slx (u)− 1 2 σ 2 S (u) du Var " ln S x (T ) S x (t) # = T t σ 2 S (u)du (4.18) We denote by F x (t,T ) =E Q T x [S x (T )|F t ] ,x∈{c,d} (4.19) the forward price of commodity evaluated under equivalent martingale measuresQ T x ,x = {c,d} 8 . From Equation 4.18 forward commodity price is given by F x (t,T ) = exp ( T t (μ S (u) +μ Slx (u))du ) = F (t,T ) exp ( T t μ Slx (u)du ) (4.20) If the correlation factors ρ Slx (t) are non-zero, the transformed spot price dynamics have 8 The information setH t includes all information except default itself, hence the conditional expectation prior to default stays invariant. 74 4.1 Forward Contract Valuation new associated drifts under Q T x ,x ∈ {c,d} measures. Under the new measures, the defaultable caplet and floorlet prices are given by Black’s formula with modified forward commodity prices, and defaultable zero-bonds as the discount rates. Π t,χ D cplt =P c (t,T ) (F (t,T ) exp (μ c (t,T )) Φ (d c1 )−KΦ (d c2 )) Π t,χ D fllt =P d (t,T ) (KΦ (−d d2 )−F (t,T ) exp (μ d (t,T )) Φ (−d d1 )) (4.21) where Φ denotes the standard normal distribution function, and d x1 = μ x (t,T ) +σ 2 (t,T )/2 σ (t,T ) d x2 = μ x (t,T )−σ 2 (t,T )/2 σ (t,T ) (4.22) μ x (t,T ) = T t μ Slx (u)du σ 2 (t,T ) = T t σ 2 F (u)du (4.23) where spot volatility σ S is replaced with forward volatility σ F . Finally, risky contract value prior to default is given by substituting Equation 4.21 in Equation 4.12. Π t,χ D fwd = F (t,T ) A + c (T ) +A − d (T ) −K B + c (T ) +B − d (T ) (4.24) where A ± x (T ) = P x (t,T ) exp (μ x (t,T )) Φ (±d x1 ) B ± x (T ) = P x (t,T ) Φ (±d x2 ) ,x{c,d} (4.25) 75 Chapter 4 Defaultable Energy Derivatives Valuation When right/wrong way risk is absent we have: μ Slc =μ Sl d = 0 d c1 =d d1 =d 1 d c2 =d d2 =d 2 d 1 =−d 2 Commodity spot dynamics collapses to the original martingale process under Q T x mea- sures. In this case, the prices of defaultable caplet and floorlet are given by Π t,χ D cplt =P c (t,T ) (F (t,T ) Φ (d 1 )−KΦ (d 2 )) Π t,χ D fllt =P d (t,T ) (KΦ (−d 2 )−F (t,T ) Φ (−d 1 )) (4.26) where d 1 = σ (t,T ) 2 , d 2 =−d 1 (4.27) and the price of a forward contract with no wrong way risk prior to default is given by Π t,χ D fwd = F (t,T ) (P c (t,T ) Φ (d 1 ) +P d (t,T ) Φ (d 2 )) −K (P c (t,T ) Φ (d 2 ) +P d (t,T ) Φ (d 1 )) (4.28) 4.2 Swap Contract Valuation A commodity swap contract is an agreement to exchange fixed for float rate multiple times in the future. The floating rate is the spot price of the commodity and the fixed rate is a rate agreed upon at the time of the contract. A swap is in fact a portfolio of 76 4.2 Swap Contract Valuation forward contracts with different maturities. Denote the exchange times in the future by T ={T 1 ,T 2 ,...,T n }, and the fixed price agreed upon to be paid by swap buyer to swap seller by K. The swap buyer is the fixed-payer party that agrees to buy the commodity at a fixed price, and the swap seller is the fixed-receiver 9 . Without loss of generality, we focus on the valuation from the point of view of the swap buyer. The argument for swap seller is completely analogous. The value of a payer swap contract at time t with no default risk is given by Π (t,χ swp ) = E Q T " n X i=1 P (t,T i ) (S (T i )−K) # = n X i=1 P (t,T i ) (F (t,T i )−K) = n X i=1 Π (t,χ fwd i ) (4.29) where F (t,T i ) are commodity forward prices for maturity T i , and Π (t,χ fwd i ) denotes the price of forward contract i, 1≤ i≤ n. Swap strike is the value of K that sets the price of swap contract at inception equal to zero, and is given by K = P n i=1 P (t,T i )F (t,T i ) P n i=1 P (t,T i ) (4.30) The value of a swap contract for a general strike K can be written in terms of swap rate denoted by S n (t) as Π (t,χ swp ) = n X i=1 P (t,T i ) (S n (t)−K) =M (t) (S n (t)−K) (4.31) where the term M (t) := P n i=1 P (t,T i ) is a portfolio of zero-coupon bonds equivalent to 9 Payer swap is also a term used to refer to the fact that the buyer pays the fixed rate. 77 Chapter 4 Defaultable Energy Derivatives Valuation annuity term in interest rate terminology. The payoff of a swap at expiry is given by Equation 4.31, nonetheless, it is not paid at any single time. The swap rate payments effectively happen at all payment dates T ={T 1 ,T 2 ,...,T n }. The payoff of a defaultable swap contract when each forward contract is independently valued as a defaultable T- claim is given by χ D swp (T) =E Q T t " n X i=1 P (t,T i ) R c + (1−R c )1 {τc>T i } (S (T i )−K) + # −E Q T t " n X i=1 P (t,T i ) R d + (1−R d )1 {τ d >T i } (S (T i )−K) − # (4.32) Following the same steps as previous section results in the following formula for a de- faultable swap valuation prior to default χ D swp (T) = n X i=1 P c (t,T i ) E Q T i c t [S (T i )|H t ]−K + − n X i=1 P d (t,T i ) E Q T i d [S (T i )|H t ]−K − (4.33) This is essentially a portfolio of options that are valued under risky probability measure. We emphasize that the expectations are taken underQ T i c measures. Substituting for risky forward rates (Equation 4.24, and Equation 4.25) risky swaption price is given by Π t,χ D swp = n X i=1 n F (t,T i ) A + c (T i ) +A − d (T i ) −K B + c (T i ) +B − d (T i ) o (4.34) where A ± x (T i ), and B ± x (T i ) x∈{c,d}, i∈{1,...,n} are given by Equation 4.25, with T replaced with T i . 78 4.3 Risky Swap Rate 4.3 Risky Swap Rate The strike price that sets the price of a risky swap equal to zero is given by K = P n i=1 F (t,T i ) A + c (T i ) +A − d (T i ) P n i=1 B + c (T i ) +B − d (T i ) (4.35) IngeneralA ± x (T )6=B ± x (T ). Intheabsenceofwrongwayriskcorrelationtermsdisappear and Equation 4.35 simplifies to K = P n i=1 F (t,T i ) (P c (t,T i ) Φ (d i1 ) +P d (t,T i ) Φ (d i2 )) P n i=1 (P c (t,T i ) Φ (d i2 ) +P d (t,T i ) Φ (d i1 )) (4.36) When dealer and counterparty have same term structure of default probabilities, or P c (t,T i ) = P d (t,T i ) ,i = 1,...,n, Equation 4.36 defines a new annuity term equivalent to that of Equation 4.30. 4.4 Risky Swap as a Basket of Forwards The above argument prices each risky forward contract independently, as a defaultable T-claim. The value of the portfolio of the forwards is given by the sum of the risky value of each forward contract. On the other hand, a second view is to price swap as a basket of underlying forward contracts and assume the dynamics of swap rates as the underlying instruments, similar to pricing of swaption contracts. Pricing problem is different here from the conventional swap pricing, in that our methodology required pricing positive and negative parts of each future payoff which occur at different future times, separately. Let’s consider a swap contract with swap rate S n which pays off at n future times{T 1 ,...,T n } with the fixed rate K. A swap contract can be valued as the 79 Chapter 4 Defaultable Energy Derivatives Valuation difference between its positive and negative parts, i.e. Π t,χ D swp = Π + t,χ D swp − Π − t,χ D swp (4.37) where the positive part of a defaultable swap contract is given by Π + t,χ D swp = ( n X i=1 P (t,T i ) (S n (t)−K) ) + ={M (t) (S n (t)−K)} + (4.38) whereswappayoffiscalculatedasthepayoffofthewholeportfolioandthendiscountedat the appropriate risky zero-bond. The payoff of the swap portfolio at timeT i <T <T i+1 , where T i , and T i+1 are swap payment dates, is calculated as payoff of a swaption with payments from timeT i+1 ,...,T n . As noted in the literature (see [50] for example) a payer swaption price with expiry T where T <T 1 <...