Multi asset option pricing with levy process

MULTI-ASSET OPTION PRICING ´ WITH LEVY PROCESS ZHOU JINGHUI (MSc., South China University of Technology) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2009 Acknowledgements A thesis is almost always a product not only of its author, but also of the environment where the author works, of the encouragements and critics gathered from colleagues and teachers, conversations after seminars and many analogous events. While I cannot justice to all of the above, I thank explicitly my supervisor Prof. Lin Ping and co-supervisor Dr. Li Xun for their intellectual support, invaluable guidance and indispensable encouragement throughout the years. I would like to thanks Prof. Olivier Pironneau, whose software FreeFem++ and book Computational Methods for Option Pricing inspired me to many deep and stimulating thoughts when applying numerical approach into finance. My perception of mathematics and finance has developed under the influence of many people. The experience as a Ph.D student with the Department of Mathematics at NUS was challenging and enjoyable. I believe that many lessons I learned here will enlighten me to the right direction in my career and in my life. Finally, my ultimate gratitude goes towards my parents whose tremendous help and support made this work possible, my fellow friends for their help in my research and life. ii Contents Acknowledgements ii Summary vi Notations ix List of Tables x List of Figures xi Introduction 1.1 1.2 Black-Scholes-Merton Framework and its Numerical Approaches . . 1.1.1 Black-Scholes-Merton Framework . . . . . . . . . . . . . . . 1.1.2 Numerical Approaches . . . . . . . . . . . . . . . . . . . . . Jump Diffusion Model . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.1 Assets with Jumps . . . . . . . . . . . . . . . . . . . . . . . 16 1.2.2 2-Dimensional Jump Diffusion Model . . . . . . . . . . . . . 17 iii Contents 1.3 iv Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . Preliminary 22 27 2.1 L´evy Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Examples of L´evy Process . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 Some Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 PIDE for Pricing Multi-Asset Option 35 3.1 Exponential L´evy Model . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 Derivation of Pricing Equation . . . . . . . . . . . . . . . . . . . . . 39 3.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Finite Element Method for PIDE 55 4.1 Variational Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.2 Error Estimate for Localization to Bounded Domain . . . . . . . . . 63 4.3 Error Estimate for Time-Discretization Scheme . . . . . . . . . . . 68 4.4 Error Estimate for Finite Element Method and its Matrix Form . . 71 Finite Difference Method for PIDE 79 5.1 Method of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.2 The ADI Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Numerical Experiments 87 6.1 Case Study for Polynomial Option . . . . . . . . . . . . . . . . . . 87 6.2 Case Study for Some Multi-Asset Options . . . . . . . . . . . . . . 91 6.3 Case Study for Different Jump Densities . . . . . . . . . . . . . . . 95 Concluding Remarks and Future Work 97 Contents v 7.1 Pros and Cons of FEM . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.2 Refinement for Nonsmooth Payoff 99 7.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.4 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 . . . . . . . . . . . . . . . . . . Bibliography 103 A Property for η(x) 111 B Itˆ o’s Formula for Semimartingale 113 C Analytic Solution for Power Options 115 C.1 Power Option under Black-Scholes Model . . . . . . . . . . . . . . . 