Numerical Methods for the Solution of the HJB Equations Arising in European and American Option Pricing with Proportional Transaction Costs

Transcription

1 Numerical Metods for te Solution of te HJB Equations Arising in European and American Option Pricing wit Proportional Transaction Costs Wen Li Tis tesis is presented for te degree of Doctor of Pilosopy of Te University of Western Australia Scool of Matematics and Statistics 2010

2 Abstract Tis tesis is concerned wit te investigation of numerical metods for te solution of te Hamilton-Jacobi-Bellman (HJB) equations arising in European and American option pricing wit proportional transaction costs. We first consider te problem of computing reservation purcase and write prices of a European option in te model proposed by Davis, Panas and Zaripopoulou [19]. It as been sown [19] tat computing te reservation purcase and write prices of a European option involves solving tree different fully nonlinear HJB equations. In tis tesis, we propose a penalty approac combined wit a finite difference sceme to solve te HJB equations. We first approximate eac of te HJB equations by a quasi-linear second order partial differential equation containing two linear penalty terms wit penalty parameters. We ten develop a numerical sceme based on te finite differencing in bot space and time for solving te penalized equation. We prove tat tere exists a unique viscosity solution to te penalized equation and te viscosity solution to te penalized equation converges to tat of te original HJB equation as te penalty parameters tend to infinity. We also prove tat te solution of te finite difference sceme converges to te viscosity solution of te penalized equation. Numerical results are given to demonstrate te effectiveness of te proposed metod. We extend te penalty approac combined wit a finite difference sceme to te HJB equations in te American option pricing model proposed by Davis and Zarpopoulou [20]. Numerical experiments are presented to illustrate te teoretical findings. i

3 Acknowledgements First of all, I would like to sincerely tank my primary supervisor, Professor Song Wang, for is elpful guidance. I am deeply indebted to im for is patience and understanding trougout my P.D study. I am grateful to my oter supervisor, Adjunct Professor CJ Go, for is advice in researc and elpful suggestions. I would like to tank my parents. Witout teir support and encouragement, I would ave never been able to finis my Master and P.D study. Tis tesis is dedicated to my parents. I also tank my usband Leo den Hollander for is love and patience. Finally, I acknowledge te financial support of an Australian Postgraduate Award. ii

4 Contents Abstract Acknowledgements List of Tables List of Figures i ii iv v 1 Introduction Options and Option Prices Option Pricing Approaces Researc Metods and Outline European Option Pricing wit Proportional Transaction Costs Model of European Option Pricing Te Hamilton-Jacobi-Bellman Equations Viscosity Solution Introduction of Viscosity Solution Existence and Uniqueness of te Viscosity Solution of (2.2.4) (2.2.8) Markov Cain Approximation Conclusion Te Penalty Approac and Its Analysis Introduction Te Formulation Existence and Uniqueness of Viscosity Solution of te Penalized Equations Convergence iii

5 3.5 Conclusion Numerical Solution of Penalized Equations Te Discretization Sceme Stability Convergence Numerical Results Conclusion Pricing American Options wit Proportional Transaction Costs American Option Pricing Model and It s Analysis American Option Pricing via Utility Maximization Te Hamilton-Jacobi-Bellman Equations for (5.1.3) and (5.1.5) Existence and Uniqueness of te Solutions of (5.1.12) (5.1.13) Te Penalty Metod for Equation (5.1.12) Numerical Solution of Equations (5.2.1) (5.2.2) Te Discretization Sceme Existence and Uniqueness of te Solution of (5.3.3) (5.3.5) Numerical Results Conclusion Conclusions Summary Future Researc Appendix 83 Appendix A: Proof of (4.3.8) Appendix B: Proof of Teorem Appendix C: Numerical results of value functions and te computed reservation prices of a European call option Appendix D: Numerical results of value functions and te computed reservation prices of an American call option and its European counterpart 93 Bibliograpy 98 iv

6 List of Tables Computed reservation prices of te European call option for γ = Computed reservation prices of te European call option for α 0 = Computed reservation purcase price of te American call option for γ = 0.1 and te reservation purcase price of its European counterpart Computed reservation purcase price of te American call option for α 0 = 0 and te reservation purcase price of its European counterpart Computed reservation purcase price of te American put option for α 0 = 0 and te reservation purcase price of its European counterpart 78 v

7 List of Figures Computed reservation purcase price of te European call option for γ = Computed reservation write price of te European call option for γ = Computed reservation purcase price of te European call option for α 0 = Computed reservation write price of te European call option for α 0 = Computed reservation purcase price of te American call option for γ = 0.1 and te reservation purcase price of its European counterpart Computed reservation purcase price of te American call option for α 0 = 0 and te reservation purcase price of its European counterpart Computed reservation purcase price of te American put option for α 0 = Computed reservation purcase price of te European put option for α 0 = vi

8 Capter 1 Introduction Pricing options is one of te most important problems in financial engineering. For over tree decades, academic researcers in finance, economics, matematics and management sciences ave engaged in te study of option pricing problem. Many option pricing approaces ave been developed. One of te major metods is te utility based option pricing approac. In te tesis, we will investigate numerical metods for te solution of te Hamilton-Jacobi-Bellman (HJB) equations arising from tis pricing approac. In te following sections of tis capter, we will first introduce te basic concepts of option pricing teory, wic will be used trougout te tesis. We ten review te literature relevant to option pricing approac. After tat, we will introduce te researc metods and te arrangement of tis tesis. 1.1 Options and Option Prices In finance, an option is a contract in wic one party (called te writer) sells to anoter party (called te older) te rigt, but not te obligation, to buy or sell a specified amount of an underlying asset suc as a stock, commodity or currency, at a fixed price (called te exercise or strike price) on or before a given date (called te expiry date). If te contract gives te older te rigt to buy te underlying asset, te option is called a call option. If te contract gives te older te rigt to sell te underlying asset, te option is called a put option. Te act of making tis transaction is referred to as exercising te option. 1

