A general equilibrium model for exchange rates and asset prices in an economy subject to jump-diffusion uncertainty

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Content A GENERAL EQUILIBRIUM MODEL FOR EXCHANGE RATES AND ASSET PRICES IN AN ECONOMY SUBJECT TO JUMP-DIFFUSION UNCERTAINTY by Mathias Knape A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) August 2009 Copyright 2009 Mathias Knape Acknowledgements First and foremost, I want to express my gratitude to my advisor and co-chair of my disser- tation committee, Fernando Zapatero. Without his invaluable guidance, unwavering support, in particular in times of no progress, and patience throughout the last years, this dissertation would have not been possible. Working with him was a very enjoyable experience. I also want to thank the co-chair of my dissertation committee, Remigijus Mikulevicius. His support of my academic pursuit extends back to the time when I wrote my Diplom thesis for Ulm University while at USC. His attention to detail is unmatched. I am also grateful to the other member of my dissertation committee, Jin Ma. His insight- ful comments helped to improve the coherence of my dissertation. My gratitude also goes to Peter Baxendale and Jianfeng Zhang, both members of my qual- ifying exam committee. They are very inspiring teachers and they have greatly in uenced my teaching style. I wish to thank the department of mathematics at USC for the nancial assistance during my doctoral studies and the sta of the department for their help throughout the years, in particular to Arnold. I would like to express my gratitude to my parents, Renate and Heribert, and my broth- ers, Philipp and Clemens. I owe my parents much of what I have become today. They have always supported and encouraged my academic endeavors. I am very grateful to my friends, in particular to Aleksey, Emmet, Florian, Frank, Jay, Jens, Johnny, Michael, Sebastian, and Tushar. They have not only greatly in uenced my academic growth, but they also provided the necessary distraction. ii Last but not least, I want to express my gratitude to Melissa for sharing the ups and downs of the dissertation process. Her unfaltering support, companionship, and motivation are immea- surable. iii Table of Contents Acknowledgements ii List of Figures v Abstract vi Chapter 1: Introduction 1 Chapter 2: Model of the economy 6 2.1 Commodity Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Financial Markets and the Exchange Rate . . . . . . . . . . . . . . . . . . . . . 8 2.3 State-Price Densities and Wealth Processes . . . . . . . . . . . . . . . . . . . . 11 2.4 Agents' Individual Optimization Problem . . . . . . . . . . . . . . . . . . . . . 25 Chapter 3: Equilibrium 36 3.1 Denition of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 Implications of the Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . 38 3.3 Equilibrium Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Chapter 4: Higher Moment Properties of Equilibrium Exchange Rate Returns 54 4.1 Skewness of the Exchange Rate Return . . . . . . . . . . . . . . . . . . . . . . . 54 4.2 Kurtosis of the Exchange Rate Return . . . . . . . . . . . . . . . . . . . . . . . 62 Chapter 5: Conclusion 66 References 68 Appendix 70 iv List of Figures 4.1 Skewness behavior of the return of the exchange rate log e(t+h) e(t) for dierent values of the domestic jump parameters where h =:05 . . . . . . . . . . . . . . 60 4.2 Skewness behavior of the return of the exchange rate log e(t+h) e(t) for dierent values of the foreign jump parameters where h = 0:05 . . . . . . . . . . . . . . 61 4.3 Excess kurtosis of the exchange rate return log e(t+h) e(t) for dierent values of the domestic jump parameters where h = 0:05 . . . . . . . . . . . . . . . . . . 64 v Abstract This dissertation examines asset prices, the exchange rate and its higher moment properties in an economy subject to both diusive and jump risk. The model used in this dissertation is an extension of Zapatero's (1995) two-good two-country intertemporal international equilib- rium model for two logarithmic representative agents. Uncertainty enters the economy through a three dimensional Brownian motion and two Poisson processes representing positive and negative jumps in the dividend process of the goods. Individual nancial markets are incom- plete, but all claims can be hedged completely in the international nancial market. From Zapatero's (1995) model it is known that the exchange rate increases with the interest rate dierential and the diusion parameter of the domestic equity market and decreases with the covariance between the domestic and foreign equity market. This dissertation shows that in a jump-diusion setting the expected equilibrium exchange rate change additionally increases (decreases), if the foreign positive and negative jump sizes are bigger (smaller) in absolute value than their domestic equivalents. This dissertation thereby provides a new explanation for the interest rate parity puzzle. It is a well-known empirical fact that exchange rates are skewed and have excess kurtosis. In contrast to traditional equilibrium models subject to diusive risk the exchange rate return exhibits these two properties in the jump-diusion setting. The sign of the skewness of the exchange rate return is dependent on the dierence between the skewness of the returns of the domestic and foreign dividend processes. vi Chapter 1 Introduction In the age of an increasingly globalized economy currency exchange rates strongly in uence investment decisions of individuals and companies. For example, vacationing in Europe in the summer of 2008 proved to be extremely expensive for the average American tourist, whereas their European associates were happy to nd how far their travel funds reached in the United States. The consequence was an increasing number of European tourists in the United States. As an example from the business world, consider the European aviation company Airbus. Airbus faced strong headwinds because its airplanes are generally priced in U.S. Dollars, but they are produced in the Euro zone. This prompted Airbus to consider opening a plant in the United States. These two examples indicate that understanding exchange rates is of vital importance. From a practitioner's point of view, the rise of the Euro during the rst part of 2008 was mainly accredited to the Euro zone's superior economic performance. One common measure of economic performance is the Gross Domestic Product (GDP). For the rst part of 2008 the European GDP posted higher gains when compared with the American GDP. But when the economic outlook in Europe suddenly turned gloomier at the end of the second quarter, accompanied by a higher than expected GDP growth in the United States, the U.S. Dollar was back on the rise. Dieckmann and Gallmeyer (2005) point out that GDP data follows a jump-diusion process. This dissertation examines how jumps in the goods market aect the exchange rate dynamics in a two-country two-good equilibrium model with logarithmic agents. To my knowledge, this is the rst study which determines the exchange rate dynamics in a general equilibrium model subject to jump-diusion risk. In an It^ o diusion setting exchange rates have been modeled using a variety of approaches. 1 Solnik (1974) and Adler and Dumas (1983) consider exchange rates in the context of an international capital asset pricing model (CAPM). Another commonly used approach is to study exchange rates in a general equilibrium model. In this model one considers several agents who derive utility from consumption of dierent goods. These goods are normally modeled by a dividend process tied to a so called productive asset. The dynamics of the dividend process are exogenously specied. It is then the objective to determine the agents' consumption rates and the asset prices using the laws of supply and demand. Mathematically speaking, since agents maximize their utility from consumption, this results in a (dynamic) optimization problem. In a general monetary equilibrium one considers money in addition to the goods. Bakshi and Chen (1997) extend the classical two country monetary model of Lucas (1982) to continuous time obtaining a general monetary equilibrium model with identical logarithmic agents. In contrast to this, Basak and Gallmeyer (1999) allow their general monetary equilibrium model to include agents who have heterogeneous tastes and hold their own money for transaction services. They provide a complete characterization of equilibrium prices for general time-additive preferences and non-Markovian exogenous processes. General equilibrium models in real terms are considered by Dumas (1992), Uppal (1993), Serrat (1995) and Zapatero (1995). Dumas (1992) and Uppal (1993) both impose restrictions on trade to prevent goods market arbitrage. Dumas (1992) investigates exchange rates in the context of deviations from the law of one price. Uppal's (1993) focus is on explaining the home bias puzzle. The home bias puzzle is the observation that investors heavily bias their portfolios towards equities in their home country. Serrat (1995) and Zapatero (1995) both consider multi-good general equilibrium models. Serrat (1995) concentrates on nding an explanation for the home bias puzzle, rather than computing the exchange rate and security prices. Zapatero's (1995) main objective is to determine asset prices and the exchange rate in equilibrium. 2 Zapatero's (1995) model constitutes the basis for the model used in this dissertation. He considers an equilibrium model with two agents, two countries and two goods. Each good is solely produced in one country. The output of the goods is subject to diusive risk. Financial markets of each country consist of one risky security and a locally riskless bond. The risky security and the output process of each country are connected as follows: Possession of a share of the risky security entitles the owner to receive the output process. Markets are individually incomplete, but since agents can trade across countries at an exchange rate, they face a complete market situation. The two agents maximize their expected intertemporal utility derived from the consumption of the two goods. Zapatero (1995) shows that in equilibrium the exchange rate is determined in such a way that the risky security of the other country is redundant for the agents. That means the agents face an incomplete market in equilibrium. Zapatero (1995) further observes that the equilibrium expected rate of change of the exchange rate increases with both the dierential of interest rates and the volatility of the domestic risky security and decreases with the covariance between the domestic and foreign security. He nds that the equilibrium volatility of the exchange rate can be decomposed into idiosyncratic factors in the nancial markets and a term re ecting the dierent weights in a common international factor. The model used in this dissertation is an extension of Zapatero's (1995). In contrast to his model, the considered economy is not only subject to risk induced by Brownian motions, but also subject to jump risk generated by two Poisson processes. The mathematical founda- tion for a general equilibrium model subject to jump-diusion uncertainty is given by Bardhan and Chao (1996). In order to guarantee market completeness, two non-dividend paying jump securities are introduced as in Dieckmann and Gallmeyer (2005). They point out that the dynamics of the jump securities are not uniquely determined. This is similar to equilibrium models involving only diusion risk, as in Karaztas et al. (1990). Therefore it is possible to choose the drift 3 coecients of the jump securities such that the agents do not hold these securities in equilib- rium. Hence, this procedure of completing the market does not aect the equilibrium allocation. In general equilibrium models subject to It^ o diusion uncertainty, as in Zapatero's (1995), optimality of a solution can be proven using a method suggested by He and Pearsons (1991) and Karatzas et al. (1991). Redundant securities are dropped from the model and replaced by other securities. The volatility coecients of these securities are chosen such that the market is complete. The drift term is then determined in such a way that the agents do not want to hold these securities in equilibrium. This approach can also be used to show the optimality of the equilibrium parameters in the jump-diusion setting considered in this dissertation. This dissertation also extends a paper by Cass and Pavlova (2003) to the jump-diusion case. They nd that under log-linear specications the Lucas (1978) tree model allows only for so-called "peculiar nancial equilibria". The equilibria are peculiar in the sense that all stocks oer the same investment opportunity in equilibrium, resulting in a completely degenerate stock market. Since it is an established empirical result that exchange rates tend to display signicant and variable skewness, and excess kurtosis, it is interesting to investigate the skewness and kurtosis behavior of the equilibrium exchange rate returns. For example, Bates (1996) ts a jump-diusion model for exchange rate returns to options data examining implicit volatility, skewness, and excess kurtosis. He nds a positively skewed implicit distribution for the $/DM exchange rate returns during the strong dollar period of 1984 and 1985. After that period he reports that the implicit skewness oscillated substantially, similar to the $/yen implicit skewness. Prichet (1991) shows that real exchange rate returns for fty-six developed countries with managed exchange rates exhibit skewness and excess kurtosis. Carr and Wu (2007) use time-changed L evy processes for exchange rates to capture the stochastic skew behavior of currency option prices. To my knowledge there has been no attempt to study skewness and kurtosis of exchange 4 rate returns in the context of a general equilibrium model in an economy subject to a jump-diusion. The framework of the general equilibrium model makes it possible to examine how the higher moments of the domestic and foreign consumption rates aect the higher moments of the exchange rate returns. The objective of this dissertation is to study how jumps in the goods market aect the exchange rate in a two country, two good equilibrium model with logarithmic agents. The model used in this dissertation is constructed in chapter two. Commodity and nancial markets are described in the rst two sections. This is followed by a detailed description of the state-price densities and of the agents' wealth processes. In the last section of chapter two the optimization problem of each agent is introduced and solved. Chapter three starts with the denition of the consumption goods market and nancial market equilibrium. The equilibrium parameters are then determined in the last section of chapter three. The nal chapter discusses the skewness and kurtosis of the exchange rate returns. 