116 C.2 Analytic Solution for Polynomial Option . . . . . . . . . . . . . . . 117 D Boundary Conditions 121 D.1 Fichera’s Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 D.2 Boundary Conditions for Some Options . . . . . . . . . . . . . . . . 122 Summary In this dissertation we will study the finite element method (FEM) which is used to solve a time dependent partial integro-differential equation (PIDE) in two dimension and an unbounded domain arising from option pricing under L´evy process. From both theoretical and numerical point of view, the most difficult part to solve the PIDE is to deal with the integral term, unbounded domain and associated cut-off boundary condition. The finite element method can treat these difficulties easily in its variational formulation and theoretical analysis framework. We will review the derivation and a few existing numerical methods for the Black-Scholes model and the jump diffusion model for the two-asset option in Chapter 1. In particular, we will provide a brief review of finite element method. Chapter includes a brief introduction of L´evy process and its examples, as well as a number of well-known inequalities which will be used in our analysis in later chapters. In Chapter 3, we provide a general derivation of pricing equations related to infinitesimal generator of L´evy process. Replacing the nonsmooth initial condition to a smooth one, we also consider its error estimate for the nonsmooth initial condition. vi Summary Chapter includes our main results. The existence and uniqueness of the solution under the weighted Soblev space are proved via G˚ arding inequality in Section 1. In Section we estimate the error of localization from the infinite domain to a finite domain. In Section 3, we consider semi-discretization in time. We obtain the error estimate for the Crank-Nicolson scheme. The remaining sections focus on error estimates of semi-discretization in spatial variables and fully discrete scheme. Chapter is relatively independent of the other chapters, where we study the pricing PIDE via a finite difference method (FDM). We discuss the idea of an alternating implicit direction (ADI) scheme for the problem. We include several examples in Chapter 6. We construct the exact solution of the first example and use it to verify the convergence of our scheme. The second and third examples are both well-known problems in option pricing. We can compare our results with benchmark solutions. In the last Chapter, we conclude this thesis and point out a few directions which we will work next. vii viii Main Notations ix Notations {Ω, F , P} The set, σ−field, measure defined for a probability space P Real-world/physical measure Q Risk-neutral/martingale measure EQ [·] Expectation under martingale measure Q Ei [·] Expectation under a general probability measure νi Wt Weiner process {St : t ∈ [0, T ]} Stochastic process for underlying assets Xt , Xt− L´evy process and its left limit LX The infinitesimal generator for L´evy process Xt Ω A domain in IR2 if use seperately ΩM The truncated domain [−M, M ] × [−M, M ] ⊂ Ω L2η (Ω) u ∈ L1loc (Ω) ueη(x) ∈ L2 (Ω) Hη1 (Ω) u ∈ L1loc (Ω) ueη(x) ∈ L2 (Ω), ∇ueη(x) ∈ (L2 (Ω)) FEM Finite Element Method FDM Finite Difference Method PIDE Partial Integro-Differential Equation GBM Geometric Brownian Motion BSM Black-Scholes-Merton List of Tables 1.1 Statistics of major indices from Jan-1999 to Dec-2008 . . . . . . . . 2.1 L´evy densities for various models . . . . . . . . . . . . . . . . . . . 