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New results on pricing Asian options

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New results on pricing Asian options

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Content NEW RESULTS ON PRICING ASIAN OPTIONS by Xiufang Li A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Ful¯llment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) May 2007 Copyright 2007 Xiufang Li Acknowledgements During the years of my study at the University of Southern California, I have bene¯tedagreatdealfrommanypeopleandhaveaccumulatedmanydebts. Here, I would like to express the gratitude that I have had in my heart. First of all, I am deeply grateful to my dissertation advisor, Professor Sergey V. Lototsky, for his guidance, support, encouragement, and for planning my overall Ph.D studies. Were it not for his great help, this dissertation would have not been completed. I cannot imagine a better advisor. I would also like to thank the members of my dissertation committee { Dr. Jianfeng Zhang, Dr. Yongheng Deng, Dr. Peter Baxendale and Dr. Remigijus Mikulevicius for their time and help. I also wish to extend my thanks to Dr. Dale Alspach, Dr. William Jaco and Dr. Ju Ning, Professors in the department of Mathematics at Oklahoma State University, for their help and cherished friendship. I thank all my good friends here at the University of Southern California for their help and friendship. They have made my graduate studies much more enjoyable and memorable. ii Iowemuchtomyparentsfortheirsupportandprotectioninhavinggrownup without having too many worries. It is impossible to acknowledge their a®ection and blessings a simple 'thank you' { my debt is eternal. I am forever indebted to my husband, who has always been there to support andencourageme,whohasbroughtmemuchhappinessandasenseofbelonging, and whose innocence and other good qualities I shall forever cherish. To him, I dedicate this dissertation, with my admiration and trust. Xiufang Li Los Angeles, California October, 2006 iii Table of Contents Acknowledgements ii List Of Tables vi List Of Figures vii Abstract viii Chapter 1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Review of Literature . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Overview of Research . . . . . . . . . . . . . . . . . . . . . . . . . 7 Chapter 2 Asian Options and the Black-Scholes Model 9 2.1 European-style and American-style Options . . . . . . . . . . . . 9 2.2 The Black-Scholes Model . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Asian Options in the Black-Scholes Model . . . . . . . . . . . . . 19 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Chapter 3 A Combinatorial Method to Price Eurasian Options 32 3.1 Binomial Tree Model . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Geometric Eurasian Options . . . . . . . . . . . . . . . . . . . . . 33 3.3 Arithmetic Eurasian Options . . . . . . . . . . . . . . . . . . . . . 52 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Chapter 4 American-style Asian Options 65 4.1 Decomposition of Amerasian Puts . . . . . . . . . . . . . . . . . . 65 4.2 The Combinatorial Method . . . . . . . . . . . . . . . . . . . . . 70 4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Chapter 5 Asymptotic Behaviors and Extensions of the Model 76 5.1 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.1.1 Zero Volatility . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.1.2 Perpetual Asian Options . . . . . . . . . . . . . . . . . . . 82 5.2 Asian Options in Foreign Markets . . . . . . . . . . . . . . . . . . 84 iv 5.2.1 Martingale Measures in Foreign Markets . . . . . . . . . . 84 5.2.2 Foreign Equity Asian Options and Quanto Asian Calls . . 88 5.2.3 Currency Asian Options . . . . . . . . . . . . . . . . . . . 94 5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Chapter 6 Summary and Conclusions 98 6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.2 Research Contributions . . . . . . . . . . . . . . . . . . . . . . . . 99 6.3 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . 100 Reference List 101 v List Of Tables 3.1 Comparison of Tree-based Algorithms . . . . . . . . . . . . . . . . 52 4.1 Geometric Eurasian Option Values . . . . . . . . . . . . . . . . . 73 4.2 Geometric Amerasian Option Values . . . . . . . . . . . . . . . . 74 vi List Of Figures 3.1 A 3-level Binomial Tree. . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Convergence of the Combinatorial Method . . . . . . . . . . . . . 46 3.3 Binomial Tree Method vs. Monte Carlo Simulation . . . . . . . . 61 5.1 The Exercise Value of the Geometric Eurasian Option vs. Time, withS 0 =40; r =0:03. Case1: K =25, Case1(a): K =S 0 =40, Case 2: K =68, Case 3: K =18; . . . . . . . . . . . . . . . . . . 79 5.2 The Exercise Value of the Arithmetic Eurasian Option vs. Time, withS 0 =40; r =0:03. Case1: K =25, Case1(a): K =S 0 =40, Case 2: K =68, Case 3: K =18; . . . . . . . . . . . . . . . . . . 80 5.3 Plots of f(¿) = S 0 +r¿S 0 +r 2 ¿ 2 K and g(¿) = S 0 e r¿ , with S 0 = 40;r =0:03. Case 1: K =45, Case 2: K =12. . . . . . . . . . . . 82 vii Abstract AnAsianoptionisapath-dependentoptionwhosepayo®dependsontheaverage price of the underlying asset during the life of the option. Asian options are very attractivetoinvestors. However,thepricingofAsianoptionsremainsachallenge. Thepurposeofthisdissertationistodevelope±cienttechniquesforpricingAsian options and to expand the existing results on the pricing of Asian options in the classic Black-Scholes model. In this dissertation, e±cient binomial tree-based algorithms have been devel- oped to price European-style Asian options and American-style geometric Asian options. Byanalyzingtheasymptoticdistributionofthearithmeticaverageprice of the underlying asset, an explicit formula for pricing European-style arithmetic Asian options has been derived. A put-call parity formula for Asian options, a decomposition formula for American-style arithmetic Asian options and pricing formulas for both European- and American-style Asian options with zero volatil- ity or with in¯nite expiration date have also been derived. Additionally, foreign equity Asian options, quanto Asian calls and currency Asian options are intro- duced and the pricing formulas presented in this dissertation have been applied to these options. viii Chapter 1 Introduction The purpose of this chapter is to introduce what is an Asian option, why we are interested in Asian options and how hard it is to price Asian options, to review the literature on pricing Asian options, and to chart the scope and purpose of this research. 1.1 Background Asian options are path-dependent options whose terminal payo® depends on the average price of the underlying asset during a speci¯ed time period before matu- rity. The two most popular ways to compute the average price of the underlying asset are arithmetic and geometric. Due to their averaging feature, Asian options are cheaper than other options and less sensitive to possible spot manipulations or extreme movements at, or near, the expiration date. From a trader's point of view, the delta of an Asian option naturally decreases since part of the average becomes known after some 1 speci¯c date. Therefore, determining the hedging strategy becomes easier. Con- sequently, Asian options are very attractive for investors. According to a survey conducted by CIBC World Business Markets and the Wharton School [9], Asian options are the most commonly used exotic options. Nonetheless, such options have turned out to be much more di±cult to price and hedge than other options, both analytically and numerically. Based on whether the owner of the option has the right to exercise the option earlier than the expiration date, Asian options can be classi¯ed into European- style Asian options and American-style Asian options. The literature on pricing European- and American-style Asian options are reviewed separately in the next section. 1.2 Review of Literature Even though pricing Asian options is NP-hard [14], a variety of techniques have been developed to price Asian options. There are three main approaches to pricing European-style Asian options. The ¯rst approach is deriving approximations of closed-form solutions. Turn- bull and Wakeman [53] and L¶ evy [42] ¯nd approximate valuation formulas by matching the ¯rst several moments of the arithmetic average. Geman and Yor [29] derive an analytical solution for pricing arithmetic Asian options in terms of the inverse Laplace transform. Numerical inversion of this transform is discussed 2 by Geman and Eydeland [28] and Shaw [51]. By using reciprocal Gamma dis- tribution to approximate the sum of lognormals, Milevsky and Posner [46] ¯nd a new approximate formula for pricing arithmetic Asian options. Using the idea that one could replicate the average of the stock price by self-¯nanced trading in stocks, a one-dimensional PDE is derived to price Asian options contingent on dividendpayingstocksbyVe· ce· r[55]. Ve· ce· randXu[56]showthatthepriceofan arithmetic Asian option satis¯es an integro-di®erential equation in the case that the underlying asset is driven by special martingale processes, of which the L¶ evy process is a special case. This method is extended to the stochastic volatility case by Fouque and Han [27]. Ju [38] derive a uni¯ed approximation formula for basket and Asian options, by considering Taylor expansion of the ratio of the characteristic function of the weighted average to that of the weighted average which is approximated by a lognormal variable. InsteadofusingtheclassicalBlack-Scholesmodel,modelsofexponentialL¶ evy type are used in [3]. In [3], Albrecher and Predota derive approximate pricing formulas for European-style Asian options under the assumptions that the asset price follows the exponential L¶ evy model and the logarithmic asset return is of normal inverse Gaussion distribution. The second approach is developing e±cient numerical algorithms for comput- ing an average payo® as given by the risk-neutral valuation formula. Kemna and Vorst [40], Boyle et al. [10] and Broadie et al. [11] evaluate arithmetic Asian 3 options by using (quasi-)Monte Carlo simulation with variance reduction tech- nique. Two methods for variance reduction, the control variable and the change of measure, are proposed by Vazquez-Abad and Dufrense [54]. A large class of numerical algorithms uses a binomial tree. In an n-level bi- nomial tree, there are 2 n averages. Because computing all the 2 n averages is very time consuming, development of e±cient tree-based algorithms is an area of active research. Hull and White [34] propose an algorithm that employs fewer averages for each node of the binomial tree. They calculate the option values corresponding to those averages only and approximate the option values corre- sponding to the missing averages by interpolation. This algorithm is e±cient, but not convergent. Klassen [41] re¯nes the Hull and White algorithm by con- sidering fewer averages at each node and by applying Richardson extrapolation. By eliminating unnecessary states and exploiting the method of Lagrange mul- tipliers to minimize the approximation error, Dai et al. [20] provide a binomial tree algorithm for European-style Asian options with running time O(n 2:5 ) and convergence rate O(n ¡1 ). Alziary et al. [4] and Dewynne and Wilmott [23] solve Partial Di®erential Equations(PDEs)numericallyvia¯nite-di®erencetechniquestopriceEuropean- styleAsianoptions. Zhang[59]deriveananalyticalapproximateformulatoprice andhedgetheEuropean-stylearithmeticAsianoptionsandevaluateitbysolving aPDEnumerically. Theerrorisatleastoforder10 ¡5 . BasedontheBlack-Scholes 4 PDE model, Foufas [26] develop a method for valuing ¯xed and °oating strike Asian options using the ¯nite element method and duality techniques. The third approach is approximating the upper and lower bounds on the optionpriceratherthanapproximatingtheoptionpriceitself. RogerandShi[49] derive range bounds of European-style Asian options. Thompson [52] provides upper and lower bounds on the value of ¯xed-strike and °oating-strike Asian options. Using the trinomial tree model, Chalasani et al. [13] propose range- bound algorithms with running time O(n 4 ). The algorithm of Aingworth et al. [1] provide a theoretical error bound of O(Kn=k), with running time O(kn 2 ), where K is the strike price, n is the number of time intervals and k is a positive integer which can be varied to adjust the tradeo® between the running time and accuracy. Tree algorithms for pricing Asian options with range bounds are also developed by Dai et al. [19]. Their algorithm runs in O(kn 2 ) time with a guaranteed error bound of O(K p n=k). By exploiting the symmetry between ¯xed and °oating-strike Asian options, Henderson et al. [31] derive bounds for °oating-strike Asian options. Dhaene et al. [24] improve the bounds for discrete arithmetic Asian options with ¯xed or °oating strike provided by Rogers and Shi [49] and Nielsen and Sandmann [48]. Hedging based on the lower and upper bound is also discussed in their papers. With few exceptions, all algorithms in this third approach are in the Black- Scholes setting. Albrecher and Predota [3] give bounds for the price of discrete Asian options in an incomplete market model with the underlying asset price 5 being of exponential L¶ evy type with variance-gamma distributed logarithmic re- turns. ThehedgingofAsianoptionsisstudiedinthefollowingpapers. Usinglognor- mal and inverse Gaussian distribution to approximate the arithmetic averages, Jacques[35]givetwoexplicitformulasforthehedgingportfolioofEuropean-style Asian options. Albrecher et al. [2] provide a static super-hedging strategy con- sisting a portfolio of European options for arithmetic Asian options under L¶ evy model. American-style Asian options are even harder to price than European-style Asian options. Longsta® and Schwartz [44] use least squares Monte Carlo simulation to estimate the conditional expected payo® of American-style options, including American-styleAsianoptions. StablenumericalPDEmethodsforpricingAmerican- styleAsianoptionsaredevelopedin[60]. In[30],theproblemofpricingAmerican- styleAsianoptionsisformulatedinthedynamicprogrammingframework. Based on the ideas in [30], Ben-Ameur et al. [5] develop a procedure based on dynamic programming combined with ¯nite-element piecewise-polynomial approximation of the payo® function for pricing American-style Asian options. Binomial tree method [17] is an intuitive and powerful technique for pricing options and has been traditionally used. Many techniques for pricing American- style Asian options are tree-based. 