5 Chapter 2 Model of the economy In this chapter the fundamentals of the model are explained. First, the probabilistic basis of the model is given, followed by a description of the commodity and nancial markets. After dening the state-price densities, the dynamics of the agents' wealth processes are derived. The chapter concludes with a description and solution to the agents' individual optimization problem. As stated in the introduction, this dissertation extends Zapatero's (1995) paper to a jump-diusion setting. The paper by Bardhan and Chao (1995) serves as a basis for this extension. They study a multi-agent dynamic production economy that is subject to jump- diusion uncertainty, but do not consider exchange rates. The model outlined in this chapter is mainly based on these two papers. Uncertainty enters the economy through a three dimensional Brownian motion and a two dimensional Poisson process. Let W (t) = (W 1 (t);W 2 (t);W 3 (t)) T be anR 3 -valued Brown- ian motion on a probability space ( W ;F W ;P W ), where W 1 , W 2 and W 3 are assumed to be independent. As usual, T denotes the transpose. Furthermore, let N(t) = (N + (t);N (t)) be a two dimensional Poisson process on a probability space ( N ;F N ;P N ). TheP W - augmentation of (W (s); 0 s t) is denoted asF W t . Similarly,F N t denotes the P N augmentation of (N(s); 0 s t). The probability space ( ;F;P ) is then dened as the product space of ( W ;F W ;P W ) and ( N ;F N ;P N ); that means = W N ,F =F W F N ,P =P W P N , and common ltrationfF t g =F W t F N t . Notice that since the probability measure is dened as the product measure of P W and P N , the Brownian motion and the Poisson processes are independent. It is assumed that N + and N do not jump at the same time and that they have only a nite number of jumps in [0;T ]. The Poisson processes N + and N haveF t -predictable stochastic intensities + (t) and 6 (t) respectively, which are bounded from above and away from zero. The time horizon is from 0 to T . This concludes the denition of the probability space of the model. In the following section the commodity markets are introduced. 2.1 Commodity Markets There are two countries: the domestic country, called D, and the foreign country, called F . Each country has an agent and produces one commodity exclusively. There are two productive assets. A share of the domestic or foreign productive asset entitles the owner to receive the dividend stream D or F , respectively, measured in terms of their associated commodity. Agents can then either consume or trade the dividend process. Agent i, where i2fD;Fg, derives utility from consumption of the domestic commodity, c iD , and the foreign commodity, c iF . Her preferences are given by a logarithmic utility function U i (t;c iD ;c iF ) = a iD log(c iD (t)) +a iF log(c iF (t)); 8(t;c iD ;c iF )2 [0;T ] (0;1) 2 ; (2.1) where a iD ;a iF 2 (0; 1) and a iD +a iF = 1. The dividend streams i (t), i 2 fD;Fg, measured in terms of the commodity of coun- try i, are governed by the following jump-diusion process d i (t) i (t) = i (t)dt + T i (t)dW (t) +K + i (t)dQ + (t) +K i (t)dQ (t); (2.2) where dQ + (t) = dN + (t) + (t)dt (2.3) dQ (t) = dN (t) (t)dt: (2.4) 7 The parameters i , i = ( i1 ; i2 ; i3 ) T , K + i and K i are exogenously given and assumed to be predictable with respect tofF t g. The drift parameter i 2 R is the instantaneous appreciation rate of the dividend process. ij 2 R is a volatility coecient representing the instantaneous intensity with which the jth Brownian motion aects the dividend process. The positive jump coecient, K + i , attains only positive values, more exactly K + i 2R + . The jump size of the negative jump, K i , takes on values in (1; 0). The value1 is excluded because otherwise the dividend processes could jump to zero and never get positive again. It is assumed that 0<c i (t)C <1 holdsdt dP almost everywhere for all (t;!)2 [0;T ] for some constants c and C; that means, the dividend processes are bounded from above and away from zero. The fact that K + i is positive and K i is negative allows for the following interpretation of N + and N . A jump in N + implies a positive jump in the dividend process. Thus a jump in N + can be understood as some good news hitting the economy by surprise. Analogously, a jump in N can be interpreted as a negative shock to the economy. The assumption that K + i and K i are predictable can be justied by the observation that in most cases it is known to what extent a shock would aect the economy, but the timing of the shock is unknown. As mentioned at the beginning of this section, agents can trade the commodities and productive assets. This may not be sucient to hedge all risks intrinsic in the market. In order to allow the agents to hedge, a complete nancial market is introduced in the next chapter. 2.2 Financial Markets and the Exchange Rate Each country's nancial market consists of four assets: the productive asset, a (locally) riskless bond and two jump securities. The introduction of the jump securities is necessary to complete the market. The jump securities are inspired by Dieckmann and Gallmeyer (2005) in which they study the equilibrium allocation of diusive and jump risks with heterogeneous agents. 8 The price of the productive asset of country i at time t is denoted by S i (t). Its dynamics are indirectly dened by the dynamics of its gain process. It is assumed that the gains process of the productive asset i, measured in terms of the commodity of country i, is given by dG i (t) G(t) = dS i (t) S i (t) + i (t) S i (t) dt = Si (t)dt + T Si (t)dW (t) +K + Si (t)dQ + (t) +K Si (t)dQ (t); (2.5) where Si , Si = ( Si1 ; Si2 ; Si3 ) T , K + Si and K Si are predictable with respect to fF t g, and are bounded uniformly in (t;!)2 [0;T ] . The drift parameter, Si 2 R, represents the instantaneous appreciation rate. The volatility coecient, Sij 2 R, represents the instantaneous intensity with which the j-th Brownian motion aects the i-th gains process. The jump parameters K + Si and K Si denote the positive and negative jump sizes, respectively. Therefore it is required that K + Si > 0 and K Si 2 (1; 0). Again, the value -1 is excluded to prevent the gains process from jumping to zero. All parameters of the gains process will be determined in equilibrium. The price process of the locally riskless bond, denominated in the commodity of the respective country, is given by dB i (t) = r i (t)B i (t)dt (2.6) for some interest rate process r i (t), which is predictable with respect tofF t g, and is bounded uniformly in (t;!)2 [0;T ] . The dynamics of the jump securities are governed by dP i (t) P i (t) = Pi (t)dt +K + Pi (t)dQ + (t) +K Pi (t)dQ (t); (2.7) where Pi , K + Pi and K Pi are predictable with respect tofF t g, and are bounded uniformly in (t;!)2 [0;T ] . The positive jump parameter, K + Pi , attains only positive values, and K Pi is 9 strictly between -1 and 0. For notational convenience, denote the price system (B D ;B F ;S D ;S F ;P D ;P F ) byP. As mentioned above these jump securities are inspired by Dieckmann and Gallmeyer (2005). In order to complete their nancial market they introduce two jump securities. In contrast to the jump securities used in this dissertation both of their jump securities only depend on one Poisson process. Therefore they can interpret their jump securities as "rare-event" insurance products, similar to catastrophe bonds in the real nancial world. In this dissertation each jump security depends on both Poisson processes to prevent any systematic asymmetries. Notice that the domestic and foreign markets are individually incomplete, but the fact that agents can trade both domestic and foreign securities allows them to hedge all risks intrinsic in the market. Since domestic and foreign assets are denoted in the domestic and foreign commodity, respectively, agents need to convert their own consumption good to the other consumption good at the prevailing exchange rate, as dened below. Strictly speaking, in order to guarantee completeness a further technical assumption on the jump-diusion coecients of the nancial and productive assets is needed. This condition will be given in section 2.3. The exchange rate is dened as the ratio of units of the foreign consumption good per units of the domestic consumption good. It is assumed to be governed by the following jump-diusion process de(t) e(t) = e (t)dt + T e (t)dW (t) +K + e (t)dQ + (t) +K e (t)dQ (t); (2.8) where e , e = ( e1 ; e2 ; e3 ) T , K + e and K e are assumed to be predictable with respect to fF t g and are bounded uniformly in (t;!)2 [0;T ] . In contrast to the jump sizes specied above, K + e and K e are only assumed to be unequal to zero. 10 After having specied the nancial market and the exchange rate in this section it is now possible to dene the state-price densities. In order to do this, it is necessary to derive the dynamics of all assets in units of the domestic and foreign commodity. 2.3 State-Price Densities and Wealth Processes This section contains the derivation of the state-price densities of both countries. It also includes the description of the agents' wealth process dynamics and a derivation of a useful relationship between the domestic and foreign market price of risk parameters. It ends by recalling a standard representation of the productive asset prices. Observe that each agent has ve risky securities to trade in: two productive assets, two jump securities and the bond of the other country. The latter security is risky due to the exchange rate. In order to dene the state-price density and to nd the dynamics of the wealth process of each agent, it is necessary to nd the dynamics of the nancial assets from the domestic and foreign point of view. The dynamics of the nancial assets, as seen by the domestic agent, are summarized in the following lemma. Lemma 2.1. From the domestic agent's point of view, the dynamics of the foreign bond, the foreign productive asset price and the foreign jump security price in terms of the domestic commodity are d B F (t) e(t) = B F (t) e(t) e (t) + T e (t) e (t) +r F (t) + (K + e (t)) 2 1 +K + e (t) + (t) + (K e (t)) 2 1 +K e (t) (t) dt T e (t)dW (t) K + e (t) 1 +K + e (t) dQ + (t) K e (t) 1 +K e (t) dQ (t) ; (2.9) 11 d S F (t) e(t) = S F (t) e(t) "" S F (t) e (t) + T e (t) e (t) T S F (t) e (t) + K + e (t)(K + e (t)K + S F (t)) 1 +K + e (t) ! + (t) + K e (t)(K e (t)K S F (t)) 1 +K e (t) ! (t) # dt + T S F (t) T e (t) dW (t) + K + S F (t)K + e (t) 1 +K + e (t) dQ + (t) + K S F (t)K e (t) 1 +K e (t) dQ (t) # 1 e(t) F (t)dt; (2.10) d P F (t) e(t) = P F (t) e(t) "" P F (t) e (t) + T e (t) e (t) + K + e (t)(K + e (t)K + P F (t)) 1 +K + e (t) ! + (t) + K e (t)(K e (t)K P F (t)) 1 +K e (t) ! (t) # dt T e (t)dW (t) + K + P F (t)K + e (t) 1 +K + e (t) dQ + (t) + K P F (t)K e (t) 1 +K e (t) dQ (t) # : (2.11) Proof: See Appendix. 2 In order to achieve a more compact notation, let D (t) = 0 B B B B B B B B B B @ S D (t) S F (t)=e(t) P D (t) P F (t)=e(t) B F (t)=e(t) 1 C C C C C C C C C C A : (2.12) D (t) represents the vector of all securities which are risky for the domestic agent. Observe that this domestic price vector is denoted in units of the domestic commodity. 12 Furthermore dene D (t) = 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B @ S D (t) S F (t) e (t) + T e (t) e (t) T S F (t) e (t) + K + e (t)(K + e (t)K + S F (t)) 1+K + e (t) + (t) + K e (t)(K e (t)K S F (t)) 1+K e (t) (t) P D (t) P F (t) e (t) + T e (t) e (t) + K + e (t)(K + e (t)K + P F (t)) 1+K + e (t) + (t) + K e (t)(K e (t)K P F (t)) 1+K e (t) (t) e (t) + T e (t) e (t) +r F (t) + (K + e (t)) 2 1+K + e (t) + (t) + (K e (t)) 2 1+K e (t) (t) 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C A (2.13) and D (t) = 0 B B B B B B B B B B B B B B B B @ T S D (t) K + S D (t) K S D (t) T S F (t) T e (t) K + S F (t)K + e (t) 1+K + e (t) K S F (t)K e (t) 1+K e (t) 0 0 0 K + P D (t) K P D (t) T e (t) K + P F (t)K + e (t) 1+K + e (t) K P F (t)K e (t) 1+K e (t) T e (t) K + e (t) 1+K + e (t) K e (t) 1+K e (t) 1 C C C C C C C C C C C C C C C C A : (2.14) From the assumptions in (2.5), (2.7), and (2.8), it follows that D and D are predictable with respect tofF t g and are bounded uniformly in (t;!)2 [0;T ] . Now it is possible to give the rst technical condition on the jump-diusion coecients, as mentioned in the previous section, to guarantee completeness. Assumption 2.2. In order to guarantee market completeness, D (t) is assumed to be non- degenerate. 13 This is a reasonable assumption because it is possible to give a condition when this matrix is invertible. Lemma 2.3. Assume that the jump coecients of the pure jump securities are given by K + Pi (t) = K + Si (t) and K Pi (t) = K Si (t) for i2fD;Fg. Then the matrix D (t) is invertible if for all t2 [0;T ] 1. the matrix 0 B B B B @ T S D (t) T S F (t) T e (t) 1 C C C C A is invertible, 2. the domestic jump parameter K + S D (t) is strictly greater than zero, and 3. K S F (t) 1 +K e (t) K S D (t) K + S D (t) K + S F (t) 1 +K + e (t) is unequal to zero. Proof: This result can be obtained by using Gaussian elimination. 2 The dividend vector D is dened as D (t) = 0 B B B B B B B B B B @ R t 0 D (s)ds R t 0 F (s) e(s) ds 0 0 0 1 C C C C C C C C C C A : Let dQ(t) = 0 B @ dQ + (t) dQ (t) 1 C A: (2.15) 14 Then the dynamics of D (t) can be written as d D (t) = diag( D (t)) 2 6 4 D (t)dt + D (t) 0 B @ dW (t) dQ(t) 1 C A 3 7 5d D (t); where diag( D (t)) denotes a matrix with D (t) on the diagonal and zeros everywhere else. Next, dene the domestic state-price density process D . The dynamics of the state price density process are given by d D (t) = D (t) " r D (t)dt T D (t)dW (t) + ~ + D (t) + (t) 1 ! dQ + (t) + ~ D (t) (t) 1 ! dQ (t) # ; (2.16) where the market price of risk process is dened as 0 B B B B @ D (t) ~ + D (t) + + (t) ~ D (t) + (t) 1 C C C C A = 1 D (t) [ D (t)r D (t)]; (2.17) where D (t)2R 3 and ~ + D (t); ~ D (t)2R. Observe that the market price of risk process has a unique solution because D is assumed to be non-degenerate. Furthermore, the processes D , ~ + D and ~ D are bounded, mea- surable and predictable with respect tofF t g, due to assumptions on D ,r D , D , + D and D . It is well-known, see for example Bremaud (1981) and Protter (1990), that D induces a new probability measure on ( ;F T ) by ~ P D (A) = E h e R T 0 r D (t)dt D (T )1 A i for all A2F T . This new probability measure will be called the domestic risk-neutral probability measure. It is also standard that under this measure ~ W D (t) = W (t) + R t 0 D (s)ds is a Brownian motion and N + (t) and N (t) are Poisson processes with intensities ~ + D (t) and ~ D (t), respectively. 