<T n , and strike K, can be calculated based on an extension of Black’s model. Swap rate S n (T ) is assumed to be lognormally distributed with forward swap rate of S n (t), and volatility σ 2 n T: dS n (t) =σ n S n (t)dW (t) (4.39) Swap rate at time T then is given by S n (T ) =S n (t)exp −σ 2 n (T−t)/2 +σ n √ T−tW (t) (4.40) A standard payer swaption payoff denoted by χ p−swptn at expiry time T is given by: χ p−swptn =M (T ) (S n (T )−K) + (4.41) 80 4.4 Risky Swap as a Basket of Forwards It follows that a payer swaption is priced similar to a regular caplet, with typical discount factor replaced by annuity term: Π (t,χ p−swptn ) =M (t) (S n (t) Φ (d 1 )−KΦ (d 2 )) (4.42) where d 1 = log (S n (t)/K) +σ 2 n (T−t)/2 σ n q (T−t) d 2 =d 1 −σ n q (T−t) (4.43) We price the risky swap through discounting the positive and negative part of the swap contract at future payoff dates at the risky curves associated with the positive and nega- tive parts of the payoff. As mentioned earlier, assumption of lognormality for swap rate is not compatible with lognormality assumption for forward rates. In our numerical results we assume forward rates are log-normally distributed with a given correlation structure and derive future simulated swap rates based on Monte Carlo simulation of forward rates into the future given the volatility and correlation structure. We then present numeri- cal results and compare them with defaultable swap value with CVA/DVA adjustments based on Monte Carlo based swaption pricing introduced in chapter 2 (see Figure 2.2). 81 5 PDE Representation of OTC and Exchange Traded Derivatives Futures contracts and futures options are instrumental for hedging future price risk for commodity producers. Futures contracts entitle the buyer to claim physical delivery of the commodity from the seller at a specific delivery point at a specified date in the fu- ture. FuturescontractsaretradedonanexchangesuchasNewYorkMercantileExchange (NYMEX) or Intercontinental Exchange (ICE) for standardized volumes for delivery of a specific commodity on predetermined dates. These contracts offer less flexibility in con- tract terms compared to forward contracts, but due to their standard nature are actively traded on the exchanges by both physical commodity producers and financial players. Exchanges mitigate default risk through clearhousing operations. Clearing houses settle contracts on a daily basis and collect and maintain margin in trading accounts of trad- ing members. Through collecting margins a clearing house assures that the contracts are settled on a daily basis and the loosing party deposits current contract value in the margin account. Futures contracts are therefore considered to have minimal counter- party default risk. Implied in the assumption is that clearing house and exchanges are default-free entities. Exchangetrading isa settingby whichfull collateralization is required. Tradingparties can avoid credit risk due to this full collateralization mechanism. Although the nature of 83 Chapter 5 PDE Representation of OTC and Exchange Traded Derivatives riskisnotchanged, clearhousingessentiallytransfersdefaultriskbacktothecounterparty by requiring margin posting. We therefore expect OTC and exchange pricing to reflect the same type of risk in different forms, one through cost of credit and the other through cost of collateral posting. In this chapter we introduce partial differential equation for pricing of a derivatives when full collateralization is required. In our general setup, we assume both parties post collateral, and the pricing equation reflects the cost of funding these margin accounts. Prior work in this domain include [11, 12, 41, 42]. [11] derives a modified partial differen- tial equation representation for a general financial derivative using a hedging argument, andcontraststhatwiththestandardBlack-ScholesPDE.ThediscreteanalogofthisPDE is the Cox-Ross-Rubinstein (CRR) binomial tree which is heavily used by practitioners. [29] develop a modified CRR binomial tree based on the PDE derived in [11]. In this chapter we rely on some of the setup in these two papers. In our setup we compare pricing PDE of an OTC contract where counterparty and dealer credit risk are priced in, with same contract when traded on the exchange with full collateralization requirement. In the first case, hedging portfolio consists of underlying asset, risk free bond, and defaultable bonds issued by the bank and the counterparty. In the second case, the setup includes a margin account which at each point in time holds the value of the derivative and attains risk free rate. Margin requirement is met by each side of the contract through borrowing from a different account, namely a funding account. This account lends cash at an unsecured funding rate. In the next section we briefly introduce cost of capital in an exchange setting where full collateralization is achieved through daily margining, and proceed into our model setup and derivations. 84 5.1 Cost of Credit vs. Cost of Capital 5.1 Cost of Credit vs. Cost of Capital Exchange traded derivatives such as futures and futures options are assumed to be fully collateralized derivatives, bearing minimal counterparty credit risk. Exchanges achieve their objective through setting margin requirement in place and daily clearing of financial transactions. The two sides of a contract are required to post cash or cash equivalent collateral to their margin account equivalent to the value of the losses on the transaction, when the transaction is out of the money from their perspective. Clearing house then essentially transfers these funds from the margin posting party to margin collecting in their daily clearing process. When the contract is in the money, the clearing house will deposit funds into the associated margin account, and when it is out of the money the funds are called out. The margin account accrues interest close to risk-free rate on the funds. However, the cost associated with depositing funds into margin account is usually far greater than the risk free rate. This inherently introduces a type of asymmetry in pricing of the contract when it is in-the-money versus out-of-the-money, essentially translating into different discount regimes for in-the-money and out-of-the-money contract. Market assessment of default probability of a firm, based on its capital structure, and the risk associated with business activities among other factors, determine the effective cost that each firm incurs for borrowing. We consider a situation where a bank has two options to trade a financial contract: • Trade over-the-counter and incur cost of credit • Trade on an exchange and incur cost of capital It is worth emphasizing that the two scenarios above are the two extremes in a whole spectrum of existing bilateral agreement settings. After the financial crisis the percep- 85 Chapter 5 PDE Representation of OTC and Exchange Traded Derivatives tion of severity and consequences of default fundamentally changed the market place. Essentially taking full credit risk for the majority of firms is deemed too costly. As a way to mitigate this risk, counterparts require the usage of bilateral collateralization mechanisms. Requiring collateral posting in bilateral contracts has become commonplace particularly under ISDA master agreements. The nature of risk taking can essentially be viewed similarly in both cases. In the first case, cost of credit is what a firm incurs for transacting with risky counterparties. In the second case, this risk is fully mitigated