6 In [32], parallel algorithms based on the binomial tree model for pricing American-style Asian options are developed. Pricing American-style Asian op- tions underlying upon multi-assets is also discussed therein. Chang et al. [15] extends the Hull and White binomial tree method to a trinomial tree model for the valuation of American-style Asian options whose strike prices could be reset to a new level. Jiang and Dai [37] show uniform convergence of the binomial tree methods for European and American path-dependent options using numer- ical analysis and the notion of viscosity solutions. Dai and Lyuu [21] propose a multiresolution lattice for pricing American-style Asian options. The basic idea of the multiresolution algorithm is to extend the precision of some of the asset prices by multiplying a positive number which will be divided in the end. The running time of this algorithm is much less than 3 n , where n is the number of time intervals. Using the notion of integrality of stock prices, Dai and Lyuu [22] develop a new trinomial tree algorithm with sub-exponential running time. 1.3 Overview of Research Our research focuses on developing fast and e±cient binomial-tree-based algo- rithms for pricing both European-style and American-style Asian options and on extending the pricing theorems for standard European and American options in the Black-Scholes model to the case of Asian options. The rest of the paper is organized as follows. Chapter 2 outlines some funda- mental results on pricing options, especially Asian options, in the Black-Scholes 7 Model. The notations and fundamental pricing theorems introduced in Chapter 2areusedthroughoutthisdissertation. InChapter3,wepresentacombinatorial methodtopriceEuropean-styleAsianoptionsandanalyzetheconvergence,error boundandcomplexityofthealgorithm. American-styleAsianoptionsarestudied in Chapter 4. In Chapter 5, asymptotic behaviors of the price of Asian options with zero volatility or in¯nite expiry and some extensions of the Black-Sholes model are presented. Chapter 6 concludes this dissertation with a summary of research contributions, and pointers to further work. 8 Chapter 2 Asian Options and the Black-Scholes Model In this chapter, fundamental pricing theorems are presented. Without specifying any model, European-style and American-style options in general setting are discussed in Section 2.1. In Section 2.2, the model is speci¯ed to be the Black- Scholes model and the existing pricing results on European- and American-style options in the Black-Scholes are talked about. Resuls on pricing Asian options in the Black-Scholes model are discussed in Section 2.3. 2.1 European-styleandAmerican-styleOptions An option gives the holder the right, not obligation, to buy or sell the underlying assetatapredeterminedprice,i.e.,thestrike price K,atorbeforeapre-speci¯ed date in the future, i.e., the expiration date T. The holder of a call (put, respec- tively)optionhastherighttobuy(sell, respectively)theunderlyingasset. There are two basic types of options: European-style options and American-style op- tions. A European-style option can only be exercised at the expiration date. An 9 American-style option which expires at time T gives the owner the right to exer- cisetheoptionatanytimeon,orbefore,theexpirationdate. Therighttoexercise at any time at will is clearly valuable. Therefore, the value of an American-style option is not less than the equivalent European-style option. In general, it is more di±cult to value American-style options than European-style options, since valuing American-style options also include deciding the optimal exercising, or stopping,time. Next,weintroducesomeimportantterminologyandfundamental theorems related to the pricing of European- and American-style options. We consider a ¯nancial market with a trading interval [0;T ¤ ]. Uncertainty in themarketismodelledbyastochasticbasis(;F;P)withtheusualassumptions [39]. Namely, is an arbitrary set,F is the ¾¡¯eld of all subsets of andP is a probability measure de¯ned on (;F). Moreover, the ¯ltrationF =(F t ) t2[0;T ¤ ] for some ¯xed T ¤ satis¯es (i)F is right continuous, i.e.,F t = T s>t F s ,8 t < T ¤ and (ii)F 0 contains all null sets. Therearetwoprimarytradedsecurities: astockandarisk-freeasset. Astock can have either negative or positive return. A risk-free asset, also called a bond, always yields a positive rate of return. Denote the stock price process S t and the risk-free asset process B t . The most popular models for the processes S t and B t are the di®usion model and the Black-Scholes model. In the di®usion model, dB t =r(t)B t dt; dS t =¹(t;S t )S t dt+¾(t;S t )S t dW t ; 10 wherer(t)istheinterestrateattimet,¹(t;S t )isthefunctionoftheappreciation rate of the stock price, ¾(t;S t ) is the function of volatility coe±cient and W t is a one-dimensional Brownian motion de¯ned on (;fF t g t¸0 ;P). The function r(t) can be either deterministic or stochastic. Stochastic model of the interest rate is discussed in [16], [33] and [43]. The Black-Scholes model will be discussed in next section. In mathematical terms, a European-style option X with expiration date T has payo® X T , which is a nonnegative F T -measurable random variable. An American-style option X a which expires at time T consists of a sequence of pay- o®s fX t g t2[0;T] , where X t is a nonnegative F t -measurable random variable for t2 [0;T]. An option is said to be path-independent if its payo® X t =g(S t ;t) for some function g, i.e., X t does not depend on the whole sample path up to time t, but only on the current value of the stock price S t . The price of an option is unique if the market is e±cient in the sense that it is free of arbitrage opportunities. De¯nition 2.1.1. A self-¯nanced portfolio Á, i.e., portfolio with no exogenous infusion or withdrawal of money, is called an arbitrage opportunity on a ¯nancial market if the value of the portfolio at time t, V t (Á), satis¯es V 0 (Á)=0; PfV T (Á)¸0g=1; and PfV T (Á)>0g>0: We say that the market is arbitrage free if there are no arbitrage possibilities. 11 By this de¯nition, if there is an arbitrage opportunity, then we could make a positive amount of money out of nothing with probability 1. It is thus a risk-free moneymakingmachine,orafreelunch,onthe¯nancialmarket. Inordertomake the market e±cient, the prices of ¯nancial derivatives have to be priced "fairly". This "fair" price of a derivative is called the arbitrage price. Next de¯nition is key to the arbitrage price of an option. De¯nition 2.1.2. A probability measureP ¤ equivalent to actual market prob- ability measureP is called a martingale measure, or risk-neutral measure, if the discounted stock price process St Bt follows aP ¤ martingale. Theorem 2.1.1 (Fundamental Theorem of Asset Pricing). The absence of arbi- trage opportunities is equivalent to the existence of a martingale measure. Proof. See chapter 10 of [47]. 2 The following two theorems state how to determine the arbitrage free prices of European- and American-style options under the martingale measure. Theorem 2.1.2 (Risk-Neutral Valuation Formula for European Options). If the market is free of arbitrage, then the arbitrage price process of the European-style option X with expiration date T is given by the risk-neutral valuation formula V t (X)=E P ¤[B t B ¡1 T g(S T )jF t ]; 8t·T; where g(S T ) is the payo® function of the option at expiry. 12 Proof. See section 3.1 and section 10.1 of [47]. 2 ThepriceofanAmerican-styleoptionisalsodeterminedbasedontheassump- tion that trading this American option would not destroy the free of arbitrage policy. Theorem 2.1.3 (Risk-Neutral Valuation Formula for American Options). To avoid arbitrage, the price of an American option X a is given by the formula V 0 (X a )= sup ¿2[0;T] © B 0 E P ¤[B ¡1 ¿ g(S ¿ ;¿)] ª ; where ¿ are all the exercisable dates and g(S ¿ ;¿) is the amount paid to the holder when the American option is exercised at time ¿. More generally, the arbitrage price at time t of an American option with payo® g equals V t (X a )=ess sup ¿2[t;T] © E P ¤[B t B ¡1 ¿ g(S ¿ ;¿)jF t ] ª : Proof. See chapter 10 of [47]. 2 2.2 The Black-Scholes Model In the Black-Scholes model, the risk free asset and the stock price have dynamics given by dB t =rB t dt; dS t =¹S t dt+¾S t dW t ; 13 where r is the constant risk-free interest rate, ¹2R is a constant appreciation rate of the stock price, ¾ > 0 is a constant volatility coe±cient, and S 0 2R + is the initial stock price and W t ; t2 [0;T], stands for a one-dimensional standard Brownian motion de¯ned on the ¯ltered probability space (;fF t g t¸0 ;P). TheuniquemartingalemeasureP ¤ ,whichisequivalenttoP,isgivenbymeans of Radon-Nikodym derivative [47] dP ¤ dP =exp · r¡¹ ¾ W ¤ T ¡ 1 2 (r¡¹) 2 ¾ 2 T ¸ ; P¡a.s. and W ¤ t =W t ¡ r¡¹ ¾ t is a standard Brownian motion on (;fF t g 0·t·T ;P ¤ ). The dynamics of the stock price underP ¤ are dS t =rS t dt+¾S t dW ¤ t ; S 0 >0: Thus, in the Black-Scholes model, we have B t =B 0 e rt ; S t =S 0 exp £ ¾W ¤ t +(r¡ 1 2 ¾ 2 )t ¤ ; and the risk-neutral valuation formulas for European-style and American-style options became more speci¯c in this model. By Theorem 2.1.2, the arbitrage price of the European-style option X in the Black-Scholes model is V t (X)=e ¡r(T¡t) E P ¤[g(S T )jF t ]; 8 t2[0;T]: (2.1) 14 In particular, the price of X at time 0 is V 0 (X) = e ¡rT E P ¤[g(S T )]: By Theorem 2.1.3, the arbitrage price of the American-style option X a in the Black-Scholes model is V t (X a )=ess sup ¿2[t;T] © e ¡r(¿¡t) E P ¤[g(S ¿ ;¿)jF t ] ª : In particular, the price of X a at time 0 is V 0 (X a )= sup ¿2[0;T] © e ¡rT E P ¤[g(S ¿ ;¿)] ª : Aside from the discounted expectations under the martingale measure, an- other way to express the value of the European- and American-style options is to use PDEs. Theorem2.2.1 (TheBlack-ScholesEquation). Ifthearbitragefreepricingfunc- tion V t (X) is of the form V t (X)=u(t;S t ); wherethefunctionu(t;x):[0;T]£R + 7!R, thenuisthesolutionofthefollowing boundary value problem @u @t + 1 2 ¾ 2 x 2@ 2 u @x 2 +rx @u @x ¡r¼ =0; u(T;S T )=g(S T ): Proof. See chapter 6 of [7]. 2 15 ThepriceprocessofAmerican-styleoptionsisrathercomplicatedandcannot be expressed by a partial di®erential equation with a boundary condition. The techniques of variational inequalities provide an adequate framework for pricing American-style options numerically. De¯neLu := 1 2 ¾ 2 x 2@ 2 u @x 2 +rx @u @x ¡ru. Then, the value of the American put option u(t;x;y) can be expressed using variational inequalities as follows. Theorem 2.2.2 (Variational Inequalities for American Puts). The arbitrage price u(t;S t ) of an American put option, whose payo® function is g(S t ;t) at time t, satis¯es @u @t +Lu·0; (1) u(t;S t )¸g(S t ;t); (2) u(T;S T )=g(S T ;T); (3) ¡ u¡g(S t ;t) ¢ ³ @u @t +Lu ´ =0: (4) or equivalently, we have max n @u @t +Lu;u¡g(S t ;t) o =0; with boundary condition u(T;S T )=g(S T ;T). Proof. ForanAmerican-styleoption,thelongandshortrelationshipisasymmet- rical{itistheholderoftheexerciserightswhocontrolstheearly-exercisefeature. 16 The writer of the American-style option can make more than the risk-free rate if the holder does not exercise optimally. Therefore, (1) holds. The right hand side of (2) represents the early exercise value. It is clear that the value of the American-style option cannot go below the early exercise value at any time to avoid arbitrage and thus (2) holds. Equation (3) is trivial. At any time t, the holder may or may not exercise his American-style option. Ifu>g(S t ;t)atthismoment,thenexercisingtheoptionisnottheleastfavorable outcomeforthewriterbecausehecanclaimanon-zeropro¯tu¡g(S t ;t)instantly. Inthiscase, @u @t +Lu=0becausethewriterhasanarbitrageopportunityif @u @t +Lu were strictly less than zero. Therefore, equation (4) holds. 