15 Before deriving the foreign state-price density, the domestic agent's wealth dynamics are considered. As stated above, the domestic agent can trade in all nancial assets. Let D (t) = ( DD (t); DF (t)) T denote the domestic agent's proportion of wealth invested in the domestic and foreign productive assets at time t. Furthermore, let D (t) = ( DD (t); DF (t); DB F (t)) T represent the domestic agent's proportion of wealth invested in the domestic jump security, the foreign jump security and the foreign bond at time t. The domestic agent's proportion of wealth invested in the domestic bond at timet is denoted by DB D (t). The allocations D (t), D (t) and DB D (t) are assumed to befF t g-adapted. The wealth process of the domestic agent can therefore be written as X D (t) = ( T D (t); T D (t))X D (t) + D (t) Z t 0 DB D (s)X D (s)r D (s)ds Z t 0 c DD (s) + c DF (s) e(s) ds: The dynamics of the domestic wealth process are then given by dX D (t) = ( T D (t); T D (t))X D (t)diag( D (t)) 1 diag( D (t)) 2 6 4 D (t)dt + D (t) 0 B @ dW (t) dQ(t) 1 C A 3 7 5 + DB D (t)X D (t)r D (t)dt c DD (t) + c DF (t) e(t) dt: The rst term represents the contribution of the risky assets to the wealth dynamics. The second term represents the dynamics of the amount invested in the domestic bond. The last term follows from the agent's consumption of the domestic and foreign commodities. Notice that the dividend process drops out of this equation. It comes into play through the dynamics of the productive assets, but is then canceled out by the dividends the agent receives. 16 Since DB D (t) = 1 ( T D (t); T D (t))1, where 1 = (1; 1; 1; 1; 1) T , this simplies to dX D (t) = ( T D (t); T D (t))X D (t)(diag( D (t))) 1 diag( D (t)) 2 6 4( D (t)r D (t)1)dt + D (t) 0 B @ dW (t) dQ(t) 1 C A 3 7 5 +X D (t)r D (t)dt c DD (t) + c DF (t) e(t) dt: (2.18) Notice that diag( D (t)) 1 diag( D (t)) = diag(1) if multiplied with a continuous dierential. This is true because N + and N are assumed to have only a nite number of jumps in [0;T ]. The next task is to derive the foreign agent's wealth process. Due to the exchange rate, the dynamics of the nancial assets from the foreign agent's point of view are dierent from the dynamics of the nancial assets from the domestic agent's point of view. The dynamics of the domestic assets in units of the foreign commodity are summarized in the next lemma. Lemma 2.4. From the foreign agent's point of view, the dynamics of the domestic bond, the domestic productive asset price and the domestic jump security price in terms of the foreign commodity are d (B D (t)e(t)) = B D (t)e(t) [ e (t) +r D (t)]dt + T e (t)dW (t) +K + e (t)dQ + (t) +K e (t)dQ (t) ; (2.19) 17 d (S D (t)e(t)) = S D (t)e(t) S D (t) + e (t) + T S D (t) e (t) +K + e (t)K + S D (t) + (t) +K e (t)K S D (t) (t) + T S D (t) + T e (t) dW (t) + K + S D (t) +K + e (t) +K + S D (t)K + e (t) dQ + (t) + K S D (t) +K e (t) +K S D (t)K e (t) dQ (t) e(t) D (t)dt; (2.20) d (P D (t)e(t)) = P D (t)e(t) hh P D (t) + e (t) +K + P D (t)K + e (t) + (t) +K P D (t)K e (t) (t) i dt + T e (t)dW (t) + K + P D (t) +K + e (t) +K + P D (t)K + e (t) dQ + (t) + K P D (t) +K e (t) +K P D (t)K e (t) dQ (t) i : (2.21) Proof: See Appendix. 2 Again, in order to enable a clear presentation, dene F (t) = 0 B B B B B B B B B B @ S D (t)e(t) S F (t) P D (t)e(t) P F (t) B D (t)e(t) 1 C C C C C C C C C C A ; (2.22) F (t) = 0 B B B B B B B B B B B B B B @ S D (t) + e (t) + T S D (t) e (t) +K + e (t)K + S D (t) + (t) +K e (t)K S D (t) (t) S F (t) P D (t) + e (t) +K + P D (t)K + e (t) + (t) +K P D (t)K e (t) (t) P F (t) e (t) +r D (t) 1 C C C C C C C C C C C C C C A (2.23) 18 and F = 0 B B B B B B B B B B @ T e + T S D K + S D +K + e (1 +K + S D ) K S D +K e (1 +K S D ) T S F K + S F K S F T e K + P D +K + e (1 +K + P D ) K P D +K e (1 +K P D ) 0 0 0 K + P F K P F T e K + e K e 1 C C C C C C C C C C A ; where the dependence of the last matrix on time t was omitted in order to facilitate the presentation. From the assumptions in (2.5), (2.7), and (2.8) it follows that F and F are predictable with respect tofF t g and are bounded uniformly in (t;!)2 [0;T ] . The second and last technical requirement announced in section 2.2 is postulated in the following assumption. Assumption 2.5. In order to guarantee completeness of the foreign market, F (t) is assumed to be non-degenerate for all t2 [0;T ]. It is possible to show under the conditions of Lemma 2.3 that the matrix F (t) is non- degenerate if and only if the matrix D (t) is invertible. Lemma 2.6. Assume that the jump coecients of the pure jump securities are given by K + Pi (t) = K + Si (t) and K Pi (t) = K Si (t) for i2fD;Fg. Then the matrix F (t) is invertible if and only if the matrix D (t) is invertible. Proof: This can be proven by Gaussian elimination. 2 The dividend vector F is dened as F (t) = 0 B B B B B B B B B B @ R t 0 D (s)e(s)ds R t 0 F (s)ds 0 0 0 1 C C C C C C C C C C A : 19 Thus, it is possible to write the dynamics of F (t) as d F (t) = diag( F (t)) 0 B @ F (t)dt + F (t) 0 B @ dW (t) dQ(t) 1 C A 1 C Ad F (t): Next, dene the foreign state-price density process F . The dynamics of the state price density process are given by d F (t) = F (t) " r F (t)dt T F (t)dW (t) + ~ + F (t) + (t) 1 ! dQ + (t) + ~ F (t) (t) 1 ! dQ (t) # ; (2.24) where the market price of risk process is dened as 0 B B B B @ F (t) ~ + F (t) + + (t) ~ F (t) + (t) 1 C C C C A = 1 F (t) [ F (t)r F (t)]; (2.25) where F (t)2R 3 and ~ + F (t); ~ F (t)2R. Observe that the market price of risk process is uniquely dened because F is assumed to be non-degenerate. Furthermore, the processes F , ~ + F , and ~ F are bounded, measurable and predictable with respect tofF t g due to assumptions on F , r F , F , + , and . Similar to the domestic state-price density, the foreign state-price density F induces a new probability measure on ( ;F T ) by ~ P F (A) =E h e R T 0 r F (t)dt F (T )1 A i for all A2F T . This new probability measure will be called the foreign risk-neutral probability measure. It is standard that under this measure ~ W F (t) =W (t)+ R t 0 F (s)ds is a Brownian motion andN + (t) and N (t) are Poisson processes with intensities ~ + F (t) and ~ F (t), respectively. As for the domestic agent, let F (t) = ( FD (t); FF (t)) T denote the foreign agent's proportion of wealth invested in the domestic and foreign productive asset at time t. The 20 vector F (t) = ( FD (t); FF (t); FB D (t)) T represents the foreign agent's proportion of wealth invested in the domestic jump security, the foreign jump security and the domestic bond at time t, respectively. The foreign agent's proportion of wealth invested in the foreign bond at timet is denoted by FB F (t). The processes F (t), F (t) and FB D (t) are assumed to befF t g-adapted. The wealth process of the foreign agent is given by X F (t) = ( T F (t); T F (t))X F (t) + F (t) + Z t 0 FB F (s)X F (s)r s (t)dt Z t 0 c FD (s) + c FF (s) e(s) ds: Then the dynamics of the foreign wealth process are given by dX F (t) = ( T F (t); T F (t))X F (t)diag( F (t)) 1 diag( F (t)) 2 6 4 F (t)dt + F (t) 0 B @ dW (t) dQ(t) 1 C A 3 7 5 + FB F (t)X F (t)r F (t)dt (c FD (t)e(t) +c FF (t))dt: Again, the rst term represents the contribution of the risky assets to the wealth dynamics. The second term represents the amount invested in the foreign bond. The last term follows from the agent's consumption of the domestic and foreign commodities. Notice that the dividend process drops out of this equation. It comes into play through the dynamics of the productive assets, but is then canceled out by the dividends the agent receives. Since FB F (t) = 1 ( T F (t); T F (t))1, where 1 = (1; 1; 1; 1; 1) T , this simplies to dX F (t) = ( T F (t); T F (t))X F (t)(diag( F (t))) 1 diag( F (t)) 2 6 4( F (t)r F (t)1)dt + F (t) 0 B @ dW (t) dQ(t) 1 C A 3 7 5 +X F (t)r F (t)dt (c FD (t)e(t) +c FF (t))dt: (2.26) 21 Notice again that diag( F (t)) 1 diag( F (t)) = diag(1) if multiplied with a continuous dierential. This is true because N + and N are assumed to have only a nite number of jumps in [0;T ]. It remains to dene an admissible policy for the domestic and foreign agents. Given an initial endowment x i0 strictly greater than zero, an admissible policy for agent i is a seven dimensional vector (c iD ;c iF ; T i ; T i ) which satises the dynamic budget constraints given in (2.18) and (2.26), respectively, for all t2 [0;T ], that the instantaneous consumption rates c iD (t) and c iF (t) are strictly positive for all t2 [0;T ], and the no bankruptcy constraint X i (T ) 0. LetA i denote the set of all admissible policies for agent i. The next Lemma contains a helpful relationship between the domestic and foreign mar- ket price of risk. Lemma 2.7. The following relationship between the domestic market price of risk and the foreign market price of risk holds: 0 B B B B @ D (t) ~ + D (t) + + (t) ~ D (t) + (t) 1 C C C C A = 0 B B B B @ F (t) ( ~ + F (t) + + (t))(1 +K + e (t)) ( ~ F (t) + (t))(1 +K e (t)) 1 C C C C A 0 B B B B @ e (t) + (t)K + e (t) (t)K e (t) 1 C C C C A : (2.27) 22 Proof: The time dependency of all parameters is dropped in this proof for notational conve- nience. Dene the matrices G = 0 B B B B B B B B B B @ 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 1 C C C C C C C C C C A (2.28) and K = 0 B B B B B B B B B B @ 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 +K + e 0 0 0 0 0 1 +K e 1 C C C C C C C C C C A : (2.29) Observe that G =G 1 . It is now possible to write F = G D K: (2.30) Therefore 0 B B B B @ D ~ + D + + ~ D + 1 C C C C A = ( F ) 1 ( F r F 1) = K 1 ( D ) 1 G 1 ( F r F 1): 23 It then follows that K 0 B B B B @ D ~ + D + + ~ D + 1 C C C C A = ( D ) 1 G 1 ( F r F 1) = 1 D G 0 B B B B B B B B B B @ S D + e + T S D e +K + e K + S D + +K e K S D r F S F r F P D + e +K + P D K + e + +K P D K e r F P F r F e +r D r F 1 C C C C C C C C C C A = 1 D 0 B B B B B B B B B B @ S D + T S D e +K + e K + S D + +K e K S D r D S F e r D P D +K + P D K + e + +K P D K e r D P F e r D e r D +r F 1 C C C C C C C C C C A = 1 D ( D r D 1) +R; where R = 0 B B B B B B B B B B B B B B @ T S D e +K + e K + S D + +K e K S D T e e + T S F e K + e K + S F + K + S F K + e 1+K + e + K e K S F + K S F K e 1+K e K + P D K + e + +K P D K e T e e K + e K + P F + K + P F K + e 1+K + e + K e +K P F + K P F K e 1+K e T e e K + e K + e 1+K + e + K e K e 1+K e 1 C C C C C C C C C C C C C C A = D 0 B B B B @ e + K + e K e 1 C C C C A : 24 Thus 0 B B B B @ D ~ + D + + ~ D + 1 C C C C A = K 0 B B B B @ F ~ + F + + ~ F + 1 C C C C A 0 B B B B @ e + K + e K e 1 C C C C A ; and the statement follows. 2 Lastly, a standard representation of the productive asset price is given. Lemma 2.8. The price S i of the productive asset can be written as S i (t) = 1 i (t) E " Z T t i (t) i (t)dtjF t # : (2.31) Proof: See Appendix. 2 In this section the agents' wealth dynamics and the state-price densities were dened. Further- more, a relationship between the domestic and foreign market price of risk and a representation of the productive asset prices were derived. In the next section the agents' optimization problem is formulated. 2.4 Agents' Individual Optimization Problem After dening the agents' individual optimization problem at the beginning of this section, a complete solution to this optimization problem is given. Agents derive utility from the consumption of the domestic and foreign commodities. Their preferences were specied in (2.1). At time t = 0 agent i has an initial endowment of X i (0) = iD (0)S D (0) + iF (0)S F (0): (2.32) 25 Agenti's goal is to maximize her expected lifetime utility from consumption over all admissible policies for which the following expectation is well dened. That means she has to solve the following optimization problem max Ai E " Z T 0 e kt (a iD log(c iD (t)) +a iF log(c iF (t)))dt # ; (2.33) where k is the agents' subjective discount rate. For simplicity, it is assumed to be the same across agents. In order to guarantee that the expected integral is well-dened, the maximum is only taken over all admissible policies (c iD ;c iF ; T i ; T i ) which fulll E " Z T 0 max(0;a iD log(c iD (t))a iF log(c iF (t)))dt # <1: (2.34) Notice that the agents act as price takers when solving their individual optimization problem. This means they take the productive and nancial asset prices as given. The key to the solution of the agents' individual optimization problems is to transform this dynamic optimization problem into a static variational problem. For this transformation, it is necessary to nd the dynamics of i X i . Since these dynamics will be needed in a separate context as well, they are derived in the following lemma. 26 Lemma 2.9. The dynamics of i X i are given by d( i X i )(t) = i (t)X i (t)( T i (t); T i (t))(diag( i (t))) 1 diag( i (t)) " i (t)diag 1; 1; 1; ~ + i (t) + (t) ; ~ i (t) (t) !# 0 B @ dW (t) dQ(t) 1 C A i (t)X i (t) 0 B B B B @ i (t) 1 ~ + i (t) + (t) 1 ~ i (t) (t) 1 C C C C A T 0 B @ dW (t) dQ(t) 1 C A i (t)C i (t)dt; (2.35) where C i (t) is the instantaneous total consumption rate of agent i in terms of the commodity produced in country i C i (t) = 8 > < > : c DD (t) + c DF (t) e(t) if i =D c FD (t)e(t) +c FF (t) if i =F : Proof: See Appendix. 2 The next proposition contains the solution to the agents' individual optimization problems and the optimal wealth process. Proposition 2.10. The optimal consumption policies which solve the domestic agent's opti- mization problem are given by ^ c DD (t) = a DD k[ DD (0)S D (0) + DF (0)S F (0)=e(0)] e kt e k(Tt) D (t) = a DD k 1e k(Tt) ^ X D (t) (2.36) ^ c DF (t) = a DF k( DD (0)S D (0) + DF (0)S F (0)=e(0)) e kt e k(Tt) D (t) e(t) = a DF k 1e k(Tt) ^ X D (t)e(t): (2.37) 27 The optimal consumption policies which solve the foreign agent's optimization problem are given by ^ c FD (t) = a FD k[ FD (0)S D (0)e(0) + FF (0)S F (0)] e kt e k(Tt) F (t)e(t) = a FD k 1e k(Tt) ^ X F (t) 1 e(t) (2.38) ^ c FF (t) = a FF k( FD (0)S D (0)e(0) + FF (0)S F (0)) e kt e k(Tt) F (t) = a FF k 1e k(Tt) ^ X F (t): (2.39) There is an optimal portfolio process ( T i ; T i ) that nances agent i's optimal consumption process. Agent i's optimal wealth process is given by ^ X i (t) = 1 i (t) E " Z T t i (s) ^ C i (s) B i (s) ds F t # = e kt e kT 1e kT X i (0) i (t) : (2.40) Proof: The proof consists of three steps: First, the optimal consumption policy is derived. It is then shown that there exists an optimal portfolio process supporting the optimal consumption policy. Lastly, the optimal wealth representation given above is proven. The main ideas of this proof are based on Bardhan and Chao (1995). 28 From (2.35) it follows that i (T )X i (T ) = X i (0) + Z T 0 i (s)X i (s)( T i (s); T i (s))(diag( i (s))) 1 diag( i (s)) " i (s)diag 1; 1; 1; ~ + i (s) + (s) ; ~ i (s) (s) !