or essentially is transferred back to the counterparty through daily margining. Pricing equation should therefore theoretically result in the same valuation in both circumstances. In the rest of this chapter we briefly discuss the mechanics of margining and collateralization fees, and contrast that with taking on full credit risk. 5.1.1 Margining Margining mechanics varies by different exchanges and instruments. [5] discusses at length differences in rules and regulations by various instrument types. Instrument payoff is settled daily through variation margin cashflows in an exchange setting. On the other hand, bilateral contracts with no collateral agreement in place are settled at contract maturity. This effectively reduces exposure volatility through accumulating wealth over time in the margin account. As outlined in Figure 5.1, time scaling of volatility effectively translates to larger risk in the final portfolio value in the first case where no margining takes place, whereas in the second case, the margin account absorbs the volatility over the lifetime of the contract. Conceptually, what is left in the valuation of the portfolio is one day volatility. 86 5.1 Cost of Credit vs. Cost of Capital Figure 5.1: Margining and Volatility 5.1.2 Collateralization In addition to monitoring credit worthiness of the counterparts, the exchanges and CCPs require an initial margin and collateral deposits in order to guarantee payments in a counterparty’s default scenario. The instruments are marked to market daily and their payoff is settled through variation margin cashflows. The exchange essentially stands between the two sides of a contract and guarantees payments of funds through this full collateralization mechanism (see Figure 5.2). As a mean to mitigate counterparty risk, bilateral counterparts have also resorted to master agreements laid out by ISDA and other regulatory bodies, with collateral requirements in place. Bilateral collateralization differs from margining largely in that daily settlement of mark to market is replaced with periodic exchange of collateral, once exposurereachesacertainthresholdsetfortandagreeduponbybothparties. Inaddition, cash collateral can be replaced with other types of collateral (such as other financial instruments or assets). In this case haircuts are assigned that discount the market value of the collateral. Haircut is intended to capture the liquidity risk should the non-cash collateral of the distressed counterparty be sold in the market to pay for losses due to default. 87 Chapter 5 PDE Representation of OTC and Exchange Traded Derivatives In the next section we derive the pricing equation for a derivatives contract in the above scenarios. In one pricing reflects cost of credit and in the other cost of capital is accounted for in the governing PDE. Figure 5.2: Exchange Traded vs. Over The Counter Derivatives 5.2 Model Setup As outlined earlier, this chapter aims at comparing cost of credit - through valuation of an OTC derivative without collateralization - with cost of capital- through valuation of a fully collateralized derivative. We differentiate the scenarios where the payoff of the derivative is positive to the bank vs. negative. In contrast with some recent works that attempt at pricing from the point of view of the seller, we take a global view on the economy and assume that the replicating portfolio should cover the aggregate costs for both parties. When the payoff of the instrument is always positive (such as a futures option or swaption), and the bank is the seller of the contract, the bank is required to post collateral when the derivative is out of the money to the bank. Cost of borrowing for the bank will appear in pricing equation in this case. On the other hand, if the bank is the buyer of the derivative, and the derivative is out of the money, the bank is not required to post any collateral in the margin account, while the counterparty will be liable to provide cash in the margin account. Our pricing equation accounts for both side 88 5.2 Model Setup costs. Similar to Black-Scholes setting, we assume that the economy is composed of a risk-free zero-couponbonddenotedbyR r , andtheunderlyingcommodityspotpricedenotedbyS. In addition, we assume that the economy includes risky zero-coupon zero-recovery bond denoted by P c issued by the counterparty, and a zero-coupon zero-recovery dealer bond denoted byP d . LetJ c , andJ d denote the default indicator functions of the counterparty and dealer respectively. Also, letλ c , andλ d denote default intensities of the counterparty and dealer. We assume the following simplified equations specify the dynamics of each of these asset classes. dP r =rP r dt dP c =P c (r c dt−dJ c ) dP d =P d (r d dt−dJ d ) dS =S (μdt +σdW ) (5.1) All parameters in the model are assumed to be deterministic functions of time where we have omitted time in the equations for the sake of simplicity. W is the standard Wiener process. Given the above dynamics, we assume funding is available to the firm at funding rate equal to unsecured bond yield ˜ r c with recovery rate R c ˜ r c (t) =R c r (t) + (1−R c )r c (t) (5.2) where r c is risky bond yield with zero recovery as defined in Equation 5.1. Recoverable proportion returns risk-free rate, and the non-recoverable portion returns higher rate which compensate for default probability 1 . Similar relationship holds for dealer bond 1 Default intensity and excess return of corporate bond have the following relationship. 89 Chapter 5 PDE Representation of OTC and Exchange Traded Derivatives with recovery R d . 5.3 Cost of Capital in Collateralized Derivative Valuation Without loss of generality, we assume that the dealer or the bank is the seller of the derivative contract in the rest of this chapter. We start with a general payoff function for the derivatives contract at a future date, and later assume that the derivative contract can take either only positive or only negative values under all states of the world. In our full collateralization scenario, both sides of the contract are required to post cash collateral when the derivative goes out of the money from their perspective. This means that there are times when the derivative goes out of the money for the dealer and the dealer is required to post margin to margin account, while when the derivative is in the money to the dealer, the counterparty should fulfill margin requirements. In this scenario we assume margin posting is done instantly and the margin account hold cash equal to total mark-to-market value of the derivative. The margin account re-invests this cash at risk-free rate. Later we will focus on caplet and floorlet valuation in the context of forward pricing, but for now we stay as general as we can. Denote the risk-free value of the contract by V, and the value of the same contract including cost of capital by ˆ V. As in the usual Black-Scholes framework, the replicating portfoliocanhedgethepayoffofthederivativethroughholdingα S unitsoftheunderlying commodity S. In our setup, the hedging portfolio consists also of a margin account M, ˜ r c −r =λ c (1−R c ) , ˜ r d −r =λ d (1−R d ) where R c , and R d are counterparty and dealer recovery rates. 