9 Tere are two basic types of options: European options and American options. A European option gives te older te rigt to buy (for a call option) or to sell (for a put option) te underlying asset at te strike price on te expiry date. An American option is an option wic in contrast to te European option can be exercised at any time before or on te expiry date. In order to acquire an option, an investor need to pay an option s price to te writer. Te value of an option depends on te price of te underlying asset, S(t), and te time t. It is well known (see, for example, [45]) tat in te market witout transaction costs, te value of an option at te expiry date T is te payoff of te option, i.e., te value of a call option at T, denoted by C, is C = (S(T) K) + = max(0,s(t) K) and te value of a put option, denoted by P, is P = (K S(T)) + = max(0,k S(T)), were S(T) is te price of te underlying asset at te expiry date and K is te strike price. However, at any time before te expiry date, te value of an option is different from te payoff. Te option pricing problem is to determine ow muc an option is wort at some earlier time t. 1.2 Option Pricing Approaces Tere are many approaces to te problem of option pricing (see a review article [10]). Te most basic approac is te Black and Scoles model. In te early 1970 s, Black and Scoles [7] used a no-arbitrage 1 valuation metod to price a European option on a stock under te assumptions: (a) te interest rate is a constant; (b) te stock price follows a geometric Brownian motion wit constant drift and volatility; 2 (c) no dividends are pay on te underlying stock; 1 Arbitrage is an opportunity to make a profit witout any risk of loss. 2 Tis will be described in Capter 2. 2

10 (d) te market is time-continuous; (e) sort selling is permitted; (f) tere are no transaction costs in buying or selling te stock. Using Ito s lemma [5, 28] and te risk-less edging principle, Black and Scoles [7] derived a partial differential equation (PDE) tat must be satisfied by te value of a European option dependent on te stock price. Tey solved te PDE and obtained a closed form solution for te value of a European call option. Te Black-Scoles model is a very effective metod for pricing options. However, in te presence of transaction costs on trading in te bond and/or stock, te Black-Scoles teory is no longer applicable. In te current literature, tere are four main approaces to te problem of option pricing wit transaction costs: imperfect replication approac (also called Delta edging approac)[22, 27, 30, 40], perfect replication approac [9, 33], super replication approac [4, 8, 21] and utility based option pricing approac [16-20, 26, 34, 46, 47]. In [16], te advantages and disadvantages of tese four approaces ave been carefully analyzed and te utility based option pricing approac is being identified to be te most reasonable and applicable pricing model. In tis tesis, we will adopt te utility based option pricing approac to te pricing of European and American options in te market wit proportional transaction costs and examine computational aspects of tis approac. Te idea of te utility based option pricing approac is to consider te optimal portfolio 3 selection problem of an investor wose objective is to maximize is/er expected utility of terminal wealt. Tis option pricing model values an option from te perspective of a singe agent and is not a model of market equilibrium. Hence, instead of a unique price te utility based option pricing model leads to two price bounds wic are called reservation purcase price and reservation write price. Te reservation purcase (respectively reservation write) price of an option, i.e., te igest (respectively lowest) price at wic te investor is willing to buy (respectively sell) te option, is te price at wic te investor as te same maximum expected utility weter e/se buys (respectively sells) te option or not. 3 A portfolio is a collection of investments eld by an investor. Tese investments often include stocks and bonds. 3

11 Te utility based option pricing approac was initiated by Hodges and Neuberger [26]. In teir work, Hodges and Neuberger [26] applied tis approac to price European options on a stock wen, in addition to te Black-Scoles assumptions (a) (e), tere are proportional transaction costs. Under an exponential utility function, tey calculated numerically te reservation prices of a European call option using a binomial lattice, witout proving te stability and convergence of te numerical metod. Davis, Panas, and Zaripopoulou [19] furter developed tis approac by presenting a rigorous matematical analysis. First, tey formulated a singular stocastic control problem for te Hodges and Neuberger model and sowed tat computing te reservation price of a European option involves solving two different stocastic control problems. Ten tey proved tat te value functions of te stocastic control problems are te unique viscosity solutions 4 of a fully nonlinear HJB equation wit different boundary conditions. Furtermore, tey proposed a discretization sceme based on te Markov cain approximation approac for solving te HJB equations and proved te convergence of te sceme. Tey also presented computational results for te reservation write price of a European call option under te assumption of exponential utility function. Te Markov cain approximation sceme suggested by Davis, Panas, and Zaripopoulou [19] as been widely used for computing te reservation prices of European options wit proportional transaction costs (see, for example, [17, 18, 46]). In [17, 18], Damgaard used tis sceme to calculate te reservation purcase and write prices of a European call option wit a HARA 5 utility function, i.e., te utility function is of te form U(W) = 1 γ γ ( aw 1 γ +b)γ, were a > 0,b R are constants and γ (, 1) \ {0} is a constant risk aversion parameter. Zakamouline [46] also used tis sceme, wit a minor modification, to compute reservation prices of a European option for an agent wit an exponential utility function. Oter studies concerning te computation of reservation prices of European options include Clewlow and Hodges [11]. Tey extended te work of Hodges and Neuberger [26] by presenting an improved computational procedure for calculating reservation prices of European options. 4 See Capter 2 for te definition of viscosity solutions. 5 HARA stands for Hyperbolic Absolute Risk Aversion. 4

12 Davis and Zarpopoulou [20] were te first to apply te utility based option pricing approac to te problem of American option pricing wit proportional transaction costs. Tey sowed tat calculating te reservation purcase price of an American option involves solving a combination of a singular stocastic control and an optimal stopping problems. Tey also proved tat te value function of te singular stocastic control problem wit an optimal stopping is te unique viscosity solution of a fully nonlinear HJB equation. Tis means tat te solution can be computed by some sopisticated discretization metods. However, tey did not suggest any discretization scemes for solving te HJB equation. Damgaard [18] and Zakamouline [47] extended te work of Davis and Zarpopoulou [20] by computing te reservation purcase prices of American options using a Markov cain approximation sceme wic is similar to tat in [19]. Te utility based option pricing approac is now considered to be an economically natural metod for valuing options wen tere are proportional transaction costs (see [16, 25, 35, 50]). However, tere is an apparent drawback of tis metod: te closed form solutions of te reservation prices are not available. Terefore, efficient and accurate numerical metods are necessary for approximating te solutions. In te literature, almost all te papers concerning te utility based option pricing approac involved numerical approximation of option prices. However, all tese papers used a binomial sceme or te simplest Markov cain approximation sceme. Bot of tese numerical scemes are explicit in time and tus computationally very expensive. Te purpose of our study is to develop a new, more efficient and accurate numerical metod for calculating reservation prices of European and American options. 1.3 Researc Metods and Outline It is known [19, 17] tat computing te reservation purcase price and write price of a European option involves solving tree difference HJB equations. Davis, Panas and Zaripopoulou [19] proposed a Markov cain approximation sceme for te equations. In tis tesis, we will present a new metod, a penalty approac combined wit a finite difference sceme, for solving tese HJB equations. First, we propose 5