# 0 B @ dW (s) dQ(s) 1 C A Z T 0 i (s)X i (s) 0 B B B B @ i (s) 1 ~ + i (s) + (s) 1 ~ i (s) (s) 1 C C C C A T 0 B @ dW (s) dQ(s) 1 C A Z T 0 i (s)C i (s)ds: Under the assumptions on the jump-diusion coecients stochastic integrals are martingales. Therefore, taking expected values one obtains that E i (T )X i (T ) = X i (0)E Z T 0 i (s)C i (s)ds: Using the no-bankruptcy condition X i (T ) 0, this yields E Z T 0 i (s)C i (s)ds X i (0): (2.41) Thus the dynamic optimization problem given in (2.33) is transformed into a static variational problem. Let ^ C i (t) = 8 > < > : ^ c DD (t) + ^ c DF (t) e(t) if i =D ^ c FD (t)e(t) + ^ c FF (t) if i =F ; 29 where ^ c DD (t) = a DD e kt D D (t) ^ c DF (t) = a DF e kt D D (t) e(t) ^ c FD (t) = a FD e kt F F (t)e(t) ^ c FF (t) = a FF e kt F F (t) with i determined by E Z T 0 i (s) ^ C i (s)ds = X i (0): (2.42) One can show that i = 1 k (1e kT ) X i (0) : Notice that i 0. Next, let c iD and c iF be any strictly positive consumption processes satisfying (2.34) and (2.41). Then E Z T 0 i (s)( ^ C i (s)C i (s))ds 0: It then follows that E Z T 0 a iD logc iD (s) +a iF logc iF (s)ds E Z T 0 a iD logc iD (s) +a iF logc iF (s)ds + i E Z T 0 i (s)( ^ C i (s)C i (s))ds E Z T 0 max c iD ;c iF [a iD logc iD (s) +a iF logc iF (s) i i (s)C i (s)]ds + i E Z T 0 i (s) ^ C i (s)ds E Z T 0 a iD log ^ c iD (s) +a iF log ^ c iF (s)ds: 30 Therefore (^ c iD ; ^ c iF ) is the optimal consumption policy for agent i. The next step is to show that there exists a portfolio strategy ( T i ; T i ) to support the optimal consumption policy. In order to show such a strategy exist, dene u i (T ) = Z T 0 1 B i (s) ^ C i (s)ds: It follows immediately from (2.42) that E Pi [u i (T )] =X i (0). Dene the martingale v i (t) = E Pi [u i (T )jF t ]: This martingale then can be represented as v i (t) = X i (0) + Z t 0 H W i (s)d ~ W i (s) + Z t 0 H Q i (s)d ~ Q i (s); where H W i and H Q i arefF t g- predictable processes satisfying Z T 0 jjH W (s)jj 2 +jjH Q (s)jj 2 ds<1: Dene the wealth process ^ X i by 1 B i (t) ^ X i (t) + Z t 0 1 B i (s) ^ C i (s)ds = v i (t): (2.43) Then d ^ X i (t) = r i (t) ^ X i (t)dt +H W i (t)B i (t)d ~ W i (t) +H Q i (t)B i (t)d ~ Q i (t) ^ C i (t)dt = r i (t) ^ X i (t)dt + (H W i (t)B i (t);H Q i (t)B i (t)) 0 B @ d ~ W i (t) d ~ Q i (t) 1 C A ^ C i (t)dt: 31 From (2.18) and (2.26) it is clear that under the probability measure P i the dynamics of the optimal wealth process corresponding to the policy (^ c iD ; ^ c iF ; ^ T i ; ^ T i ) are given by d ^ X i (t) = (^ T i (t); ^ T i (t)) ^ X i (t)(diag( i (t))) 1 diag( i (t)) i (t) 0 B @ d ~ W i (t) d ~ Q i (t) 1 C A + ^ X i (t)r i (t)dt ^ C i (t)dt: (2.44) Comparing the (d ~ W i ;d ~ Q i ) T -terms and solving for (^ T i ; ^ T i ) one obtains that ^ X i (t) 0 B @ ^ i (t) ^ i (t) 1 C A = B i (t) H W i (t);H Q i (t) ( i (t)) 1 (diag( i (t))) 1 diag( i (t)): Since i is assumed to be non-degenerate and i is positive this denition is unique. Thus, there exists a portfolio strategy which supports the consumption policy (c iD ;c iF ). It remains to prove the representation of optimal wealth. Applying It^ o on ^ Xi(t) Bi(t) gives ^ X i (t) B i (t) = Z t 0 1 B i (s) (^ T i (s); ^ T i (s)) ^ X i (s)(diag( i (s))) 1 diag( i (s)) i (s) 0 B @ d ~ W i (s) d ~ Q i (s) 1 C A + ^ X i (0) Z t 0 ^ C i (s) B i (s) ds: Evaluating at t =T and subtracting gives ^ X i (t) B i (t) ^ X i (T ) B i (T ) = Z T t 1 B i (s) (^ T i (s); ^ T i (s)) ^ X i (s)(diag( i (s))) 1 diag( i (s)) i (s) 0 B @ d ~ W i (s) d ~ Q i (s) 1 C A + Z T t ^ C i (s) B i (s) ds: 32 From (2.43) it follows immediately that ^ X i (T ) = 0 almost surely. This makes intuitively sense because the agents do not derive utility from terminal wealth. Taking conditional expectation and using the fact that stochastic integrals are martingales results in ^ X i (t) = B i (t)E Pi " Z T t ^ C i (s) B i (s) ds F t # = 1 i (t) E " Z T t i (s) ^ C i (s)ds F t # : Substitution of the optimal consumption policy into this expression shows that ^ X i (t) = 1 i (t) ( iD (0)S D (0) + iF (0)S F (0)) e kt 1e kT [e k(tT) + 1] = e kt e kT 1e kT X i (0) i (t) : 2 The last proposition gives the optimal consumption policies and proves the existence of an opti- mal portfolio process to nance the optimal consumption process. The next lemma computes the optimal portfolio process explicitly. Lemma 2.11. Agent i's optimal portfolio process is given by X i (t) 0 B @ i (t) i (t) 1 C A = X i (t) (diag( i (t)) 1 diag( i (t)) ( T i (t)) 1 0 B B B B @ i (t) + (t) ~ + i (t) 1 (t) ~ i (t) 1 1 C C C C A : (2.45) Proof: Equation (2.35) implies that i (t)X i (t) + Z t 0 i (s)C i (s)ds is a martingale because stochastic integrals are martingales. This follows because of the assump- tions on the jump-diusion coecients. Hence i (t)X i (t) + Z t 0 i (s)C i (s)ds = E " i (T )X i (T ) + Z T 0 i (s)C i (s)ds F t # : 33 From (2.40) it follows that the optimal wealth X i (T ) = 0. Using (2.36) and (2.37), (2.38) and (2.39), respectively, one can show that E " Z T 0 i (s)C i (s)ds F t # =X 0 : Thus the dynamics of i (t)X i (t)+ R t 0 i (s)C i (s)ds have to be zero. The dynamics of i (t)X i (t)+ R t 0 i (s)C i (s)ds follow immediately from (2.35). In order to be zero, the following has to hold 0 = i (t)X i (t)( T i (t); T i (t))(diag( i (t))) 1 diag( (t)) i (t) diag 1; 1; 1; ~ + i (t) + (t) ; ~ i (t) (t) ! i (t)X i (t) 0 B B B B @ i (t) 1 ~ + i (t) + (t) 1 ~ i (t) (t) 1 C C C C A T : Solving for i (t)X i (t)( T i (t); T i (t)) yields the result. 2 Using the optimal portfolio strategy given in the lemma above one can rewrite the dynamics of the optimal wealth process. Corollary 2.12. Agent i's optimal wealth process satises the following dynamics dX i (t) = X i (t)( i (t)r i (t)1) T ( T i (t)) 1 diag 1; 1; 1; 1 ~ + i (t) ; 1 ~ i (t) ! 1 i (t)( i (t)r i (t)1)dt +X i (t)( i (t)r i (t)1) T ( T i (t)) 1 diag 1; 1; 1; 1 ~ + i (t) ; 1 ~ i (t) ! 0 B @ dW (t) dQ(t) 1 C A +X i (t)r i (t)dt k 1e k(tT) X i (t)dt: Proof: The result follows immediately from (2.45), (2.36) - (2.39), (2.18) and (2:26). 2 In this chapter the solution to the agents' optimization problem was determined. Notice that each agent takes the asset prices as given when solving his individual optimization problem. 34 The laws of supply and demand will then determine the asset prices in equilibrium. This is the topic of the next chapter. 35 Chapter 3 Equilibrium In equilibrium, the supply of the domestic and foreign goods should equal their respective demand. These conditions are generally called the domestic and foreign goods market clearing conditions. Furthermore, since there is one unit of the domestic and foreign productive asset, respectively, the aggregated portfolio holdings of the domestic and foreign productive assets should be equal to one. Since the bonds and jump securities are assumed to be in zero net supply, the aggregated portfolio holdings of these securities have to be zero. These conditions are called the nancial market clearing conditions. This chapter starts with formalizing the denition of equilibrium. The next section of this chapter discusses the implications of the goods market clearing conditions and nancial market clearing conditions. The solution to the equilibrium problem is derived in the last section. 3.1 Denition of Equilibrium The formal denition of an equilibrium is as follows. See Basak and Gallmeyer (1995) for a denition of a monetary equilibrium. Denition 3.1. Assume that the agents' preferences (U D ;U F ) and initial endowments (X D (0);X F (0)) are given in (2.1) and (2.32), respectively. An equilibrium is dened to be an allocation ((^ c DD ; ^ c DF ; ^ T D ; ^ T D ); (^ c FD ; ^ c FF ; ^ T F ; ^ T F )) and a 36 price systemP such that (^ c iD ; ^ c iF ; ^ T i ; ^ T i ) is a solution to agent i's individual optimization problem, where i2fD;Fg and markets clear for t2 [0;T ]: c DD (t) +c FD (t) = D (t) (3.1) c DF (t) +c FF (t) = F (t) (3.2) DD (t)X D (t) + 1 e(t) FD (t)X F (t) = S D (t) (3.3) DF (t)X D (t)e(t) + FF (t)X F (t) = S F (t) (3.4) DD (t)X D + 1 e(t) FD (t)X F (t) = 0 (3.5) DF (t)X D e(t) + FF (t)X F (t) = 0 (3.6) DB D (t)X D (t) + FB D (t) X F (t) e(t) = 0 (3.7) DB F (t)X D (t)e(t) + FB F (t)X F = 0: (3.8) Equations (3.1) and (3.2) are called the domestic and foreign goods market clearing conditions. Likewise, equations (3.3) - (3.8) are called the nancial market clearing conditions. The components of the allocation vector given in the denition above are the domestic agent's optimal consumption rates of the domestic good, foreign good, and her optimal portfolio holdings and the foreign agent's optimal consumption rates of the domestic good, foreign good, and her optimal portfolio holdings. Notice that equations (3.1) and (3.2) guarantee clearing in the goods market. These conditions model the fact that the agents are not allowed to store the goods, but they have to consume them instantaneously. Equations (3.3)-(3.8) ensure clearing in the nancial market. Dividing (3.3) and (3.4) by S D (t) and S F (t), respectively, these equations say that the domestic and foreign agents' portfolio holdings of each productive asset have to add up to one since there is one unit of each productive asset. Equations (3.5) - (3.8) show that the domestic agent can only hold a long (short) position in the jump securities and bonds if the foreign agent is willing to take an 37 appropriate short (long) position and vice versa. The rst step in nding the equilibrium allocation is to study the implications of the goods market and nancial market clearing condition. 3.2 Implications of the Equilibrium Conditions The ultimate goal is to determine the equilibrium parameters with help of the equilibrium conditions specied in the previous section. The nancial market conditions (3.3)-(3.8) can be used directly in the search of the equilibrium parameters, whereas the goods market conditions (3.1) and (3.2) have further implications for the equilibrium parameters. The following lemma summarizes these implications. Lemma 3.2. If the consumption goods markets are in equilibrium, then the following restric- tions have to hold D (t) T D (t) = c DD (t) T D (t) +c FD (t)[ T F (t) T e (t)] (3.9) F (t) T F (t) = c FF (t) T F (t) +c DF (t)[ T D (t) + T e (t)] (3.10) D (t)K + D (t) = c DD (t) + (t) ~ + D (t) 1 ! +c FD (t) " + (t) ~ + F (t) 1 1 +K + e (t) 1 # (3.11) F (t)K + F (t) = c FF (t) + (t) ~ + F (t) 1 ! +c DF (t) " + (t) ~ + D (t) (1 +K + e (t)) 1 # (3.12) D (t)K D (t) = c DD (t) (t) ~ D (t) 1 ! +c FD (t) " (t) ~ F (t) 1 1 +K e (t) 1 # (3.13) F (t)K F (t) = c FF (t) (t) ~ F (t) 1 ! +c DF (t) " (t) ~ D (t) (1 +K e (t)) 1 # (3.14) 38 D (t)[ D (t)K + D (t) + (t)K D (t) (t)] = c DD (t) h r D (t)k + (t) + ~ + D (t) (t) + ~ D (t) + T D (t) D (t) i +c FD (t) e (t)k +K + e (t) + (t) +K e (t) (t) + T e (t) e (t) +r F (t) + (t) + ~ + F (t) (t) + ~ F (t) + T F (t) F (t) T e (t) F (t) i (3.15) F (t)[ F (t)K + F (t) + (t)K F (t) (t)] = c FF (t) h r F (t)k + (t) + ~ + F (t) (t) + ~ F (t) + T F (t) F (t) i +c DF (t) h e (t)kK + e (t) + (t)K e (t) (t) +r D (t) + (t) + ~ + D (t) (t) + ~ D (t) + T D (t) D (t) + T e (t) D (t) i : (3.16) Proof: See Appendix. 2 The conditions provided in the lemma above provide the key for solving the equilibrium. The next section solves the equilibrium problem. 3.3 Equilibrium Parameters This section determines all equilibrium parameters. Furthermore, it discusses the implications of the equilibrium dynamics of the exchange rate for the uncovered interest rate parity puzzle. The following theorem gives the solution for the equilibrium parameters. 39 Theorem 3.3. If the consumption goods market equilibrium and nancial market equilibrium hold, then D (t) = D (t) F (t) = F (t) ~ + D (t) = + (t) K + D (t) + 1 ~ D (t) = (t) K D (t) + 1 ~ + F (t) = + (t) K + F (t) + 1 ~ F (t) = (t) K F (t) + 1 r D (t) = D (t) + (t) (K + D (t)) 2 K + D (t) + 1 (t) (K D (t)) 2 K D (t) + 1 T D (t) D (t) +k r F (t) = F (t) + (t) (K + F (t)) 2 K + F (t) + 1 (t) (K F (t)) 2 K F (t) + 1 T F (t) F (t) +k e (t) = F (t) D (t) + T D (t)( D (t) F (t)) K + D K + F (t)K + D (t) K + D (t) + 1 + (t)K D K F (t)K D (t) K D (t) + 1 (t) e (t) = F (t) D (t) K + e (t) = K + F (t)K + D (t) K + D (t) + 1 K e (t) = K F (t)K D (t) K D (t) + 1 Si (t) = i (t) +k Si (t) = i (t) K + Si (t) = K + i (t) K Si (t) = K i (t); where i2fD;Fg. Recall that i is the expected appreciation rate of the ith dividend process, i is the associated volatility parameter of the process, K + i andK i are the positive and negative jump sizes of the process. The agents' subjective discount rate is denoted by k. The parameters + i 40 and i are the intensities of the Poisson processes N + and N , respectively. The parameters i , + i , and i denote the market price of risk for agent i. The domestic and foreign interest rates are given by r D and r F . The exchange rate parameters are denoted by e , e , K + e , and K e . Similarly, the parameters of productive asset i are represented by Si , Si ,K + Si andK Si . Proof: The proof consists of several parts: First, Lemma 2.7, Lemma 3.2, and the denition of the market price of risk make it possible to determine the market price of risk parameters, the domestic and foreign interest rate and the exchange rate parameters. The productive asset price parameters then follow with the help of Lemma 2.8. It turns out that the equilibrium parameters result in an incomplete market. Finally, it is proven that the equilibrium parameters are optimal for this incomplete market setting using a technique rst proposed by Karatzas et al (1991). Recall from (2.27) that D (t) = F (t) e (t): Using this, (3.1) and (3.2) in (3.9) and (3.10) shows that i (t) = i (t): It then immediately follows that e (t) = F (t) D (t): The parameters ~ + i and ~ i can be determined similarly. Notice from (2.27) it follows that ~ + D (t) = (1 +K + e (t)) ~ + F (t) (3.17) ~ D (t) = (1 +K e (t)) ~ F (t): (3.18) 41 From equations (3.11) - (3.14), (3.1) and (3.2) it is then immediately clear that ~ + i (t) = + (t) K + i (t) + 1 ~ i (t) = (t) K i (t) + 1 : The values K + e (t) and K e (t) can then be easily found from (3.17) and (3.18). Next, determine the domestic and foreign interest rate and the drift parameter of the exchange rate process. From the fth row of the denition of the domestic market price of risk, given in (2.17), we obtain that e (t) = K + e (t) 1 +K + e (t) ~ + D (t) + K e (t) 1 +K e (t) ~ D (t) e (t) T F (t)K + e (t) + (t)K e (t) (t)r F (t) +r D (t): (3.19) Plugging this in (3.15) and (3.16) gives D (t)[ D (t)K + D (t) + (t)K D (t) (t)] = D (t)[r D (t)k + (t) + ~ + D (t) (t) + ~ D (t) + T D (t) D (t)] and F (t)[ F (t)K + F (t) + (t)K F (t) (t)] = F (t)[r F (t)k + (t) + ~ + F (t) (t) + ~ F (t) + T F (t) F (t)]: The domestic and foreign interest rate can then be determined from these two equations. Equation (3.19) then gives the drift of the exchange rate. Therefore, all market price of risk parameters and all parameters of the exchange rate are determined. It remains to compute the parameters of the productive asset price processes. To this end recall (2.31), S i (t) = 1 i (t) E " Z T t i (t) i (t)dtjF t # : 42 It is now possible to compute the stock price explicitly. Apply It^ o to compute the dynamics of i (t) i (t), d( i (t) i (t)) = i (t) i (t) " i (t)r i (t) T i (t) i (t) +K + i (t) ~ + i (t) + (t) 1 ! + (t) +K i (t) ~ i (t) (t) 1 ! (t) ! dt + ( T i (t) T i (t))dW (t) + K + i (t) + ~ + i (t) + (t) 1 ! +K + i (t) ~ + i (t) + (t) 1 !! dQ + (t) + K i (t) + ~ i (t) (t) 1 ! +K i (t) ~ i (t) (t) 1 !! dQ (t) # : Substituting the predetermined parameters into this equation, implies d( i (t) i (t)) = i (t) i (t)kdt: Thus i (t) i (t) = i (0)e kt (3.20) is deterministic. Hence S i (t) = i (0) i (t) e kt e kT k = i (t) 1e k(Tt) k : (3.21) Notice that S i (t) is the product of the dividend process and a deterministic factor which can be interpreted as a discount factor. At terminal time T the price of the productive asset i is zero. This is reasonable since the agents derive no utility from terminal wealth. 