90 5.3 Cost of Capital in Collateralized Derivative Valuation and a cash account γ. The value of hedging portfolio Π at time t is given by Π t =α S S t +M t +γ t (5.3) The margin account holds the margin amount equivalent to derivative’s value and accrues risk free rate r. M t = ˆ V t (5.4) Ifthedealerispostingmargin,thecosttoborrowthisreflectthedealer’scostofborrowing capital. On the other hand, if the counterparty is posting collateral, the cost to borrow is counterparty’s cost of borrowing. Cash account can be decomposed to four components: γ R , or repo account which supplies the cash required to maintain the underlying position, counterparty’s funding account denoted by γ Fc , dealer’s funding account γ F d , and a risk free account, γ r , that holds all excess cash and accrues risk free rate. γ t =γ R +γ Fc +γ F d +γ r (5.5) The rate paid on a repurchase agreement is usually fairly close to risk free rate. To keep the argument general, we denote this rate by r R . On the other hand, the funding accounts accrue unsecured funding rate ˜ r c , and ˜ r d . Replicating portfolio Π t offsets the value of the derivatives contract at all time − ˆ V t = Π t (5.6) Self-financing condition implies that the evolution of the hedge portfolio after dt and 91 Chapter 5 PDE Representation of OTC and Exchange Traded Derivatives before re-balancing satisfies dΠ t =α S dS t +dM t +dγ t (5.7) where dM t =rM t dt =r ˆ V t dt (5.8) and dγ t =dγ R +dγ Fc +dγ F d +dγ r (5.9) Since γ R finances α S units of the underlying, the cost to finance purchasing of the un- derlying equals the change in repo account, accruing interest at repo rate r R dγ R =−α S S t r R dt (5.10) For the counterparty, the cost of posting on margin account when the derivative is out of the money is dγ Fc =−M + t ˜ r c (t)dt =− ˆ V + t ˜ r c (t)dt (5.11) 92 5.3 Cost of Capital in Collateralized Derivative Valuation Similarly, dealer funding account accrues ˜ r d when cash is borrowed dγ F d =−M − t ˜ r d (t)dt =− ˆ V − t ˜ r d (t)dt (5.12) Substituting Equation 5.10, Equation 5.11, and Equation 5.12 in Equation 5.7 yields −d ˆ V =dΠ t =α S dS t +r ˆ V t dt−α S r R S t dt− ˜ r c ˆ V + t dt− ˜ r d ˆ V − t dt = (μ−r q )α S S t +r ˆ V t − ˜ r c ˆ V + t − ˜ r d ˆ V − t dt +α S σS t dW (5.13) From Ito’s lemma, for a general claim ˆ V (t,S) we have d ˆ V = ∂ ˆ V ∂t + ∂ ˆ V ∂S μS t + ∂ 2 ˆ V ∂S 2 σ 2 S 2 t ! dt + ∂ ˆ V ∂S σS t dW (5.14) From Equation 5.13 and Equation 5.14, hedge ratio should satisfy α S = ∂ ˆ V ∂S (5.15) It follows that ˆ V is the solution to the following PDE ∂ ˆ V ∂t + ∂ ˆ V ∂S r q S t + ∂ 2 ˆ V ∂S 2 σ 2 S 2 t − ˜ r c ˆ V + t − ˜ r d ˆ V − t = 0 (5.16) V (t,S) on the other hand is the solution to Black-Scholes PDE in the form of ∂V ∂t + ∂V ∂S r q S t + ∂ 2 V ∂S 2 σ 2 S 2 t −rV = 0 (5.17) 93 Chapter 5 PDE Representation of OTC and Exchange Traded Derivatives 5.3.1 Derivative price is always positive If the derivative takes only positive value, the pricing PDE can be simplified to ∂ ˆ V ∂t + ∂ ˆ V ∂S r q S t + ∂ 2 ˆ V ∂S 2 σ 2 S 2 t − ˜ r c ˆ V t = 0 (5.18) Equation 5.18 is in the form of Black-Scholes PDE with the risk free rater replaced by a new discount rate ˜ r c defined above. This is the rate of return of a bond with recoveryR c . Next, focusing on a forward contract, assume that the terminal payoff of the derivative is given by ˆ V (T,S) = (S T −K) + (5.19) Fayman-Kac representation of ˆ V (t,S) satisfying the above PDE equation and terminal condition is given by ˆ V (t,S) =E Q T t h P c (t,T ) (S T −K) + i (5.20) where P c (t,T ) is the discount factor with rate ˜ r c over time [t,T ]. 5.3.2 Derivative price is always negative If the derivative takes only negative value, the pricing PDE can be simplified to ∂ ˆ V ∂t + ∂ ˆ V ∂S r q S t + ∂ 2 ˆ V ∂S 2 σ 2 S 2 t − ˜ r d ˆ V t = 0 (5.21) Similar to prior example, let’s consider the negative of the payoff of a floorlet at expiry ˆ V (T,S) = (S T −K) − (5.22) 94 5.4 Cost of Credit in Uncollateralized Derivative Valuation Fayman-Kac representation of ˆ V (t,S) satisfying the above PDE equation and terminal condition is given by ˆ V (t,S) =E Q T t h P d (t,T ) (S T −K) − i (5.23) where P d (t,T ) is the discount factor with rate ˜ r d over time [t,T ]. 5.4 Cost of Credit in Uncollateralized Derivative Valuation In this section, we consider an OTC like derivative without collateralization. We will use the framework developed in [11] to construct the hedging portfolio. The seller of the derivative will incur a loss if the value of the contract is positive to the seller and the counterparty defaults. In order to hedge against this loss, the seller can take a short position α d in counterparty’s defaultable bond P d . The seller will benefit from its own default if the value of the contract is negative and the seller defaults. In this case, the seller can take a long positionα c in it’s own bondP c . We assume that once the possibility of default is fully accounted for through the positions in defaultable bonds, the rest of the cash can be borrowed or invested in the cash account at risk free rate. This account does not consider any default probabilities through higher spreads because the rest of the replicating portfolio prices default risk. Similar to previous section, letV denote the value of the contract when cost of credit is not considered, and denote by ˜ V the value of the same contract including cost of credit. The replicating portfolio consists of α S units of the underlying commodity S, α d units of bank’s defaultable bond P d , α c units of counterparty’s defaultable bond P c , and cash 95 Chapter 5 PDE Representation of OTC and Exchange Traded Derivatives account γ. The value of hedging portfolio Π at time t is given by Π t =α S S t +α c P c +α d P d +γ t (5.24) Similar to prior setting, we have the following composition for the cash account: dγ t =dγ R +dγ Pc +dγ P d +dγ r (5.25) where γ R finances α S units of the underlying and incurs repo rate, γ Pc , and γ P d are borrowing or lending to maintain bond positions and γ r is the remainder of replicating portfolio. We assume borrowing and lending rates are the same and equal to risk free rate r, sincecounterpartyriskisaccountedforbyholdingapositionindealerandcounterparty bonds. Afterdt, the change in cash account before re-balancing the portfolio is given by dγ t =−α S r q Sdt−α c P c rdt−α d P d rdt−r ˜ Vdt (5.26) From self financing condition hedge portfolio should also satisfy the following dΠ t =α S dS t +α c dP c +α d dP d +dγ t (5.27) The hedge portfolio should always offset the value of the derivative. Consequently we can rewrite d ˜ V as follows −d ˜ V =dΠ t =α S dS t +α c dP c +α d dP d +dγ t (5.28) The first three terms are expressed in the dynamics of fundamental assets in the economy 96 5.4 Cost of Credit in Uncollateralized Derivative Valuation in Equation 5.1. After substitution and rearranging Equation 5.28 we have −d ˜ V =α S S (μdt +σdW ) +α c P c (r c dt−dJ c ) +α d P d (r d dt−dJ d ) −α S r q Sdt−α c P c rdt−α d P d rdt−r ˜ Vdt = n α S S (μ−r q ) +α c (r c −r)P c +α d (r d −r)P d −r ˜ V o dt +α S σSdW−α c P c dJ c −α d P d dJ d (5.29) A modification of Ito’s lemma when the derivative ˜ V (t,S,J c ,J d ) is a jump diffusion process is given by the following d ˆ V = ∂ ˆ V ∂t + ∂ ˆ V ∂S μS t + ∂ 2 ˆ V ∂S 2 σ 2 S 2 t ! dt + ∂ ˆ V ∂S σS t dW + Δ ˜ V c dJ c + Δ ˜ V d dJ d (5.30) where the last two terms correspond to the value of the contract that is lost in the event of default. If the contract is in the money to the bank, only R c portion of the value is recovered. Thus we have Δ ˜ V c = ˜ V (t,S, 1, 0)− ˜ V (t,S, 0, 0) =R c ˜ V + + ˜ V − − ˜ V =− (1−R c ) ˜ V + (5.31) similarly Δ ˜ V d =− (1−R d ) ˜ V − (5.32) 97 Chapter 5 PDE Representation of OTC and Exchange Traded Derivatives It follows from Equation 5.29, and Equation 5.30 that the weights must satisfy α S =− ∂ ˜ V ∂S α c =− (1−R c ) ˜ V + P c α d =− (1−R d ) ˜ V − P d (5.33) ˜ V is the function that holds the following PDE true. ∂ ˜ V ∂t + ∂ ˜ V ∂S r q S t + ∂ 2 ˜ V ∂S 2 σ 2 S 2 t −r ˜ V = (1−R c ) (r c −r) ˜ V + + (1−R d ) (r d −r) ˜ V − (5.34) 5.4.1 Derivative price is always positive Whenthederivativevalueisalwaysnon-negativethesecondtermontheRHSofEquation 5.34 is eliminated and the positive part of the derivative equals its value. Under this scenario Equation 5.34 simplifies to ∂ ˜ V ∂t + ∂ ˜ V ∂S r q S t + ∂ 2 ˜ V ∂S 2 σ 2 S 2 t −r ˜ V = (1−R c ) (r c −r) ˜ V (5.35) After re-arranging terms ∂ ˜ V ∂t + ∂ ˜ V ∂S r q S t + ∂ 2 ˜ V ∂S 2 σ 2 S 2 t − ˜ r c ˜ V = 0 (5.36) where ˜ r c denotes unsecured funding rate given by Equation 5.2. Comparing Equation 5.36 and the PDE derived when cost of capital is included in Equation 5.18 reveals that under the setup of the economy assumed in Equation 5.1, the governing PDE in this case is identical to the one in Equation 5.18, which represents cost of capital. 