13 a penalty approac to te HJB equations. Ten, we will develop a finite difference discretization sceme for solving te penalized equations. Te penalty metod and te finite difference sceme will be also used to solve te American option pricing problem. Te tesis is organized as follows. In Capter 2, we present a utility based European option pricing model using te framework of Davis, Panas, and Zaripopoulou [19]. We first define te reservation purcase price and write price of a European call option based on te utility maximization teory. Since te definitions of reservation option prices involve te value functions of tree different stocastic control problems, we ten present te HJB equations governing tese value functions. Furtermore, we introduce te definition of viscosity solution and sow, along te lines of Davis, Panas, and Zaripopoulou [19], tat te value functions are te unique viscosity solutions of teir associated HJB equations. In addition, we will introduce te Markov cain approximation approac, wic was used for solving te HJB equations by Davis, Panas, and Zaripopoulou [19]. In Capter 3, we present a penalty approac to te HJB equations. We first approximate eac of te HJB equations by a quasi-linear 2nd-order partial differential equation (PDE) containing two linear penalty terms wit penalty parameters λ 1 and λ 2. Ten, we sow tat tere exists a unique viscosity solution to te penalized PDE. Finally, we prove tat te viscosity solution to te penalized PDE converges to tat of te corresponding original HJB equation as te penalty parameters λ 1 and λ 2 approac infinity. In Capter 4, we develop a new numerical metod based on finite difference scemes in bot space and time for solving te penalized equation. We sow tat te sceme is stable and te solution of te sceme converges to te viscosity solution of te penalized equation. Numerical results are also given to demonstrate te effectiveness and accuracy of te metod. In Capter 5, we consider te problem of computing reservation prices of an American option. In Section 5.1, we will first present an American option pricing model using te utility based option pricing approac and matematically analyze te model. We sow tat computing te reservation purcase price of an American option involves solving two different fully nonlinear HJB equations. In Section 5.2 6

14 and Section 5.3 we will propose a penalty metod combined wit a finite difference sceme to solve tese HJB equations and present some numerical results of te reservation purcase prices of American call and put options. Finally, in Section 5.4, we summarize main conclusions of tis capter. In Capter 6, we conclude te tesis and suggest directions for furter researc. 7

15 Capter 2 European Option Pricing wit Proportional Transaction Costs Te purpose of tis capter is to introduce and analyze te utility based European option pricing approac. 2.1 Model of European Option Pricing In tis section, we will present a utility based option pricing model for valuing a European call option in te market wit proportional transaction costs. Te construction of te model follows tose developed in [19] and [18]. Consider a time interval [0,T] and a continuous-time economy wit a risky stock and a risk-less bond. Assume tat te price of te stock at time u, S u, evolves according to te following geometric Brownian motion: ds u S u = µdu + σdz u, (2.1.1) were µ and σ are respectively te constant drift rate and volatility, Z u wic represents te single source of uncertainty in te market, is a standard Brownian motion on a filtered probability space (Ω, F, (F u ) 0 u T,P). 1 Te price of te bond, B(u), evolves according to te ordinary differential equation were r 0 is a constant interest rate. db(u) = rb(u)du, 1 (F u ) 0 u T is te augmentation under P of F Z u = {Z s 0 s u} for 0 u T. 8

16 We suppose tat te investors in te economy must pay transaction costs wen buying or selling te stock and te transaction costs are proportional to te amount transferred from te stock to te bond. Let β u denote te amount te investors old in te bond and α u te number of sares of te stock eld by te investors at time u [0,T], ten te evolution equations for β u and α u are dβ u = rβ u du (1 + θ)s u dl u + (1 θ)s u dm u, (2.1.2) dα u = dl u dm u, (2.1.3) were θ [0, 1) is te fraction of te traded amount in te stock, L u and M u are respectively te cumulative number of sares bougt and sold up to u. Let c(α u,s u ) denote te liquidated cas value of te stock and W u te investor s wealt at time u, we ave c(α u,s u ) = α u S u θs u α u, W u = β u + S u (α u θ α u ). We now describe te utility based option pricing approac. Te basic idea of te utility based option pricing approac is to consider te optimal portfolio problem of an investor wose objective is to coose an admissible trading strategy to maximize is utility of terminal wealt. To use tis approac to value reservation purcase price and write price of a European call option, we need to define tree different utility maximization problems: (i) Utility maximization problem for an investor witout an option. (ii) Utility maximization problem for a buyer of a European call option. (iii) Utility maximization problem for a writer of a European call option. Before introducing te utility maximization problems, we first define te terminal wealt for different investors. 1. Te terminal wealt of an investor witout an option. Suppose tat at te terminal time T, an investor olds β T dollars in te bond and α T sares of te stock wose price is S T, ten te investor s terminal wealt is β T + S T (α T θ α T ). 9

17 2. Te terminal wealt of an investor buying a European call option. We consider an investor wo purcases a European call option written on te stock wit strike price K and expiry date T. Assume tat te European option is cassettling, i.e., te option writer will pay te option older an amount of cas equal to te payoff of te option, (S T K) +, at te expiry date wen te option is exercised, were S T is te stock price at te expiry date. Suppose tat at time T, an option older as β T dollars in te bond and α T sares of te stock, ten is terminal wealt after exercising te option is β T + S T (α T θ α T ) + (S T K) Te terminal wealt of an investor selling a European call option. Consider an investor wo sells a cas-settling European call option written on te stock wit strike price K and expiry date T. Suppose tat at time T, te option writer old β T dollars in te bond and α T sares of te stock at price S T, ten is terminal wealt after e as met is liabilities is β T + S T (α T θ α T ) (S T K) +. Here we sall point out tat in our model, te investor s wealt is required to be positive at any time u [0,T], i.e., tere is no bankruptcy in our economy. Based on te above definitions, we now define te utility maximization problems. Problem Utility maximization for an investor witout an option Suppose tat an investor trades only in te underlying stock and te bond. At time t [0,T], te investor olds β dollars in te bond and α sares of te stock wose price is S. Te objective of te investor is to maximize te expected utility of terminal wealt over all admissible strategies, i.e., V 0 (t,α,β,s) = sup E t [U(β T + S T (α T θ α T ))] (0 t T), (2.1.4) Λ 0 (t,α,β,s) were V 0 (t,α,β,s) denotes te investor s time t maximum expected utility of terminal wealt, also known as value function. E t denotes te expectation operator conditional on te time t information (α,β,s). U( ) is te utility function, to be defined later. β T + S T (α T θ α T ) is te investor s wealt at te terminal time T and Λ 0 (t,α,β,s) is te set of admissible strategies available to te investor, defined 10