2 The following theorem is for American call options. Theorem 2.2.3 (Variational Inequalities for American Calls). The arbitrage price u(t;S t ) of an American call option, whose payo® function is g(S t ;t) at time t, satis¯es @u @t +Lu¸0; u(t;S t )¸g(S t ;t); u(T;S T )=g(S T ;T); ¡ u¡g(S t ;t) ¢ ³ @u @t +Lu ´ =0: or equivalently, we have min n @u @t +Lu;u¡g(S t ;t) o =0; 17 with boundary condition u(T;S T )=g(S T ;T). Proof. Similar to the proof of Theorem 2.2.2. 2 A solution to the variational inequality problems associated to American op- tionsaboveisnotknownexplicitly. Numericalapproachestosolvetheseproblems were developed by Jaillet et al [36]. The fundamental pricing results stated above can be easily extended to the case that dividends are paid during the option's lifetime. If the stock continuously pays dividends at some ¯xed rate k, then the dy- namics of the stock price is given by dS t = (¹+k)S t dt+¾S t dW t : In this case, the pricing PDE for European-style options becomes [47] @u @t + 1 2 ¾ 2 x 2 @ 2 u @x 2 +(r¡k)x @u @x ¡ru=0: De¯ne ~ L = 1 2 ¾ 2 x 2@ 2 u @x 2 +(r¡k)x @u @x ¡ru, then the arbitrage price u(S t ;t) of an American-style call option with continuous dividend rate k satis¯es @u @t + ~ Lu¸0; u(t;S t )¸g(S t ;t); u(T;S T )=g(S T ;T); ¡ u¡g(S t ;t) ¢ ³ @u @t + ~ Lu ´ =0: Similar variational inequalities for American put options with dividend can be derived easily. 18 The standard Black-Scholes valuation results are also extended in some other ways. To list a few, the valuation formula for futures options is established by Black [8]; stochastic volatility models are examined in [57] and [50]. 2.3 Asian Options in the Black-Scholes Model Asianoptionsarepath-dependentoptionswhosepayo®dependsontheaverageof the stock price from time 0 to the expiration date T. Based on the types of aver- age, Asian options can be classi¯ed into geometric Asian options and arithmetic Asian options. Based on the early-exercise feature, there are European-style Asian options, which are also called Eurasian options, and American-style Asian options, which are also referred to Amerasian options. At expiry, the payo® function for a geometric Eurasian call option is g(fS t g t2[0;T] ;T)=maxfG T ¡K;0g; whereG T =exp ³ 1 T R T 0 logS(t)dt ´ isthegeometricaverageofthestockpriceover the period [0;T]. The payo® function for an arithmetic Eurasian put option is g(fS t g t2[0;T] ;T)=maxfK¡A T ;0g; where A T = 1 T R T 0 S(t)dt is the arithmetic average of the stock price over the period [0;T]. 19 In the Black-Scholes model, based on the risk-neutral formula (2.1), the arbi- trage price of a Eurasian option X is,8t2[0;T], V t (X)=e ¡r(T¡t) E P ¤[g(fS t g t2[0;T] ;T)jF t ]; =e ¡r(T¡t) E P ¤[g(A T ;T)jF t ]; (2.2) where A T is the geometric averageG T or the arithmetic averageA T . The relationship between the arbitrage price of a Eurasian call option and that of the corresponding Eurasian put option, namely, the put-call parity for Eurasian options, can be derived using pricing formula (2.2). Theorem 2.3.1. The arbitrage prices C t ; P t of Eurasian call and put options with the same expiration date T and strike price K satisfy C t ¡P t =e ¡r(T¡t) E P ¤[A T jF t ]¡Ke ¡r(T¡t) ;8 t2[0;T]; (2.3) where A T is the average stock price at expiry. Proof. Clearly, the terminal payo®s of Asian call and put options satisfy C T ¡P T =(A T ¡K) + ¡(K¡A T ) + =A T ¡K: 20 Using this equality and the risk-neutral valuation formula for Eurasian options (2.2), we have C t ¡P t =e ¡r(T¡t) E P ¤[C T jF t ]¡e ¡r(T¡t) E P ¤[P T jF t ] =e ¡r(T¡t) E P ¤[A T ¡KjF t ]: Therefore,8t2[0;T], C t ¡P t =e ¡r(T¡t) E P ¤[A T jF t ]¡e ¡r(T¡t) K: 2 Usingtheput-callparityforEurasianoptions(2.3),thepriceofaEurasianput option can be calculated, provided that the price of the corresponding Eurasian call option is known. Thus, only Eurasian call options are emphasized in the following chapters. For Amerasian options, we have the following theorem. Theorem 2.3.2 (Pricing Theorem for Amerasian Options). To avoid arbitrage, the price of an Amerasian option X a is given by the formula V 0 (X a )= sup ¿2[0;T] © E P ¤[e ¡r¿ g(A ¿ ;S ¿ ;¿)] ª ; (2.4) where ¿ are all the possible exercisable dates and g(A ¿ ;S ¿ ;¿), with S ¿ the stock price at time ¿ and A ¿ the average stock prices up to time ¿, is the amount paid 21 to the holder if the Amerasian option is exercised at time ¿. More generally, the arbitrage price at time t of an Amerasian option with payo® function g equals V t (X a )=e rt ess sup ¿2[t;T] © E P ¤[e ¡r¿ X a ¿ jF t ] ª In particular, for a geometric Amerasian call option, g(A ¿ ;S ¿ ;¿)=maxfG ¿ ¡K;0g; and for an arithmetic Amerasian put, g(A ¿ ;S ¿ ;¿)=maxfK¡A ¿ ;0g: In [40], it is proved that the logarithm of the geometric average of the stock priceslogG T isnormallydistributedwithmean 1 2 (r¡ 1 2 ¾ 2 )T+logS 0 andvariance 1 3 ¾ 2 T, i.e., logG T »N µ 1 2 (r¡ 1 2 ¾ 2 )T +logS 0 ; 1 3 ¾ 2 T ¶ ; and the price of a continuously sampled geometric Eurasian call option, denoted as C G 0 (X), is C G 0 (X)=e ¡rT E P ¤[maxfG T ¡K;0g] =e ¡rT · e d ¤ 0 S 0 N(d 0 )¡KN µ d 0 ¡¾ q 1 3 T ¶¸ ; (2.5) 22 whereN is the cumulative standard normal distribution function and d ¤ 0 and d 0 can be written as d ¤ 0 = 1 2 ¡ r¡ 1 6 ¾ 2 ¢ T and d 0 = log(S 0 =K)+ 1 2 ¡ r+ 1 6 ¾ 2 ¢ T ¾ q 1 3 T : It is then easy to prove that the arbitrage price of the geometric Eurasian option at time t is C G t (X)=e ¡r(T¡t) E P ¤[maxfG T ¡K;0gjF t ] =e ¡r(T¡t) E P ¤ £ maxfe logG T ¡K;0gjF t ¤ =e ¡r(T¡t) · e d ¤ t G t N(d t )¡KN ³ d t ¡¾ q 1 3 (T ¡t) ´ ¸ ; (2.6) where d ¤ t and d t are d ¤ t = 1 2 ¡ r¡ 1 6 ¾ 2 ¢ (T ¡t) and d t = log(G t =K)+ 1 2 ¡ r+ 1 6 ¾ 2 ¢ (T ¡t) ¾ q 1 3 (T ¡t) : Note that logG T »N µ 1 2 (r¡ 1 2 ¾ 2 )T +logS 0 ; 1 3 ¾ 2 T ¶ implies G T is lognormally distributed with mean 1 2 (r¡ 1 6 ¾ 2 )T + logS 0 . Based on this fact and the put-call parity formula (2.3), we obtain the price of the geometric Eurasian put option: 23 P G t =C G t +e ¡r(T¡t) K¡e ¡r(T¡t) E P ¤[G T jF t ] =C G t +e ¡r(T¡t) h K¡G t e 1 2 (r¡ 1 6 ¾ 2 )(T¡t) i =e ¡r(T¡t) · e d ¤ t G t N(d t )¡KN ³ d t ¡¾ q 1 3 (T ¡t) ´ ¸ +e ¡r(T¡t) £ K¡G t e d ¤ t ¤ ; by (2.5) =e ¡r(T¡t) · KN ³ ¡d t +¾ q 1 3 (T ¡t) ´ ¡G t e d ¤ t N(¡d t ) ¸ : Asidefromtheexplicitformulaswediscussedabove, pricingPDEforgeomet- ric Eurosian options is also available. Theorem 2.3.3. De¯ne I t = R t 0 logS(¿)d¿ and let f : R 7! R be a Borel- measurable function, such that the random variable. For geometric Eurasian call and put options, X = (G T ¡K) + and X = (K¡G T ) + , respectively. X = f(G T ) is integrable under P ¤ . Then the arbitrage price in Black-Scholes model of the geometric Eurasian option X which settles at time T is given by the equality V t (X) = u(t;S t ;I t ), where the function u(t;x;y) : [0;T]£R + £R + 7!R solves the partial di®erential equation @u @t + 1 2 ¾ 2 x 2 @ 2 u @x 2 +logx @u @y +rx @u @x ¡ru=0; 0·t<T with boundary condition u(T;S T ;I T )=f(e 1 T I T ). 24 Proof. Please also see chapter 18 of [58]. By the risk-neutral valuation formula (2.2) and the de¯nition of I t , the arbitrage price of the geometric Asian option X at time t is V t (X)=e ¡r(T¡t) E P ¤[f(G T )jF t ] =e ¡r(T¡t) E P ¤ · f ³ 1 T Z T 0 logS(¿)d¿ ´¯ ¯ ¯F t ¸ =u(t;S t ;I t ); for some function u(t;x;y):[0;T]£R + £R + 7!R. Then,weconsideraportfolio,denotedasY,containingX andshortanumber ¢ofthestock,i.e.,Y =u(t;S t ;I t )¡¢S t . Thechangeofthevalueofthisportfolio is given by dY = µ @u @t + 1 2 ¾ 2 x 2 @ 2 u @x 2 ¶ dt+ @u @y dy+ µ @u @x ¡¢ ¶ dS t : Choosing ¢= @u @x to hedge the risk and using dy =dI =logSdt give us dY = µ @u @t + 1 2 ¾ 2 x 2 @ 2 u @x 2 +logx @u @y ¶ dt: This change is risk-free, i.e., dY = rYdt = r(u¡¢S)dt = r(u¡ @u @x S)dt. Thus, we have @u @t + 1 2 ¾ 2 x 2 @ 2 u @x 2 +logx @u @y +rx @u @x ¡ru=0: 25 The boundary condition for this PDE is u(T;S T ;I T )=maxfG T ¡K;0g for geo- metricEurasiancalls, andu(T;S T ;I T )=maxfK¡G T ;0gforgeometricEurasian puts. 2 De¯neLu:= 1 2 ¾ 2 x 2@ 2 u @x 2 +logx @u @y +rx @u @x ¡ru. Then,thevalueofthegeometric Amerasianputu(t;x;y)canbeexpressedusingvariationalinequalitiesasfollows. Theorem 2.3.4 (Variational Inequalities for geometric Amerasian Puts). To avoid arbitrage, the price of the geometric Amerasian put options u(t;S t ;I t ) sat- is¯es @u @t +Lu·0; u(t;S t ;I t )¸(K¡G t ) + ; u(T;S T ;I T )=(K¡G T ) + ; ¡ u¡(K¡G t ) + ¢ ³ @u @t +Lu ´ =0: or equivalently, we have max n @u @t +Lu;u¡(K¡G t ) + o =0; with boundary condition u(T;S T ;I T )=(K¡G T ) + . Proof. Follow the same reasoning as in Theorem 2.2.2. 2 The following theorem is for geometric Amerasian call options. 26 Theorem 2.3.5 (Variational Inequalities for geometric Amerasian Calls). For Amerasian call option, we have @u @t +Lu¸0; u(t;S t ;I t )¸(G t ¡K) + ; u(T;S T ;I T )=(G T ¡K) + ; ¡ u¡(G t ¡K) + ¢ ³ @u @t +Lu ´ =0: or equivalently, we have min n @u @t +Lu;u¡(G t ¡K) + o =0; with boundary condition u(T;S T ;I T )=(G T ¡K) + . Unlike the geometric averageG T , since the distribution of the arithmetic av- erage A T is unknown thus far, there is no explicit formula for arithmetic Asian options. There are pricing PDE for arithmetic Eurasian options and variational inequalities for arithmetic Amerasian options. De¯neI = R t 0 S(¿)d¿. Then,bythesamereasoningasthatusedforgeometric Eurasian options, we obtain that the arbitrage price of an arithmetic Eurasian option is V t = u(t;S t ;I t ), for some function u(t;x;y) : [0;T]£R + £R + 7! R, which satis¯es @u @t + 1 2 ¾ 2 x 2 @ 2 u @x 2 +x @u @y +rx @u @x ¡ru=0; 27 with the boundary condition u(T;S T ;I T ) = maxf 1 T I T ¡ K;0g for calls, and u(T;S T ;I T ) = maxfK ¡ 1 T I T ;0g for puts. For details, the reader is referred to see chapter 18 of [58]. De¯neLu:= 1 2 ¾ 2 x 2@ 2 u @x 2 +x @u @y +rx @u @x ¡ru. Then, the pricingofthe arithmetic Amerasian put option can be formulated using variational inequalities as follows. Theorem 2.3.6 (Variational Inequalities for arithmetic Amerasian Puts). To avoid arbitrage, the price of the arithmetic Amerasian put options u(t;S t ;I t ) sat- is¯es @u @t +Lu·0; u(t;S t ;I t )¸(K¡A t ) + ; u(T;S T ;I T )=(K¡A T ) + ; ¡ u¡(K¡A t ) + ¢ ³ @u @t +Lu ´ =0: or equivalently, we have max n @u @t +Lu;u¡(K¡A t ) + o =0; with boundary condition u(T;S T ;I T )=(K¡A T ) + . Proof. Follow the same reasoning as in Theorem 2.2.2. 2 The following theorem is for arithmetic Amerasian call options. 28 Theorem 2.3.7 (Variational Inequalities for arithmetic Amerasian Calls). For arithmetic Amerasian call option, we have @u @t +Lu¸0; u(t;S t ;I t )¸(A t ¡K) + ; u(T;S T ;I T )=(A T ¡K) + ; ¡ u¡(A t ¡K) + ¢ ³ @u @t +Lu ´ =0: or equivalently, we have min n @u @t +Lu;u¡(A t ¡K) + o =0; with boundary condition u(T;S T ;I T )=(A T ¡K) + . AsolutiontothevariationalinequalityproblemsassociatedtoAmerasianop- tionsaboveisnotknownexplicitly. Numericalapproachestosolvetheseproblems were developed by Jaillet et al [36]. In pricing Asian options, the Black-Scholes model can also be extended. The cases of Asian options contingent on futures and Asian options written on a dividend-paying stock are discussed below. Let f S (t;T);t2 [0;T]; stand for the futures price of a certain stock S on the date T. The evolution of the futures price f t =f S (t;T) is given by df t =¹ f f t dt+¾ f f t dW t ; 29 where ¹ f , ¾ > 0 are real numbers, and W t , for t 2 [0;T], is a one-dimensional standard Brownian motion de¯ned on (;F;P). In the Black-Scholes model, by the rule of absence of arbitrage and It^ o's Lemma, we have df t =(¹¡r)f t dt+¾f t dW t : The unique martingale measureP ¤ for the process f de¯ned on (;F) is given by means of Radon-Nikodym derivative dP ¤ dP =exp µ ¡ ¹ ¾ W T ¡ 1 2 ¹ 2 ¾ 2 T ¶ ; and W ¤ t =W t + ¹t ¾ . Thus, underP ¤ , f t =f 0 expf¾W t +(¡ 1 2 ¾ 2 )tg; where f 0 =S 0 e rT . More details can be seen in [47]. Following the same reasoning as that in the standard Black-Scholes model, we obtain the pricing PDE for arithmetic Eurasian option @u @t + 1 2 ¾ 2 x 2 @ 2 u @x 2 +x @u @y +rx @u @x ¡r¼ =0; (2.7) withtheboundaryconditionu(T;f T ;I T )=maxf 1 T I T ¡K;0g,whereI t = R t 0 f(x)dx. 30 Iftheunderlyingstockcontinuouslypaysdividendsatsome¯xedrate k, then the dynamics of the stock price is given by dS t = (¹+k)S t dt+¾S t dW t : In this case, it is clear that the pricing PDE of arithmetic Eurasian options becomes @u @t + 1 2 ¾ 2 x 2 @ 2 u @x 2 +x @u @y +(r¡k)x @u @x ¡ru=0; since dY =rYdt+D, where D is the dividend payed, i.e., D =kS¢=kS @u @S : Some authors derive other pricing formulas in this extended model, for in- stance, Ve· ce· r [55]. In [27], Fouque and Han generalize Ve· ce· r's technique [55] and derive a pricing PDE for pricing arithmetic Eurasian options to the case of stochastic volatility. 2.4 Summary Thischapterintroducesfundamentalde¯nitionsandnotationson¯nancialderiva- tives, including standard Euroopean, American options and various kinds of AsianoptionsintheBlack-Scholessetting. Thesede¯nitionsandnotationswillbe used throughout this dissertation. This chapter also presents main pricing theo- remsforstandardEuropeanandAmericanoptionsandEuropean-andAmerican- style Asian options in the Black-Scholes model. 