98 5.4 Cost of Credit in Uncollateralized Derivative Valuation 5.4.2 Derivative price is always negative Similarlywhenthederivative’svalueisalwaysnegativetotheseller,thePDEinEquation 5.33 simplifies to ∂ ˜ V ∂t + ∂ ˜ V ∂S r q S t + ∂ 2 ˜ V ∂S 2 σ 2 S 2 t − ˜ r d ˜ V = 0 (5.37) where ˜ r d denotes dealer’s unsecured funding rate of the dealer. ˜ r d =R d r + (1−R d )r d (5.38) Similarly, this equation is identical to the one which accounts for cost of capital in Equation 5.21. Above derivations show that cost of credit and cost of capital are two different but closely related mechanisms that account for losses due to default in pricing equation when complete market conditions are not satisfied. This is essentially a departure from the Black-Scholes world which assumes that both parties honor their obligations at maturity. The pricing PDE however is closely related to Black-Scholes PDE with the risk free discount rate replaces with higher discount rates representing default probabilities. 99 6 Numerical Results In previous chapters we formulated defaultable derivatives prices based on discounting future cashflows under risky curves that reflect default probabilities. In our discount based approach the positive and negative parts of the contract are discounted under two discount curves with the yields that reflect market perception of riskiness of the two sides of the contract. In this chapter we present numerical results that compare derivative risky value derived from our formulation and contrast it with adjustment based approach which relies on CVA/DVA for fair value pricing. We consider two case studies of valuing crude oil forward and swap contracts between a AA rated hypothetical bank and a crude oil producer with lower credit rating. As described earlier, if a futures contract is considered risk-free and is striked at K to result in zero contract value at inception, the price of an equivalent forward contract with the same value of K results in a value different from zero at inception. The price is given by the difference between a cap and a floor contract as described in prior chapters. Our numerical analysis set includes 24 Brent futures contract prices traded on Inter Continental Exchange (ICE) for expiry months of May 2015 through April 2017. One year of historical evolution of these prices (observed prior to April 15, 2015) are shows in Figure 6.1, where Contract 1 indicated the futures contract with closest expiry observed on April 15, 2015 i.e. May2015 contract and so on 1 . Another view of the same data 1 The plot does not show all 24 contracts for better readability. 101 Chapter 6 Numerical Results is to observe the commodity forward curve on different dates. Figure 6.2 shows the forward curve observed on April 15, 2015, where thex axis plots contract expiry starting with May2015 and ending April2017, and the y axis plot the observed prices for those contracts. Figure 6.3 shows the discount factors associated with risk-free zero-bond, prototypical BBB rated crude oil producer, and prototypical AA rated banking sector entity 2 . The producer with a BBB rating has the largest spread with an upward sloping yield curve. When single name spreads are available they can be used to reflect exact market infor- mation. Implied in these rates are market view on recovery rates and the probability of default. Table 6.1 shows model parameter assumptions including forward contract volatilities and volatility assumption for discount factors. In the first case study we calculate the price of forward contracts equivalent to the futures contracts in our dataset, based on the two methods mentioned earlier. Larger counterparty spread when compared to the bank’s spread implies that the forward con- tractsatinceptionshouldhavenegativevaluetothebank; thebankshouldonlybewilling to enter the forward contract at a fee which reflects higher counterparty risk compared to the bank’s own. A second look at the same results can be to compare the commodity forward curve derived from futures contracts with the commodity forward curve implied by equivalent forward contracts. These results and the discussions are presented in the following section. We have carried out similar comparison for swap contracts with payment frequencies from 1 to 12 months, in the second case study. The swap contract with a single payment is essentially identical to May2015 forward contract. The swap with 2 payments is a weighted average of May2015 and June2015 contract, and so on. The prices of these 2 The curves are extracted from Bloomberg and where the terms were not available, the initial values were interpolated linearly to estimate spreads at missing terms. 102 6.1 Discussion swap contracts based on the two methodologies above are presented and discussed in the following section. Finally we have examined different right and wrong way risk scenarios and compared risky contract values in each scenario with the value when correlation is assumed negligi- ble. The correlation parameter in the formulation derived in previous chapters essentially dictates the extent of future payoff and default correlation and provide us with a simple way to quickly assess the impact in valuation. 6.1 Discussion The first series of results compare adjustment based risky forward contract value with the resultsfromtheproposeddiscountingmethod. Futurerealizationofcommoditypricesare simulated by a Monte Carlo simulation engine. Monte Carlo method relies on historical covariance matrix of asset prices and assumes future price realizations are governed by the same covariance matrix. Therefore, the first step in Monte Carlo simulation involves estimating covariance matrix of these forward contracts based on historical return data. The Monte Carlo engine designed here is slightly different from the conventional ones to accommodate the fact that the 24 assets present in covariance matrix calculation each have a different expiry date. In other words, the engine should have a built-in logic to stop price evolution after each contract’s expiry date. The Cholesky decomposition of the covariance matrix is then used to simulate the evolution of commodity forward rates where simulation stops after the expiry date. Figure 6.4 shows sample simulated future paths of contract 24 which expires after 24 months following simulation date. Simulation iteration is set to 1000 paths with 24 future expiry dates. The commodity forward rates are assumed to have log-normal distribution with their covariance matrix derived from the historical data as previously described. 