43 Computing the dynamics of the productive asset price process results in dS i (t) = S i (t)[( i (t) +k)dt + T i (t)dW (t) +K + i (t)dQ + (t) +K i (t)dQ (t)] i (t)dt: Comparing coecients yields the parameters of the productive asset price processes. This choice of parameters results in an incomplete market for both agents. This can be seen as follows. Consider the dynamics of the foreign asset as seen by the domestic agent. In Lemma 2.1 it was shown that d S F (t) e(t) = S F (t) e(t) "" S F (t) e (t) + T e (t) e (t) T S F (t) e (t) + K + e (t)(K + e (t)K + S F (t)) 1 +K + e (t) ! + (t) + K e (t)(K e (t)K S F (t)) 1 +K e (t) ! (t) # dt + T S F (t) T e (t) dW (t) + K + S F (t)K + e (t) 1 +K + e (t) dQ + (t) + K S F (t)K e (t) 1 +K e (t) dQ (t) # 1 e(t) F (t)dt: Using the equilibrium parameters this simplies to d S F (t) e(t) = S F (t) e(t) " ( D (t) +k)dt + T D (t)dW (t) +K + D (t)dQ + (t) +K D (t)dQ (t) # 1 e(t) F (t)dt: Observe that the dynamics of the domestic productive asset in equilibrium are given by d (S D (t)) = S D (t) ( D (t) +k)dt + T D (t)dW (t) +K + D (t)dQ + (t) +K D (t)dQ (t) D (t)dt: 44 Therefore, if the gains process of the foreign productive asset is measured in the domestic asset, it follows the same jump-diusion process as the domestic productive asset. This fact makes the market for the domestic agent incomplete. Similarly, an analogous result for the foreign agent can be proven. From Lemma 2.2 and using the equilibrium parameters determined above it follows that d (S D (t)e(t)) = S D (t)e(t) " ( F (t) +k)dt + T F (t)dW (t) +K + F (t)dQ + (t) +K F (t)dQ (t) # 1 e(t) D (t)dt: Compare this to the dynamics of the foreign productive asset in equilibrium which are given by d (S F (t)) = S F (t) ( F (t) +k)dt + T F (t)dW (t) +K + F (t)dQ + (t) +K F (t)dQ (t) F (t)dt: Therefore the domestic and foreign gain processes measured in terms of the foreign commodity follow identical jump-diusion dynamics. Thus the foreign agent faces an incomplete market as well. Notice that the domestic and foreign jump security are also redundant. Thus, the rst and second row of the matrices D and F are linearly dependent in equilibrium. Further- more, the third and fourth row of these matrices are linearly dependent. Therefore, these matrices are not invertible in equilibrium. It now remains to show that the parameters given in the theorem are equilibrium val- ues for an incomplete market setting. Notice that the optimal investment process obtained in (2.45) cannot be used because i is no longer invertible. Since the domestic and foreign productive asset oer the same investment opportunity, 45 it can be assumed without loss of generality that both agents only invest in the productive asset of their own country. Notice that agents derive utility from consumption of the domestic and foreign consumption good, but they can satisfy these needs by trading the consumption good directly. From the nancial equilibrium conditions in (3.3) and (3.4) it is immediate that X D (t) = S D (t) X F (t) = S F (t): Recall that the drift term of the jump securities will be determined such that the agents do not want to invest in these assets. Therefore, the dynamics of the domestic wealth process are given by dX D (t) = (:::)dt + T S D (t)dW (t) + DB F (t) T e (t)dW (t) + (:::)dQ(t): But since the dynamics of the domestic wealth process has to agree with the domestic productive asset and the Brownian motion parameters of the domestic productive asset and the exchange rate are given in equilibrium by S D (t) = D (t) e (t) = F (t) D (t); DB F (t) has to equal zero. Notice that e (t) has to have at least one non-zero component because otherwise the matrix D (t) is not invertible. Similarly, it can be shown that the foreign investor does not invest in the domestic bond. Therefore, the agents only invest in their own productive asset. Karatzas et al (1991) propose a solution to a similar problem in the Brownian motion setting. Redundant securities are dropped from i and replaced by ctitious assets. The Brownian motion coecients and jump coecents of these ctitious assets are chosen in such a way that the nancial markets are complete. The appreciation rate of these assets is then determined in such a way that it is optimal for the agents to invest nothing in them. 46 In order to formalize this argument in our setting dene the asset price vector for the domestic investor as D (t) = 0 B B B B B B B B B B @ S D (t) P D (t) S F1 (t) S F2 (t) B F (t)=e(t) 1 C C C C C C C C C C A ; where S F1 , and S F2 are the ctitious assets. Choose the coecients of the volatility matrix D (t) = 0 B B B B B B B B B B @ T S D (t) K + S D (t) K S D (t) 0 0 0 K + S D (t) K S D (t) T S F1 (t) K + S F1 (t) K S F1 (t) T S F2 (t) K + S F2 (t) K S F2 (t) T e (t) K + e (t) 1+K + e (t) K e (t) 1+K e (t) 1 C C C C C C C C C C A such that it is invertible. The drift of the asset price vector is given by D (t) = 0 B B B B B B B B B B @ S D (t) P D (t) S F1 (t) S F2 (t) e (t) + T e (t) e (t) +r F (t) + (K + e (t)) 2 1+K + e (t) + (t) + (K e (t)) 2 1+K e (t) (t) 1 C C C C C C C C C C A : The dividend vector is dened as D (t) = 0 B B B B B B B B B B @ R t 0 D (s)ds 0 0 0 0 1 C C C C C C C C C C A 47 and the domestic market price of risk parameters are: 0 B B B B @ D (t) ~ + D (t) + + (t) ~ D (t) + (t) 1 C C C C A = 1 D (t) D (t)r D (t) : Similarly, dene the asset price vector of the foreign investor as F (t) = 0 B B B B B B B B B B @ S F (t) P D (t)e(t) S F1 (t)e(t) S F2 (t)e(t) B D (t)e(t) 1 C C C C C C C C C C A : The volatility matrix of the foreign asset price vector is then given by F = 0 B B B B B B B B B B @ S F K + S F K S F T e K + S D +K + e (1 +K + S D ) K S D +K e (1 +K S D ) T e + T S F1 K + S F1 +K + e (1 +K + S F1 ) K S F1 +K e (1 +K S F1 ) T e + T S F2 K + S F2 +K + e (1 +K + S F2 ) K S F2 +K e (1 +K S F2 ) T e K + e K e 1 C C C C C C C C C C A and the vector of drift parameters of the foreign asset price vector is dened as F = 0 B B B B B B B B B B @ S F P D + e +K + e K + S D + +K e K S D S F1 + e + T S F1 e +K + e K + S F1 + +K e K S F1 S F2 + e + T S F2 e +K + e K + S F2 + +K e K S F2 e +r D 1 C C C C C C C C C C A ; where the time dependence of the parameters is omitted. 48 Since the domestic and foreign productive assets are redundant in equilibrium, the fol- lowing equations hold S F (t) = T e (t) + T S D (t) K + S F (t) = K + S D (t) +K + e (t)(1 +K + S D (t)) K S F (t) = K S D (t) +K e (t)(1 +K S D (t)) S F (t) = S D (t) + e (t) + T S D (t) e (t) +K + e (t)K + S D (t) + (t) +K e (t)K S D (t) (t): Thus, F (t) = G D (t)K; where G and K are dened in (2.28) and (2.29). Therefore, the matrix F is invertible because it is the product of invertible matrices. Notice that the matrix K is invertible. If the jump coecients K + e (t) and K e (t) were equal to negative one for a t2f0;Tg, then the matrix D (t) would be not invertible which contradicts the assumption above. The dividend vector of the foreign asset price vector is dened as F (t) = 0 B B B B B B B B B B @ R t 0 F (s)ds 0 0 0 0 1 C C C C C C C C C C A : Finally, the foreign market price of risk parameters are given by 0 B B B B @ F (t) ~ + F (t) + + (t) ~ F (t) + (t) 1 C C C C A = 1 F (t) F (t)r F (t) : 49 It is possible to show that an analogous relationship to (2.27) still holds. Thus, all computations to obtain the equilibrium parameters can be computed as before. In particular, it holds that i (t) = i (t) ~ + i (t) = ~ + i (t) ~ i (t) = ~ i (t): It remains to choose the drift parameters of the ctitious assets in such a way that the agents do not want to trade in these assets. Denote by i the ve dimensional column vector of wealth proportions the agent invests in her productive asset, the jump security, the two ctitious securities, and the foreign bond. From (2.45) it follows that these proportions are given by X i (t) i (t) = X i (t) (diag( i (t)) 1 diag( i (t)) ( T i (t)) 1 0 B B B B @ i (t) + (t) ~ + i (t) 1 (t) ~ i (t) 1 1 C C C C A = X i (t) (diag( i (t)) 1 diag( i (t)) ( T i (t)) 1 diag 1; 1; 1; 1 ~ + i (t) ; 1 ~ i (t) ! 0 B B B B @ i (t) + (t) ~ + i (t) (t) ~ i (t) 1 C C C C A : Each agent invests all her wealth in their own productive asset and nothing in the foreign bond. Therefore, set 1 = (1; 0; 0; 0; 0) T . Notice that from the nancial equilibrium conditions it is then known that X i (t) = S i (t). Thus, using the denition of the market price of risk, the domestic drift term is determined by D (t) = r D (t)1 + D (t)diag 1; 1; 1; ~ + D (t); ~ D (t) T D (t) 1 = r D (t)1 + D (t)diag 1; 1; 1; ~ + D (t); ~ D (t) 0 B B B B @ S D (t) K + S D (t) K S D (t) 1 C C C C A : 50 Componentwise this means that S D (t) = r D (t) + T S D (t) S D (t) + (K + S D (t)) 2 ~ + D (t) + (K S D (t)) 2 ~ D (t) P D (t) = r D (t) + (K + S D (t)) 2 ~ + D (t) + (K S D (t)) 2 ~ D (t) S F1 (t) = r D (t) + T S F1 (t) S D (t) +K + S F1 (t)K + S D (t) ~ + D (t) +K S F1 (t)K S D (t) ~ D (t) S F2 (t) = r D (t) + T S F2 (t) S D (t) +K + S F2 (t)K + S D (t) ~ + D (t) +K S F2 (t)K S D (t) ~ D (t) e (t) = r F (t)r D (t) + T e (t) e (t) + T e (t) S D (t) + (K + e (t)) 2 1 +K + e (t) + (t) + K + e (t) 1 +K + e (t) K + S D (t) ~ + D (t) + (K e (t)) 2 1 +K e (t) (t) + K e (t) 1 +K e (t) K S D (t) ~ D (t): All values on the right hand side of the equations are known in equilibrium. Hence, if the drift parameters of the jump security and the two ctitious assets are chosen as postulated above, the domestic agent will not invest in these assets in equilibrium. Since these assets are in zero net supply, it is also optimal for the foreign agent to not invest in these assets. Notice that the rst and last component equations are automatically fullled in equilibrium because they are part of the the market price of risk equation. This completes the proof. 2 A few comments are in order. First, the theorem above provides a complete solution to the problem. As in the Brownian motion case solved by Zapatero (1995), it is possible to completely endogenize all parameters. If the dividend processes do not have any jumps, then the results obtained here completely agree with the ndings of Zapatero (1995) in a pure diusion framework. Secondly, the results obtained here can also be seen as an extension of a paper by Cass and Pavlova (2003) to a jump-diusion setting. As stated in the introduction, they nd that under log-linear specications the Lucas (1978) tree model allows only for so-called "peculiar nancial equilibria". The equilibria are peculiar in the sense that all stocks oer the same investment opportunity in equilibrium, resulting in a completely degenerate stock market. As shown in the proof above the stock market is also completely degenerate in a jump-diusion setting. 51 The third comment is in regard the domestic and foreign interest rate. Notice that the domestic and foreign interest rate are only in uenced by domestic and foreign parameters, respectively. Furthermore, the presence of jumps in the model reduces the interest rates. It is interesting that both negative and positive jumps lead to a reduction in the interest rates. This can be explained by the fact that adding jumps to the model increases the uncertainty for the investors. Therefore, there is an increased demand for the risk-free security leading to a reduced interest rate. This behavior is similar to the traditional Brownion motion case. The diusion parameter also reduces the interest rate. The fourth observation concerns the dynamics of the exchange rate, the main object of interest. The initial question was to determine the eect on exchange rates of jumps in aggregate consumption. The dynamics of the exchange rate are summarized in the next corollary. Corollary 3.4. The parameters of the exchange rate are e (t) = r F (t)r D (t) + T F (t) T F (t) T D (t) (3.22) + K + F (t) K + F (t)K + D (t) K + D (t) + 1 (K + F (t) + 1) + (t) + K F (t) K F (t)K D (t) K D (t) + 1 K F (t) + 1 (t) e (t) = F (t) D (t) K + e (t) = K + F (t)K + D (t) K + D (t) + 1 K e (t) = K F (t)K D (t) K D (t) + 1 : Proof: The representation of the drift term of the exchange rate follows immediately from Theorem 3.3 after a few algebraic manipulations. 2 There is a very large body of empirical literature studying exchange rates. Numerous papers have been written in the attempt to explain the uncovered interest rate parity puzzle; for a survey see for example Froot and Thaler (1990), Lewis (1995), and Engel (1996). Uncovered interest rate parity demands that if you borrow at the domestic interest rate and invest 52 the proceeds in the foreign bond, then the expected return on this portfolio should be zero. Expressed dierently, it says that the expected exchange rate change should be equal to the dierence between the domestic and foreign interest rates. Therefore running a regression of exchange rate returns on the interest rate dierential should result in a zero intercept and a slope coecient of unity. Empirical data shows that this hypothesis can be rejected. According to Froot (1990) the average coecient across some 75 published estimates is -.88. Assuming the foreign interest rate is higher than the domestic interest rate, the uncovered interest parity principle predicts that the exchange rate has to rise. Froot's (1990) survey results, in contrast, imply that the exchange rate is expected to fall. This indicates that other factors besides the interest rate dierential determine the expected exchange rate change. In the model considered here, the expected exchange rate change is given by (3.22). Zapatero (1995) already observed that the drift term of the exchange rate should not only depend on the interest rate dierential, but also on the volatility of the domestic and foreign stock market. Incorporating jumps into Zapatero's (1995) model adds additional terms to the drift term of the exchange rate which are determined by the domestic and foreign jump parameters, and the jump intensities. For example, if the domestic stock market exhibits larger positive and larger negative jump coecients than the foreign stock market, this leads to a decrease in the drift of the exchange rate. Therefore, equation (3.22) provides a new testing hypothesis and a new explanation for the uncovered interest rate parity puzzle. This chapter started with the denition of the goods market equilibrium and the nan- cial market equilibrium. Next, the equilibrium parameters were determined. All model parameters, apart from the exogenously given dividend processes, were endogenized. This was followed by a discussion of the equilibrium dynamics of the exchange rate and its implication for the uncovered interest parity puzzle. In the next chapter the higher moment properties of the equilibrium exchange rate will be discussed. 