18 as te set of rigt-continuous, measurable, F-adapted, increasing processes, L u and M u (t u T), suc tat te following conditions are satisfied: 1. Te associated processes (α Lu,Mu,β Lu,Mu,S u ) satisfy (2.1.1) to (2.1.3) in [t,t] wit te initial state (t,α,β,s). 2. β Lu,Mu + S u α Lu,Mu S u θ α Lu,Mu > 0, u [t,t]. Remark Condition 2 is an no-bankruptcy restriction. It ensures tat te investor s wealt is positive at any time u [t,t]. Tis will be used in te proofs of relevant teorems to be presented later in te tesis. Problem Utility maximization for te buyer of a European call option Assume tat te investor trades in te market for te stock and te bond, and in addition, purcases a cas-settling European call option written on te stock wit strike price K and expiry date T. Ten, te investor s time t expected utility of terminal wealt is to be maximized over te set of feasible strategies, i.e., V b (t,α,β,s) = sup E t [U(β T + S T (α T θ α T ) + (S T K) + )] Λ b (t,α,β,s) were Λ b (t,α,β,s) = Λ 0 (t,α,β,s). (0 t T), (2.1.5) Problem Utility maximization for te writer of a European call option If te investor trades in te market for te stock and te bond, and in addition, sells a cas-settling European call option written on te stock wit strike price K and expiry date T. Ten, te investor wises to maximize te expected utility of terminal wealt over te set of feasible strategies, i.e., V w (t,α,β,s) = sup E t [U(β T + S T (α T θ α T ) (S T K) + )] Λ w (t,α,β,s) (0 t T), (2.1.6) were Λ w (t,α,β,s) denotes te writer s admissible strategies wic are defined as te set of rigt-continuous, measurable, F-adapted, increasing processes, L u and M u (t u T), suc tat te following conditions are satisfied: 1. Te associated processes (α Lu,Mu,β Lu,Mu,S u ) satisfy (2.1.1) to (2.1.3) in [t,t] wit te initial state (t,α,β,s). 11

19 ( ) 2. β Lu,Mu + S u α L u,m u 1 1 θ Su θ α L u,m u 1 1 θ > 0, u [t,t]. Remark Condition 2 in Problem is also an no-bankruptcy restriction. In [37], Sreve, Soner and Critanic sowed tat, in order to keep te writer s wealt positive, it is imperative to keep at least one sare of te stock at all trading times. Note tat in our model te option is cas-settling. Damgarrd [17] ave sowed tat in tis case te ceapest strategy to keep te writer s terminal wealt positive is buying 1 1 θ sares of te stock and old it to time of expiry of te option. Condition 2 above allows te writer to pursue te ceapest strategy at any time (see [17] for details). Using te above definitions, we define te reservation purcase and write prices of a European call option as follows. Definition (reservation purcase price) Consider an investor wo starts trading at time t = 0 wit olding β dollars in te bond and α sares of te stock of price S. Assume tat te investor only can buy te option at te initial time t = 0. Ten te investor s reservation purcase price of a European call option is defined as te amount, P b, suc tat V b (0,α,β P b,s) = V 0 (0,α,β,S). Definition (reservation write price) Consider an investor wo starts trading at time t = 0 wit olding β dollars in te bond and α sares of te stock wose price is S. Assume tat te investor can only sell te option at te initial time t = 0. Ten te investor s reservation write price of a European call options is defined as te amount, P w, suc tat V w (0,α,β + P w,s) = V 0 (0,α,β,S). We comment tat, from te above definitions, te reservation purcase and write prices are defined as, respectively, te amounts required to provide te same maximum expected utilities as not buying and selling te option. From te above definitions we also see tat computing reservation purcase or write price of an option involves two of te tree value functions defined in (2.1.4) (2.1.6). By te dynamic programming principle, Davis, Panas and Zaripopoulou [19] ave sowed 12

20 tat te value functions are viscosity solutions of a set of HJB equations given later. To guarantee tat te solution is unique, wic means we can find te value functions by solving te respective HJB equations, te utility function U : [0, ) R is required to satisfy te sublinear growt condition defined as follows. Definition A function U( ) is sublinear growt if tere exist constants κ > 0 and λ (0, 1) suc tat for all x 0,U(x) κ(1 + x) λ. 2 We comment tat, from te above definition, if te utility function U( ) is bounded in [0, ), ten it is sublinear growt. Following Davis, Panas and Zaripopoulou [19], in tis tesis we assume tat te utility function U : [0, ) R is te exponential function of te following form: U(W) = 1 exp( γw), (2.1.7) were γ > 0 is a constant risk aversion parameter. Now our goal is to find te value functions (2.1.4) (2.1.6). As stated above, it as been proved in [19] tat te value functions are te unique viscosity solutions of a set of HJB equations. So our task is to solve tese HJB equations. In te following sections of tis capter, we will describe te HJB equations satisfied by te value functions (2.1.4) (2.1.6) and sow, along te lines of Davis, Panas and Zaripopoulou [19], tat te value functions are te unique viscosity solutions of teir associated HJB equations. 2.2 Te Hamilton-Jacobi-Bellman Equations In tis section, we will present an HJB equation wit a set of appropriate terminal conditions governing te value functions in (2.1.4) (2.1.6). A detailed deduction of tese equations can be found in [19]. In wat follows, we only state te results witout proofs. 2 Te sublinear growt condition will be used in Lemma and Lemma

21 Let L k,k = 1, 2, 3 be te linear differential operators defined respectively by L 1 ( = t + rβ β + µs S + 1 ) 2 σ2 S 2 2, (2.2.1) S 2 L 2 = + (1 + θ)s α β, (2.2.2) L 3 = (1 θ)s α β. (2.2.3) Te HJB equation for te value functions V i, i = 0,b,w, is min {L 1 V, L 2 V, L 3 V } = 0, (t,α,β,s) [0,T) Ω i (2.2.4) wit te following terminal conditions, respectively, V (T,α,β,S) = V i (T,α,β,S), (α,β,s) Ω i (2.2.5) for i = 0,b,w, were V 0 (T,α,β,S) = U(β + S(α θ α )), (2.2.6) V b (T,α,β,S) = U(β + S(α θ α ) + (S K) + ), (2.2.7) V w (T,α,β,S) = U(β + S(α θ α ) (S K) + ), (2.2.8) and Ω 0 = Ω b = {(α,β,s) R R R + : β + Sα Sθ α > 0}, (2.2.9) ( Ω w = {(α,β,s) R R R + : β+s α 1 ) Sθ 1 θ α 1 1 θ > 0}. (2.2.10) Note tat equation (2.2.4) is nonlinear. It as in general no classical solution. Terefore, te notion of solution to te HJB equation (2.2.4) must be relaxed. Using te notion of viscosity solution, Davis, Panas, and Zaripopoulou [19] sowed tat te value functions V i, i = 0,b,w, are unique viscosity solutions to te HJB equation (2.2.4) wit satisfying, respectively, te terminal conditions (2.2.5) (2.2.8). We will discuss tis in te next section. We comment tat (2.2.4) is equivalent to te following linear complementarity problem: L k V 0 for k = 1, 2, 3 and L 1 V L 2 V L 2 V = 0. (2.2.11) 14

22 2.3 Viscosity Solution In tis section, we present some backgrounds on te teory of viscosity solution and sow tat te value functions defined by (2.1.4) (2.1.6) are unique constrained viscosity solutions of teir respective HJB equations (2.2.4) (2.2.8) Introduction of Viscosity Solution Te concept of viscosity solution was first introduced by Crandall and Lions [14] for andling weak solutions of nonlinear first order PDEs, and was ten extended to te second order PDEs by Lions [32]. For a general introduction to viscosity solution teory, we refer to Crandall, Isii and Lions [15] and Fleming and Soner [23]. Te concept of constrained viscosity solution was introduced by Soner [39] and Katsoulakis [29] to andle control problems wit state constrains. Viscosity solutions were first employed in matematical finance by Zaripopoulou [49]. In er P.D tesis [49], Zaripopoulou applied viscosity solution teory to stocastic optimization problems arising in optimal investment and consumption models. In te recent years, viscosity solutions ave become a standard tool in te study of stocastic control problems arising in models of matematical finance (see [48]). Before defining viscosity solution, we first recall te definitions of semicontinuous functions wic will be used later. Definition (upper semi-continuous function (USC)) A function f : Ω R n R is USC if for any x Ω and ε > 0, tere exists a neigborood B of x suc tat f(y) < f(x) + ε for all y B. Definition (lower semi-continuous function (LSC)) A function f : Ω R n R is LSC if for any x Ω and ε > 0, tere exists a neigborood B of x suc tat f(y) > f(x) ε for all y B. 15

23 In order to introduce te notion of viscosity solution, let us consider a fully non-linear second order PDE of te following form: F(X,W(X),DW(X),D 2 W(X)) = 0, for X [0,T) Ω (2.3.1a) wit te terminal condition W(X) = g(x), for X {T } Ω, (2.3.1b) were Ω R n be an open set, W : [0,T] Ω R is an unknown function, and DW and D 2 W denote respectively te gradient and Hessian of W wit respect to (t,x 1,...,x n ). Let S n denote te set of n n symmetric matrices, ten F is a mapping F : [0,T] Ω R R n S n R. Te teory of viscosity solution applies to PDE (2.3.1a)-(2.3.1b) were F is continuous in all its arguments and satisfies te following degenerate elliptic condition: F(X,Y,Z,A) F(X,Y,Z,B) wenever B A for X [0,T) Ω,Y R,Z R n, and A,B S n. For any A,B S n, we say A B if (Aξ,ξ) (Bξ,ξ) for all ξ R n, were (, ) is te Euclidean inner product on R n. To motivate te definition of viscosity solutions, we first give te following proposition regarding classical solutions. Proposition Let F : [0,T] Ω R R n S n R satisfy te degenerate elliptic condition. If a function W C 1,2 ([0,T] Ω) is a classical solution of (2.3.1a), ten 1. W is a classical sub-solution of (2.3.1a), tat is, for all ψ C 1,2 ([0,T) Ω) and every local maximum point, X 0 [0,T) Ω of W ψ, we ave F(X 0,W(X 0 ),Dψ(X 0 ),D 2 ψ(x 0 )) 0. (2.3.2) 2. W is a classical super-solution of (2.3.1a), tat is, for all ψ C 1,2 ([0,T) Ω) and every local minimum point, X 0 [0,T) Ω of W ψ, we ave F(X 0,W(X 0 ),Dψ(X 0 ),D 2 ψ(x 0 )) 0. (2.3.3) 16

24 PROOF. Te proof is very simple. If W ψ as a local maximum at X 0, ten DW(X 0 ) = Dψ(X 0 ), D 2 W(X 0 ) D 2 ψ(x 0 ). Since F satisfies te degenerate elliptic condition, we ave 0 = F(X 0,W(X 0 ),DW(X 0 ),D 2 W(X 0 )) F(X 0,W(X 0 ),Dψ(X 0 ),D 2 ψ(x 0 )) wic is (2.3.2). In te same way, we can obtain (2.3.3). It is wort noting tat te definition of sub-solution or super-solution given above does not require te existence of te first and second derivatives of W (te derivation is performed on test function ψ). Tus, te above proposition motivates te following definition. Definition (Viscosity sub- and super-solutions) 1. An upper semi-continuous function W : [0,T) Ω R is a viscosity subsolution of (2.3.1a) on [0,T) Ω if, for every ψ C 1,2 ([0,T) Ω) and every local maximum point, X 0 [0,T) Ω of W ψ, we ave F(X 0,W(X 0 ),Dψ(X 0 ),D 2 ψ(x 0 )) A lower semi-continuous function W : [0,T) Ω R is a viscosity supersolution of (2.3.1a) on [0,T) Ω if, for every ψ C 1,2 ([0,T) Ω) and every local minimum point, X 0 [0,T) Ω of W ψ, we ave F(X 0,W(X 0 ),Dψ(X 0 ),D 2 ψ(x 0 )) 0. Remark In te above definition, we can replace local maximum (minimum) point by strict local maximum (minimum) point, or global maximum (minimum) point or strict global maximum (minimum) point. We can also assume tat te extremum of W ψ as te value zero. Remark It is clear from Proposition tat if W is a classical sub-solution (respectively super-solution) of (2.3.1a), ten it is a viscosity sub-solution (respectively, viscosity super-solution) of (2.3.1a). 17

25 Using te above definition, we ave te definition of constrained viscosity solution as follows. Definition (Constrained viscosity solution) A continuous function W : [0,T) Ω R is a constrained viscosity solution of (2.3.1a) if it is bot a viscosity subsolution of (2.3.1a) on [0,T) Ω and a supersolution of (2.3.1a) on [0,T) Ω. Furtermore, a continuous function W : [0,T] Ω R is a constrained viscosity solution of (2.3.1a) (2.3.1b) if it is a constrained viscosity solution of (2.3.1a) satisfying W(X) = g(x) for X {T } Ω Existence and Uniqueness of te Viscosity Solution of (2.2.4) (2.2.8) Having introduced te definitions of viscosity solutions, we can now establis te following teorem: Teorem Let i {0,b,w} and assume tat te value function V i is continuous on [0,T] Ω i. Ten, V i is a constrained viscosity solution of (2.2.4) on [0,T) Ω i, were Ω i are defined in (2.2.9) and (2.2.10) respectively for i = 0,b,w. PROOF. See te proof of Teorem 2 in [19]. To prove te uniqueness of te solution to (2.2.4) (2.2.8), we need te following comparison result wic was proved in [20](see also [19]). Lemma Let u i be a bounded upper semi-continuous viscosity subsolution of (2.2.4) on [0,T) Ω i and v i be a lower semi-continuous function wic is bounded from below, exibits sublinear growt and is a viscosity supersolution of (2.2.4) in [0,T) Ω i suc tat u i (T,α,β,S) v i (T,α,β,S), (α,β,s) Ω i, Ten u i v i on [0,T] Ω i. Wit te above lemma, we now state our main teorem of tis section: Teorem Let i {0,b,w} and assume tat te value function V i is continuous on [0,T] Ω i, ten te value function V i is te unique constrained viscosity 18

26 solution of (2.2.4) wit te terminal condition V (T,α,β,S) = V i (T,α,β,S), for (α,β,s) Ω i, (2.3.4) were V i (T,α,β,S) are defined in (2.2.6) (2.2.8) respectively for i = 0,b,w. PROOF. Te proof is based on te comparison result and te definition of viscosity solution. Suppose tat tere are two viscosity solutions, V i 1 and V i 2, to (2.2.4) and (2.3.4). Since V i 1 and V i 2 are te subsolution and supersolution respectively, ten V i 1 V i 2 by Lemma On te oter and, V i 1 V i 2, since V i 1 and V i 2 are te supersolution and subsolution respectively. Hence, V i 1 = V i Markov Cain Approximation Tis section gives a sort introduction to te Markov cain approximation approac wic was employed to solve (2.2.4)-(2.2.5) by Davis, Panas, and Zaripopoulou [19]. Te starting point of Markov cain approximation is constructing a discrete-time price process approximating te continuous-time one presented in Section 2.1. Te discrete-time program is solved by using a backward recursive algoritm. Consider te discrete time variable t {t 0,t 1,t 2,...,t n } and assume tat t i = i t, were i = 0, 1,...,n and t = T. Let β n i denote te amount in te bond at te discrete time t i, ten te discrete time equation for te bond is defined by B (i+1) = B i e r t. Te discrete stock price process at time t i, denoted by S i, is modeled by 1 S i k u wit probability S (i+1) =, 2 (2.4.1) 1 S i k d wit probability, 2 were k u = e (µ t+σ t),k d = e (µ t σ t). According to Cox and Rubinstein [13, Capter 5], te discrete time process (2.4.1) converges in distribution to its continuous counterpart (2.1.1) as n tends to. Using te above discrete price processes, Davis, Panas, and Zaripopoulou [19] proposed te following discretization sceme for (2.2.4): V t = O( t)v t (2.4.2) 19

27 were te operator O( t)v ( t) is defined by O( t)v ( t) (t i,α i,β i,s i ) = max {V ( t) (t i,α i + λ t,β i (1 + θ)s i λ t,s i ), V ( t) (t i,α i λ t,β i + (1 θ)s i λ t,s i ), E{V ( t) (t i + t,α i,β i e r t,s i e (µ t+σε t) )}}, were λ > 0 is a constant and ε is a random variable taking values ±1 wit probability 1 2 eac. Tey proved tat te solution V ( t) of (2.4.2) converges to te constrained viscosity solution of (2.2.4) as t 0 (see Teorem 4, [19]). Te proof followed a fairly standard structure wic was described in Barles and Souganidis [3]. For completeness of te presentation, we now introduce te main steps of te proof. PROOF. (Sketc) 1. Let V t (t v t i,α i,β i,s i ) if t [t i,t i + t),α [α i,α i + λ t), (t,α,β,s) = V i (T,α,β,S) if t = T, were i = 0,w and V 0,V w are defined in (2.2.6) and (2.2.8), respectively Define te functions υ(x) = lim sup υ t (Y ),υ(x) = lim inf Y X, t 0 Y X, t 0 υ t (Y ), (2.4.3) were X = (t,α,β,s) and Y = (t i,α i,β i,s i ). 3. Prove tat υ and υ are a viscosity subsolution and supersolution of (2.2.4), respectively and υ(t,α,β,s) υ(t,α,β,s) Combining 3 wit te comparison result implies υ υ. On te oter and, te definitions of υ and υ yield υ υ. Tus, υ = υ. Since (2.2.4) (2.2.5) as unique solution V, υ = υ = V wic, togeter wit (2.4.3), implies te convergence of v ( t) to V. 3 In [15], Davis, Panas, and Zaripopoulou considered only reservation write price. 4 In [15], Davis, Panas, and Zaripopoulou only proved tat υ and υ are a viscosity subsolution and supersolution of (2.2.4), respectively. Tey did not sow υ(t, α, β, S) υ(t, α, β, S) 20

28 2.5 Conclusion In tis capter, we presented a utility based option pricing model for a European call option wit proportional transaction costs. According to te model, computing te reservation purcase price or write price is equivalent to finding te constrained viscosity solutions to te HJB equations (2.2.4) (2.2.8). We introduced te Markov cain approximation approac, wic was used for solving tese HJB equations by Davis, Panas, and Zaripopoulou [19] and many oters (see, for example, [17, 18, 26]). In te following two capters we will present a new metod, penalty approac combined wit a finite difference sceme, for solving te HJB equations. 21

29 Capter 3 Te Penalty Approac and Its Analysis Te purpose of tis capter is to propose a penalty approac to te HJB equation (2.2.4). We begin wit a brief review of penalty metods wic ave been used to andle te option pricing problem. Motivated by tese metods, we ten present a penalty formulation for (2.2.4). We sow tat tere exists a unique constrained viscosity solution to te penalized equation wit satisfying te terminal condition (2.2.5). Finally, we prove tat te solution to te penalty problem converges to tat of (2.2.4) and (2.2.5) as penalty parameters tend to infinity. Te results from tis capter ave been publised in [31]. 3.1 Introduction Te penalty metod as long been used successfully for constrained optimization problems (see, for example, [36]). More recently, it as been used to solve te problem of American option pricing witout transaction costs [24, 43, 52, 53]. In tis section, we will review two basic penalty metods wic ave been used for solving American option pricing problem. We start by introducing te American option pricing model. Consider an American put option wit strike price K and expiry date T. Assume tat te price of te underlying asset, S, satisfies te following stocastic process: ds = µsdt + σsdz, 22

30 were µ is te drift rate, σ is volatility, and Z is a standard Brownian motion. It is known [44, 45] tat te value of an American option, denoted by V (S,t), satisfies te following form of linear complementarity problem: LV (S, t) 0 (3.1.1) V (S,t) V (S) 0 LV (S,t).(V (S.t) V (S)) = 0 were LV = V t 1 2 σ2 S 2 2 V S 2 rs V S + rv, (3.1.2) r is te risk-free interest rate and V (S) is te payoff function given by V (S) = V (S,T) = max{k S, 0}. (3.1.3) Te boundary conditions are V (0,t) = K; (3.1.4) V (S,t) = 0; S. (3.1.5) Te system of (3.1.1)-(3.1.5) is called American option pricing model. In te literature, tere are two basic penalty approaces, linear (l 1 ) penalty approac and power penalty approac, to te complementarity problem (3.1.1)- (3.1.5)(see, for example, [24, 43]). In [24], Forsyt and Vetzal proposed a linear penalty approac to (3.1.1)-(3.1.5). Tey replaced problem (3.1.1) by te following nonlinear PDE: LV = ρ max{v (S) V (S,t), 0}, (3.1.6) were ρ is a positive penalty parameter. Intuitively, equation (3.1.6) can be explained as follows. 1. If V (S,t) V (S), LV (S,t) = 0, and tus (3.1.1) is satisfied. 2. If V (S,t) < V (S), ten max{v (S) V (S,t), 0} = V (S) V (S,t). Using (3.1.6) yields V (S) V (S,t) = 1 (LV ). ρ 23

31 Terefore, if ρ and LV is bounded, V (S) V (S,t) 0. Tus, (3.1.1) is satisfied witin a tolerance depending on ρ. Forsyt and Vetzal [24] ave sowed tat te above penalty metod is efficient for pricing American options. In [43], Wang, Yang and Teo proposed a power penalty approac to (3.1.1). Tey replaced problem (3.1.1) by te following nonlinear PDE: LV λ (S,t) + λ[v (S) V λ (S,t)] 1 k + = 0, (3.1.7) were k > 0 is a parameter, [a] + = max{a, 0} and λ > 1 is te penalty parameter. Wang, Yang and Teo [43] sowed tat te rate of convergence in a weigted Sobolev norm for te power penalty approac (3.1.7) is of order O(λ k 2). Motivated by te above penalty approaces for a linear complementarity problem, we will propose a penalty formulation for (2.2.4) in te following section. 3.2 Te Formulation We consider te following problem approximating (2.2.4), or te linear complementarity problem (2.2.11): L 1 V λ1,λ 2 + λ 1 [L 2 V λ1,λ 2 ] + λ 2 [L 3 V λ1,λ 2 ] = 0 (3.2.1) for (t,α,β,s) [0,T) Ω i wit te terminal condition V λ1,λ 2 (T,α,β,S) = V i (T,α,β,S), for (α,β,s) Ω i, (3.2.2) were L k are te differential operators defined in (2.2.1) (2.2.3), V i (T,α,β,S) is te boundary condition given in (2.2.6) (2.2.8) for eac i, λ 1,λ 2 > 1 are penalty parameters, i {0,b,w} and [v] = min{v, 0} for any function v. Tis PDE contains two penalty terms. Te intuition of tis penalty formulation is as follows. Te inequalities L k V λ1,λ 2 0, k = 2, 3 define two bounds on te solutions to L 1 V λ1,λ 2 = 0. Wen tese constraints are violated by V λ1,λ 2 in some subregions of [0,T) Ω i, te negative parts of L 2 V λ1,λ 2 and L 3 V λ1,λ 2 are penalized. 24

32 3.3 Existence and Uniqueness of Viscosity Solution of te Penalized Equations In tis section, we will sow tat tere exists a unique constrained viscosity solution to te penalized equations (3.2.1) (3.2.2). We prove tat te viscosity solution to te penalized equations converges to tat of te HJB equations (2.2.4) (2.2.8) wen te penalty parameters λ 1 and λ 2 approac infinity. Before proving te existence of viscosity solution of te penalized equations (3.2.1) (3.2.2), we first consider te following HJB equation: L 1 V λ1,λ 2 + min m [0,λ 1 ] ml 2V λ1,λ 2 + min n [0,λ 2 ] nl 3V λ1,λ 2 = 0. (3.3.1) It is easy to prove tat (3.2.1) and (3.3.1) are equivalent as given in te following teorem. Teorem Equation(3.3.1) is equivalent to (3.2.1). PROOF. Since [L 2 V λ1,λ 2 ] = min{l 2 V λ1,λ 2, 0}, we ave λ λ 1 [L 2 V λ1,λ 2 ] 1 L 2 V λ1,λ 2 if L 2 V λ1,λ 2 0, = 0 if L 2 V λ1,λ 2 > 0. On te oter and, min ml 2 V λ1,λ 2 = m [0,λ 1 ] Tus, In te same way, we can prove tat Tus, (3.2.1) and (3.3.1) are equivalent. λ 1 L 2 V λ1,λ 2 if L 2 V λ1,λ 2 0(m = λ 1 ), 0 if L 2 V λ1,λ 2 > 0(m = 0). λ 1 [L 2 V λ1,λ 2 ] = min m [0,λ 1 ] ml 2V λ1,λ 2. λ 2 [L 3 V λ1,λ 2 ] = min nl 3 V λ1,λ 2. n [0,λ 2 ] Now we are going to prove tat tere exists a unique viscosity solution to (3.3.1) satisfying te terminal condition (3.2.2). For brevity, we only consider te unique solvability of te HJB equation for V b, i.e., te case tat i = b in (3.2.2). Te proofs for te oter two cases are essentially te same as tat for i = b and tus omitted. 25

33 We consider a control problem similar to Problem in Capter 2. Assume tat te market consists of a stock and a bond. Te stock price evolves according to (2.1.1). Let β u denote te amount invested in te bond and α u te number of sares of te stock eld by te investor at time u [0,T]. Te evolution equations for β u and α u are respectively dβ u = rβ u du (1 + θ)s u m u du + (1 θ)s u n u du, (3.3.2) dα u = m u du n u du, (3.3.3) were r 0 is a constant interest rate, θ [0, 1) is te fraction of traded amount in te stock, and m u and n u denote te rates of buying and selling te stock, respectively. Let te set of admissible trading strategies for an investor wo starts in te state (t,α,β,s) be denoted by Λ λ1 λ 2 (t,α,β,s). Ten, Λ λ1 λ 2 (t,α,β,s t ) is defined as te set of feasible controls, m u and n u (t u T), suc tat 1. m u and n u are rigt-continuous, measurable, F-adapted, nonnegative processes m u λ 1, 0 n u λ Te associated process (α mu,nu,β mu,nu,s u ) satisfies (3.3.3), (3.3.2) and (2.1.1), respectively, in [t,t] wit te initial condition (α t,β t,s t ) = (α,β,s). 4. (αu mu,nu,βu mu,nu,s u ) Ω b, u [t,t], were Ω b is defined in (2.2.9). Te objective of te problem is to maximize te investor s expected utility of terminal wealt. For eac fixed pair (λ 1,λ 2 ) and a state (t,α,β,s), we define te value function as V λ1,λ 2 (t,α,β,s) = sup E t [U(β T +S T (α T θ α T )+(S T K) + )] Λ λ1,λ 2 (t,α,β,s) (0 t T), (3.3.4) were E t denotes te expectation operator conditional on te time t information (α,β,s) We next sow tat te value function V λ1,λ 2 is a bounded constrained viscosity solution of equation (3.3.1) satisfying an appropriate terminal condition. Tis is given in te following teorem. 26

34 Teorem Assume tat te value function V λ1,λ 2 defined in (3.3.4) is continuous on [0,T] Ω b. Ten, V λ1,λ 2 is a constrained viscosity solution of equation (3.3.1) satisfying te terminal condition V λ1,λ 2 (T,α,β,S) = U(β + S(α θ α ) + (S K) + ), (α,β,s) Ω b. (3.3.5) Furtermore, V λ1,λ 2 is bounded uniformly in λ 1 and λ 2. PROOF. In view of te definition of constrained viscosity solution, we need to prove tat V λ1,λ 2 is a viscosity subsolution of (3.3.1) on [0,T) Ω b and a viscosity supersolution of (3.3.1) in [0,T] Ω b. We now prove te former. Let ψ(t,α,β,s) C 1,2 ([0,T) Ω b ) and suppose X 0 = (t 0,α 0,β 0,S 0 ) [0,T] Ω b is a local maximum point of V λ1,λ 2 (X) ψ(x) for X [0,T) Ω b. Witout loss of generality, we assume tat V λ1,λ 2 (X) ψ(x) for all X [0,T) Ω b wit V λ1,λ 2 (X 0 ) = ψ(x 0 ). We need to sow tat (L 1 ψ)(x 0 ) + min m(l 2 ψ)(x 0 ) + min n(l 3 ψ)(x 0 ) 0. (3.3.6) m [0,λ 1 ] n [0,λ 2 ] Let u ε = t 0 + ε and (m ε u ε,n ε u ε ) Λ λ1,λ 2 (t 0,α 0,β 0,S 0 ) be an ε 2 -optimal control. By te dynamic programming principle, we ave V λ1,λ 2 (X 0 ) ε 2 + E t0 [V λ1,λ 2 (X ε u ε )], were E t0 denotes te expectation operator conditional on te time t 0 information (α 0,β 0,S 0 ). Xu ε ε = (u ε,αu ε ε,βu ε ε,su ε ε ) is an ε 2 -optimal trajectory corresponding to te controls (m ε u ε,n ε u ε ), according to (3.3.2), (3.3.3) and (2.1.1). Since V λ1,λ 2 (X 0 ) = ψ(x 0 ) and X 0 is a local maximum point of V λ1,λ 2 (X) ψ(x), we ave ψ(x 0 ) ε 2 + E t0 [ψ(x ε u ε )]. Applying Ito s formula to te last term of te above inequality yields uε ( ψ 0 ε 2 +E t0 t 0 t (Xε u) + rβu ε ψ β (Xε u) + µsu ε ψ S (Xε u) σ2 (Su) ε 2 2 ψ ) S u) du 2(Xε uε ( ψ + E t0 t 0 α (Xε u) (1 + θ)su ε ψ ) β (Xε u) m ε udu uε ( + E t0 ψ α (Xε u) + (1 θ)su ε ψ ) β (Xε u) n ε udu. t 0 27