103 Chapter 6 Numerical Results Table 6.2 lists commodity forward rates derived from risk-free futures contracts, along with the risky value of the forward contracts based on the two methodologies. As can be seen from the results, the two methods closely track one another. Next we analyze the price of swap contracts with twelve different maturities listed in Table 6.3. These are the contracts whereby the bank agreed to buy crude oil from the producer each month up to the month of the expiry, at a fixed swap rate. The risk free swap rate is calculated from risk free futures contract values at inception. Monte Carlo simulation is again used to compare the value of a swap striked at these risk free swap rates based on CVA/DVA adjustment and contrasts the results with the proposed discounting method. The calculated price of the contract can also be expressed as a new “risky” swap rate through simply shifting the original swap rate by the new value of the contract divided by annuity term. Table 6.3 lists the two adjusted swap rates calculated through this approach. As the table shows the two methodologies result in very close adjustments to the swap rates. These results demonstrate that discount based method essentially reflects the same contract value as adjusting the price by CVA/DVA. Discount based method however greatly simplifies pricing through reliance on existing pricing infrastructure of the bank, where the risk-free rates are replaced with risky curves, and the positive and negative values of the contracts are calculated through existing option pricing models. Next, we demonstrate the impact of WWR and RWR in contract valuation. Figure 6.5 shows the risky value of each forward contract at inception under different correlation scenarios, andFigure 6.6showsthesameresultstranslatedintocommodityforwardcurve for the same correlation factors. Figure 6.6 can also be interpreted as the implied strike prices that set the value of the risky forward contracts equal to zero at inception. The difference between strike prices of risky forward and futures contracts increase as contract 104 6.1 Discussion expiry increases. In Figure 6.6 the solid red line shows the original commodity forward curve derived from the futures contracts, where each contract is priced in a default risk-free world. The second scenario withρ Slc = 0, ρ Sl d = 0 in dashed blue line demonstrates the commodity forward curve derived from risky forward contracts as defined in Equation 4.24, when correlation between commodity spot price and credit risk is assumed zero 3 . As the graph shows, due to substantial difference between the discount curve of the producer and the bank’s own, the bank assumes credit risk and therefore the futures contract is valued less than it’s risk-free equivalent. This is reflected in the lower commodity forward curve. Large positive value of correlation between commodity price and discount curve have different interpretations in terms of their impact on the value of the contract. A positive ρ Slc implies that higher default probability of the counterparty coincides with higher commodity price, and ultimately higher contract value to the bank. This is an instance of wrong way risk from the point of view of the bank. The third scenario withρ Slc = 0.2, ρ Sl d = 0 compares commodity forward curve when wrong way risk is present. A further downward shift in the curve shows that the contracts are valued less to the bank due to WWR. A second instance of WWR is in fact when there is a positive correlation between commodity price and the bank’s own credit risk. This implies that higher commodity spot price, and consecutively lower values of the floor coincide with higher probability of the bank’s default. The bank benefits less from its own default if correlation between commodityspotpriceanddefaultprobabilityispositive. Contrastingforwardcurvewhen ρ Slc = 0, ρ Sl d = 0.2 versus ρ Slc = 0, ρ Sl d = 0 case illustrates contracts are less valuable to the bank in the former case. Finally ρ Slc = 0.2, ρ Sl d = 0.2 shows the double impact of both counterparty and bank’s correlation with spot price dynamics. 3 Commodity future price is then the sum of contract value and original futures price. 105 Chapter 6 Numerical Results The next series of results in Figure 6.7, and Figure 6.8 are closer proxies to the real- world, wherethecorrelationbetweenbank’sdefaultandcommodityspotpriceisassumed zero, while the producer’s default is strongly correlated with commodity spot price. A bank with a diversified portfolio can be assumed to have a credit quality independent from the value of the contract, i.e. zero correlation. On the other hand, It is expected thatwhenthefix-payerofacommodityswapcontractisthecommodityproducer, thesit- uations where the forward contract is in the money to the bank (future commodity prices are higher than the fix price), the producer is generating larger than average cash-flows, hence has better credit quality (right-way risk). Forcomparisonpurposeswehaveincludedbothpositiveandnegativecorrelationcases, while as previously explained, in real world a producer’s default is likely negatively cor- related with commodity prices (ρ Slc =−0.5, ρ Sl d = 0, andρ Slc =−0.8, ρ Sl d = 0). Com- paring the results show that the bank benefits from negative counterparty-commodity price, and the stronger the correlation, the higher the benefit to the bank. When corre- lations are assumed to beρ Slc =−0.8, andρ Sl d = 0, the contracts further out are valued almost $1.5 (2%) higher than their risk-free counterparts. Finally, Figure 6.9, and Figure 6.10 contrast different scenarios of right and wrong way risk for both the bank and producer. The forward contracts are valued the highest when the correlation between spot price and both bank’s credit risk and counterparty’s credit risk are assumed negative. In this scenario, bank’s exposure to the counterparty is lower when the counterparty is more likely to default. In addition bank’s likelihood of default is highest when the contract is more likely to be out of the money to the bank. Under such circumstances the bank should be willing to pay a premium to enter the forward contracts with the producer, as reflected in the upward sloping curve in Figure 6.9. 106 6.2 Results 6.2 Results Table 6.1: Model Parameters σ S σ lc σ l d 37% front contract to 22% back contract 20% 20% Figure 6.1: ICE Brent futures contracts historical prices 107 Chapter 6 Numerical Results Table 6.2: Risky forward contract value - adjustment vs. discounting method Risk-free commodity forward Forward value - Ad- justment Method ($) Forward value - Discount Method ($) Adjustment Method Diff (%) Discount Method Diff (%) May’15 60.32 -0.047 -0.047 0.1% 0.1% Jun’15 63.32 -0.066 -0.065 0.1% 0.1% Jul’15 64.13 -0.092 -0.091 0.1% 0.1% Aug’15 64.81 -0.13 -0.13 0.2% 0.2% Sep’15 65.38 -0.31 -0.30 0.5% 0.5% Oct’15 65.89 -0.34 -0.32 0.5% 0.5% Nov’15 66.37 -0.40 -0.41 0.6% 0.6% Dec’15 66.81 -0.40 -0.38 0.6% 0.6% Jan’16 67.18 -0.48 -0.45 0.7% 0.7% Feb’16 67.5 -0.63 -0.58 0.9% 0.9% Mar’16 67.8 -0.66 -0.60 1.0% 0.9% Apr’16 68.14 -0.69 -0.64 1.0% 0.9% May’16 68.42 -0.70 -0.64 1.0% 0.9% Jun’16 68.68 -0.72 -0.66 1.0% 1.0% Jul’16 68.96 -0.74 -0.66 1.1% 1.0% Aug’16 69.22 -0.81 -0.74 1.2% 1.1% Sep’16 69.47 -0.81 -0.73 1.2% 1.0% Oct’16 69.72 -096 -0.86 1.4% 1.3% Nov’16 69.93 -0.98 -0.87 1.4% 1.2% Dec’16 70.14 -0.98 -0.86 1.4% 1.2% Jan’17 