31 Chapter 3 A Combinatorial Method to Price Eurasian Options Inspiredbythedistributionoftheaveragestockpricesandthenicecharacteristics of the binomial tree model, we found a rather simple way to approximately price Eurasian options. 3.1 Binomial Tree Model In the binomial tree model, (k;m) is used to represent the node at depth k, with m denoting the number of up-ticks from the root node to this node. Clearly, the root node can be represented by (0;0). Its up- and down-successors are represented by (1;1) and (1;0) respectively. In general, the nodes at depth k are denotedby(k;k); (k;k¡1);:::;(k;0). Thehighestlevel,orthelevelcorresponds to the expiration date, is n. 32 (0,0) (1,1) (1,0) (2,2) (2,1) (2,0) (3,3) (3,2) (3,1) (3,0) Figure 3.1: A 3-level Binomial Tree Moreover, S(k;m) represents the stock price at the node (k;m) andG(k;m) and A(k;m) represent the mean of the geometric and arithmetic average stock price at this node, respectively. In particular, G(0;0) and A(0;0) stand for the geometric and arithmetic average stock price at time 0, which is known to be S 0 . Atthenode(k;m),itiseasytoseethatthestockpriceisS 0 u m d k¡m ,itsfrequency of occurrence is ¡ k m ¢ , and the probability of occurrence is ¡ k m ¢ p m q k¡m , where u and d represent the up- and down- factor, p and q represent the probabilities of going up and down, respectively. In the risk-neutral probability measure with n time steps, u = e ¾ p T=n , d = 1 u , p = 1+rT=n¡d u¡d and q = 1¡p [7]. Figure 3.1 shows a 3-level binomial tree in such notation. 3.2 Geometric Eurasian Options The following is the main idea of the combinatorial method for pricing geometric Eurasian options. Instead of calculating all the possible geometric averages of 33 the stock price at each node or all special forms of the geometric averages of the stock price, only the ¯rst and the second moment of the geometric averages of the stock price at each node are calculated. Then, the geometric average of the stock price is proved to be asymptotically lognormally distributed. Based on this distribution, the expected payo® at expiry can be computed. Finally, the value of the geometric Eurasian option can be obtained by discounting the expected payo® at expiry back to the initial time. Let g(k;m) be the mean of the logarithm of the geometric average at node (k;m) and g 2 (k;m) the mean of the square of the logarithm of the geometric averageatnode(k;m). E[g n ]andVar(g n )representtheexpectationandvariance of the logarithm of the geometric average at level n, or at expiry, respectively. By the structure of the binomial tree, the node (k;m) can only be reached from two possible nodes, assuming 0 < m < k. More speci¯cally, the node (k;m) can be reached either by taking a down-tick from the node (k¡ 1;m), or by taking an up-tick from the node (k¡1;m¡1). Therefore, by using the information from these two nodes (k¡1;m) and (k¡1;m¡1), the arithmetic average of the geometric averages of the stock price at the node (k;m) can be obtained by the following formula g(k;m)= a 1 ¡ k¡1 m ¢ +b 1 ¡ k¡1 m¡1 ¢ ¡ k¡1 m ¢ + ¡ k¡1 m¡1 ¢ = a 1 (k¡m)+b 1 m k (3.1) 34 where a 1 = k k+1 g(k¡1;m)+ 1 k+1 log(S 0 u m d k¡m ); b 1 = k k+1 g(k¡1;m¡1)+ 1 k+1 log(S 0 u m d k¡m ): Note that a 1 is the mean of the logarithm of the geometric averages at node (k;m) obtained from the node (k¡1;m) and b 1 is the mean of the logarithm of the geometric averages at node (k;m) obtained from the node (k¡1;m¡1). There are two special cases, i.e., when m=0 and m=k: (i) When m = 0, the node (k;0) can only be reached from the node (k¡1;0). Thus, the geometric average stock price at the node (k;0) is g(k;0)= k k+1 g(k¡1;0)+ 1 k+1 log(S 0 d k ): (3.2) (ii) When m = k, since the node (k;k) can only be reached from the node (k¡1;k¡1), the formula to compute the geometric average stock price at the node (k;k) is g(k;m)= k k+1 g(k¡1;k¡1)+ 1 k+1 log(S 0 u k ): (3.3) Thus, for each node in the binomial lattice, the expected value of the logarithm of the geometric average stock price at that node can be computed by formulas (3.1), (3.2) or (3.3). By the same reasoning, we can compute the mean of the square of the loga- rithm of the geometric average stock price g 2 (k;m) by the following formulas, 35 g 2 (k;m)= 8 > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > : a 2 ¡ k¡1 m ¢ +b 2 ¡ k¡1 m¡1 ¢ ¡ k¡1 m ¢ + ¡ k¡1 m¡1 ¢ = a 2 (k¡m)+b 2 m k ; if 0<m<k; [g(k;m)] 2 = k 2 (k+1) 2 g 2 (k¡1;0)+ 1 (k+1) 2 [log(S 0 d k )] 2 +2 k (k+1) 2 g(k¡1;0)log(S 0 d k ); if m=0; [g(k;m)] 2 = k 2 (k+1) 2 g 2 (k¡1;k¡1)+ 1 (k+1) 2 [log(S 0 u k )] 2 +2 k (k+1) 2 g(k¡1;k¡1)log(S 0 u k ); if m=k: (3.4) where a 2 =a 2 1 = k 2 (k+1) 2 g 2 (k¡1;m)+ 1 (k+1) 2 [log(S 0 u m d k¡m )] 2 +2 k (k+1) 2 g(k¡1;m)log(S 0 u m d k¡m ); and b 2 =b 2 1 = k 2 (k+1) 2 g 2 (k¡1;m¡1)+ 1 (k+1) 2 [log(S 0 u m d k¡m )] 2 +2 k (k+1) 2 g(k¡1;m¡1)log(S 0 u m d k¡m ): Usingtheformulasabove,theexpectedvaluesofthelogarithmofthegeomet- ricaverageandthesquareofthelogarithmofthegeometricaverageateachnode at the expiration date, or at level n, can be calculated. The expected values of thelogarithmofthegeometricaverageanditssquareatterminallevel n,denoted by ¹ g n =E[g n ] andE[g 2 n ], can thus be calculated as follows: 36 ¹ gn =E[g n ] = n X m=1 g(n;m) µ n m ¶ p m q n¡m ; E[g 2 n ] = n X m=1 g 2 (n;m) µ n m ¶ p m q n¡m : The variance of the geometric average at the expiration date is ¾ 2 g n =E[g 2 n ]¡(E[g n ]) 2 : Using this combinatorial method, higher moments of the geometric average can also be computed exactly and rapidly. Based on the combinatorial algorithm and the assumption that at level n, when n is su±ciently large, the geometric average is lognormally distributed, the price of geometric Asian options can be computed by C G 0 =e ¡rT Z +1 logK (e x ¡K) 1 p 2¼¾ g n e ¡ (x¡¹g n ) 2 2¾ 2 gn dx: (3.5) Using the put-call parity formula (2.3), the price of the corresponding geometric Asian put option can be computed by P G 0 =C G 0 +e ¡rT K¡S 0 : 37 Next, weshowthattheasymptoticdistributionofthelogarithmofthegeometric average at terminal level, g n , is indeed normally distributed. De¯ne the sequence of i.i.d. random variablesfX i g n i=1 to be X i (n)= 8 > > > < > > > : u=1+¾ p T=n; with probability p; d=1¡¾ p T=n; with probability q; where p = (1+rT=n)¡d u¡d = rT=n+¾ p T=n 2¾ p T=n = r p T=n 2¾ + 1 2 , and q = 1¡ p, and thus p¡q =2p¡1= r p T=n ¾ . Let Y i = logX i . Then, fY i g n i=1 is also a sequence of i.i.d. random variables and Y i (n)= 8 > > > < > > > : logu=log ³ 1+¾ p T=n ´ ; with probability p; logd=log ³ 1¡¾ p T=n ´ ; with probability q: When n is su±ciently large, the mean and variance of Y i are E[Y i (n)]=plog ³ 1+¾ p T=n ´ +qlog ³ 1¡¾ p T=n ´ =p · ¾ p T=n¡ 1 2 ³ ¾ p T=n ´ 2 + 1 3 ³ ¾ p T=n ´ 3 ¡O(n ¡2 ) ¸ +q · ¡¾ p T=n¡ 1 2 ³ ¾ p T=n ´ 2 ¡ 1 3 ³ ¾ p T=n ´ 3 ¡O(n ¡2 ) ¸ =(p¡q)¾ p T=n¡ 1 2 ³ ¾ p T=n ´ 2 +(p¡q) 1 3 ³ ¾ p T=n ´ 3 +O(n ¡5=2 ) = rT n ¡ ¾ 2 T 2n + r¾ 2 T 2 3n 2 +O(n ¡5=2 ) (3.6) 38 and Var(Y i (n))=E[Y i (n) 2 ]¡ ¡ E[Y i (n)] ¢ 2 =p · ¾ p T=n¡ 1 2 ³ ¾ p T=n ´ 2 + 1 3 ³ ¾ p T=n ´ 3 ¡O(n ¡2 ) ¸ 2 +q · ¡¾ p T=n¡ 1 2 ³ ¾ p T=n ´ 2 ¡ 1 3 ³ ¾ p T=n ´ 3 ¡O(n ¡2 ) ¸ 2 ¡ · rT n ¡ ¾ 2 T 2n +O(n ¡2 ) ¸ 2 = ¾ 2 T n + ¾ 2 T 2 4n 2 ¡ r¾ 2 T 2 n 2 + 2¾ 4 T 2 3n 2 ¡ (2rT ¡¾ 2 T) 2 4n 2 +O(n ¡3 ) = ¾ 2 T n + c 0 n 2 +O(n ¡3 ); (3.7) where c 0 = 1 4 ¾ 2 T 2 ¡r¾ 2 T 2 + 2 3 ¾ 4 T 2 ¡(2rT ¡¾ 2 T) 2 : The geometric average in n-time-period binomial tree model G n is G n = Ã n Y i=0 S i ! 1 n+1 = n S 0 ¢[S 0 ¢X 1 (n)]¢[S 0 ¢X 1 (n)¢X 2 (n)]¢:::¢[S 0 ¢X 1 (n)¢:::¢X n (n)] o 1 n+1 =S 0 ¢[X 1 (n)] n n+1 ¢[X 2 (n)] n¡1 n+1 ¢:::¢[X n (n)] 1 n+1 : Let g n =logG n . Then g n =log n S 0 ¢[X 1 (n)] n n+1 ¢[X 2 (n)] n¡1 n+1 ¢:::¢[X n (n)] 1 n+1 o =logS 0 + n n+1 logX 1 (n)+ n¡1 n+1 logX 2 (n)+:::+ 1 n+1 logX n (n) =logS 0 + n X i=1 n+1¡i n+1 Y i (n): 39 Theorem 3.2.1 (The Lindeberg-Feller Theorem, [25]). For each n, let X n;i , 1·i·n, be independent random variables withE[X n;i ]=0. Suppose (i) lim n!1 P n i=1 E[X 2 n;i ]=¾ 2 >0; (ii) For all ²>0, lim n!1 P n i=1 E[jX n;i j 2 ; jX n;i j>²]=0. ThenS n =X n;1 +:::+X n;n d ¡ !N(0;¾ 2 ),asn!1,where d ¡ !denotesconvergence in distribution. In what follows, we apply this theorem to ¯nd the limiting distribution of g n , i.e., logG n , as n!1. Let X n;i = n+1¡i n+1 [Y i (n)¡E[Y i (n)]]. Then, for each n, X n;i , with 1 · i· n, are independent,E[X n;i ]=0 and n X i=1 E[X 2 n;i ]= n X i=1 µ n+1¡i n+1 ¶ 2 E[Y i (n)¡E[Y i (n)]] 2 = n X i=1 µ n+1¡i n+1 ¶ 2 · ¾ 2 T n + c 0 n 2 +O(n ¡3 ) ¸ ; by (3.7) = n(n+1)(2n+1) 6(n+1) 2 · ¾ 2 T n + c 0 n 2 +O(n ¡3 ) ¸ = ¾ 2 T 3 + ¾ 2 T +2c 0 6n +O(n ¡2 ) ! ¾ 2 T 3 >0; as n!1: Thus, condition (i) of Theorem 3.2.1 is satis¯ed. To check (ii), 8 ² > 0, consider P n i=1 E[jX n;i j 2 ; jX n;i j>²]: 40 n X i=1 E[jX n;i j 2 ; jX n;i j>²] · n X i=1 q EjX n;i j 4 E[1 2 fjX n;i j>²g ]; by Cauchy-Schwarz inequality = n X i=1 s µ n+1¡i n+1 ¶ 4 E[Y i (n)¡E[Y i (n)]] 4 ¢P(jX n;i j>²): (3.8) Since E[Y i (n)¡E[Y i (n)]] 4 =p h ¾ p T=n+O(n ¡1 ) i 4 +q h ¡¾ p T=n+O(n ¡1 ) i 4 =p ³ ¾ p T=n ´ 4 +q ³ ¡¾ p T=n ´ 4 +O(n ¡5=2 )= ³ ¾ p T=n ´ 4 +O(n ¡5=2 ) = ¾ 4 T 2 n 2 +O(n ¡5=2 ); and81·i·n, P(jX n;i j>²)=P µ n+1¡i n+1 jY i (n)¡E[Y i (n)]j>² ¶ · µ n+1¡i n+1 ¶ 2 Var(Y i (n)) ² 2 = µ n+1¡i n+1 ¶ 2 · ¾ 2 T n² 2 + O(n ¡2 ) ² 2 ¸ ; by (3.7) · ¾ 2 T n² 2 +O(n ¡2 ); inequality (3.8) becomes 41 n X i=1 E[jX n;i j 2 ; jX n;i j>²] · n X i=1 s µ n+1¡i n+1 ¶ 4 ¢ · ¾ 4 T 2 n 2 +O(n ¡5=2 ) ¸ ¢ · ¾ 2 T n² 2 +O(n ¡2 ) ¸ = n(n+1)(2n+1) 6(n+1) 2 · ¾ 3 T 3=2 n 3=2 ² +O(n ¡7=4 ) ¸ = ¾ 3 T 3=2 3n 1=2 ² +O(n ¡3=4 )!0; as n!1. Hence, condition (ii) in Theorem 3.2.1 is also satis¯ed. Thus, by Theorem 3.2.1, n X i=1 X n;i d ¡ !N µ 0; ¾ 2 T 3 ¶ ; as n!1; i.e., n X i=1 n+1¡i n+1 Y i (n) d ¡ !N µ ¹; ¾ 2 T 3 ¶ ; as n!1; where ¹= lim n!1 n X i=1 n+1¡i n+1 E[Y i (n)] = lim n!1 n X i=1 n+1¡i n+1 µ rT n ¡ ¾ 2 T 2n + r¾ 2 T 2 3n 2 +O(n ¡3 ) ¶ = lim n!1 µ rT n ¡ ¾ 2 T 2n + r¾ 2 T 2 3n 2 +O(n ¡3 ) ¶ n X k=1 k n+1 = lim n!1 µ rT n ¡ ¾ 2 T 2n + r¾ 2 T 2 3n 2 +O(n ¡3 ) ¶ n(n+1) 2(n+1) = rT 2 ¡ ¾ 2 T 4 + lim n!1 µ r¾ 2 T 2 6n +O(n ¡2 ) ¶ = rT 2 ¡ ¾ 2 T 4 : (3.9) 42 Therefore,thelogarithmofthegeometricaveragestockpriceisasymptotically normally distributed: g n =logS 0 + n X i=1 n+1¡i n+1 Y i (n)»N µ logS 0 + rT 2 ¡ ¾ 2 T 4 ; ¾ 2 T 3 ¶ ; as n!1: Namely, the distribution of the geometric average stock price is log-normal. Next, we show that the price of the geometric Eurasian option computed by the combinatorial method (3.5) converges to the exact price (2.5) derived by Kemna and Vorst. Let ¹ g =logS 0 + rT 2 ¡ ¾ 2 T 4 and ¾ 2 g = ¾ 2 T 3 . Then, as n!1, g n »N(¹ g ;¾ 2 g ): We need to show that C g 0 =e ¡rT Z +1 logK (e x ¡K) 1 p 2¼¾ gn e ¡ (x¡¹ gn ) 2 2¾ 2 gn dx n!1 ¡ ¡¡ !e ¡rT Z +1 logK (e x ¡K) 1 p 2¼¾ g e ¡ (x¡¹ g ) 2 2¾ 2 g dx: (3.10) To show (3.10), we need to show the that the sequence of the random variables fe g n g is uniformly integrable. To check uniform integrability of fe g n g, it su±ces to show that there exists ¸>1 such that sup n E[je g n j ¸ ]=sup n E[e ¸g n ]<1: 43 Since E[e ¸g n ]=E h e ¸(logS 0 + P n i=1 n+1¡i n+1 Y i (n)) i =c 1 E[e ¸ P n i=1 n+1¡i n+1 Y i (n) ]; where c 1 =e ¸logS 0 ; =c 1 E " n Y i=1 e ¸(n+1¡i) n+1 Y i (n) # =c 1 E " n Y i=1 (X i (n)) ¸(n+1¡i) n+1 # ; since8i; Y i =logX i ; =c 1 n Y i=1 h pu ¸(n+1¡i) n+1 +qd ¸(n+1¡i) n+1 i =c 1 n Y i=1 " µ 1 2 + c 2 p n ¶µ 1+ c 3 p n ¶ ¸(n+1¡i) n+1 + µ 1 2 ¡ c 2 p n ¶µ 1¡ c 3 p n ¶ ¸(n+1¡i) n+1 # ; where c 2 = r p T 2¾ >0 and c 3 =¾ p T > 0; <c 1 n Y i=1 " µ 1 2 + c 2 p n ¶µ 1+ c 3 p n ¶ ¸ + µ 1 2 ¡ c 2 p n ¶ c 4 µ 1¡ c 3 p n ¶ ¸ # ; for some constant 1·c 4 <1; <c 1 c 4 n Y i=1 " µ 1 2 + c 2 p n ¶µ 1+ c 3 p n ¶ ¸ + µ 1 2 ¡ c 2 p n ¶µ 1¡ c 3 p n ¶ ¸ # ; taking ¸=2 gives us E[e ¸g n ]<c 1 c 4 n Y i=1 " µ 1 2 + c 2 p n ¶µ 1+ c 3 p n ¶ 2 + µ 1 2 ¡ c 2 p n ¶µ 1¡ c 3 p n ¶ 2 # =c 1 c 4 n Y i=1 · 1 2 µ 2+ 2c 2 3 n ¶ + c 2 p n µ 2 c 3 p n ¶¸ =c 1 c 4 n Y i=1 ³ 1+ c n ´ ; where c=c 2 3 +2c 2 c 3 =c 1 c 4 ³ 1+ c n ´ n <c 1 c 4 e c <1: 44 Therefore, e g n is uniformly integrable and thus, as n!1, C g 0 !e ¡rT Z +1 logK (e x ¡K) 1 p 2¼¾ g e ¡ (x¡¹ g ) 2 2¾ 2 g dx =e ¡rT Z +1 logK¡¹g ¾g (e ¾gy+¹g ¡K) 1 p 2¼ e ¡ y 2 2 dy; by changing of variables y = x¡¹ g ¾ g =e ¡rT " Z +1 logK¡¹g ¾ g e ¾ g y+¹ g 1 p 2¼ e ¡ y 2 2 dy¡K Z +1 logK¡¹g ¾ g 1 p 2¼ e ¡ y 2 2 dy # ,e ¡rT [I 1 +I 2 ]: For I 1 : I 1 = Z +1 logK¡¹g ¾g e ¾gy+¹g 1 p 2¼ e ¡ y 2 2 dy =e ¹g+ 1 2 ¾ 2 g Z +1 logK¡¹g ¾g 1 p 2¼ e ¡ (y¡¾ g ) 2 2 dy =e ¹ g + 1 2 ¾ 2 g Z +1 logK¡¹ g ¾ g ¡¾g 1 p 2¼ e ¡ z 2 2 dz; by letting z =y¡¾ g ; =e ¹ g + 1 2 ¾ 2 g N µ ¾ g ¡ logK¡¹ g ¾ g ¶ =S 0 e 1 2 r¡ ¾ 2 6 T N 0 @ log(S 0 =K)+ 1 2 ³ r+ ¾ 2 6 ´ T ¾ p T=3 1 A : For I 2 : I 2 =¡K Z +1 logK¡¹g ¾g 1 p 2¼ e ¡ y 2 2 dy =¡KN µ ¡ logK¡¹ g ¾ g ¶ =¡KN 0 @ log(S 0 =K)+ 1 2 ³ r¡ ¾ 2 2 ´ T ¾ p T=3 1 A : Thus, we obtain C G 0 ! e ¡rT · S 0 e 1 2 r¡ ¾ 2 6 T N µ log(S 0 =K)+ 1 2 r+ ¾ 2 6 T ¾ p T=3 ¶ ¡KN µ log(S 0 =K)+ 1 2 r¡ ¾ 2 2 T ¾ p T=3 ¶¸ ; 45 and this agrees with the explicit formula (2.5) derived by Kemna and Vorst. The following graph compares the combinatorial approximation for di®erent numbers of time intervals n with the exact value computed by (2.5). The parameters are S 0 =40;K =40;T =1=10;¾ =0:2;r =0:03: 10 15 20 25 30 35 40 45 50 0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 0.66 0.68 0.7 The Number of Time Intervals Prices of Geometric Asian Options GeoKemnaVorst GeoCombinatorial Figure 3.2: Convergence of the Combinatorial Method Next, we analyze the error bound of the combinatorial method. The asymp- totic distribution of the logrithm of the geometric average is g»N(¹ g ;¾ g ); where ¹ g =logS 0 + rT 2 ¡ ¾ 2 T 4 and ¾ 2 g = ¾ 2 T 3 . 