53 Chapter 4 Higher Moment Properties of Equilibrium Exchange Rate Returns It is an established empirical result that exchange rates tend to display signicant and variable skewness, and excess kurtosis. For example, Bates (1996) ts a jump-diusion model for exchange rate returns to options data examining implicit volatility, skewness, and excess kurtosis. He nds a positively skewed implicit distribution for the $/DM exchange rate returns during the strong dollar period of 1984 and 1985. After that period he reports that the implicit skewness oscillated substantially, similar to the $/yen implicit skewness. Prichet (1991) shows that real exchange rate returns for 56 developed countries with managed exchange rates exhibit skewness and excess kurtosis. Carr and Wu (2007) use time-changed L evy processes for exchange rates to capture the stochastic skew behavior of currency option prices. To my knowledge there has been no attempt to study skewness and kurtosis of exchange rates in the context of a general equilibrium model in an economy subject to a jump-diusion. The purpose of this chapter is to understand how the skewness and kurtosis of the aggregate consumption processes aect the skewness and kurtosis of the exchange rate. This chapter starts with a discussion of the skewness of the logarithmic returns of the dividend and the exchange rate process. It also analyzes the relationship between the skewness of these processes. The second part of this chapter considers the kurtosis of the logarithmic returns of the dividend and exchange rate process. 4.1 Skewness of the Exchange Rate Return For reasons of tractability it has to be assumed that all coecients of the aggregate consump- tion processes are constant and the positive and negative jumps are independent. Random coecients of the aggregate consumption processes are clearly desirable because this would 54 enable a study of the stochastic skew behavior of the exchange rate in an equilibrium setting. Therefore, the assumption of constant coecients is rather stringent, but it allows to gain a rst understanding how the skewness of the returns of the dividend processes in uence the skewness of the exchange rate returns. The skewness of the consumption process is derived in the next lemma. Lemma 4.1. Assume that the process i (t) is governed by the following jump-diusion equation d i (t) = i (t) i dt + T i dW (t) +K + i dQ + (t) +K i dQ (t) ; where dQ + (t) = dN + (t) + dt dQ (t) = dN (t) dt: Assume further that the Poisson processes N + and N are independent. Then the variance of log i(t+h) i(t) is given by Var log i (t +h) i (t) = T i i h + + h log 1 +K + i 2 + h log 1 +K i 2 : The third centralized moment is given by E log i (t +h) i (t) E log i (t +h) i (t) 3 = + h log 1 +K + i 3 + h log 1 +K i 3 : The skewness is then given by log i (t+h) i (t) = + h log 1 +K + i 3 + h log 1 +K i 3 T i i h + + h log 1 +K + i 2 + h log 1 +K i 2 3 2 : (4.1) 55 Proof: Since all parameters are constant, the dividend process is given by i (t) = i (0) exp i K + i + K i 1 2 T i i t + T i W (t) 1 +K + i N + (t) 1 +K i N (t) : Thus, it follows that i (t +h) i (t) = exp i K + i + K i 1 2 T i i h + T i W (h) log(1 +K + i ) N + (h) log(1 +K + i ) N (h) : The log-dierenced dividend process can then be written as log i (t +h) i (t) = i K + i + K i 1 2 T i i h + T i W (h) +N + (h) log(1 +K + i ) +N (h) log(1 +K + i ): It is then immediately clear that the third centralized moment can be expressed as E log i (t +h) i (t) E log i (t +h) i (t) 3 = E T i W (h) +N + (h) log(1 +K + i ) +N (h) log(1 +K i ) + h log(1 +K + i ) h log(1 +K i ) 3 : Since the Wiener and Poisson processes are all independent, it follows that E log i (t +h) i (t) E log i (t +h) i (t) 3 = E T i W (h) 3 +E N + (h) log(1 +K + i ) + h log(1 +K + i ) 3 +E N (h) log(1 +K i ) h log(1 +K i ) 3 = + h log 1 +K + i 3 + h log 1 +K i 3 : In the last equation the fact was used that the third centralized moment of the Poisson Process N j (t) is given by j t, where j2f+;g. 56 The variance then follows similarly Var log i (t +h) i (t) = E T i W (h) +N + (h) log(1 +K + i ) +N (h) log(1 +K i ) + h log(1 +K + i ) h log(1 +K i ) 2 = T i i h + + h log 1 +K + i 2 + h log 1 +K i 2 : The skewness is then dened as log i (t+h) i (t) = E h log i(t+h) i(t) E log i(t+h) i(t) i 3 (Var log i(t+h) i(t) ) 3 2 : 2 Notice that in a pure diusion model the log-dierenced dividend process is not skewed at all. Since one purpose of this section is to understand how the skewness of the logarithmic return of the dividend processes in uence the skewness of the logarithmic exchange rate return, assume for example that the logarithmic return of the domestic dividend process is not skewed and the logarithmic return of the foreign consumption process is positively skewed. It is expected that the exchange rate is positively skewed in this situation. This can be explained from an equilibrium perspective as follows. Since the logarithmic return of the foreign consumption process is positively skewed and the logarithmic return of the domestic process is not skewed, it will happen at cer- tain times that the return of the foreign consumption process is signicantly higher than the return of the domestic consumption process. Notice that in equilibrium aggregate consumption equals aggregate demand. Therefore, the demand for the foreign consump- tion good will be higher than for the domestic consumption good at these times. This results in a higher relative price for the foreign consumption good, and consequently in a higher exchange rate. Thus, this setting creates a large return of the exchange rate which lies above the mean return. Hence, the logarithmic return of the exchange rate is positively skewed. 57 A similar intuition can be developed for all other possible situations. In summary, if the return of the domestic consumption process is more (less) positively skewed than the return of the foreign consumption process, then the return of the exchange rate is expected to be negatively (positively) skewed. The computation and the numerical studies of the skewness of the logarithmic return on the exchange rate will conrm this intuition. The skewness of the logarithmic exchange rate return can be derived as in the case of the dividend processes. Lemma 4.2. Under the assumption that the dividend processes have constant coecients and the Poisson processesN + andN are independent, the variance of the log-dierenced exchange rate process e(t) is given by Var log e(t +h) e(t) = T e e h + + h log 1 +K + e 2 + h log 1 +K e 2 : The third centralized moment is given by E log e(t +h) e(t) E log e(t +h) e(t) 3 = + h log 1 +K + e 3 + h log 1 +K e 3 : The skewness is then given by log( e(t+h) e(t) ) = + h (log (1 +K + e )) 3 + h (log (1 +K e )) 3 T e e h + + h log 1 +K + e 2 + h log 1 +K e 2 3 2 : Proof: This can be proven in a very similar way as in the case of the dividend process. 2 58 Using the equilibrium values for the jump parameters of the exchange rate, the skewness of the log-dierenced exchange rate can be expressed as log( e(t+h) e(t) ) = + h log 1+K + F 1+K + D 3 + h log 1+K F 1+K D 3 T e e h + + h log 1+K + F 1+K + D 2 + h log 1+K F 1+K D 2 !3 2 : This representation shows that the Brownian motion parameters of the dividend processes only in uence the magnitude of the skewness of the log-dierenced exchange rate, but not the sign of the skewness. The sign of the skewness is determined by the domestic and foreign jump parameter. A bigger positive jump parameter of the foreign dividend process increases the skewness and a bigger negative jump parameter of the foreign dividend process decreases the skewness of the log-dierenced exchange rate. Figures 4.1 and 4.2 illustrate this behavior. The region where the skewness of the return of the domestic dividend process is higher than the foreign dividend process agrees mostly with the region where the exchange rate returns are negatively skewed. It is not an exact one to one rule since there are deviations from this rule when the skewness of the domestic and foreign dividend returns are almost equal, but overall it is a good predictor of the behavior of the skewness of the exchange rate returns. The conclusion of this section is that exchange rate returns are skewed in a jump- diusion model. This is an advantage over traditional diusion models. Furthermore, the skewness of the exchange rate returns can be explained by the relationship between the skewness of the domestic and foreign dividend returns. A shortcoming of the model considered here is that the skewness of the exchange rate process is not stochastic. A rst study of this issue could be done by simulation, but this is beyond the scope of this dissertation. The last section explores the kurtosis behavior of the exchange rate returns. 59 Figure 4.1: Skewness behavior of the return of the exchange rate log e(t+h) e(t) for dierent values of the domestic jump parameters where h =:05 60 Figure 4.2: Skewness behavior of the return of the exchange rate log e(t+h) e(t) for dierent values of the foreign jump parameters where h = 0:05 61 4.2 Kurtosis of the Exchange Rate Return A distribution with a high kurtosis has a sharper peak and longer, fatter tails. A distribution exhibiting a low kurtosis has a more rounded peak and shorter thinner tails. Since an intuitive relationship between the kurtosis of the return on the dividend processes and the exchange rate is not obvious from an equilibrium perspective, this section discusses this relationship only from an analytical and numerical standpoint. Under the same assumptions as for the skewness, it is possible to derive an expression for the kurtosis of the logarithmic exchange rate return. Lemma 4.3. Under the assumption that the dividend processes have constant coecients and the Poisson processes N + and N are independent, the excess kurtosis of the log-dierenced exchange rate process e(t) is given by log e(t +h) e(t) = + h (log (1 +K + e )) 4 + h (log (1 +K e )) 4 T e e h + + h log 1 +K + e 2 + h log 1 +K e 2 2 : Proof: The fourth centralized moment can be calculated as follows. E log e(t +h) e(t) E log e(t +h) e(t) 4 = E T e W (h) +N + (h) log(1 +K + e ) +N (h) log(1 +K e ) + h log(1 +K + e ) h log(1 +K e ) 4 = E T e W (h) 4 + 6 T e e h Var N + (h) log 1 +K + e +Var N (h) log 1 +K e +E N + (h) log 1 +K + e + h log 1 +K + e 4 +6Var N + (h) log 1 +K + e Var N (h) log 1 +K e +E N (h) log 1 +K e h log 1 +K e 4 = ( T e e ) 2 3h 2 + 6 T e e h h log 1 +K + e 2 + h + log 1 +K e 2 h i + + h 1 + 3 + h log 1 +K + e 4 + 6 log 1 +K + e 2 log 1 +K e 2 + h 2 + h 1 + 3 h log 1 +K e 4 ; 62 where in the second equality the fact was used that the Brownian motion and Poisson processes are all independent. A simple calculation then shows that the excess kurtosis is given by log e(t +h) e(t) = ( T e e ) 2 3h 2 + 6 T e e h h (log (1 +K + e )) 2 + h + (log (1 +K e )) 2 h i T e e h + + h log 1 +K + e 2 + h log 1 +K e 2 2 + + h (1 + 3 + h) (log (1 +K + e )) 4 + 6 (log (1 +K + e )) 2 (log (1 +K e )) 2 + h 2 1 + h (1 + 3 h) (log (1 +K e )) 4 1 3 = + h (log (1 +K + e )) 4 + h (log (1 +K e )) 4 T e e h + + h log 1 +K + e 2 + h log 1 +K e 2 2 : 2 The excess kurtosis of the log-dierenced exchange rate is always positive in a jump-diusion model. This agrees with empirical results which show that log-dierenced exchange rates are leptokurtic, see for example Westereld (1977) and Boothe and Glassman (1987). A pure diusion model in contrast has zero excess kurtosis. In the case of a jump-diusion model, the Brownian motion parameter only in uences the magnitude of the excess kurtosis. If the domestic and foreign jump parameters are similar, then the excess kurtosis of the exchange rate return is very small. Figure 4.3 shows how the kurtosis of the exchange rate return varies in dependence of the domestic jump parameters. Evaluating the kurtosis with respect to the foreign jump parameters produces similar results. The next corollary contains the expression for the excess kurtosis of the dividend pro- cesses. 63 Figure 4.3: Excess kurtosis of the exchange rate return log e(t+h) e(t) for dierent values of the domestic jump parameters where h = 0:05 64 Corollary 4.4. Under the assumption that the dividend processes have constant coecients and the Poisson processes N + and N are independent, the excess kurtosis of the log-dierenced dividend process i (t) is given by log i (t +h) i (t) = + h log 1 +K + i 4 + h log 1 +K i 4 T i i h + + h log 1 +K + i 2 + h log 1 +K i 2 2 : Proof: This can be proved analogously to the previous lemma. 2 65 Chapter 5 Conclusion The purpose of this dissertation is to examine the eects of jumps in economic fundamentals on the exchange rate in the framework of Zapatero's (1995) two-good two-country intertemporal international equilibrium model for logarithmic representative agents. The assumption of logarithmic preferences is essential to obtain an explicit equilibrium solution. The agents face ve sources of risk, one international and two country specic diusive sources, and positive and negative jump risks modeled by two Poisson processes. Individual nancial markets are incomplete, but the agents can hedge their risks completely by trading in the international nancial market at the stochastic exchange rate. It is possible to endogenize all model parameters in equilibrium, except the parameters of the dividend process which are assumed to be given. In equilibrium it turns out that the domestic and foreign productive assets oer the same investment opportunity, thus resulting in an incomplete market. Therfore, this dissertation also extends the research of Cass and Pavlova (2003) to a jump-diusion setting. The expected equilibrium exchange rate change exhibits all properties of Zapatero's (1995) model. It increases with the interest rate dierential and the diusion parameter of the domestic equity market and decreases with the covariance between the diusion parameters of the domestic and foreign equity market. Furthermore, the expected equilibrium exchange rate change reacts to dierences in the domestic and foreign jump sizes. If the foreign positive and negative jump sizes are bigger (smaller) in absolute value than their domestic equivalents, then the expected equilibrium exchange rate change increases (decreases). Therefore, the addition of jumps oers a possible explanation for the interest rate parity puzzle. It would be an interesting task for future research to test this explanation empirically. Traditional equilibrium models subject to diusive risk have one serious shortcoming. 66 It is a well-known empirical fact that exchange rate returns are skewed and have positive excess kurtosis, but the exchange rate return in diusive models does not show these properties. The addition of jump risk remedies this shortcoming. The exchange rate return is skewed and has positive excess kurtosis. The sign of the skewness of the exchange rate return is dependent on the dierence between the skewness of the returns of the domestic and foreign dividend processes. Due to tractability issues it is only possible to derive analytic expressions for the skewness and excess kurtosis under the assumption of constant coecients and independence between the positive and negative jumps. This assumption is rather stringent since Carr and Wu (2007) point out that the skew of exchange rate returns is stochastic. Future research should examine the stochastic skew behavior of exchange rate returns in equilibrium models. 67 References [1] Adler, M., Dumas, B., 1983, International Portfolio Choice and Corporation Finance: A Synthesis, Journal of Finance, 38, 925-984. [2] Bakshi, G. S., Chen, Z., 1997, Equilibrium Valuation of Foreign Exchange Claims, Journal of Finance, 52, 799-826. 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[22] Protter, P., 1990, Stochastic Integration and Dierential Equations, Springer-Verlag, New York, NY. [23] Serrat, A., 2001, A Dynamic Equilibrium Model of International Portfolio Holdings, Econo- metrica, 69, 1467-1489. [24] Solnik, B., 1974, An Equilibrium Model of the International Capital Market, Journal of Economic Theory, 8, 500-524. [25] Uppal, R., 1993, A General Equilibrium Model of International Porfolio Choice, Journal of Finance, 48, 529-553. [26] Westereld, J. M., 1997, An Examination of Foreign Exchange Risk Under Fixed and Floating Rate Regimes, Journal of International Economics, 7, 181 - 200. [27] Zapatero, F., 1995, Equilibrium Asset Prices and Exchange Rates, Journal of Economic Dynamics and Control, 19, 787-811. 69 Appendix Proof Lemma 2.1: Use It^ o to obtain 1 e(t) = 1 e(0) Z t 0 1 e 2 (s) de c (s) + 1 2 Z t 0 2 e 3 (s) <de c (s);de c (s)> + X 0<st 1 e(s) 1 e(s) ; where e c (t) denotes the continuous part of the exchange rate process e(t). In order to compute P 0<st 1 e(s) 1 e(s) , notice that e(t) = exp Z t 0 e (s)K + e (s) + (s)K e (s) 1 2 T e (s) e (s) ds exp Z t 0 T e (s)dW (s) Y 0<st (1 +K + e (s)N + (s))(1 +K e (s)N (s)): Therefore it follows that 1 e(t) 1 e(t) = 1 e(t) 1 (1 +K + e (t)N + (t))(1 +K e (t)N (t)) 1 = 1 e(t) 1 1 +K + e (t) 1 N + (t) + 1 1 +K e (t) 1 N (t) : The last equality is due to the assumption that N + and N do not jump at the same time. Hence, the dynamics of the exchange rate in dierential form are d 1 e(t) = 1 e(t) h e (t) +K + e (t) + (t) +K e (t) (t) + T e (t) e (t) dt T e (t)dW (t) K + e (t) 1 +K + e (t) dN + (t) K e (t) 1 +K e (t) dN (t) : 70 Applying It^ o's product formula it immediately follows that B F (t) e(t) = B F (0) e(0) + Z t 0 B F (s)d 1 e(s) + Z t 0 1 e(s) dB F (s) + B F ; 1 e (t): Since the foreign bond does not have a jump-diusion part, the quadratic variation term is zero. Thus d B F (t) e(t) = B F (t) e(t) e (t) +K + e (t) + (t) +K e (t) (t) + T e (t) e (t) +r F (t) dt T e (t)dW (t) K + e (t) 1 +K + e (t) dN + (t) K e (t) 1 +K e (t) dN (t): The dynamics of S F e can be derived similarly. It^ o's product rule gives S F (t) e(t) = S F (0) e(0) + Z t 0 1 e(s) dS F (s) + Z t 0 S F (s)d 1 e(s) + [S F ; 1 e ](t): The quadratic variation term is given by S F ; 1 e (t) = Z t 0 S F (s) e(s) T S F (s) e (s)ds + X 0<st S F (s) 1 e = Z t 0 S F (s) e(s) T S F (s) e (s)ds + X 0<st S F (s)[K + S F (s)N + (s) +K S F (s)N (s)] 1 e(s) K + e (s) 1 +K + e (s) N + (s) K e (s) 1 +K e (s) N (s) = Z t 0 S F (s) e(s) T S F (s) e (s)ds + X 0<st S F (s) e(s) " K + S F (s)K + e (s) 1 +K + e (s) N + (s) K S F (s)K e (s) 1 +K e (s) N (s) # ; where the last equality follows from the assumption thatN + andN do not jump at the same time. 71 Hence d S F (t) e(t) = S F (t) e(t) S F (t)dt + T S F (t)dW (t) +K + S F (t)dQ + (t) +K S F (t)dQ (t) +[ e (t) +K + e (t) + (t) +K e (t) (t) + T e (t) e (t)]dt T e (t)dW (t) K + e (t) 1 +K + e (t) dN + (t) K e (t) 1 +K e (t) dN (t) T S F (t) e (t)dt K + S F (t) K + e (t) 1 +K + e (t) dN + (t)K S F (t) K e (t) 1 +K e (t) dN (t) 1 e(t) F (t)dt: Collecting terms and simplifying results in d S F (t) e(t) = S F (t) e(t) "" S F (t) e (t) + e (t) e (t) T S F (t) e (t) + K + e (t)(K + e (t)K + S F (t)) 1 +K + e (t) ! + (t) + K e (t)(K e (t)K S F (t)) 1 +K e (t) ! (t) # dt + T S F (t) T e (t) dW (t) + K + S F (t)K + e (t) 1 +K + e (t) dQ + (t) + K S F (t)K e (t) 1 +K e (t) dQ (t) # 1 e(t) F (t)dt: It remains to determine the dynamics of P F (t) e(t) . It^ o's formula gives P F (t) e(t) = P F (0) e(0) + Z t 0 1 e(s) dP F (s) + Z t 0 P F (s)d 1 e(s) + 1 e ;P F (t); where the quadratic variation term is given by 1 e ;P F (t) = X 0<st 1 e(s) P F (s) = X 0<st P F (s) e(s) " K + P F (s)K + e (s) 1 +K + e (s) N + (s) K P F (s)K e (s) 1 +K e (s) N (s) # : 72 Thus d P F (t) e(t) = P F (t) e(t) P F (t) + (t)K + P F (t) (t)K P F (t) dt +K + P F (t)dN + (t) +K P F (t)dN (t) + e (t) +K + e (t) + (t) +K e (t) (t) + T e (t) e (t) dt T e (t)dW (t) K + e (t) 1 +K + e (t) dN + (t) K e (t) 1 +K e (t) dN (t) K + P F (t) K + e (t) 1 +K + e (t) dN + (t)K P F (t) K e (t) 1 +K e (t) dN (t): After simple algebraic manipulations the result follows d P F (t) e(t) = P F (t) e(t) "" P F (t) e (t) +jj e (t)jj 2 + K + e (t)(K + e (t)K + P F (t)) 1 +K + e (t) + (t) + K e (t)(K e (t)K P F (t)) 1 +K e (t) (t) # dt T e (t)dW (t) + K + P F (t)K + e (t) 1 +K + e (t) dQ + (t) + K P F (t)K e (t) 1 +K e (t) dQ (t) # : 2 Proof of Lemma 2.4: Similar to the proof above, this proof consists mainly of an application of It^ o's lemma. Applied to B D (t)e(t) it follows that B D (t)e(t) = B D (0)e(0) + Z t 0 B D (s)de(s) + Z t 0 e(s)dB D (s) + [B D ;e](t): Since the quadratic variation term is zero, it is then obvious that d (B D (t)e(t)) = B D (t)e(t) [ e (t) +r D (t)]dt + T e (t)dW (t) +K + e (t)dQ + (t) +K e (t)dQ (t) : 73 Analogously, S D (t)e(t) = S D (0)e(0) + Z t 0 S D (s)de(s) + Z t 0 e(s)dS D (s) + [e;S D ](t): Notice that the quadratic variation term is given by [e;S D ](t) = Z t 0 S D (s)e(s) T S D (s) e (s)ds + X 0<st e(s)S D (s) = Z t 0 S D (s)e(s) T S D (s) e (s)ds + X 0<st S D (s)e(s)[K + e (s)N + (s) +K e (s)N (s)] [K + S D (s)N + (s) +K S D (s)N (s)] = Z t 0 S D (s)e(s) T S D (s) e (s)ds + X 0<st S D (s)e(s)[K + e (s)K + S D (s)N + (s) +K e (s)K S D (s)N (s)]: Therefore d(S D (t)e(t)) = S D (t)e(t) e (t)dt + T e (t)dW (t) +K + e (t)dQ + (t) +K e (t)dQ (t) + S D (t)dt + T S D (t)dW (t) +K + S D (t)dQ + (t) +K S D (t)dQ (t) + T S D (t) e (t)dt +K + e (t)K + S D (t)dN + (t) +K e (t)K S D (t)dN (t) e(t) D (t)dt = S D (t)e(t) S D (t) + e (t) + T S D (t) e (t) +K + e (t)K + S D (t) + (t) +K e (t)K S D (t) (t) dt + T S D (t) + T e (t) dW (t) + K + S D (t) +K + e (t) +K + S D (t)K + e (t) dQ + (t) + K S D (t) +K e (t) +K S D (t)K e (t) dQ (t) e(t) D (t)dt: Another application of It^ o's lemma yields P D (t)e(t) = P D (0)e(0) + Z t 0 P D (s)de(s) + Z t 0 e(s)dP D (s) + [e;P D ](t); 74 where the quadratic variation term is given by [P D ;e](t) = X 0<st P D (s)e(s) = X 0<st P D (s)e(s)[K + P D (s)K + e (s)N + (s) +K P D (s)K e (s)N (s)]: Consequently d (P D (t)e(t)) P D (t)e(t) = h P D (t) + e (t) +K + P D (t)K + e (t) + (t) +K P D (t)K e (t) (t) i dt + T e (t)dW (t) + K + P D (t) +K + e (t) +K + P D (t)K + e (t) dQ + (t) + K P D (t) +K e (t) +K P D (t)K e (t) dQ (t): 2 Proof Lemma 2.8: Use It^ o to get i (t)S i (t) = Z t 0 i (u) i (u)du + Z t 0 i (u)S i (u) Si (u)r i (u) T Si (u) i (u) +K + Si (u)( ~ + i (u) + (u)) +K Si (u)( ~ i (u) (u)) i du + Z t 0 i (u)S i (u) T Si (u) T i (u) dW (u) + Z t 0 i (u)S i (u) " ~ + i (u) + (u) 1 ! +K + Si (u) ~ + i (u) + (u) # dQ + (u) + Z t 0 i (u)S i (u) " ~ i (u) (u) 1 ! +K Si (u) ~ i (u) (u) # dQ (u) = Z t 0 i (u) i (u)du + Z t 0 i (u)S i (u) T Si (u) T i (u) dW (u) + Z t 0 i (u)S i (u) " ~ + i (u) + (u) 1 ! +K + Si (u) ~ + i (u) + (u) # dQ + (u) + Z t 0 i (u)S i (u) " ~ i (u) (u) 1 ! +K Si (u) ~ i (u) (u) # dQ (u); 75 where the last equality holds due to the denition of the domestic and foreign market price of risk. Evaluating the equation above at time t = T and subtracting this equation from the equation above shows that i (t)S i (t) i (T )S i (T ) = Z T t i (u) i (u)du Z T t i (u)S i (u) T Si (u) T i (u) dW (u) Z T t i (u)S i (u) " ~ + i (u) + (u) 1 ! +K + Si (u) ~ + i (u) + (u) # dQ + (u) Z T t i (u)S i (u) " ~ i (u) (u) 1 ! +K Si (u) ~ i (u) (u) # dQ (u); Observe that the terminal productive asset price S i (T ) is equal to zero because holding the asset provides no further benet to the agents. Furthermore, notice that due to the uniform integrability and predictability of all parameters stochastic integrals are martingales. Taking the conditional expectation with respect toF t leads to i (t)S i (t) = E " Z T t i (u) i (u)dujF t # : 2 Proof Lemma 2.9: The proof only considers the case of the foreign agent. The proof for the domestic agent is very similar. Using It^ o's product rule for jump processes gives F (t)X F (t) = F (0)X F (0) + Z t 0 F (s)dX F (s) + Z t 0 X F (s)d F (s) + [ F ;X F ] (t): 76 At rst compute the cross-variation term. Dene F W (t) = 0 B B B B B B B B B B B B B B @ T e (t) + T S D (t) T S F (t) T e (t) 0 T e (t) 1 C C C C C C C C C C C C C C A : From (2.24) and (2.26) it then follows that [ F ;X F ] (t) = Z t 0 F (s)X F (s)( T F (s); T F (s))(diag( F (s))) 1 diag( F (s)) F W (s) F (s)ds + X 0<st X F (s) F (s): The sum over the product of the jump component of X F and F simplies as follows X 0<st X F (s) F (s) = X 0<st F (s)X F (s)( T F (s); T F (s))(diag( F (s))) 1 diag( F (s)) " ~ + F (s) + (s) 1 ! N + (s) + ~ F (s) (s) 1 ! N (s) # 2 6 6 6 6 6 6 6 6 6 6 4 0 B B B B B B B B B B @ K + S D (s) +K + e (s) +K + S D (s)K + e (s) K + S F (s) K + P D (s) +K + e (s) +K + P D (s)K + e (s) K + P F (s) K + e (s) 1 C C C C C C C C C C A N + (s) + 0 B B B B B B B B B B @ K S D (s) +K e (s) +K S D (s)K e (s) K S F (s) K P D (s) +K e (s) +K P D (s)K e (s) K P F (s) K e (s) 1 C C C C C C C C C C A N (s) 3 7 7 7 7 7 7 7 7 7 7 5 77 = X 0<st F (s)X F (s)( T F (s); T F (s))(diag( F (s))) 1 diag( F (s)) 2 6 6 6 6 6 6 6 6 6 6 4 ~ + F (s) + (s) 1 ! 0 B B B B B B B B B B @ K + S D (s) +K + e (s) +K + S D (s)K + e (s) K + S F (s) K + P D (s) +K + e (s) +K + P D (s)K + e (s) K + P F (s) K + e (s) 1 C C C C C C C C C C A N + (s) + ~ F (s) (s) 1 ! 0 B B B B B B B B B B @ K S D (s) +K e (s) +K S D (s)K e (s) K S F (s) K P D (s) +K e (s) +K P D (s)K e (s) K P F (s) K e (s) 1 C C C C C C C C C C A N (s) 3 7 7 7 7 7 7 7 7 7 7 5 : In order to achieve a clearer presentation, dene and F Q (t) = 0 B B B B B B B B B B @ K + S D (t) +K + e (t) +K + S D (t)K + e (t) K S D (t) +K e (t) +K S D (t)K e (t) K + S F (t) K S F (t) K + P D (t) +K + e (t) +K + P D (t)K + e (t) K P D (t) +K e (t) +K P D (t)K e (t) K + P F (t) K P F (t) K + e (t) K e (t) 1 C C C C C C C C C C A : Notice that F (t) = ( F W (t); F Q (t)). Thus the quadratic variation can be written in dierential form as d[ F ;X F ](t) = F (t)X F (t)( T F (t); T F (t))(diag( F (t))) 1 diag( F (t)) F W (t) F (t)dt + F (t)X F (t)( T F (t); T F (t))(diag( F (t))) 1 diag( F (t)) F Q (t) diag ~ + F (t) + (t) 1; ~ F (t) (t) 1 ! 0 B @ dN + (t) dN (t) 1 C A: 78 Then d F (t)X F (t) = F (t)X F (t)( T F (t); T F (t))(diag( F (t))) 1 diag( F (t)) 2 6 4( F (t)r F (t)1)dt + F (t) 0 B @ dW (t) dQ(t) 1 C A 3 7 5 + F (t)X F (t)r F (t)dt F (t) (c FD (t)e(t) +c FF (t))dt +X F (t) F (t) " r F (t)dt T F (t)dW (t) +diag ~ + F (t) + (t) 1; ~ F (t) (t) 1 ! dQ(t) # F (t)X F (t)( T F (t); T F (t))(diag( F (t))) 1 diag( F (t)) F W (t) F (t)dt + F (t)X F (t)( T F (t); T F (t))(diag( F (t))) 1 diag( F (t)) F Q (t) diag ~ + F (t) + (t) 1; ~ F (t) (t) 1 ! dQ(t) + F (t)X F (t)( T F (t); T F (t))(diag( F (t))) 1 diag( F (t)) F Q (t) 0 B @ ~ + F (t) + (t) ~ F (t) (t) 1 C Adt = F (t)X F (t)( T F (t); T F (t)) 2 6 6 6 6 4 F (t)r F 1 F 0 B B B B @ F (t) ~ + F (t) + + (t) ~ F (t) + (t) 1 C C C C A 3 7 7 7 7 5 dt + F (t)X F (t)( T F (t); T F (t))(diag( F (t))) 1 diag( F (t)) " F (t)diag 1; 1; 1; ~ + F (t) + (t) ; ~ F (t) (t) !# 0 B @ dW (t) dQ(t) 1 C A F (t)X F (t) 0 B B B B @ F (t) 1 ~ + F (t) + (t) 1 ~ F (t) (t) 1 C C C C A T 0 B @ dW (t) dQ(t) 1 C A F (t)C F (t)dt; 79 Using the denition of ( F (t); ~ + F (t) + + (t); ~ F (t) + (t)) T , given in (2.25), the result follows. 2 Proof Lemma 3.2: The principal idea of this proof is as follows: Dierentiating both sides of (3.1) and (3.2) and comparing coecients gives the restrictions postulated in the lemma. In order to dierentiate c DD ;c DF ;c FD and c FF in (3.1) and (3.2) the representation given in (2.36), (2.37), (2.38) and (2.39) will be used. This requires the dynamics of e(t) D (t) and 1 F (t)e(t) . In order to derive these nd the dynamics of 1 i(t) at rst. Applying It^ o to 1 i(t) gives 1 i (t) = 1 i (0) Z t 0 1 2 i (s) d c i (s) + 1 2 Z t 0 2 3 i (s) <d c i (s);d c i (s)> + X 0<st 1 i (s) 1 i (s) ; where c i (s) denotes the continuous part of i (s). It can be shown that i (t) = exp Z t 0 h r i (s) + + (s) ~ i (s) + (s) ~ i (s) i ds exp Z t 0 T i (s)dW (s) 1 2 Z t 0 T i (s) i (s)ds Y 0<st 1 + ~ + i (s) + (s) 1 ! N + (s) ! 1 + ~ i (s) (s) 1 ! N (s) ! : Then 1 i (t) 1 i (t) = 1 i (t) 0 @ 1 1 + ~ + i (t) + (t) 1 N + (t) + ~ i (t) (t) 1 N (t) 1 1 A = 1 i (t) " + (t) ~ + i (t) 1 ! N + (t) + (t) ~ i (t) 1 ! N (t) # ; where the second equality holds true due to the assumption that the processes N + andN do not jump at the same time. 80 Hence d 1 i (t) = 1 i (t) h r i (t) + (t) + ~ + i (t) (t) + ~ i (t) + T i (t) i (t) dt + T i (t)dW (t) + + (t) ~ + i (t) 1 ! dN + (t) + (t) ~ i (t) 1 ! dN (t) # : The dynamics of e(t) D (t) can then be determined by using It^ o's product rule e(t) D (t) = e(0) D (0) + Z t 0 e(s)d 1 D (s) + Z t 0 1 D (s) de(s) + e; 1 D (t): The quadratic variation term is given by e; 1 D (t) = Z t 0 e(s) D (s) T e (s) D (s)ds + X 0<st e(s) 1 D (s) = Z t 0 e(s) D (s) T e (s) D (s)ds + X 0<st e(s) D (s) " K + e (s) + (s) ~ D (s) 1 ! N + (s) +K e (s) (s) ~ D (s) 1 ! N (s) # : Consequently d e(t) D (t) = e(t) D (t) " h r D (t) + (t) + ~ + D (t) (t) + ~ D (t) + T D (t) D (t) + e (t)K + e (t) + (t)K e (t) (t) + T e (t) D (t) dt +[ T D (t) + T e (t)]dW (t) + " + (t) ~ + D (t) 1 ! +K + e (t) +K + e (t) + (t) ~ + D (t) 1 !# dN + (t) + " (t) ~ D (t) 1 ! +K e (t) +K e (t) (t) ~ D (t) 1 !# dN (t) # : 81 The dynamics of 1 F (t)e(t) can also be determined by It^ o's product rule. Notice that the dynamics of 1 e(t) were determined in the proof of lemma 1.1. Thus 1 F (t)e(t) = 1 F (0)e(0) + Z t 0 1 F (s) d 1 e(s) + Z t 0 1 e(s) d 1 F (s) + 1 e ; 1 F (t); where the quadratic variation is given by 1 e ; 1 F (t) = Z t 0 1 F (s)e(s) T e (s) F (s)ds + X 0<st 1 e(s) 1 F (s) = Z t 0 1 F (s)e(s) T e (s) F (s)ds + X 0<st 1 F (s)e(s) " + (s) ~ + F (s) 1 ! K + e (s) 1 +K + e (s) N + (s) + (s) ~ F (s) 1 ! K e (s) 1 +K e (s) N (s) # : Hence the dynamics of 1 F (t)e(t) are given by d 1 F (t)e(t) = 1 F (t)e(t) " e (t) +K + e (t) + (t) +K e (t) (t) + T e (t) e (t) +r F (t) + (t) + ~ + F (t) (t) + ~ F (t) + T F (t) F (t) T e (t) F (t)]dt +[ T e (t) + T F (t)]dW (t) + " K + e (t) 1 +K + e (t) + + (t) ~ + F (t) 1 ! + (t) ~ + F (t) 1 ! K + e (t) 1 +K + e (t) # dN + (t) + " K e (t) 1 +K e (t) + (t) ~ F (t) 1 ! (t) ~ F (t) 1 ! K e (t) 1 +K e (t) # dN (t) # : It is now possible to dierentiate (3.1) and (3.2). Using (2.36) and (2.38) shows that (3.1) is equivalent to e kt e k(Tt) D (t) = a DD kX D (0) 1 D (t) +a FD kX F (0) 1 F (t)e(t) : 82 Dierentiating both sides results in k e kt e k(Tt) D (t)dt + e kt e k(Tt) d D (t) = a DD kX D (0)d 1 D (t) +a FD kX F (0)d 1 F (t)e(t) : It follows that D (t) D (t)dt + T D (t)dW (t) +K + D (t)dQ + (t) +K D (t)dQ (t) = c DD (t) " r D (t) + (t) + ~ + D (t) (t) + ~ D (t) + T D (t) D (t) dt + T D (t)dW (t) + + (t) ~ + D (t) 1 ! dN + (t) + (t) ~ D (t) 1 ! dN (t) # +c FD (t) " e (t) +K + e (t) + (t) +K e (t) (t) + T e (t) e (t) +r F (t) + (t) + ~ + F (t) (t) + ~ F (t) + T F (t) F (t) T e (t) F (t)]dt +[ T e (t) + T F (t)]dW (t) + " K + e (t) 1 +K + e (t) + + (t) ~ + F (t) 1 ! + (t) ~ + F (t) 1 ! K + e (t) 1 +K + e (t) # dN + (t) + " K e (t) 1 +K e (t) + (t) ~ F (t) 1 ! (t) ~ F (t) 1 ! K e (t) 1 +K e (t) # dN (t) # k D (t)dt: Simplifying, using the domestic goods market clearing condition, and comparing coecients proves equations (3.9), (3.11), (3.13) and (3.15). Equations (3.10), (3.12), (3.14) and (3.16) can then be obtained by dierentiating (3.2). 2 83

Abstract (if available)

Abstract This dissertation examines asset prices, the exchange rate and its higher moment properties in an economy subject to both diffusive and jump risk. The model used in this dissertation is an extension of Zapatero's (1995) two-good two-country intertemporal international equilibrium model for two logarithmic representative agents. Uncertainty enters the economy through a three dimensional Brownian motion and two Poisson processes representing positive and negative jumps in the dividend process of the goods. Individual financial markets are incomplete, but all claims can be hedged completely in the international financial market. From Zapatero's (1995) model it is known that the exchange rate increases with the interest rate differential and the diffusion parameter of the domestic equity market and decreases with the covariance between the domestic and foreign equity market. This dissertation shows that in a jump-diffusion setting the expected equilibrium exchange rate change additionally increases (decreases), if the foreign positive and negative jump sizes are bigger (smaller) in absolute value than their domestic equivalents. This dissertation thereby provides a new explanation for the interest rate parity puzzle. It is a well-known empirical fact that exchange rates are skewed and have excess kurtosis. In contrast to traditional equilibrium models subject to diffusive risk the exchange rate return exhibits these two properties in the jump-diffusion setting. The sign of the skewness of the exchange rate return is dependent on the difference between the skewness of the returns of the domestic and foreign dividend processes.

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University of Southern California Dissertations and Theses

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Essays on interest rate determination in open economies

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Creator Knape, Mathias (author)

Core Title A general equilibrium model for exchange rates and asset prices in an economy subject to jump-diffusion uncertainty

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Applied Mathematics

Publication Date 08/05/2011

Defense Date 06/19/2009

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

Tag exchange rate,incomplete markets,intertemporal international equilibrium model,jump-diffusion,logarithmic agents,OAI-PMH Harvest,skewness and kurtosis of exchange rate returns,uncovered interest rate parity puzzle

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Mikulevicius, Remigijus (committee chair), Zapatero, Fernando (committee chair), Ma, Jin (committee member)

Creator Email knape@usc.edu,mathias.knape@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m2486

Unique identifier UC1213663

Identifier etd-Knape-3040 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-180414 (legacy record id),usctheses-m2486 (legacy record id)

Legacy Identifier etd-Knape-3040.pdf

Dmrecord 180414

Document Type Dissertation

Rights Knape, Mathias

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

Repository Email cisadmin@lib.usc.edu