70.34 -0.98 -0.86 1.3% 1.2% Feb’17 70.54 -0.99 -0.87 1.4% 1.4% Mar’17 70.76 -0.96 -0.84 1.4% 1.4% Apr’17 70.99 -1.08 -0.97 1.5% 1.4% 108 6.2 Results Table 6.3: Risky swap contract value - adjustment vs. discounting method Swap Term (months) Risk- free Swap Rate Annuity Term Swap value - Adjust- ment Method ($) Swap value - Dis- count Method ($) Risky Swap Rate - Adjust- ment Method Risky Swap Rate - Dis- count Method 1 60.32 1.0 -0.02 -0.02 60.30 60.30 2 61.82 2.01 -0.14 -0.14 61.75 61.75 3 62.59 2.98 -0.14 -0.16 62.54 62.53 4 63.14 3.98 -0.40 -0.45 63.04 63.03 5 63.59 4.97 -0.41 -0.48 63.51 63.49 6 63.97 5.96 -0.88 -0.99 63.83 63.81 7 64.31 6.96 -1.30 -1.46 64.13 64.11 8 64.63 7.95 -1.56 -1.79 64.43 64.40 9 64.92 8.94 -2.48 -2.74 64.63 64.60 10 65.17 9.93 -2.66 -2.97 64.90 64.87 11 65.42 10.92 -3.13 -3.53 65.12 65.08 12 65.63 11.91 -3.56 -4.01 65.34 65.30 109 Chapter 6 Numerical Results Table 6.4: Risky commodity forward curve Risk-free commodity forward ρ Slc = 0 ρ Sl d = 0 ρ Slc = 0.2 ρ Sl d = 0 ρ Slc = 0 ρ Sl d = 0.2 ρ Slc = 0.2 ρ Sl d = 0.2 May’15 60.32 60.29 60.24 60.24 60.20 Jun’15 63.32 63.25 63.17 63.18 63.10 Jul’15 64.13 64.04 63.92 63.93 63.81 Aug’15 64.81 64.66 64.51 64.53 64.37 Sep’15 65.38 65.17 64.98 65.00 64.82 Oct’15 65.89 65.60 65.39 65.41 65.20 Nov’15 66.37 66.04 65.80 65.82 65.58 Dec’15 66.81 66.43 66.17 66.20 65.93 Jan’16 67.18 66.74 66.44 66.48 66.18 Feb’16 67.5 66.99 66.67 66.71 66.39 Mar’16 67.8 67.25 66.91 66.95 66.61 Apr’16 68.14 67.55 67.20 67.23 66.87 May’16 68.42 67.81 67.43 67.47 67.09 Jun’16 68.68 68.04 67.65 67.69 67.29 Jul’16 68.96 68.30 67.89 67.93 67.52 Aug’16 69.22 68.54 68.11 68.16 67.73 Sep’16 69.47 68.72 68.28 68.32 67.88 Oct’16 69.72 68.90 68.45 68.49 68.04 Nov’16 69.93 69.09 68.62 68.66 68.19 Dec’16 70.14 69.28 68.80 69.83 68.35 Jan’17 70.34 69.47 68.97 69.01 68.51 Feb’17 70.54 69.66 69.14 69.18 68.69 Mar’17 70.76 69.87 69.35 69.39 68.86 Apr’17 70.99 70.11 69.56 69.61 69.06 110 6.2 Results Figure 6.2: Crude oil forward curve observed on April 15, 2015 Figure 6.3: Risk-free and risky discount curves 111 Chapter 6 Numerical Results Figure 6.4: Monte Carlo simulation of commodity forward rates Figure 6.5: Risky forward contract values and correlation effect 112 6.2 Results Figure 6.6: Commodity forward curve associated with above scenarios Figure 6.7: Risky forward contract values right vs. wrong way risk - counterparty 113 Chapter 6 Numerical Results Figure 6.8: Commodity forward curve associated with above scenarios Figure 6.9: Risky forward contract values right vs. wrong way risk - dealer and coun- terparty 114 6.2 Results Figure 6.10: Commodity forward curve associated with above scenarios 115 Chapter 6 Numerical Results Table 6.5: Risky forward contract values right vs. wrong way risk - counterparty Risk-free commodity forward ρ Slc = 0.5 ρ Sl d = 0 ρ Slc = −0.5 ρ Sl d = 0 ρ Slc = 0.8 ρ Sl d = 0 ρ Slc =−0.8 ρ Sl d = 0 May’15 60.32 60.18 60.38 60.13 60.44 Jun’15 63.32 63.06 63.46 62.94 63.58 Jul’15 64.13 63.75 64.34 63.58 64.53 Aug’15 64.81 64.29 65.06 64.07 65.30 Sep’15 65.38 64.71 65.64 64.46 65.95 Oct’15 65.89 65.08 66.16 64.79 66.51 Nov’15 66.37 65.45 66.67 65.12 67.08 Dec’15 66.81 65.78 67.14 65.41 67.59 Jan’16 67.18 66.03 67.51 65.63 68.00 Feb’16 67.5 66.23 67.82 65.80 68.36 Mar’16 67.8 66.43 68.15 65.98 68.73 Apr’16 68.14 66.68 68.51 66.21 69.13 May’16 68.42 66.89 68.82 66.39 69.49 Jun’16 68.68 67.08 69.12 66.56 69.81 Jul’16 68.96 67.30 69.42 66.75 70.15 Aug’16 69.22 67.51 69.71 66.94 70.46 Sep’16 69.47 67.66 69.92 67.08 70.71 Oct’16 69.72 67.81 70.14 67.23 70.95 Nov’16 69.93 67.97 70.36 67.36 71.20 Dec’16 70.14 68.12 70.59 67.50 71.46 Jan’17 70.34 68.27 70.83 67.50 71.73 Feb’17 70.54 68.42 71.07 67.77 72.00 Mar’17 70.76 68.61 71.33 67.93 72.30 Apr’17 70.99 68.81 71.60 68.12 72.60 116 6.2 Results Table 6.6: Risky forward contract values right vs. wrong way risk - dealer and counterparty Risk-free commodity forward ρ Slc = 0.5 ρ Sl d = 0.5 ρ Slc = 0.5 ρ Sl d = −0.5 ρ Slc = −0.5 ρ Sl d = 0.5 ρ Slc =−0.5 ρ Sl d =−0.5 May’15 60.32 60.09 60.29 60.27 60.46 Jun’15 63.32 62.87 63.27 63.24 63.64 Jul’15 64.13 63.48 64.07 64.00 64.60 Aug’15 64.81 63.94 64.71 64.62 65.39 Sep’15 65.38 64.30 65.23 65.11 66.04 Oct’15 65.89 64.60 65.68 65.54 66.62 Nov’15 66.37 64.90 66.13 65.97 67.19 Dec’15 66.81 65.17 66.54 66.36 67.72 Jan’16 67.18 65.37 66.85 66.65 68.14 Feb’16 67.5 65.51 67.11 66.90 68.50 Mar’16 67.8 65.67 67.38 67.16 68.87 Apr’16 68.14 65.87 67.69 67.45 69.28 May’16 68.42 66.02 67.96 67.70 69.64 Jun’16 68.68 66.17 68.21 67.94 69.97 Jul’16 68.96 66.35 68.47 68.19 70.31 Aug’16 69.22 66.52 68.72 68.43 70.63 Sep’16 69.47 66.63 68.90 68.61 70.88 Oct’16 69.72 66.75 69.08 68.80 71.12 Nov’16 69.93 66.87 69.27 68.99 71.38 Dec’16 70.14 66.99 69.46 69.17 71.64 Jan’17 70.34 67.10 69.66 69.36 71.91 Feb’17 70.54 67.21 69.86 69.54 72.18 Mar’17 70.76 67.36 70.08 69.76 72.48 Apr’17 70.99 67.53 70.32 69.98 72.78 117 Appendix I Let PE (t) denote the positive exposure, and h i denote the discount factor for a value h i ≥ 0. Let R denote the discounted positive exposure integrated from t = 0 to infinity. We call this value total discounted positive exposure. R = ∞ 0 PE (t)e −h i t dt (6.1) We will show that the expected total discounted positive exposure is equal to the expected undiscounted total positive exposure up to an exponentially distributed random time with rate h i [47]. E [R] =E " η i 0 PE (t)dt # (6.2) Let η i be an exponential random variable with rate h i , that is independent of all the random variables PE (t). We want to show that E [R] =E ∞ 0 PE (t)e −h i t dt = ∞ 0 E (PE (t))e −h i t dt = ∞ 0 EPE (t)e −h i t dt =E " η i 0 PE (t)dt # for all i (6.3) 119 Chapter 6 Numerical Results Define for each t≥ 0, a random variable I (t) by I (t) =          1, if t≤η i 0, if t>η i (6.4) and note that η i 0 PE (t)dt = ∞ 0 PE (t)I (t)dt (6.5) Thus, E " η i 0 PE (t)dt # =E " ∞ 0 PE (t)I (t)dt # = ∞ 0 E [PE (t)I (t)dt] = ∞ 0 E [PE (t)]E [I (t)]dt = ∞ 0 E [PE (t)]P [η i ≥t]dt = ∞ 0 E [PE (t)]e −h i x dt = ∞ 0 EPE (t)e −h i x dt (6.6) Therefore, the expected total discounted positive exposure is equal to the expected total undiscounted positive exposure by a random time that is exponentially distributed with a rate equal to the discount factor. 120 Acknowledgments I wish to express my sincere gratitude to Dr. Fernando Zapatero, Professor of Finance and Business Economics at the University of Southern California, for his guidance and valuable advise in carrying out this research. I am most grateful to Dr. Sheldon Ross, Epstein Chair Professor of Industrial and Systems Engineering at the University of Southern California, for providing me an op- portunity to carry out this research and being a great source of encouragement. I also sincerely thank Dr. Reza Simchi for his valuable advise and insight throughout this research. Finally, I wish to express my deepest appreciation to Dr. Amin Saeedfar without whom this work would not have been possible. 121 Bibliography [1] S. Alavian. Fair value and fva–a quant’s perspective. 7th Annual Credit Risk in Banking Summit, Berlin, 2014. [2] C. Albanese, L. B. Andersen, and S. Iabichino. The fva puzzle: Accounting, risk management and collateral trading. Risk Management and Collateral Trading (Oc- tober 31, 2014), 2014. [3] C. Albanese, G. Pietronero, and S. White. Optimal funding strategies for counter- party credit risk liabilities. Available at SSRN 1844713, 2011. [4] M. Bianchetti. 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Abstract (if available)