32 3.1 Payoff functions for some multi-asset options . . . . . . . . . . . . . 40 6.1 Parameters for polynomial option: (S1 + S2 )2 . . . . . . . . . . . . 88 6.2 Results for polynomial option with positive correlation: ρ = 0.3. . . 88 6.3 Results for polynomial option with negative correlation: ρ = −0.3 . 90 6.4 Parameters for pricing basket put option: K − . . . . 91 6.5 Results for some two-asset put options . . . . . . . . . . . . . . . . 93 i=1 wi Si + x Bibliography 108 [49] T. 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Berridge and J.M. Schumacher. An irregular grid approach for pricing high-dimensional american options. Discussion Paper 18, Tilburg University, Center for Economic Research, 2004. [67] Reisinger Christoph and Wittum Gabriel. On multigrid for anisotropic equations and variational inequalities “pricing multi-dimensional european and american options”. Computing and Visualization in Science, 7:189–197, 2004. [68] D. W. Peaceman and Jr. Rachford. The numerical solution of parabolic and elliptic differential equations. Journal of the Society for Industrial and Applied Mathematics, 3(1):28–41, March 1955. [69] J. Douglas. Alternating direction methods for three space variables. Numerische Mathematik, 4:41–63, 1962. Appendix A Property for η(x) Suppose    η1 (|x1 | + |x2 |)      η |x | + η |x | 1 2 η(x) =   η2 |x1 | + η1 |x2 |      η (|x | + |x |) 2 if x1 < 0, x2 < 0, if x1 < 0, x2 > 0, (A.0.1) if x1 > 0, x2 < 0, if x1 > 0, x2 > 0. where ≤ η1 ≤ G1 and ≤ η2 ≤ G2 . Then η(x) satisfies η ∈ L1loc (R), η ∈ L∞ (R) and ∆θ η = η(x + θy) − η(x) ≤ η(y), ∀x, y ∈ R2 , ||θ|| ≤ 1. (A.0.2) We discuss it in the following 16(= 24 ) cases: Case 1. If xi + θyi > 0, xi > for i = 1, 2, then ∆θ η = η2 θ(y1 + y2 ), thus    −η1 (y1 + y2 ) if y1 < 0, y2 < 0,      −η y + η y if y < 0, y > 0, 1 2 ∆θ η <   η2 y − η1 y if y1 > 0, y2 < 0,      η (y + y ) if y > 0, y > 0. which means ∆θ η < η(y). 111 Property for η(x) 112 Case 2. If xi + θyi > for i = 1, and x1 > 0, x2 < 0, then ∆θ η = η2 θ(y1 + y2 ) + (η1 + η2 )x2 ≤ η2 θ(y1 + y2 ) ≤ η2 (y1 + y2 ). Note that y2 > in this case and   −η y + η y if y < 0, 1 2 η(y) =  η (y + y ) if y1 > 0. 2 Thus ∆θ η < η(y). Case 3. If xi + θyi > for i = 1, and x1 < 0, x2 > 0, then ∆θ η = η2 θ(y1 + y2 ) + (η1 + η2 )x1 ≤ η2 θ(y1 + y2 ) ≤ η2 (y1 + y2 ). Note that y1 > in this case and   η y − η y if y < 0, 1 2 η(y) =  η (y + y ) if y > 0. 2 Thus ∆θ η < η(y). Case 4. If xi + θyi > 0, xi < for i = 1, 2, then ∆θ η = η2 θ(y1 + y2 ) + (η1 + η2 )(x1 + x2 ) ≤ η2 θ(y1 + y2 ) ≤ η(y1 + y2 ). Note that y1 > 0, y2 > in this case, which implies η(y) = η2 (y1 + y2 ). Thus ∆θ η < η(y). Case 5-16. These remaining 12 cases can be grouped by changing the signs of xi + θyi for i = 1, 2. They are similar to the above four cases. we omit here. More generally, we can get the following result: ± η(x + θy) − η(x) ≤ η(±y), ∀x, y ∈ R2 , ||θ|| ≤ 1. (A.0.3) Appendix B Itˆo’s Formula for Semimartingale A real valued process X defined on the filtered probability space {Ω, F, (Ft )t≥0 , P} is called semimartingale if it can be decomposed as Xt = Mt + At , where M is a local martingale and A is a c`adl`ag adapted process of locally bounded variation. An IRd valued process X = (X , X , ., X d ) is a semimartingale if each of its components X i is a semimartingale. Itˆo’s formula applies to both to continuous and discontinuous semimartingales. Theorem B.1 (Itˆo’s Formula for Continuous Semimartingales). Let X = (X , X , ., X d ) be a d-dimensional continuous semimartingale and let f (X) be a twice continuously differentiable real valued function on IRd . Then f(X) is again a semimartingale satisfying d df (Xt ) = i=1 d ∂f (Xt ) ∂ f (Xt ) i dX + d[X i , X j ]t , t ∂X i i,j=1 ∂X i ∂X j where [X i , X j ]t is the quadratic covariation of X i and X j . Theorem B.2 (Itˆo’s Formula for Non-continuous Semimartingales). Let X = (X , X , ., X d ) be a d-dimensional semimartingale and let f (X) be is a twice 113 Itˆ o’s Formula for Semimartingale 114 continuously differentiable real valued function on IRd , Then f(X) is again a semimartingale satisfying d f (Xt ) = f (X0 ) + i=1 d ∂ f (Xs− ) ∂f (Xs− ) i dX + d[X i , X j ]s s ∂X i i,j=1 ∂X i ∂X j d f (Xs ) − f (Xs− ) − + 0K} −Ke−r(T −t) Q(S1p (T )S2q (T ) > K), and the change of measure defined by the Radon-Nikodym derivative ˜ S1p (T )S2q (T ) dQ = dQ EQ [S1p (T )S2q (T )] one obtains: ˜ t (S p (T )S q (T ) > K) C(K, t) = e−r(T −t) EtQ [S1p (T )S2q (T )] Q −Ke−r(T −t) Qt (S1p (T )S2q (T ) > K) In a way, the above equation extends the well-known formula for a European call generally obtained via changes of numeraire. 115 C.1 Power Option under Black-Scholes Model C.1 116 Power Option under Black-Scholes Model Assume we have an equivalent martingale measure, Q, such that the discounted prices at the constant interest rate r are Q-martingales. More exactly, dSi /Si = rdt + σi dWi , i = 1, 2, dW1 dW2 = ρdt, where dW1 , dW2 are standard Brownian motions. For a given p, q , it is easy to S1p (T )S2q (T ) follows a log-normal law. Indeed, as ln(S1 (T )), ln(S2 (T )) is normal, then ln (S1p (T )S2q (T )) = p · ln(S1 (T )) + q · ln(S2 (T )) is normal too, but with parameters: • mean - p ln(S1 (t)) + (r − 21 σ12 (T − t)) + q ln(S2 (t)) + (r − 21 σ22 (T − t)) • variance - [p2 σ12 + q σ22 p2 + 2pqρσ1 σ2 ] (T − t) By Itˆo’s Lemma, the stochastic differential equation of S = S1p S2q is given as follows, dS = rˆSdt + σSdW, where 1 rˆ = (p + q)r + (p2 − p)σ12 + (q − q)σ22 + ρσ1 σ2 pq 2 σ = dW = p2 σ12 + q σ22 + 2pqρσ1 σ2 (pσ1 dW1 + qσ2 dW2 ) σ Then, using above equation, the (p, q)-power call price and put price in BlackScholes model is obtained as follows: C(S, t) = SN (d1 ) − Ke−ˆrT N (d2 ) P (S, t) = Ke−ˆrT N (−d2 ) − SN (−d1 ) C.2 Analytic Solution for Polynomial Option 117 where d1 d2 ln(St /K) + (ˆ r + σ /2)(T − t) √ = σ T −t √ ln(St /K) + (ˆ r − σ /2)(T − t) √ = = d1 − σ T − t. σ T −t and N (x) is the cumulative function at x of a standard Gaussian random variable. This formula is valid for any pq = 0, and when p = 1, q = we can express the classic Black-Scholes formula. C.2 Analytic Solution for Polynomial Option Consider the terminal payoff of a polynomial option: H(S1 , S2 ) = (S1 + S2 )2 , which is a differentiable function w.r.t S1 , S2 . Before deriving the exact solution of PIDE to pricing the option under exponential L´evy models, let’s consider the exact solution of the following pricing equation:  ∂V 2 ∂ 2V 2 ∂ 2V ∂ 2V ∂V ∂V   + σ S + σ σ ρS S + σ2 S2 + rS1 + rS2 = rV,  2 1  ∂t ∂S1 ∂S1 ∂S2 ∂S2 ∂S1 ∂S2      V (T, S1 , S2 ) = (S1 + S2 )2 , for all (S1 , S2 ) ∈ R2+    (C.2.1) V (t, S1 , 0) = S12 e(r+σ1 )(T −t) , for all t ∈ [0, T ),      V (t, 0, S ) = S e(r+σ22 )(T −t) , for all t ∈ [0, T ),   2      V (t, S , S ) = V (t, S , S ), if S → ∞ or S → ∞ for all t ∈ [0, T ). BS where VBS (t, S1 , S2 ) = S12 e(r+σ1 )(T −t) + S22 e(r+σ2 )(T −t) + 2S1 S2 e(r+ρσ1 σ2 )(T −t) . (C.2.2) Obviously the equation (C.2.1) is to price the polynomial option under Black and Scholes’ framework, i.e., the case when no jumps exist (λ1 = λ2 = 0). By using Itˆo’s lemma, we could verify that VBS (t, S1 , S2 ) is the analytic solution to the pricing equation (C.2.1). C.2 Analytic Solution for Polynomial Option 118 If this polynomial option is priced under exponential Levy model, then the pricing equation becomes as follows: ∂V ∂t + rS1 + R + R ∂ 2V ∂ 2V ∂V ∂V 1 ∂ 2V + rS2 + σ12 S12 + ρσ1 σ2 S1 S2 + σ22 S22 − rV ∂S1 ∂S2 ∂S1 ∂S1 ∂S2 ∂S2 ∂V V (t, S1 ex , S2 ) − V (t, S) − S1 (ex − 1) (t, S) ν1 (dx) ∂S1 ∂V V (t, S1 , S2 ex ) − V (t, S) − S2 (ex − 1) (t, S) ν2 (dx) = 0, (C.2.3) ∂S2 where νi (dx) = λi ex pi (ex )dx, λ1 = 0, λ2 = 0. From the exact solution (C.2.2) of the polynomial option in Black-Scholes model, we know S12 e(r+σ1 )(T −t) , S22 e(r+σ2 )(T −t) , S1 S2 e(r+ρσ1 σ2 )(T −t) are all functions satisfying the first equation of (C.2.1). So it is not feasible to use their linear combination to construct the exact solution of (C.2.3). Since the jump component of the above (C.2.3) will increase the volatility of the stock price, it is natural to assume that the solution of (C.2.3) as follows: V (t, S1 , S2 ) = S12 e(r+aσ1 )(T −t) + S22 e(r+bσ2 )(T −t) + 2S1 S2 e(r+cρσ1 σ2 )(T −t) , (C.2.4) where constants a, b, c are needed to specify. Substituting this solution in equation (C.2.3) and rearranging the terms, then the differential terms can be rewritten as: ∂V ∂t + rS1 ∂V ∂ 2V ∂ 2V ∂ 2V ∂V + rS2 + σ12 S12 + ρσ1 σ2 S1 S2 + σ22 S22 − rV ∂S1 ∂S2 ∂S2 ∂S1 ∂S2 ∂S2 = (1 − a)σ12 S12 e(r+aσ1 )(T −t) + (1 − b)σ2 S22 e(r+bσ2 )(T −t) +2(1 − c)σ1 σ2 S1 S2 e(r+cρσ1 σ2 )(T −t) . C.2 Analytic Solution for Polynomial Option 119 And the jump term in the equation (C.2.3) can also be rearranged as following: ∂V (t, S1 , S2 ) ν1 (dx) ∂S1 ∂V V (t, S1 , S2 ex ) − V (t, S1 , S2 ) − S2 (ex − 1) (t, S1 , S2 ) ν2 (dx) ∂S2 V (t, S1 ex , S2 ) − V (t, S1 , S2 ) − S1 (ex − 1) R + R (ex − 1)2 S12 e(r+aσ1 )(T −t) ex p1 (ex )dx = λ1 R +λ2 = (ex − 1)2 S22 e(r+bσ2 )(T −t) ex p2 (ex )dx R (r+aσ12 )(T −t) Λ1 S1 e where Λi = λi + Λ2 S22 e(r+bσ2 )(T −t) , (C.2.5) R+ (y − 1)2 pi (y)dy = λi e2(νi +γi ) − 2e(νi + γi ) − , i = 1, 2. Thus combining (C.2.5) and (C.2.5) together,    (1 − a)σ12 + Λ1 =   (1 − b)σ22 + Λ2 =     2(1 − c)σ σ + = we have : 0 then aσ12 = σ12 + Λ1 , bσ22 = σ22 + Λ2 , c = 1. Based on the known constants a, b, c, we can get the exact solution (C.2.6) for the equation (C.2.3): CELM (t, S1 , S2 ) = S12 e(r+σ1 +Λ1 )(T −t) + S22 e(r+σ2 +Λ2 )(T −t) + 2S1 S2 e(r+ρσ1 σ2 )(T −t) , (C.2.6) with the following boundary condition:   V (T, S1 , S2 ) = (S1 + S2 )2 ,        V (S1 , 0, t) = S12 e(r+σ12 +Λ1 )(T −t) , for all t ∈ [0, T ), (C.2.7)  (r+σ22 +Λ2 )(T −t)  V (0, S , t) = S e , for all t ∈ [0, T ),  2      V (S , S , t) = C if S1 → ∞ or S2 → ∞, for all t ∈ [0, T ). ELM (S1 , S2 , t), C.2 Analytic Solution for Polynomial Option 120 Therefore by means of variable transformation (3.3.1): xi = ln Si , τ = T − t, it is easy to derive the solution V (τ, x) for the following pricing PIDE, which is similar to (3.3.6).  ∂V   (τ, x)   ∂τ      V (0, x1 , x2 )     V (τ, x1 , x2 )        V (τ, x)       V (τ, x , x ) = ∇ · (κ∇V ) + ∇ · (αV ) + J [V ](τ, x), = (ex1 + ex2 )2 , (x) ∈ Ω, x2 → −∞, τ ∈ (0, T ] x1 → −∞, τ ∈ (0, T ] = e2x1 +(r+σ1 +Λ1 )τ , = e2x2 +(r+σ2 +Λ2 )τ , = VELM (τ, x), if x1 → ∞ or x2 → ∞, τ ∈ (0, T ]. And its exact solution is: VELM (τ, x) = e2x1 +(r+σ1 +Λ1 )τ + e2x2 +(r+σ2 +Λ2 )τ + 2ex1 +x2 +(r+ρσ1 σ2 )τ . (C.2.8) Appendix D Boundary Conditions D.1 Fichera’s Condition Consider a linear equation m aij (x) i,j=1 ∂ 2u + ∂xi ∂xj m bi (x) i=1 ∂u + c(x)u = 0, x ∈ Ω ∂xi with non-negative characteristic form m aij (x)ξi ξj ≥ 0, ξ ∈ Ω Γ, i,j=1 where Γ = ∂Ω. Define Γ3 = {x ∈ Γ| m i,j=1 aij (x)ξi ξj > 0}. Let n = (n1 , n2 , · · · , nm ) be the inward normal on Γ − Γ3 , which is the part of boundary Γ on which the left-hand side of above inequality vanishes. We define Fichera function m m bi (x) − f (x) = i=1 j=1 ∂aij (x) ∂xj ni It is shown that data must be prescribed on Γ − Γ3 if the Fichera function is negative on it. In other words, we must specified the boundary condition on m x ∈ ∂Ω m bi (x) − aij (x)xi xj = and i,j=1 m i=1 j=1 ∂aij (x) ∂xj ni < . 121 D.2 Boundary Conditions for Some Options D.2 122 Boundary Conditions for Some Options We denote CE (t, S, K, σ), PE (t, S, K, σ) the values of European call option and put option (expiring at T ) at time t under Black-Scholes model. We consider basket option with payoff max(K − (w1 S1 + w2 S2 ), 0), w1 , w2 > 0. When S1 = 0, the payoff could be reduced to w2 max K w2 − S2 , , which is the payoff of w2 units of put option on the second underlying with strike When S2 = 0, the payoff could be reduced to w1 max K w1 K . w2 Similarly − S1 , , which is the payoff of w1 units of put option on the first underlying with strike K . w1 Thus we have the following PDE to price European basket put.  2 ∂ 2V 2 ∂ 2V ∂V ∂ 2V ∂V ∂V   + σ + ρσ σ S S + σ2 S2 + rS1 + rS2 = rV S  2 1  ∂t ∂S1 ∂S1 ∂S2 ∂S2 ∂S1 ∂S2      V (T, S1 , S2 ) = max(K − (w1 S1 + w2 S2 ), 0), for all (S1 , S2 ) ∈ R2+    (D.2.1) V (t, S1 , 0) = w1 P (t, S1 , wK1 , σ1 ), for all t ∈ [0, T ),      V (t, 0, S ) = w P (t, S , K , σ ), for all t ∈ [0, T ),   2 w2      V (t, S , S ) = 0, if S → ∞ or S → ∞ for all t ∈ [0, T ). Similarly the spread option is obtained by solving the following PDE:  2 ∂ 2V ∂V 2 ∂ 2V ∂ 2V ∂V ∂V   + + rS2 = rV + σ S + ρσ σ S S σ2 S2 + rS1 2  1  ∂t ∂S ∂S ∂S ∂S ∂S ∂S 2      V (T, S1 , S2 ) = max(S1 − S2 , 0), for all (S1 , S2 ) ∈ R2+       V (t, S , 0) = S , for all t ∈ [0, T ), 1 (D.2.2)    V (t, 0, S2 ) = 0, for all t ∈ [0, T ),        V (t, S1 , ∞) = 0, for all t ∈ [0, T ),      ∂V (t, ∞, S2 ) = 1, for all t ∈ [0, T ). ∂S1 And its analytic solution is given as V (t, S1 , S2 ) = S1 N (d1 ) − S2 N (d2 ), D.2 Boundary Conditions for Some Options 123 where d1 d2 ˆ (T − t) ln( SS12 ) + 12 σ √ , = σ ˆ T −t √ = d1 − σ ˆ T − t, σ ˆ = σ12 − 2ρσ1 σ2 + σ22 . The power option with payoff H(S1 , S2 ) = max(K − S1p S2q , 0), p, q > is obtained by solving the following PDE:  ∂V 2 ∂ 2V ∂ 2V 2 ∂ 2V ∂V ∂V   + σ S + ρσ σ S S + σ2 S2 + rS1 + rS2 = rV 2  1  ∂t ∂S ∂S ∂S ∂S ∂S ∂S 2      V (T, S1 , S2 ) = max(K − S1p S2q , 0), for all (S1 , S2 ) ∈ R2+       V (t, S , 0) = Ke−r(T −t) , for all t ∈ [0, T ), (D.2.3)  −r(T −t)   V (t, 0, S2 ) = Ke , for all t ∈ [0, T ),        V (t, S1 , ∞) = 0, for all t ∈ [0, T ),      V (t, ∞, S2 ) = 0, for all t ∈ [0, T ). And its analytic solution is given in Appendix (C.1). [...]... single underlying Using this approach, the problem of pricing an option on a basket is reduced to price an option on a single equity Accordingly, the model for pricing options with exotic features can also be applied to options on baskets Precise error estimates are generally not provided Here, however we price options on multi- asset option using a multidimensional setting Following the no-arbitrage... European option using a binomial tree method.The binomial model is an option pricing technique in which the underlying asset price is assumed to follow a multiplicative binomial process over discrete periods For n discrete periods, a lattice with n + 1 ending asset values is formed Given the ending asset value at each lattice point, the value of the option at maturity can be calculated The price of the option. .. stronger assumption The single asset model for asset prices, given in equation (1.1.5), can easily be generalized to deal with an option with multiple underlying assets Each asset price, Si , is driven by a geometric Brownian motion under risk neutral world Q dSi = (r − qi )dt + σi dWi , Si (1.1.6) The random variables Wi are standard Brownian motions that are correlated, with the correlation between... model, we consider multi- asset option on risky assets with jump diffusion processes   dS (t) = S (t){µ (t)dt + σ (t)dW + (J − 1)dq }, t ∈ [0, T ], i i i i i i i (1.2.1)  S (0) = s > 0, i i together with a riskless bond whose value B(t) satisfies dB(t) = r(t)B(t)dt, where µi (t) > 0 is the drift rate, r(t) is the interest rate, (W1 , W2 ) is a 2-dimensional correlated Brownian motion with ρ EQ [dW1 dW2... = EQ [e−r(T −t) h(ST )], (1.1.4) ST is the value of the underlying asset at the option expiry date T This risk neutral valuation approach to option pricing was suggested by [9] The option price is the expectation of the discounted payoff at maturity under risk neutral probability Q And the asset price S follows the risk-neutral price process, dS = (r − q)dt + σdW, S (1.1.5) where the expected return... European options can only be exercised on the maturity date of the contract American style options, however, can be exercised at any time between the start of the contract and the maturity date, which makes it much harder to find the price 1.1 Black-Scholes-Merton Framework and its Numerical Approaches The theory of option pricing could be traced back to [7], who revolutionized option pricing with the... Wj denoted ρij The PDE for the value, V , of an option that depends on the evolution of m different underlying assets, all in the same country, with price 0 < Si < ∞, where i = 1, · · · , m, is − where m ∂V = D[V ] − rV, ∂t 1 ∂2 D[·] = ρij σi σj Si Sj + 2 i,j=1 ∂Si ∂Sj (1.1.7) m (r − qi )Si i=1 ∂ ∂Si The boundary conditions for some multi- asset options pricing problem are required according to Fichera’s... Poisson process dqi is defined by   0 with probability 1 − λ dt, i dqi =  1 with probability λ dt, i the parameter λi is the mean arrival time of the Poisson process And the jump size Ji follows normal distribution with mean mi and standard deviation γi Here we assume the jump size are uncorrelated 1.2 Jump Diffusion Model 19 We assume that the assets and the bond can be bought and sold without... not usually be integrated into the transformed pricing equation In the following years, several papers were published using FE for various pricing problems: convertibles ([34], [35]) and various exotic options ([36], [37]) A more general study about application of finite element to option pricing is detailed in [38] and [39] Problems arising from option pricing are usually of the following form, ∂u =... stochastic processes, which generalize Brownian motion Looking at the definition of Brownian motion, we would like to have a similar, i.e., with independent and stationary increments, process, based on a more general distribution than the Normal distribution However, in order to define such a stochastic process with independent and stationary increments, the distribution has to be infinitely divisible such processes . Nu- merical Approaches The theory of option pricing could be traced back to [7], who revolutionized option pricing with the introduction of the first modern option pricing model. In the same year,. functions for some multi-asset options . . . . . . . . . . . . . 40 6.1 Parameters for polynomial option: (S 1 + S 2 ) 2 . . . . . . . . . . . . 88 6.2 Results for polynomial option with positive. for polynomial option with negative correlation: ρ = −0.3 . 90 6.4 Parameters for pricing basket put option:  K −  2 i=1 w i S i  + . . . . 91 6.5 Results for some two-asset put options . . .