46 The exact price of the geometric Eurasian call option is C G 0 =e ¡rT · e ¹g+ 1 2 ¾ 2 g N µ ¾ g ¡ logK¡¹ g ¾ g ¶ ¡KN µ ¡ logK¡¹ g ¾ g ¶¸ =e ¡rT S 0 e 1 2 r¡ ¾ 2 6 T N 0 @ log(S 0 =K)+ 1 2 ³ r+ ¾ 2 6 ´ T ¾ p T=3 1 A ¡e ¡rT KN 0 @ log(S 0 =K)+ 1 2 ³ r¡ ¾ 2 2 ´ T ¾ p T=3 1 A ,S 0 e a 0 N(a)¡e ¡rT KN(b); where a = log(S 0 =K)+ 1 2 ³ r+ ¾ 2 6 ´ T ¾ p T=3 , b = log(S 0 =K)+ 1 2 ³ r¡ ¾ 2 2 ´ T ¾ p T=3 , and a 0 =e ¡ 1 2 (r+ ¾ 2 6 )T . When n is su±ciently large, the approximate logarithm of the average satis¯es g n »N(¹ gn ;¾ gn ); where ¹ g n =logS 0 + rT 2 ¡ ¾ 2 T 4 + r¾ 2 T 2 6n +O(n ¡2 ); ¾ 2 gn = ¾ 2 T 3 + 2¡3¾ 2 T 6n +O(n ¡2 ): Here, only O(n ¡1 ) terms are considered and O(n ¡2 ) and higher order terms are omitted. ThepriceofthegeometricEurasianoptionapproximatedbythecombinatorial algorithm is 47 C G n 0 =e ¡rT · e ¹ gn + 1 2 ¾ 2 g n N µ ¾ g n ¡ logK¡¹ g n ¾ gn ¶ ¡KN µ ¡ logK¡¹ g n ¾ gn ¶¸ =e ¡rT S 0 e 1 2 r¡ ¾ 2 6 T+ c 1 n N 0 @ log(S 0 =K)+ 1 2 ³ r+ ¾ 2 6 ´ T + c 2 n p ¾ 2 T=3+ c 3 n 1 A ¡e ¡rT KN 0 @ log(S 0 =K)+ 1 2 ³ r¡ ¾ 2 2 ´ T + c 4 n p ¾ 2 T=3+ c 3 n 1 A ,S 0 e a 0 + c 1 n N(a n )¡e ¡rT KN(b n ); where a 0 =e ¡ 1 2 (r+ ¾ 2 6 )T ; c 1 = 2r¾ 2 T 2 +2¡3¾ 2 T 12 ; c 2 = r¾ 2 T 2 +2¡3¾ 2 T 6 ; c 3 = 2¡3¾ 2 T 6 ; c 4 = r¾ 2 T 2 6 ; a n = log(S 0 =K)+ 1 2 ³ r+ ¾ 2 6 ´ T + c 2 n p ¾ 2 T=3+ c 3 n ; and b n = log(S 0 =K)+ 1 2 ³ r¡ ¾ 2 2 ´ T + c 4 n p ¾ 2 T=3+ c 3 n : Denote ² n the error of the combinatorial algorithm based on n-level binomial tree. Then,j² n j=jC Gn 0 ¡C G 0 j. It satis¯es j² n j=jC Gn 0 ¡C G 0 j = ¯ ¯ ¯S 0 e a 0 N(a)¡e ¡rT KN(b)¡S 0 e a 0 + c 1 n N(a n )+e ¡rT KN(b n ) ¯ ¯ ¯ ·e a 0 S 0 ¯ ¯ ¯N(a)¡e c 1 n N(a n ) ¯ ¯ ¯+e ¡rT KjN(b)¡N(b n )j ·e a 0 S 0 ³ jN(a)¡N(a n )j+ ¯ ¯ ¯N(a n )¡e c 1 n N(a n ) ¯ ¯ ¯ ´ +e ¡rT KjN(b)¡N(b n )j: 48 Now, forjN(a)¡N(a n )j, ja¡a n j= ¯ ¯ ¯ ¯ ¯ ¯ log(S 0 =K)+ 1 2 ³ r+ ¾ 2 6 ´ T p ¾ 2 T=3 ¡ log(S 0 =K)+ 1 2 ³ r+ ¾ 2 6 ´ T + c 2 n p ¾ 2 T=3+ c 3 n ¯ ¯ ¯ ¯ ¯ ¯ · ¯ ¯ ¯ ¯ ¯ ¯ log(S 0 =K)+ 1 2 ³ r+ ¾ 2 6 ´ T p ¾ 2 T=3 ¡ log(S 0 =K)+ 1 2 ³ r+ ¾ 2 6 ´ T + c 2 n p ¾ 2 T=3 ¯ ¯ ¯ ¯ ¯ ¯ + ¯ ¯ ¯ ¯ ¯ ¯ log(S 0 =K)+ 1 2 ³ r+ ¾ 2 6 ´ T + c 2 n p ¾ 2 T=3 ¡ log(S 0 =K)+ 1 2 ³ r+ ¾ 2 6 ´ T + c 2 n p ¾ 2 T=3+ c 3 n ¯ ¯ ¯ ¯ ¯ ¯ ; by triangle inequality, , ¯ ¯ ¯ ¯ ¯ c 2 n p ¾ 2 T=3 ¯ ¯ ¯ ¯ ¯ + ¯ ¯ ¯f n (¾ 2 T=3)¡f n ³ ¾ 2 T=3+ c 3 n ´¯ ¯ ¯; where f n (x)= log(S 0 =K)+ 1 2 ³ r+ ¾ 2 6 ´ T + c 2 n p x , · ¯ ¯ ¯ ¯ ¯ c 2 n p ¾ 2 T=3 ¯ ¯ ¯ ¯ ¯ + c 3 n ¢ max x2[0;1] jf 0 n (¾ 2 T=3+x)j = ¯ ¯ ¯ ¯ ¯ c 2 n p ¾ 2 T=3 ¯ ¯ ¯ ¯ ¯ + c 3 n ¢ max x2[0;1] ¯ ¯ ¯ ¯ ¯ ¯ ¡ log(S 0 =K)+ 1 2 ³ r+ ¾ 2 6 ´ T + c 2 n 2 p (¾ 2 T=3+x) 3 ¯ ¯ ¯ ¯ ¯ ¯ · c 2 n p ¾ 2 T=3 + c 3 n ¢ max x2[0;1] ¯ ¯ ¯ ¯ ¯ ¯ ¡ log(S 0 =K)+ 1 2 ³ r+ ¾ 2 6 ´ T 2 p (¾ 2 T=3+x) 3 ¯ ¯ ¯ ¯ ¯ ¯ ; since f n is a decreasing function and c 2 n >0; = c 2 n p ¾ 2 T=3 + c 3 n ¢M 1 ; where M 1 = log(S 0 =K)+ 1 2 ³ r+ ¾ 2 6 ´ T 2 p (¾ 2 T=3) 3 ; , ~ c 1 n ; with ~ c 1 = c 2 p ¾ 2 T=3 +c 3 ¢M 1 , 49 we thus have jN(a)¡N(a n )j·ja¡a n j¢max x jN 0 (x)j · ~ c 1 n ¢max x ¯ ¯ ¯ ¯ 1 p 2¼ e ¡ 1 2 x 2 ¯ ¯ ¯ ¯ ; · ~ c 1 n ¢ 1 p 2¼ By the same reasoning, we could also get jb¡b n j= ¯ ¯ ¯ ¯ ¯ ¯ log(S 0 =K)+ 1 2 ³ r¡ ¾ 2 2 ´ T p ¾ 2 T=3 ¡ log(S 0 =K)+ 1 2 ³ r¡ ¾ 2 2 ´ T + c 4 n p ¾ 2 T=3+ c 3 n ¯ ¯ ¯ ¯ ¯ ¯ · c 4 n p ¾ 2 T=3 + c 3 n ¢M 2 ; where M 2 = log(S 0 =K)+ 1 2 ³ r¡ ¾ 2 2 ´ T 2 p (¾ 2 T=3) 3 ; , ~ c 2 n ; with ~ c 2 = c 4 p ¾ 2 T=3 +c 3 ¢M 2 , and jN(b)¡N(b n )j· ~ c 2 n ¢ 1 p 2¼ : For ¯ ¯ ¯N(a n )¡e c 1 n N(a n ) ¯ ¯ ¯, we have ¯ ¯ ¯N(a n )¡e c 1 n N(a n ) ¯ ¯ ¯ =N(a n )j1¡e c 1 n j·j1¡e c 1 n j · c 1 n ¢ max x2[0;1] je x j= c 1 e n : Therefore, we have j² n j=jC Gn 0 ¡C G 0 j· c n ; 50 where c=c(S 0 ;K;r;¾;T) =e a 0 S 0 µ 1 p 2¼ ~ c 1 +c 1 e ¶ +e ¡rT K µ 1 p 2¼ ~ c 2 ¶ =e a 0 S 0 " 1 p 2¼ Ã c 2 p ¾ 2 T=3 +c 3 M 1 ! +c 1 e # +e ¡rT K Ã c 4 p ¾ 2 T=3 +c 3 M 2 ! : Hence, the error bound of the combinatorial method is of O(n ¡1 ). Forann¡periodbinomial tree, thereare2 n averagesand thusthenaivealgo- rithm of pricing Asian options runs O(2 n ) time. Many improved this O(2 n )-time bound substantially. A very successful paradigm by Hull-White [34] is e±cient, with a running time of O(kn 2 ), where k is the average number of the averages at eachnode. Chalasanietal. [13]proposeO(n 4 )-timealgorithmsforAsianoptions. Aingworth et al. [1] develop an O(kn 2 )-time algorithm with an error bound of O(Kn=k), where K is the strike price and k is a positive number and can be varied. The algorithm of Dai, Huang and Lyuu [20] runs in O(kn 2 ) time with a guaranteed error bound of O(K p n=k). To make the error bound of O(n ¡1 ), it su±ces to choose k to be proportional to n 3=2 , which makes the running time O(n 7=2 ). In our combinatorial algorithm, only two values, i.e., g(k;m) and g 2 (k;m), are evaluated at the node (k;m). Since there are k nodes at each level, for k =1;:::;n, the running time is proportional to P n k=1 2k, which is of O(2n 2 ). 51 Table 3.1 compares all the newly developed tree-based algorithms in terms of complexity and error bound. In Table 3.1, k represents any positive number and K represents the strike price of the option. Table 3.1: Comparison of Tree-based Algorithms - Combinatorial Naive Aingworth et al. Dai et al. Complexity O(2n 2 ) O(2 n ) O(kn 2 ) O(kn 2 ) Error Bound O(n ¡1 ) 0 O(Kn=k) O(K p n=k) 3.3 Arithmetic Eurasian Options The combinatorial algorithm can also be applied to approximate the price of the arithmetic Eurasian options. Based on the fact that node (k;m) can be reached either by taking a down-tick from the node (k¡1;m), or by taking an up-tick fromthenode(k¡1;m¡1), if0<m<k, theexpectedarithmeticaveragestock price at the node (k;m),A(k;m), can be obtained by the following formula: A(k;m)= a 1 ¡ k¡1 m ¢ +b 1 ¡ k¡1 m¡1 ¢ ¡ k¡1 m ¢ + ¡ k¡1 m¡1 ¢ ; or A(k;m)= a 1 (k¡m)+b 1 m k ; (3.11) where a 1 = A(k¡1;m)k+S 0 u m d k¡m k+1 , b 1 = A(k¡1;m¡1)k+S 0 u m d k¡m k+1 . Note that a 1 is the mean of the arithmetic averages at node (k;m) obtained from 52 the node (k¡1;m) and b 1 is the mean of the arithmetic averages at node (k;m) obtained from the node (k¡1;m¡1). When m = 0, the node (k;0) can only be reached from the node (k¡1;0). Thus, the arithmetic average stock price at the node (k;0) is A(k;0)= A(k¡1;0)k+S 0 d k k+1 : (3.12) Likewise, when m = k, since the node (k;k) can only be reached from the node (k¡1;k¡1), the formula to compute the arithmetic average stock price at the node (k;k) is A(k;k)= A(k¡1;k¡1)k+S 0 u k k+1 : (3.13) Thus,foreachnodeinthebinomiallattice,theexpectedarithmeticaveragestock price at that node can be computed by (3.11), (3.12) or (3.13). Following the same reasoning, we can compute the expected value of the square of the average stock priceA 2 (k;m) by the formulas below, A 2 (k;m)= a 2 ¡ k¡1 m ¢ +b 2 ¡ k¡1 m¡1 ¢ ¡ k¡1 m ¢ + ¡ k¡1 m¡1 ¢ ; if 0<m<k where a 2 =a 2 1 = A 2 (k¡1;m)k 2 + ¡ S 0 u m d k¡m ¢ 2 +2A(k¡1;m)kS 0 u m d k¡m (k+1) 2 b 2 =b 2 1 = A 2 (k¡1;m¡1)k 2 + ¡ S 0 u m d k¡m ¢ 2 +2A(k¡1;m¡1)kS 0 u m d k¡m (k+1) 2 : 53 When m=0, A 2 (k;0)= A 2 (k¡1;0)k 2 +(S 0 d k ) 2 +2A(k¡1;0)kS 0 d k (k+1) 2 ; and when m=k, A 2 (k;k)= A 2 (k¡1;k¡1)k 2 +(S 0 u k ) 2 +2A(k¡1;k¡1)kS 0 u k (k+1) 2 : Clearly, the expectation of the arithmetic average stock priceE[A n ] and the expectation of the square of the arithmetic averageE[A 2 n ] are E[A n ]= n X m=1 A(n;m) µ n m ¶ p m q n¡m ; E[A 2 n ]= n X m=1 A 2 (n;m) µ n m ¶ p m q n¡m : The variance of the arithmetic average stock price is Var(A n )=E[A 2 n ]¡(E[A n ]) 2 : Higher moments of the arithmetic average stock price can also be computed exactly using the combinatorial method. The random variables X i and Y i are de¯ned as in Section 3.2. The arithmetic average in n-time-period binomial model,A n , can be written as A n = 1 n+1 Ã n X i=0 S i ! ; 54 where S 0 ´s 0 and S i =s 0 Q i j=1 X j (n), for i=1;2;:::;n . Let t2[0;1) and de¯ne S [nt] =s 0 [nt] Y i=1 X i (n); where [nt] is the largest integer part of nt. Then, logS [nt] =logs 0 + P [nt] i=1 Y i (n). Theorem 3.3.1 (Extension of Donsker's Theorem, [6]). For each n, let X n;i with 1· i· n, be independent random variables with mean 0 and variance ¾ 2 n;i . Suppose (i) P n i=1 ¾ 2 n;i =1; (ii) For all ²>0, lim n!1 P n i=1 E[jX n;i j 2 ; jX n;i j>²]=0: Then as n!1, P [nt] i=1 X n;i L ¡ ! W(t), where W(t) is the Wiener process and L ¡ ! means weak convergence of stochastic processes. In what follows, we apply this theorem to ¯nd the asymptotic distribution of A n . Let X n;i = 1 ¾ p T (Y i (n)¡E[Y i (n)]). Then, for each n, X n;i , 1 · i · n, are independentrandomvariableswithmean0andvariance¾ 2 n;i = 1 ¾ 2 T Var(Y i (n))= 1 n +O(n ¡2 ). 55 Clearly, condition (i) in Theorem 3.3.1 is satis¯ed, since P n i=1 ¾ 2 n;i = 1 + O(n ¡1 ). To check condition (ii), consider P n i=1 E[jX n;i j 2 ; jX n;i j>²], for8 ²>0: n X i=1 E[jX n;i j 2 ; jX n;i j>²] · n X i=1 q EjX n;i j 4 E[1 2 fjX n;i j>²g ]; by Cauchy-Schwarz inequality = n X i=1 r 1 ¾ 4 T 2 E[Y i (n)¡E[Y i (n)]] 4 ¢P(jX n;i j>²): (3.14) Since E[Y i (n)¡E[Y i (n)]] 4 =p h ¾ p T=n+O(n ¡1 ) i 4 +q h ¡¾ p T=n+O(n ¡1 ) i 4 =p ³ ¾ p T=n ´ 4 +q ³ ¡¾ p T=n ´ 4 +O(n ¡5=2 )= ³ ¾ p T=n ´ 4 +O(n ¡5=2 ) = ¾ 4 T 2 n 2 +O(n ¡5=2 ); and8 1·i·n, P(jX n;i j>²)=P µ 1 ¾ p T jY i (n)¡E[Y i (n)]j>² ¶ · µ 1 ¾ p T ¶ 2 Var(Y i (n)) ² 2 ; by Chebyshev's inequality = µ 1 ¾ p T ¶ 2 µ ¾ 2 T n² 2 + O(n ¡2 ) ² 2 ¶ = 1 n² 2 +O(n ¡2 ); inequality (3.14) becomes n X i=1 E[jX n;i j 2 ; jX n;i j>²]· n X i=1 s 1 ¾ 4 T 2 ¢ · ¾ 4 T 2 n 2 +O(n ¡5=2 ) ¸ ¢ · 1 n² 2 +O(n ¡2 ) ¸ = n X i=1 r 1 n 3 ² 2 +O(n ¡7=2 )=O(n ¡ 1 2 )!0; as n!1. 56 Therefore, condition (ii) is also satis¯ed. Thus, by Theorem 3.3.1, P [nt] i=1 X n;i ! W(t), as n!1. Since as n!1, [nt] X i=1 X n;i = 1 ¾ p T [nt] X i=1 £ Y i (n)¡E[Y i (n)] ¤ = 1 ¾ p T [nt] X i=1 · Y i (n)¡ rT n + ¾ 2 T 2n +O(n ¡2 ) ¸ ! 1 ¾ p T [nt] X i=1 Y i (n)¡ 1 ¾ p T µ rT ¡ ¾ 2 T 2 ¶ [nt] n +O(n ¡1 ); and lim n!1 sup 0·t·1 ¯ ¯ ¯ [nt] n ¡t ¯ ¯ ¯ =0, we have, as n!1; 1 ¾ p T [nt] X i=1 Y i (n)¡ 1 ¾ p T µ rT ¡ ¾ 2 T 2 ¶ t L ¡ !W(t); i.e., 1 ¾ p T [nt] X i=1 Y i (n) L ¡ !W(t)+ 1 ¾ p T µ rT ¡ ¾ 2 T 2 ¶ t; or, logS [nt] L ¡ !logs 0 +¾ p TW(t)+ µ rT ¡ ¾ 2 T 2 ¶ t: When n is su±ciently large, we could approximateA n by R 1 0 S [nt] dt, since Z 1 0 S [nt] dt= n X i=0 Z i+1 n+1 i n+1 S [nt] dt = n X i=0 Z i+1 n+1 i n+1 S i dt= n X i=0 S i 1 n+1 =A n : 57 Therefore, by the continuous mapping theorem [6], as n!1; S [nt] L ¡ !exp ½ logs 0 +¾ p TW(t)+ µ rT ¡ ¾ 2 T 2 ¶ t ¾ ; or S [nt] L ¡ !s 0 exp ½ ¾ p TW(t)+ µ rT ¡ ¾ 2 T 2 ¶ t ¾ ; and A n L ¡ !s 0 Z 1 0 exp ½ ¾ p TW(t)+ µ rT ¡ ¾ 2 T 2 ¶ t ¾ dt; or A n L ¡ !® Z 1 0 e ¾ p TW(t) dt; with ® = s 0 rT ¡ ¾ 2 T 2 ³ e rT¡ ¾ 2 T 2 ¡1 ´ : TochecktheuniformintegrabilityofA n ,itsu±cestoshowthatthereexistssome ¸>1 such that sup n E[jA n j ¸ ]<1: For ¸=2, E[jA n j ¸ ]=E[jA n j 2 ]=E " ¯ ¯ ¯ ¯ P n i=0 S i n+1 ¯ ¯ ¯ ¯ 2 # = 1 (n+1) 2 E[s 0 +s 0 ¢X 1 +:::+s 0 X 1 ¢:::¢X n ] 2 · 1 (n+1) 2 (n+1)E £ s 2 0 +(s 0 ¢X 1 ) 2 +:::+(s 0 X 1 ¢:::¢X n ) 2 ¤ = 1 n+1 s 2 0 ¡ 1+E[X 2 1 ]+:::+E[X 2 1 ]¢:::¢E[X 2 n ] ¢ ; by independence, = s 2 0 n+1 ¡ 1+E[X 2 1 ]+:::+(E[X 2 1 ]) n ¢ = s 2 0 n+1 1¡(u 2 p+d 2 q) n+1 1¡(u 2 p+d 2 q) : 58 Since u=1+ c 1 p n , d=1¡ c 1 p n , with c 1 =¾ p T, and p= 1 2 + c 2 p n , q = 1 2 ¡ c 2 p n , with c 2 = r p T 2¾ , we have u 2 p+d 2 q = µ 1+ c 1 p n ¶ 2 µ 1 2 + c 2 p n ¶ + µ 1¡ c 1 p n ¶ 2 µ 1 2 ¡ c 2 p n ¶ = 1 2 µ 2+ 2c 2 1 n ¶ + c 2 p n µ 4c 1 p n ¶ =1+ c 3 n ; where c 3 =c 2 1 +4c 1 c 2 >0: Therefore, E[jA n j 2 ]= s 2 0 n+1 1¡(u 2 p+d 2 q) n+1 1¡(u 2 p+d 2 q) = s 2 0 n+1 ¡ 1+ c 3 n ¢ n+1 ¡1 c 3 n <s 2 0 (1+e c 3 )<1: Hence, E[(A n ¡K)1 fAn¸Kg ]! Z fx¸Kg (x¡K)L(x)dx; (3.15) where L(x) is the density function of the random variable ® R 1 0 e ¾ p TW(t) dt: Ac- cordingtoMatsumotoandYor[45],foranyt>0,theprobabilitydensityfunction for R t 0 e 2W(s) ds is P µZ t 0 e 2W(s) ds2du ¶ = du p 2¼u 3 1 p 2¼t Z R cosh(y)e ¡(cosh(y)) 2 =2u¡(y+ p ¡1¼=2) 2 =2t dy: (3.16) By the scaling property of Brownian motion, aW(t=a 2 ) = W(t) in law and then the following equation holds Z 1 0 e ¾ p TW(t) dt= Z 1 0 e ¾ p T¢ 2 ¾ p T W t 4=¾ 2 T dt= Z 1 0 e 2W(t¾ 2 T=4) dt = 4 ¾ 2 T Z ¾ 2 T=4 0 e 2W(s) ds; by change of variables; 59 i.e., ¾ 2 T 4 Z 1 0 e ¾ p TW(t) dt= Z ¾ 2 T=4 0 e 2W(s) ds: Thus, by (3.16), the probability density function for R 1 0 e ¾ p TW(t) dt is P µ ¾ 2 T 4 Z 1 0 e ¾ p TW(t) dt2du ¶ =P Ã Z ¾ 2 T=4 0 e 2W(s) ds2du ! = du p 2¼u 3 1 p 2¼(¾ 2 T=4) Z R cosh(y)e ¡(cosh(y)) 2 =2u¡(y+ p ¡1¼=2) 2 =(2¾ 2 T=4) dy = 1 ¼¾ du p Tu 3 Z R cosh(y)e ¡(cosh(y)) 2 =2u¡2(y+ p ¡1¼=2) 2 =(¾ 2 T) dy: Therefore, the density function of ® R 1 0 e ¾ p TW(t) dt,L(x), is L(x)= 1 ¼¾ (¾ 2 T=4)=® p T 3 [(¾ 2 T=4)x=®] 3 Z R cosh(y)e ¡®(cosh(y)) 2 =[2(¾ 2 T=4)x]¡2(y+ p ¡1¼=2) 2 =(¾ 2 T) dy; i.e., L(x)= 2 p ® ¼¾ 2 T 2 1 p x 3 Z R cosh(y)e ¡2®(cosh(y)) 2 =(¾ 2 Tx)¡2(y+ p ¡1¼=2) 2 =(¾ 2 T) dy: (3.17) Hence, the price of arithmetic Eurasian call options, denoted as C A 0 , is C A 0 =e ¡rT Z +1 K (x¡K)L(x) dx; (3.18) withL(x) given by (3.17). Next, we compare the prices of arithmetic Eurasian call options obtained by Monte Carlo simulation with variance reduction technique and the binomial 60 tree method. The following graph illustrates how Monte Carlo simulation and the binomial tree method behave. The parameters are S 0 = 40;K = 40;T = 1=10;¾ =0:2;r =0:03: 50 60 70 80 90 100 0.2 0.3 0.4 0.5 0.6 0.7 0.8 The Number of Time Intervals Prices of Arithmetic Asian Options Binomial Tree Method Monte Carlo Method Figure 3.3: Binomial Tree Method vs. Monte Carlo Simulation Next,we¯ndtheerrorboundofthebinomialtreemethod. Thearithmeticav- erageAhasthesamedistributionas® R 1 0 e ¾ p TW(t) dt,where® = s 0 ³ e rT¡ ¾ 2 T 2 ¡1 ´ rT ¡ ¾ 2 T 2 . When n su±cient large, A n has the same distribution as ® n R 1 0 e ¾ p TW(t) dt, with ® n = s 0 ³ e rT¡ ¾ 2 T 2 + r¾ 2 T 2 3n ¡1 ´ rT ¡ ¾ 2 T 2 + r¾ 2 T 2 3n , by (3.6). The exact price of the arithmetic Eurasian call option is C A 0 =e ¡rT Z +1 K (x¡K)L(x)dx; whereL(x)=L(®;x)= 2 p ® ¼¾ 2 T 2 1 p x 3 R R cosh(y)e ¡2®(cosh(y)) 2 =(¾ 2 Tx)¡2(y+ p ¡1¼=2) 2 =(¾ 2 T) dy is the density function of ® R 1 0 e ¾ p TW(t) dt. 61 The approximate price of the arithmetic Eurasian call option is C A n 0 =e ¡rT Z +1 K (x¡K)L n (x)dx; whereL n (x)=L(® n ;x)= 2 p ® n ¼¾ 2 T 2 1 p x 3 R R cosh(y)e ¡2®n(cosh(y)) 2 =(¾ 2 Tx)¡2(y+ p ¡1¼=2) 2 =(¾ 2 T) dy is the density function of ® n R 1 0 e ¾ p TW(t) dt. Let L(z;x)= 2 p z ¼¾ 2 T 2 p x 3 Z R cosh(y)e ¡2z(cosh(y)) 2 =(¾ 2 Tx)¡2(y+ p ¡1¼=2) 2 =(¾ 2 T) dy; (3.19) andf(z)=e ¡rT R +1 K (x¡K)L(z;x)dx. Then,f 0 (z)=e ¡rT R +1 K (x¡K) @L(z;x) @z dx, C A n 0 =f(® n ) and C A 0 =f(®). Therefore, the error bound of the arithmetic Eurasian call option satisfy j² n j=jC A n 0 ¡C A 0 j =jf(® n )¡f(®)j·j® n ¡®j¢ max z2[®¡1;®+1] jf 0 (z)j: (3.20) Let g(x)= S 0 e rT¡ ¾ 2 T 2 +x ¡1 rT¡ ¾ 2 T 2 +x , then j® n ¡®j= ¯ ¯ ¯ ¯ g µ r¾ 2 T 2 3n ¶ ¡g(0) ¯ ¯ ¯ ¯ · r¾ 2 T 2 3n max x2[0;1] jg 0 (x)j, ~ c 1 n ; with ~ c 1 = r¾ 2 T 2 3 ¢M 1 . Thus,j² n j· ~ c 1 n ¢max z2[®¡1;®+1] jf 0 (z)j. 62 Note that @L(z;x) @z = 1 2z L(z;x) + 2 p z ¼¾ 4 T 3 p x 5 Z R (cosh(y)) 3 e ¡2z(cosh(y)) 2 =(¾ 2 Tx)¡2(y+ p ¡1¼=2) 2 =(¾ 2 T) dy , 1 2z L(z;x)+ 2 p z ¼¾ 4 T 3 p x 5 I 1 ; we have max z2[®¡1;®+1] jf 0 (z)j= max z2[®¡1;®+1] ¯ ¯ ¯ ¯ e ¡rT Z +1 K (x¡K) @L(z;x) @z dx ¯ ¯ ¯ ¯ = max z2[®¡1;®+1] ¯ ¯ ¯ ¯ e ¡rT Z +1 K (x¡K) 1 2z L(z;x)dx ¯ ¯ ¯ ¯ + max z2[®¡1;®+1] ¯ ¯ ¯ ¯ e ¡rT Z +1 K (x¡K) 2 p z ¼¾ 4 T 3 p x 5 I 1 dx ¯ ¯ ¯ ¯ ,I 2 +I 3 : Clearly, I 2 is bounded by a ¯nite positive constant, M 2 , since z is bounded and L(z;x), which is de¯ned by (3.19), is a density function. To claim I 3 < 1, we consider I 1 ¯rst. Since cosh(y)= e y +e ¡y 2 , we have I 1 = Z R (cosh(y)) 3 e ¡2z(cosh(y)) 2 =(¾ 2 Tx)¡2(y+ p ¡1¼=2) 2 =(¾ 2 T) dy · Z R µ e y +e ¡y 2 ¶ 3 e (¡2y 2 + ¼ 2 2 )=(¾ 2 T) dy =e ¼ 2 =(2¾ 2 T) Z R e 3y +3e y +3e ¡y +e ¡3y 8 e ¡2y 2 =(¾ 2 T) dy ·M 3 <1: 63 Therefore, I 3 = max z2[®¡1;®+1] ¯ ¯ ¯ ¯ e ¡rT Z +1 K (x¡K) 2 p z ¼¾ 4 T 3 p x 5 I 1 dx ¯ ¯ ¯ ¯ · max z2[®¡1;®+1] ¯ ¯ ¯ ¯ e ¡rT Z +1 K (x¡K) 2 p z ¼¾ 4 T 3 p x 5 M 3 dx ¯ ¯ ¯ ¯ · e ¡rT 2 p ®¡1M 3 ¼¾ 4 T 3 Z +1 K µ 1 p x 3 ¡ K p x 5 ¶ dx = e ¡rT 2 p ®¡1M 3 ¼¾ 4 T 3 4 3 p K <1: Hence, by (3.20), ² n · c n , where c=c(S 0 ;K;r;¾;T)= r¾ 2 T 2 M 1 3 µ M 2 + e ¡rT 2 p ®¡1M 3 ¼¾ 4 T 3 4 3 p K ¶ : 3.4 Summary This chapter presents a combinatorial algorithm based on binomial tree model forpricingEuropean-stylegeometricAsianoptions. Thisalgorithmisconvergent witherrorboundO(n ¡1 )andcomplexityO(2n 2 ). Inthischapter,weestablishthe asymptotic distribution of geometric and arithmetic averages of the underlying assets. Based on the distribution of the arithmetic average, we derive an explicit pricing formula for European-style arithmetic Asian options. Thecombinatorialalgorithmdescribedinthischapterwillbeextendedtothe pricing of American-style geometric Asian options in next chapter. 64 Chapter 4 American-style Asian Options European-style options can be exercised at the expiration date only, whereas American-style options o®er earlier exercise opportunities. Since American-style options give the holder more right than European-style options, they are more expensive than their European counterparts. In Section 4.1, a theorem, which providesadecompositionformulaofthepriceofanAmerican-styleAsianputop- tion, is stated and proved. In Section 4.2, the combinatorial algorithm described in Chapter 3 is applied to the pricing of American-style geometric Asian options. 4.1 Decomposition of Amerasian Puts We introduce some notations ¯rst. Denote by C and D the continuation region andstoppingregion, respectively. Inthestoppingregion, itisoptimaltoexercise 65 the American options. D is de¯ned as that subset ofR + £[0;T] for which the stopping time ¿ ¤ t satis¯es,8 t2[0;T], ¿ ¤ t =inffu2[t;T]j (S u ;u)2Dg: The continuation region C is the complement of D in R + £ [0;T]. Let P a t = P a (S t ;T¡t) be the price of the standard American put option with strick price K and expiration date T at time t and P t = P(S t ;T ¡t) the price of the corre- sponding standard European put at time t. Then, we have D =f(S t ;t)2R + £[0;T]j P a (S t ;T ¡t)=(K¡S t ) + g and C =f(S t ;t)2R + £[0;T]j P a (S t ;T ¡t)>(K¡S t ) + g: De¯ne the function of critical stock price b ¤ :[0;T]7!R + by b ¤ (T ¡t)=supfS t 2R + j P a (S t ;T ¡t)=(K¡S t ) + g: Clearly, the graph of b ¤ is contained in the stopping region D. Therefore, it is optimal to exercise the put option at time t only if the current stock price S t is at or below the level b ¤ (T ¡t). Now, we state the theorem of early exercise premium representation of the standard American put option using the notations above. 66 Theorem 4.1.1. The price of the standard American put option may be repre- sented as the sum of the arbitrage price of the corresponding standard European putoptionandtheso-calledearlyexercisepremium. Moreexplicitly, thefollowing decomposition of the price of an American-style put option is valid P a t =P t +E P ¤ ·Z T t e ¡r(u¡t) 1 fSu<b ¤ (T¡u)g rKdu ¯ ¯ ¯F t ¸ : Proof. See paper [12]. 2 This theorem illustrates why, and how much more, the price of a standard Amer- ican put option is expensive than the corresponding European put. Corollary 4.1.1. For t=0, we have P a 0 =P 0 +E P ¤ ·Z T 0 e ¡ru 1 fSu<b ¤ (T¡u)g rKdu ¸ ; where P 0 is the price at time 0 of the standard European put. Early exercise premium representation of arithmetic Amerasian put option can also be derived. Let V a t be the time t price of an arithmetic Amerasian put option with strike price K and expiration date T and V t the time t price of the corresponding arithmetic Eurasian put option. De¯ne the function of critical arithmetic average stock price c ¤ :[0;T]7!R + by c ¤ (T ¡t)=supfA t 2R + j V a t =(K¡A t ) + g: 67 Theorem 4.1.2. The following decomposition of the price at time 0 of an arith- metic Amerasian put option is valid V a 0 =V 0 +E P ¤ ·Z T 0 e ¡rt 1 fAt·c ¤ (T¡t)g £ r(K¡A t )+ 1 t (S t ¡A t ) ¤ dt ¸ : (4.1) Proof. LetthediscountedpricingfunctionofthearithmeticAmerasianputoption be Z t =Z(S t ;I t ;t):=e ¡rt V a t (S t ;I t ;t)=e ¡rt V a t ; (4.2) where I t = R t 0 S u du. By It^ o 's lemma, we have Z T =Z 0 + Z T 0 @Z t @S dS t + Z T 0 · @ 2 Z t @S 2 ¾ 2 S 2 t 2 + @Z t @t ¸ dt+ Z T 0 @Z t @I dI t : Therefore from (4.2) and dI t =S t dt : e ¡rt V a T =V a 0 + Z T 0 e ¡rt @V a t @S dS t + Z T 0 · e ¡rt @ 2 V a t @S 2 ¾ 2 S 2 t 2 ¡re ¡rt V a t +e ¡rt @V a t @t ¸ dt+ Z T 0 e ¡rt @V a t @I S t dt; i.e., e ¡rt V a T =V a 0 + Z T 0 e ¡rt @V a t @S dS t + Z T 0 e ¡rt · @ 2 V a t @S 2 ¾ 2 S 2 t 2 ¡rV a t + @V a t @t + @V a t @I S t ¸ dt: Since V a T = maxf0;K ¡A T g = maxf0;K ¡ 1 T I T g and recall that under mar- tingale measure P ¤ , we have dS t = rS t dt + ¾S t dW ¤ t , where W ¤ t = W t ¡ ¹¡r ¾ t 68 is a standard Brownian motion. Separating the put value into the two regions, V a t =1 fA t >c ¤ (T¡t)g V a t +1 fA t ·c ¤ (T¡t)g (K¡A t ), whereA t = 1 t I t , we have e ¡rt maxf0;K¡A T g =V a 0 + Z T 0 e ¡rt · 1 fA t >c ¤ (T¡t)g @V a t @S +1 fA t ·c ¤ (T¡t)g ¢0 ¸ [rS t dt+¾S t dW ¤ t ] + Z T 0 e ¡rt 1 fA t >c ¤ (T¡t)g · @ 2 V a t @S 2 ¾ 2 S 2 t 2 ¡rV a t + @V a t @t + @V a t @I S t ¸ dt + Z T 0 e ¡rt 1 fAt·c ¤ (T¡t)g · 0¡r(K¡A t )+ 1 t 2 I t ¡ 1 t S t ¸ dt: On 1 fA t >c ¤ (T¡t)g , by (2.7), the pricing function V a t satis¯es the following pricing PDE: @V a t @t + 1 2 ¾ 2 S 2 @ 2 V a t @S 2 +S @V a t @I +rS @V a t @S ¡rV a t =0: Consequently, the terms multiplying 1 fA t >c ¤ (T¡t)g sum to zero, leaving: e ¡rt maxf0;K¡A T g=V a 0 + Z T 0 e ¡rt ¾S t @V a t @S dW ¤ t ¡ Z T 0 e ¡rt 1 fA t ·c ¤ (T¡t)g £ r(K¡A t )+ 1 t (S t ¡A t ) ¤ dt: Taking the expectation with respect toP ¤ gives us: V 0 ´E P ¤[e ¡rt maxf0;K¡A T g] =V a 0 ¡E P ¤ ·Z T 0 e ¡rt 1 fAt·c ¤ (T¡t)g £ r(K¡A t )+ 1 t (S t ¡A t ) ¤ dt ¸ : Thus, (4.1) holds. 2 Next theorem is a more general version of this theorem. 69 Theorem 4.1.3. The following decomposition of the price of an arithmetic Am- erasian put option is valid V a t =V t +E P ¤ ·Z T 0 e ¡rt 1 fAt·b ¤ (T¡t)g £ r(K¡A t )+ 1 t (S t ¡A t ) ¤ dt ¯ ¯ ¯F t ¸ : Proof. Follow similar reasoning as that of Theorem 4.1.2. 2 By virtual of the previous two theorems, the price of an arithmetic Am- erasian put option may be represented as the sum of the arbitrage price of the correspondingarithmeticEurasianputoptionandanothertermconsideredasthe early exercise premium. Similar decomposition theorems for the price of arith- metic Eurasian call options can be derived easily, following similar arguments used to prove Theorem 4.1.2. 4.2 The Combinatorial Method The combinatorial method described in Chapter 3 can be applied to the pricing of geometric Amerasian options. By de¯nition, the value of Amerasian options equals the maximum of the exercise value and holding value. Exercise value is the option value if the option is exercised at that time. Holding value is the option value if it is not exercised. Denote C Ga k;m the price of geometric Amerasian call option at node (k;m), C Ga;e k;m 70 the exercise value and C Ga;h k;m the holding value of the geometric Amerasian call option at node (k;m). Then, we have C Ga k;m =maxfC Ga;e k;m ;C Ga;h k;m g: (4.3) Recall that g(k;m) represent the expected value of the logarithm of the ge- ometric average stock price at node (k;m) and g 2 (k;m) the expected value of the square of the logarithm of the geometric average. Using the combinatorial method, g(k;m) and g 2 (k;m) can be computed by (3.1), (3.2), (3.3) and (3.4). Let ¹(k;m)=g(k;m) and ¾ 2 (k;m)=g 2 (k;m)¡[g(k;m)] 2 : Then, ¹(k;m) and ¾ 2 (k;m) are the expectation and variance of the logarithm of the geometric average stock price at node (k;m), respectively. Assumption 1. We assume that when k is su±ciently large, the distribution of the geometric average stock prices at node (k;m), for m=1;:::;k¡1, is normal. Based on Assumption 1, at node (k;m), the exercise value of geometric Am- erasian call options C Ga;e k;m can be computed by C Ga;e k;m = 8 > > > > > > > > < > > > > > > > > : maxfe g(k;0) ¡K;0g; if m=0, R +1 logK (e x ¡K) 1 p 2¼¾(k;m) e ¡ [x¡¹(k;m)] 2 2¾ 2 (k;m) dx; if m=1;:::;k¡1, maxfe g(k;k) ¡K;0g if m=k. (4.4) 71 The holding value at node (k;m), can be obtained by C Ga;h k;m =e ¡rT=n ¡ pC Ga k+1;m+1 +qC Ga k+1;m ¢ : (4.5) In particular, at the expiration date, the value of the Amerasian call options equals the exercise value. Namely, C Ga n;m =C Ga;e n;m = 8 > > > > > > > > < > > > > > > > > : maxfe g(n;0) ¡K;0g; if m=0, R +1 logK (e x ¡K) 1 p 2¼¾(n;m) e ¡ [x¡¹(n;m)] 2 2¾ 2 (n;m) dx; if m=1;:::;n¡1, maxfe g(n;n) ¡K;0g; if m=n. (4.6) Using the formulas (4.3), (4.4), (4.5) and (4.6) above, the arbitrage price of the Amerasian option can be computed by backward iteration. Beforeweshowthenumericalresultsofthecombinatorialmethodforgeomet- ric Amerasian option, we verify the accuracy of Assumption 1 ¯rst: we compare thepriceofthegeometricEurasianoptionbasedonAssumption1withtheexact price derived by Kemna and Vorst. By Assumption 1, the price of the geometric Eurasian option can be computed by backward iteration. At expiry, C G n;m = 8 > > > > > > > > < > > > > > > > > : maxfe g(n;0) ¡K;0g; if m=0, R +1 logK (e x ¡K) 1 p 2¼¾(n;m) e ¡ [x¡¹(n;m)] 2 2¾ 2 (n;m) dx; if m=1;:::;n¡1, maxfe g(n;n) ¡K;0g; if m=n. (4.7) 72 For k =0;1;:::;n¡1 and m=0;1;:::;k, C G k;m =e ¡rT=n ¡ pC G k+1;m+1 +qC G k+1;m ¢ : (4.8) At time 0, the price of geometric Eurasian put option is C G 0;0 . Table 4.1 compares the price of the geometric Eurasian put option computed by (4.7) and (4.8) with the exact price. In Table 4.1, the parameters used are: S 0 = 40, r = 3%, ¾ = 0:2, T = 12 and K varies. True value represents the option value at time 0 computedbytheexplicitformuladerivedbyKemnaandVorst(2.5). nrepresents the number of time intervals in the binomial tree and C G 0;0 represents the time 0 option value computed by the combinatorial method based on Assumption 1. Table 4.1: Geometric Eurasian Option Values Parameters True Value Option Price Based on Assumption 1 n 10 20 30 40 K =18 19.5678 C G 0;0 19.4085 19.4839 19.5109 19.5247 n 10 20 30 40 K =25 14.9170 C G 0;0 14.7626 14.8351 14.8613 14.8748 n 10 20 30 40 K =40 7.1370 C G 0;0 6.9684 7.0471 7.0757 7.0905 From Table 4.1, we can see that Assumption 1 is a very good approxima- tion. Thus, we could use the combinatorial method described above based on Assumption 1 to compute the price of geometric Amerasian option. Table 4.2 compares the geometric Amerasian option prices computed by the combinatorial method based on Assumption 1 with those computed by Monte Carlo simulation 73 with variance reduction technique. In Table 4.2, the parameters are: S 0 = 40, r =3%, ¾ =0:2, T =12 and K varies. C Ga MC represents the geometric Amerasian optionvalueattime0computedbyMonteCarlomethodwithvariancereduction technique. n represents the number of time intervals in the binomial tree and C Ga 0;0 represents the time 0 geometric Amerasian option value computed by the combinatorial method based on Assumption 1. Table 4.2: Geometric Amerasian Option Values Parameters C Ga MC Option Price Based on Assumption 1 n 10 20 30 K =18 22.8782 C Ga 0;0 22.0000 22.0000 22.0000 n 10 20 30 K =25 16.3562 C Ga 0;0 15.5021 15.5631 15.5833 n 10 20 30 K =40 7.8660 C Ga 0;0 7.0010 7.0891 7.1216 From the combinatorial algorithm discussed above, we can see that only four values need to be computed at each node. More speci¯cally, at node (k;m), only ¹(k;m),¾ 2 (k;m),C Ga;e k;m andC Ga;h k;m arecomputed. Therearek nodesateachlevel, for k = 1;:::;n. Therefore, the running time of the combinatorial algorithm for geometric Amerasian option is proportional to P n k=1 4k, which is of O(4n 2 ). 74 4.3 Summary This chapter discusses the pricing of American-style Asian options. More specif- ically, this chapter presents a decomposition theorem for the price of American- style arithmetic Asian put options. By virtue of this theorem, the price of an arithmetic Amerasian put option may be represented as the sum of the arbitrage price of the corresponding arithmetic Eurasian put option and another term con- sideredastheearlyexercisepremium. Thischapteralsopresentsacombinatorial algorithm, the complexity of which is O(4n 2 ), for pricing American-style geomet- ric Asian options. 75 Chapter 5 Asymptotic Behaviors and Extensions of the Model Inthischapter, theclassicBlack-Scholesmodelisextendedtothefollowingcases - zero volatility, no expiry and trading in both domestic and foreign security market. Foreignrisk-freebonds,foreignstocks,theirderivatives,includingforeign equity options, quanto calls and currency options, and the valuation formulas for the derivatives are discussed. 5.1 Asymptotic Analysis Inthissection,theasymptoticbehaviorsofthepriceofAsianoptionsforthecase of (i) zero volatility coe±cient ¾!0 and (ii) in¯nite expiration date T !1 are studied. 76 5.1.1 Zero Volatility When ¾!0, there's no stochastic term in the dynamics of the underlying stock. Therefore, the option price equals the discounted sure payo®. The prices of European option, geometric and arithmetic Eurasian and Amerasian option in case that the volatility is zero are discussed below. In the Black-Scholes model, when ¾ = 0, the dynamics of the stock price become dS t =rS t dt+0; i.e., S t =S 0 e rt ; and thus, the price of the vanilla European call option becomes e ¡rT max © S 0 e rT ¡K;0 ª ; or, max © S 0 ¡Ke ¡rT ;0 ª : (5.1) Based on the dynamics of S t , the geometric average at expiry becomes G T =exp µ 1 T Z T 0 logS t dt ¶ =exp µ 1 T Z T 0 (logS 0 +rt)dt ¶ =exp µ logS 0 + rT 2 ¶ =S 0 e rT 2 ; and so the price of the geometric Eurasian call option when ¾!0 is e ¡rT max n S 0 e rT 2 ¡K;0 o ; or, max n S 0 e ¡rT 2 ¡Ke ¡rT ;0 o : (5.2) 77 We could get the same result by passing ¾ ! 0 to the limit in the following pricing formula derived by Kemna and Vorst, C G 0 (X)=e ¡rT · e d ¤ 0 S 0 N(d 0 )¡KN µ d 0 ¡¾ q 1 3 T ¶¸ ; where d ¤ 0 = 1 2 ¡ r¡ 1 6 ¾ 2 ¢ T and d 0 = log(S 0 =K)+ 1 2 ¡ r+ 1 6 ¾ 2 ¢ T ¾ q 1 3 T : When ¾ ! 0, d ¤ 0 ! rT 2 , d 0 !1 and therefore (5.2) follows. By Theorem 2.3.2, the price of the geometric Amerasian call option when ¾!0 is sup ¿2[0;T] n max n S 0 e ¡r¿ 2 ¡Ke ¡r¿ ;0 oo ; which is no longer a stochastic function and thus the optimal stopping time can be found relatively easily. Let f(¿) = max n S 0 e ¡r¿ 2 ¡Ke ¡r¿ ;0 o ; for ¿ 2 [0;T], be the exercise value of the geometric Eurasian call option at time ¿. To ¯nd the maximum of the non-di®erentiable function f(¿), we ¯rst look at its graph. The graphs of f(¿) with di®erent S 0 and K are shown in Figure 5.1 below. It is can be seen from the plot of f(¿) that at the non-di®erentiable points, f(¿) = 0 and thus the maxima is never achieved at those points. Therefore, only the positive di®erentiable part of f(¿), i.e., S 0 e ¡r¿ 2 ¡ Ke ¡r¿ , needs to be considered. Solving0= ³ S 0 e ¡r¿ 2 ¡Ke ¡r¿ ´ 0 = ¡r 2 S 0 e ¡r¿ 2 +rKe ¡r¿ andtakingthe end points ¿ = 0 and ¿ = T into account give us that the optimal exercise time of the geometric Amerasian option ¿ ¤ =min n T;max n 2 r log 2K S 0 ;0 oo : Therefore, The price of the geometric Amerasian option is max n S 0 e ¡r¿ ¤ 2 ¡Ke ¡r¿ ¤ ;0 o . 78 0 50 100 150 0 5 10 15 20 t f(t) Case 1 0 50 100 150 0 2 4 6 t f(t) Case 2 0 50 100 150 0 2 4 6 8 10 t f(t) Case 1(a) 0 50 100 150 0 5 10 15 20 25 t f(t) Case 3 Figure5.1: TheExerciseValueoftheGeometricEurasianOptionvs. Time, with S 0 = 40; r = 0:03. Case 1: K = 25, Case 1(a): K = S 0 = 40, Case 2: K = 68, Case 3: K =18; For arithmetic Asian options, we have derived that A T =® Z 1 0 e ¾ p TW(t)dt ; with ® = S 0 (e rT¡ ¾ 2 T 2 ¡1) rT ¡ ¾ 2 T 2 : When ¾ =0,A T = S 0 (e rT ¡1) rT and the price of the arithmetic Eurasian option is e ¡rT max ½ S 0 (e rT ¡1) rT ¡K;0 ¾ ; i.e.; max ½ S 0 (1¡e ¡rT ) rT ¡Ke ¡rT ;0 ¾ : (5.3) Same result holds in the Black-Scholes model. Based on the dynamics of S t , we have A T = 1 T Z T 0 S t dt= 1 T Z T 0 (S 0 e rt )dt= S 0 (e rT ¡1) rT ; 79 and thus (5.3) follows. Comparing (5.1), (5.2) and (5.3), it is clear that when ¾ ! 0, the prices of the geometric and arithmetic Eurasian option are lower than that of the vanilla European option. This is because, when ¾ ! 0, the stock price S t = S 0 e rt is a monotonically increasing function and so the stock price at T is higher than the geometric or arithmetic average of the stock price from time 0 up to time T. 0 50 100 150 200 5 10 15 20 t h(t) Case 1 0 50 100 150 200 0 2 4 6 8 10 t h(t) Case 2 0 50 100 150 200 0 5 10 15 20 25 t h(t) Case 3 0 50 100 150 200 0 5 10 15 t h(t) Case 1(a) Figure 5.2: The Exercise Value of the Arithmetic Eurasian Option vs. Time, with S 0 = 40; r = 0:03. Case 1: K = 25, Case 1(a): K = S 0 = 40, Case 2: K =68, Case 3: K =18; By Theorem 2.3.2, the price of the arithmetic Amerasian option when ¾!0 is sup ¿2[0;T] ½ max ½ S 0 (1¡e ¡r¿ ) r¿ ¡Ke ¡r¿ ;0 ¾¾ : 80 Leth(¿)=max n S 0 (1¡e ¡r¿ ) r¿ ¡Ke ¡r¿ ;0 o . The graphs ofh(¿)areshowninFigure 5.2. To ¯nd the price of the arithmetic Amerasian option, it is su±ce to ¯nd the the maximum of the positive di®erentiable part of h(¿), by the same reasoning used in ¯nding the geometric Amerasian option price. Let h 1 (¿)= S 0 (1¡e ¡r¿ ) r¿ ¡Ke ¡r¿ . Then h 0 1 (¿)= [S 0 (re ¡r¿ )¡(rKe ¡r¿ ¡r 2 ¿Ke ¡r¿ )](r¿)¡[S 0 (1¡e ¡r¿ )¡r¿Ke ¡r¿ ]¢r r 2 ¿ 2 = r¿S 0 e ¡r¿ +r 2 ¿ 2 Ke ¡r¿ ¡S 0 +S 0 e ¡r¿ r¿ 2 = e ¡r¿ (r¿S 0 +r 2 ¿ 2 K¡S 0 e r¿ +S 0 ) r¿ 2 : To make h 0 1 (¿) zero, we need S 0 +r¿S 0 +r 2 ¿ 2 K =S 0 e r¿ : There are two solutions to this nonlinear equation. ¿ =0 is always one solution. The other solution, which might be positive or negative, varies with di®erent parameters S 0 and K. See Figure 5.3 for illustration. Since ¿ 2 [0;T], we are only interested in the nonnegative solutions. Denote the positive solution to this equation as ¿ 0 . Then, it is optimal to exercise the arithmetic Amerasian option at ¿ ¤ = minfT;maxf¿ 0 ;0gg and the value of this option is max n S 0 (1¡e ¡r¿ ¤ ) r¿ ¤ ¡Ke ¡r¿ ¤ ;0 o : 81 −20 0 20 40 60 80 0 50 100 150 200 250 300 350 400 450 t Case 1 S 0 +S 0 rt+Kr 2 t 2 S 0 e rt Intersection Intersection −80 −60 −40 −20 0 20 0 10 20 30 40 50 60 70 80 t Case 2 S 0 +S 0 rt+Kr 2 t 2 S 0 e rt Intersection Intersection Figure 5.3: Plots of f(¿) = S 0 + r¿S 0 +r 2 ¿ 2 K and g(¿) = S 0 e r¿ , with S 0 = 40;r =0:03. Case 1: K =45, Case 2: K =12. 5.1.2 Perpetual Asian Options First, we consider Eurasian options with extremely large expiry. Recall that the pricing formulas for the geometric and arithmetic Asian options are C G 0 (X)=e ¡rT · e d ¤ 0 S 0 N(d 0 )¡KN µ d 0 ¡¾ q 1 3 T ¶¸ ; with d ¤ 0 = 1 2 ¡ r¡ 1 6 ¾ 2 ¢ T, and d 0 = log(S 0 =K)+ 1 2 ¡ r+ 1 6 ¾ 2 ¢ T ¾ q 1 3 T ; and C A 0 =e ¡rT Z +1 K (x¡K)L(x)dx; with L(x) = 2 p ® ¼¾ 2 T 2 1 p x 3 R R cosh(y)e ¡2®(cosh(y)) 2 =(¾ 2 Tx)¡2(y+ p ¡1¼=2) 2 =(¾ 2 T) dy: Passing T !1 to the limit in the pricing formulas above, it is clear that the prices of the Eurasian options C G 0 !0 and C A 0 !0. 82 Let T !1 in the Black-Scholes European call formula C 0 =S 0 N(d 1 )¡e ¡rT KN(d 2 ); where d 1;2 = log S 0 K +(r§ ¾ 2 2 )T ¾ p T . It is clear that the value of the European call without expiry is S 0 , since N(d 1 ) ! 1 and e ¡rT ! 0, as T ! 1. Thus, the European-style Asian options with extremely large expiration date are di®erent from the vanilla European options with extremely large expiration date. This is because the expected value of the geometric and arithmetic averages increase at a rate lower than the discounting factor and the value of the averages, and thus the price of the Asian options with extremely large expiration date, are killed by the discounting factor eventually. Those American options that have no expiration date or can be exercised at any time are also referred to the perpetual American options [58]. Clearly, per- petualAmericanoptionsgivetheownermoreright,andthusaremoreexpensive, than American options. According to Wilmott, the price of the perpetual Amer- ican options does not depend on time t [58]. And it is proved, by using PDE approach, that even though by exercising the perpetual American option, the owner is likely to make some money, the exercise region of perpetual American call is empty, i.e., it is never optimal to exercise the perpetual American call [58]. The perpetual Amerasian options are discussed in [29] and [18]. Same as perpetual American options, it is never optimal to exercise perpetual Amerasian 83 options. But di®erent from perpetual American options, the prices of perpetual Amerasian options have dependence on t [18]. 5.2 Asian Options in Foreign Markets Inthissection, martingalemeasuresinforeignmarketsarediscussed. Foreigneq- uityAsianoptions,quantoAsiancallsandcurrencyAsianoptionsareintroduced. Their pricing formulas are also derived. 5.2.1 Martingale Measures in Foreign Markets We ¯rst introduce some new concepts and notations that are required in foreign markets. Let Q t be the exchange rate, i.e., the price at time t of one unit of foreign currency in the domestic currency. The exchange rate process, which is de¯ned on the ¯ltered probability space (;fF t g t¸0 ;P), is modelled by dQ t =Q t (¹ Q dt+¾ Q dW t ); Q 0 >0; (5.4) where ¹ Q 2R is a constant drift coe±cient, ¾ Q 2R denotes a constant volatility factor. Clearly, the solution of equation (5.4) is Q t =Q 0 expf¾ Q W t +(¹ Q ¡ 1 2 ¾ 2 Q )tg: 84 Let B d t =exp(r d t) and B f t =exp(r f t), where r d and r f are the domestic and foreign interest rates, respectively. De¯ne Q ¤ t := B f t Q t =B d t = e (r f ¡r d )t Q t . Then Q ¤ t represents the value of the foreign savings account at time t when converted indomesticcurrencyanddiscountedbythecurrentvalueofthedomesticsavings account. Moreover, Q ¤ t satis¯es Q ¤ t =e (r f ¡r d )t Q t =Q 0 expf¾ Q W t +(¹ Q +r f ¡r d ¡ 1 2 ¾ 2 Q )tg; or equivalently, dQ ¤ t =Q ¤ t [(¹ Q +r f ¡r d )dt+¾ Q dW t ]: Thus, Q ¤ is a martingale underP if and only if ¹ Q =r d ¡r f . Assume that Q ¤ is a martingale under an equivalent probability measureP ¤ . P ¤ is called the domestic martingale measure. Intuitively, P ¤ is a risk-neutral probability measure as seen from the perspective of an investor who trades in domestic currency. Under the domestic martingale measureP ¤ , we have dQ t =Q t [(r d ¡r f ) dt+¾ Q dW ¤ t ]; Q 0 >0; where W ¤ is a 1-dimensional Brownian motion underP ¤ . The arbitrage price, V t (X), expressed in domestic currency, of any contingent claim X that expires at time T and is dominated in the domestic currency, is 85 V t (X)=e ¡r d (T¡t) E P ¤[XjF t ]: IfacontingentclaimY withexpirationdateT ispricedinunitsofforeigncur- rency, then its arbitrage price at time t, expressed in units of domestic currency, is given by V t (Y)=e ¡r d (T¡t) E P ¤[Q T YjF t ]: Since Q t is the exchange rate, i.e., the price at time t of one unit of foreign currency in the domestic currency, it is evident that the price at t of one unit of the domestic currency, expressed in units of foreign currency, equals R t = Q ¡1 t . By It^ o's formula, we have dQ ¡1 t =¡Q ¡2 t dQ t +Q ¡3 t dQ 2 t ; or, dR t =¡R 2 t h Q t [(r d ¡r f ) dt+¾ Q dW ¤ t ] i +R 3 t (Q 2 t ¾ 2 Q dt) =¡R t h (r d ¡r f ) dt+¾ Q dW ¤ t +¾ 2 Q dt i =R t h (r f ¡r d ) dt¡¾ Q (dW ¤ t ¡¾ Q dt) i ; i.e., dR ¤ t =¡R ¤ t ¾ Q (dW ¤ t ¡¾ Q dt); (5.5) where R ¤ t :=R t e (r d ¡r f )t . 86 From (5.5), we can see that there exist a measure ~ P such that ~ P is equivalent toP ¤ andR ¤ is a martingale under ~ P. ~ P is called the foreign martingale measure. Under the foreign martingale measure ~ P, we have dR ¤ t =¡R ¤ t ¾ Q d ~ W t ; where ~ W t =W ¤ t ¡¾ Q t, and thus dR t =R t h (r f ¡r d ) dt¡¾ Q d ~ W t i : The arbitrage price at time t, in units of foreign currency, of a contingent claim X with expiration date T is ~ V t (X)=e ¡r f (T¡t) E ~ P [R T XjF t ]: Let S f t be the price at time t of the foreign stock price expressed in units of foreign currency. To avoid arbitrage, assume that the dynamics of S f t under the foreign martingale measure ~ P are dS f t =S f t (r f dt+¾ S f d ~ W t ); S f 0 >0; 87 with a constant volatility coe±cient ¾ S f 2R. Equivalently, under the domestic martingale measureP ¤ , S f t satis¯es dS f t =S f t h (r f ¡¾ Q ¾ S f) dt+¾ S f d ~ W ¤ t i : (5.6) De¯ne ~ S f t := Q t S f t , then ~ S f t represents the price at time t of a foreign stock S f expressedinunitsofdomesticcurrency. UsingIt^ o'sformulaandthedynamics of Q t under the domestic martingale measureP ¤ , we could get d ~ S f t = ~ S f t h r d dt+(¾ S f +¾ Q ) dW ¤ t i : 5.2.2 Foreign Equity Asian Options and Quanto Asian Calls A foreign equity option is an option whose terminal payo®, in units of domestic currency, depends not only on the future behavior of the exchange rate, but also on the price of a certain foreign stock. Denote K f as the strike price of an option in units of foreign currency. Then, the payo® of a foreign equity call struck in foreign currency with expiration date T equals C f T =Q T (S f T ¡K f ) + : 88 The arbitrage price of this option at time t is C f t =e ¡r d (T¡t) E P ¤[Q T (S f T ¡K f ) + jF t ] =e ¡r f (T¡t) Q t E ~ P [(S f T ¡K f ) + jF t ] =Q t h S f t N(d 1 )¡K f e ¡r f (T¡t) N(d 2 ) i ; by the Black-Scholes formula. where d 1;2 = log(S f t =K f )+(r f § 1 2 ¾ 2 S f )(T ¡t) ¾ S f p T ¡t : Now, we consider foreign equity Asian options. A geometric foreign equity Asian option has payo® at expiry C Gf T =Q T (G f T ¡K f ) + ; where G f T stands for the geometric average of the stock price in units of foreign currency over the period [0,T]. Using the explicit pricing formula (2.6), we obtain C Gf t =e ¡r d (T¡t) E P ¤[Q T (G f T ¡K f ) + jF t ] =e ¡r f (T¡t) Q t E ~ P [(G f T ¡K f ) + jF t ] =Q t · e d ¤ t G f t N(d t )¡K f N µ d t ¡¾ S f q 1 3 (T ¡t) ¶¸ ; where d ¤ t and d t are d ¤ t = 1 2 (r f ¡ 1 6 ¾ 2 S f )(T ¡t) and d t = log(G f t =K f )+ 1 2 (r f + 1 6 ¾ 2 S f )(T ¡t) ¾ S f q 1 3 (T ¡t) : 89 Likewise, an arithmetic foreign equity Asian call has payo® at expiry C Af T =Q T (A f T ¡K f ) + ; whereA f T is the arithmetic average of the stock price in units of foreign currency over the period [0,T]. The price of the arithmetic foreign equity Asian call is C Af t =e ¡r d (T¡t) E P ¤[Q T (A f T ¡K f ) + jF t ] =e ¡r f (T¡t) Q t E ~ P [(A f T ¡K f ) + jF t ] =Q t n e ¡r f (T¡t) E ~ P [(A f T ¡K f ) + jF t ] o : In particular, at the initial time, C Af 0 =Q 0 n e ¡r f T E ~ P [(A f T ¡K f ) + ] o ; where the part inside the fg can be computed based on the explicit formula (3.18): C Af 0 =Q 0 e ¡r f T Z +1 K f ¡ x¡K f ¢ L(x)dx; whereL(x) is given by L(x)= 2 p ® ¼¾ 2 S f T 2 1 p x 3 Z R cosh(y)e ¡2®(cosh(y)) 2 =(¾ 2 S f Tx)¡2(y+ p ¡1¼=2) 2 =(¾ 2 S f T) dy; with ® = Q 0 r f T ¡ ¾ 2 S f T 2 Ã e r f T¡ ¾ 2 S f T 2 ¡1 ! : 90 DenoteK d asthestrikepriceofanoptioninunitsofdomesticcurrency. Then, thepayo®ofaforeignequitycallstruckindomesticcurrency withexpirationdate T equals C d T =(S f T Q T ¡K d ) + =( ~ S f T ¡K d ) + : Thearbitragepriceoftheoptionattimetis,bytherisk-neutralvaluationformula and the Black-Scholes formula, C d t =e ¡r d (T¡t) E P ¤ h ( ~ S f T ¡K d ) + jF t i = ~ S f t N(d 1 )¡K d e ¡r d (T¡t) N(d 2 ); where d 1;2 = log( ~ S f t =K d )+[r d § 1 2 (¾ S f +¾ Q ) 2 ](T ¡t) (¾ S f +¾ Q ) p T ¡t : ConsiderageometricAsianforeignequityoptionstruckindomesticcurrency, whose payo® is C Gd T =(G f T Q T ¡K d ) + =( ~ G f T ¡K d ) + : Then, by the same reasoning as before, we obtain the arbitrage price at time t of this geometric Asian option: C Gd t =e ¡r d (T¡t) E P ¤ h ( ~ G f T ¡K d ) + jF t i =e d ¤ ~ G f t N(d)¡K d N ³ d¡(¾ S f +¾ Q ) q 1 3 (T ¡t) ´ ; by (2.6), with d ¤ = 1 2 [r d ¡ 1 6 (¾ S f+¾ Q ) 2 ](T¡t); d= log( ~ G f t =K d )+ 1 2 [r d + 1 6 (¾ S f +¾ Q ) 2 ](T ¡t) (¾ S f +¾ Q ) q 1 3 (T ¡t) : 91 An arithmetic Asian foreign equity option struck in domestic currency has payo® C Ad T =(A f T Q T ¡K d ) + =( ~ A f T ¡K d ) + : Its arbitrage price at time t is C Ad t =e ¡r d (T¡t) E P ¤ h ( ~ A f T ¡K d ) + jF t i ; and the option price at time 0 can be be computed based on the explicit formula (3.18) or approximated by the combinatorial method, withA f T instead ofA T : C Ad 0 =e ¡r d T Z +1 K d ¡ x¡K d ¢ L(x)dx; whereL(x) is given by L(x)= 2 p ® ¼(¾ S f +¾ Q ) 2 T 2 1 p x 3 ¢ Z R cosh(y)e ¡2®(cosh(y)) 2 =((¾ S f +¾ Q ) 2 Tx)¡2(y+ p ¡1¼=2) 2 =((¾ S f +¾ Q ) 2 T) dy; with ® = S f 0 r d T ¡ (¾ S f +¾ Q ) 2 T 2 Ã e r d T¡ (¾ S f +¾ Q ) 2 T 2 ¡1 ! : A quanto call underlying a certain foreign stock with expiry date T has a payo® that does not depend on the exchange risk, more speci¯cally, C q T = ¹ Q(S f T ¡K f ) + ; 92 where ¹ Q is the pre-speci¯ed exchange rate at T. Since the payo® of the quanto option is expressed in units of domestic currency, its arbitrage price equals C q t = ¹ Qe ¡r d (T¡t) E P ¤ h (S f T ¡K f ) + jF t i = ¹ Qe (±¡r d )(T¡t) E P ¤ h e ¡±(T¡t) (S f T ¡K f ) + jF t i ; where ± = r d ¡¾ Q ¾ S f. Note that ± is the drift coe±cient the dynamics of S f t , by (5.6). Using the classic form of Black-Scholes formula, we can get C q t = ¹ Qe (±¡r d )(T¡t) h S f t e ±(T¡t) N(d 1 )¡K f N(d 2 ) i ; where d 1;2 = log(S f t =K f )+(±§ 1 2 ¾ 2 S f )(T ¡t) ¾ S f p T ¡t : The arbitrage price of a geometric Asian quanto call at time t can be derived, by using (2.6), as follows: C Gq t = ¹ Qe ¡r d (T¡t) E P ¤ h (G f T ¡K f ) + jF t i = ¹ Qe (±¡r d )(T¡t) E P ¤ h e ¡±(T¡t) (G f T ¡K f ) + jF t i = ¹ Qe (±¡r d )(T¡t) · e d ¤ ~ G f t N(d)¡K f N ³ d¡¾ S f q 1 3 (T ¡t) ´ ¸ ; where d ¤ and d can be written as d ¤ = 1 2 (±¡ 1 6 ¾ 2 S f )(T ¡t); d= log( ~ G f t =K f )+ 1 2 (±+ 1 6 ¾ 2 S f )(T ¡t) ¾ S f q 1 3 (T ¡t) : 93 The arbitrage price of an arithmetic Asian quanto call at time t is C Aq t = ¹ Qe ¡r d (T¡t) E P ¤ h (A f T ¡K f ) + ¯ ¯ F t i ; At time 0, the price can be computed based on the pricing formula (3.18): C Aq 0 =e ¡r d T Z +1 K f ¡ x¡K f ¢ L(x)dx; whereL(x) is given by L(x)= 2 p ® ¼¾ 2 S f T 2 1 p x 3 Z R cosh(y)e ¡2®(cosh(y)) 2 =(¾ 2 S f Tx)¡2(y+ p ¡1¼=2) 2 =(¾ 2 S f T) dy; with ® = Q 0 ±T ¡ ¾ 2 S f T 2 Ã e ±T¡ ¾ 2 S f T 2 ¡1 ! : 5.2.3 Currency Asian Options A currency European call option has payo® at the expiration date T C c T =(Q T ¡K) + ; where Q T is the spot exchange rate at T and K is the strike price in units of domestic currency per foreign unit. 94 The risk-neutral valuation formula for this currency European call option is, for all t, C c t =e ¡r d (T¡t) E P ¤[(Q T ¡K) + jF t ] =e ¡r f T e ¡r d (T¡t) E P ¤ · ³ e r f T Q T ¡e r f T K ´ + ¯ ¯ ¯F t ¸ : Recallthatindomesticmarket,ifthedynamicsofS t underamartingalemeasure P ¤ satis¯es dS t = S t (rdt+¾dW ¤ t ), then the Black-Scholes formula tells us that the arbitrage price of a European call option with strike price K and expiration date T is given by C t =e ¡r(T¡t) E P ¤[(S T ¡K) + jF t ]=S t N(d 1 )¡Ke ¡r(T¡t) N(d 2 ); where d 1;2 = log(S t =K)+(r§ 1 2 ¾ 2 )(T ¡t) ¾ p T ¡t : Using the Black-Scholes formula, we have C c t =e ¡r f T e ¡r d (T¡t) E P ¤ · ³ e r f T Q T ¡e r f T K ´ + ¯ ¯ ¯F t ¸ =Q t e ¡r f (T¡t) N(d 1 )¡Ke ¡r d (T¡t) N(d 2 ); where d 1;2 = log(Q t =K)+(r d +r f § 1 2 ¾ 2 Q )(T ¡t) ¾ Q p T ¡t : At expiry, a geometric currency Asian call option has payo® C Gc T =(G Q T ¡K) + ; 95 where G Q T is the geometric average of the exchange rate from time 0 to time T and K is the strike price in units of domestic currency per foreign unit. The arbitrage price of this geometric currency Asian call option is, for all t, C Gc t =e ¡r d (T¡t) E P ¤[(G Q T ¡K) + jF t ] =e ¡r f T e ¡r d (T¡t) E P ¤ · ³ e r f T G Q T ¡e r f T K ´ + ¯ ¯ ¯F t ¸ : By the explicit pricing formula for geometric Asian options (2.6), we have C Gc t =e ¡r f T e ¡r d (T¡t) E P ¤ · ³ e r f T G Q T ¡e r f T K ´ + ¯ ¯ ¯F t ¸ =e ¡r f T · e d ¤ e r f t G Q t N(d)¡e r f T Ke ¡r d (T¡t) N ³ d¡¾ Q q 1 3 (T ¡t) ´ ¸ =e ¡r f (T¡t) e d ¤ G Q t N(d)¡Ke ¡r d (T¡t) N ³ d¡¾ Q q 1 3 (T ¡t) ´ ; where d ¤ and d can be written as d ¤ = 1 2 (r d +r f ¡ 1 6 ¾ 2 Q )(T ¡t); d= log(G Q t =K)+ 1 2 (r d +r f + 1 6 ¾ 2 Q )(T ¡t) ¾ Q q 1 3 (T ¡t) : An arithmetic currency Asian call option has payo® at the expiration date T C Ac T =(A Q T ¡K) + ; whereA Q T is the arithmetic average of the exchange rate from time 0 to time T. 96 The arbitrage price of an arithmetic Asian currency call at t is C Ac t =e ¡r d (T¡t) E P ¤[(A Q T ¡K) + jF t ] =e ¡r f T ½ e ¡r d (T¡t) E P ¤ · ³ e r f T A Q T ¡e r f T K ´ + ¯ ¯ ¯F t ¸¾ : The price at time 0 can be computed based on the pricing formula (3.18): C Ac 0 =e ¡(r d +r f )T Z +1 K ¡ x¡e r f T K ¢ L(x)dx; whereL(x) is given by L(x)= 2 p ® ¼¾ 2 Q T 2 1 p x 3 Z R cosh(y)e ¡2®(cosh(y)) 2 =(¾ 2 Q Tx)¡2(y+ p ¡1¼=2) 2 =(¾ 2 Q T) dy; with ® = Q 0 (r d +r f )T ¡ ¾ 2 Q T 2 · e (r d +r f )T¡ ¾ 2 Q T 2 ¡1 ¸ : 5.3 Summary This chapter is an extension of the previous chapters. The pricing theorems, formulas and algorithms derived in Chapter 2 and Chapter 3 are extended to the pricing of European-style and American-style options with zero volatility or with in¯niteexpirationdateandtothepricingofforeignequityAsianoptions, quanto Asian calls and currency Asian options. 97 Chapter 6 Summary and Conclusions This chapter summarizes the major contributions of this research, and outlines directions for future research that will expand the results on the pricing of Asian options obtained in this work. 6.1 Summary This dissertation focuses on developing fast and e±cient methods for pricing AsianoptionsandextendingthepricingtheoremsforstandardEuropeanoptions to Asian options. Chapter 1 outlines the concept of Asian options, their attractiveness to in- vestors/traders, and the challenges associated with pricing the same. A review of all relevant research is also presented in Chapter 1. Chapter 2 reviews some fundamental pricing theorems for standard European and American options in the Black-Scholes model and extends those pricing theorems to the cases of geo- metric and arithmetic European- and American-style Asian options. Chapter 3 98 presents a combinatorial algorithm in the Binomial tree model for pricing both geometric and arithmetic European-style Asian options. The convergence, error bound and complexity of this combinatorial algorithm is also discussed. Chapter 4 presents a theorem that decomposes an arithmetic American-style Asian op- tion into its European counterpart and another part considered as early-exercise premium and thus builds a bridge between arithmetic European- and American- style Asian put options. An algorithm in the Binomial tree model for geometric American-style Asian options is also presented in Chapter 4. Chapter 5 presents asymptotic analysis for the cases of (i) zero volatility coe±cient, and (ii) in¯nite expiration date. Chapter 5 also introduces foreign equity Asian options, quanto Asian calls, currency Asian options and their pricing formulas. 6.2 Research Contributions Themostsigni¯cantcontributionofthisdissertationisacombinatorialalgorithm in the binomial model for pricing European-style Asian options and geomet- ric American-style Asian options and an explicit formula for pricing arithmetic European-style Asian options. By this algorithm, the expectation and variance, or even higher moments, of the geometric or arithmetic average of the prices of the underlying assets at each node in the binomial tree can be calculated. 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Abstract (if available)

Abstract An Asian option is a path-dependent option whose payoff depends on the average price of the underlying asset during the life of the option. Asian options are very attractive to investors. However, the pricing of Asian options remains a challenge. The purpose of this dissertation is to develop efficient techniques for pricing Asian options and to expand the existing results on the pricing of Asian options in the classic Black-Scholes model.

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Asset Metadata

Creator Li, Xiufang (author)

Core Title New results on pricing Asian options

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Applied Mathematics

Defense Date 01/25/2007

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

Tag Asian Options,binomial tree method,combinatorial method,OAI-PMH Harvest

Language English

Advisor Lototsky, Sergey V. (committee chair), Deng, Yongheng (committee member), Zhang, Jianfeng (committee member)

Creator Email xiufangl@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m301

Unique identifier UC1439611

Identifier etd-Li-20070228 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-309399 (legacy record id),usctheses-m301 (legacy record id)

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Document Type Dissertation

Rights Li, Xiufang

Type texts

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

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

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