Abstract In this work we develop a framework for pricing bilateral contracts subject to both counterparty and dealer default risk, with the main focus on forward and swap contracts predominantly traded in energy markets. Unlike futures contracts that are traded on an Exchange and settled daily through clear-housing process, forward contracts can bear full risk of default by both parties. Market practice to account for associated credit risk has been to add adjustments to risk-neutral value of the contract, based on expected loss due to possibility of default. Calculation of expected loss relies on deriving incremental default probabilities based on assumptions on hazard rate models, and become computationally intensive. ❧ In our approach, pricing formula is based on valuation of future payoffs discounted under discount factors reflective of dealer and counterparty credit spreads. Default risk is essentially accounted for through discounting in our framework. Technical derivations of the valuation formulas in our methodology is based on change of numeraire and using a market measure suitable for the specific pricing problem under consideration. The new measure accounts for wrong/right way risk through correlation of payoff process with the probability measure. This methodology integrates Credit and Debit Value Adjustments into a single pricing formula, accounting for cost of credit in pricing equation, where credit risk is warehoused internally. ❧ In contrast with a bilateral setting that in extreme case of zero collateralization results in bearing full counterparty default risk, exchanges have traditionally relied on full collateralization by all participants in order to virtually eliminate default risk. Margining mechanism assures that adequate amount of cash is deposited in margin accounts on a daily basis by all counterparties, equivalent to the value of the contract. Posting collateral in the form of cash requires participants to access funds available to them only at unsecured funding rate, which is higher than risk free rate, while the margin account accrues interest close to risk-free rate. This asymmetry results in cost of capital to the market participant, and has historically not been addressed in pricing of exchange traded derivatives that are traditionally assumed to be risk-free. ❧ In the second part of this work we derive partial differential equations for the two scenarios where cost of credit and cost of capital are considered in the pricing equation. The results show that under certain generic assumptions on funding rate, both over-the-counter and exchange trading result in identical pricing PDEs. Our main finding here is that margining is essentially a mechanism by which credit risk is transferred back to the counterparties, while the inherent nature of risk is preserved. ❧ In the final chapter of this work, we illustrate numerical results with case studies based on crude oil forward and swap contracts and examine the impact of counterparty and dealer credit spread and right/wrong way risk on contract value.

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Asset Metadata

Creator Nejatbakhsh Esfahani, Naemeh (author)

Core Title Pricing OTC energy derivatives: credit, debit, funding value adjustment, and wrong way risk

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Industrial and Systems Engineering

Publication Date 10/28/2017

Defense Date 10/19/2015

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag cost of capital,cost of credit,CVA,derivatives,DVA,OAI-PMH Harvest,OTC,Pricing,wrong way risk

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Ross, Sheldon M. (committee chair), Dessouky, Maged (committee member), Zapatero, Fernando (committee member)

Creator Email naomi.esfahani@gmail.com,nejatbak@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-193586

Unique identifier UC11279340

Identifier etd-Nejatbakhs-3998.pdf (filename),usctheses-c40-193586 (legacy record id)

Legacy Identifier etd-Nejatbakhs-3998.pdf

Dmrecord 193586

Document Type Dissertation

Format application/pdf (imt)

Rights Nejatbakhsh Esfahani, Naemeh

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA