Numerical valuation for option pricing under jump-diffusion models by finite differences

Transcription

1 Numerical valuation for option pricing under jump-diffusion models by finite differences YongHoon Kwon Younhee Lee Department of Mathematics Pohang University of Science and Technology June 23, 2010

2 Table of contents 1 Introduction 2 Numerical method for option pricing Implicit method with three time levels Numerical analysis 3 Numerical results 4 Conclusions

3 Introduction Exponential jump-diffusion model We assume that the stock price process S t in a risk-neutral world follows an exponential jump-diffusion model where ds t /S t = (r λζ)dt + σdw t + ηdn t, (1) r : the riskfree interest rate, σ : the volatility, W t : the Wiener process, N t : the Poisson process with intensity λ, η : a random variable of jump size from S t to (η + 1)S t, ζ : the expectation E[η] of the random variable η.

4 Introduction Exponential jump-diffusion model Example (Jump-diffusion model) (1) Merton model ln(η + 1) N(µ J, σj 2 ). (2) (2) Kou model f (x) = pλ + e λ+x 1 x 0 + (1 p)λ e λ x 1 x<0, (3) where f (x) is a density function of ln(η + 1).

5 Introduction PIDE under jump-diffusion models Under exponential jump-diffusion model, the price of a European call option C(t, S) satisfies the PIDE below. C t (t, S) + σ2 S R 2 C C (t, S) + rs S 2 S (t, S) rc(t, S) ] ν(dx) = 0 [ C(t, Se x ) C(t, S) S(e x 1) C (t, S) S on [0, T ) (0, ) with the terminal condition where K is a strike price. C(T, S) = (S K) + for all S > 0,

6 Let Introduction PIDE under jump-diffusion models τ = T t, x = ln(s/s 0 ). By the change of variables, u(τ, x) = C(T τ, S 0 e x ) satisfies u σ2 2 u σ2 (τ, x) = (τ, x) + (r τ 2 x 2 2 λζ) u (τ, x) x (r + λ)u(τ, x) + λ u(τ, z)f (z x)dz (4) on (0, T ] (, ) with the initial condition u(0, x) = (S 0 e x K) + for all x (, ), (5) where ζ = R (ex 1)f (x)dx with the distribution function of jumps f (x) and λ is the intensity of jumps. R

7 Introduction Survey of option pricing In the sense of the viscosity solution, Briani, Chioma, and Natalini (2004) - An explicit difference method. Cont and Voltchkova (2005) - An explicit-implicit method.

8 Introduction Survey of option pricing In the sense of the viscosity solution, Briani, Chioma, and Natalini (2004) - An explicit difference method. Cont and Voltchkova (2005) - An explicit-implicit method. As using an iterative method, d Halluin, Forsyth, and Vetzal (2005) - An implicit method of the Crank-Nicolson type. Almendral and Oosterlee (2005) - A backward differentiation formula (BDF2).

9 Numerical method for option pricing Implicit method with three time levels We shall construct a numerical method with finite differences to solve the following initial-valued PIDE where u (τ, x) = Lu(τ, x) on (0, T ] R, (6) τ u(0, x) = (S 0 e x K) +, (7) Lu(τ, x) =Du(τ, x) + Iu(τ, x) (r + λ)u(τ, x), Du(τ, x) = σ2 2 u σ2 (τ, x) + (r 2 x 2 2 λζ) u (τ, x), x Iu(τ, x) =λ u(τ, z)f (z x)dz. R

10 Numerical method for option pricing Implicit method with three time levels At first, we have to restrict the domain R of the space variable to a bounded interval. The asymptotic behavior of the price of a European call option is described by lim u(τ, x) = 0, lim x u(τ, x) = S 0e x Ke rτ. (8) x So, there exists an interval Ω := [ X, X ], X > 0 such that we can divide the integral term into two parts u(τ, z)f (z x)dz = u(τ, z)f (z x)dz+ u(τ, z)f (z x)dz R Ω R\Ω

11 Numerical method for option pricing Implicit method with three time levels Let us define R(τ, x, X ) by R(τ, x, X ) = R\Ω u(τ, z)f (z x)dz. In the case of Merton model R(τ, x, X ) ( = S 0 e x+µ J+ σ 2 J x X + µj + σ 2 2 Φ J σ J where In the case of Kou model Φ(y) = 1 2π y ) Ke rτ Φ e x2 2 dx. ( x X + µj R(τ, x, X ) = S 0 pλ + λ + 1 eλ+x (λ+ 1)X Kpe rτ λ+(x x). σ J ),

12 Numerical method for option pricing Implicit method with three time levels On the truncated domain [0, T ] [ X, X ], let τ = T /N and x = 2X /M for M, N > 0. And let τ n = n τ for n = 0, 1,..., N and x m = X + m x for m = 0, 1,..., M. Let u n m = u(τ n, x m ) and f m,j = f (x j x m ). Ω u(τ n, z)f (z x m )dz x 2 u n 0f m,0 + 2 M 1 j=1 uj n f m,j + um n f m,m.

13 Numerical method for option pricing Implicit method with three time levels ( u Dum n n+1 m D where D u n m = σ2 2 I u n m = λ x 2 + um n 1 ), Ium n I u n 2 m, Lum n L um, n um+1 n 2un m + um 1 n x 2 + (r σ2 2 λζ)un m+1 un m 1, 2 x u n 0f m,0 + 2 M 1 j=1 uj n f m,j + um n f m,m + λr(τ n, x m, X ), D u L um n m n + I um n (r + λ)um n for n = 0, = ( ) D u n+1 m +un 1 m + I um n (r + λ)um n for n 1. 2

14 Numerical method for option pricing Implicit method with three time levels Algorithm of the implicit method with three time levels Initial condition: U 0 m = max(s 0 e xm K, 0) for 0 m M, Boundary condition: for m = 0, M and for 1 n N U n m = max(0, S 0 e xm Ke rτn ), (S1) For n = 0 and for 1 m M 1 Um n+1 Um n τ = D Um n + I Um n (r + λ)um, n (S2) For 1 n N 1 and for 1 m M 1 Um n+1 Un 1 m ( 2 τ = D U n+1 m +Un 1 m 2 ) + I U n m (r + λ)u n m.

15 Theorem (Consistency) Numerical method for option pricing Numerical analysis Let v C ((0, T ] R) satisfy the asymptotic behavior (8). If τ and x are sufficiently small, Then for any ɛ > 0 there exists a truncated interval [ X, X ] such that v τ (τ n, x m ) Lv(τ n, x m ) ( v(τn+1, x m ) v(τ n, x m ) τ ) L v(τ n, x m ) = O( τ + x 2 + ɛ) for n = 0, (9) ( ) v τ (τ v(τn+1, x m ) v(τ n 1, x m ) n, x m ) Lv(τ n, x m ) L v(τ n, x m ) 2 τ where (τ n, x m ) (0, T ] [ X, X ]. = O( τ 2 + x 2 + ɛ) for n 1, (10)

16 Theorem (Consistency) Numerical method for option pricing Numerical analysis Let v C ((0, T ] R) satisfy the asymptotic behavior (8). If τ and x are sufficiently small, Then for any ɛ > 0 there exists a truncated interval [ X, X ] such that v τ (τ n, x m ) Lv(τ n, x m ) ( v(τn+1, x m ) v(τ n, x m ) τ ) L v(τ n, x m ) = O( τ + x 2 + ɛ) for n = 0, (9) ( ) v τ (τ v(τn+1, x m ) v(τ n 1, x m ) n, x m ) Lv(τ n, x m ) L v(τ n, x m ) 2 τ where (τ n, x m ) (0, T ] [ X, X ]. Theorem (Stability) = O( τ 2 + x 2 + ɛ) for n 1, (10) The finite difference method (S1)-(S2) is stable in the sense of the Von Neumann analysis if τ < 1 2(r+2λ).

17 Numerical method for option pricing Numerical analysis We shall use a discrete vector norm x l 2 defined by x l 2 = x j 1/2 x j 2. Let ξ n be the error vector on the n-th time level by ξ n m = u n m U n m for 1 m M 1, where u is the unique solution of the initial-valued PIDE in (6)-(7) and U is the solution of the finite difference approximation in (S1)-(S2).

18 Numerical method for option pricing Numerical analysis Lemma Let {a n } n 0 be a nonnegative sequence such that for n 2 a n a n 2 + K τa n 1 + d, where τ, K, d are positive constants. If a 0 = 0, then for n 2 n 2 a n (1 + K τ) n 1 a 1 + d (1 + K τ) j. j=0

19 Numerical method for option pricing Numerical analysis Theorem (Convergence) If τ and x are sufficiently small, then there exists a positive constant K independent of τ and x such that for 1 n N ξ n l 2 K( τ 2 + x ɛ). (11) x 3/2

20 Numerical method for option pricing Numerical analysis Theorem (Convergence) If τ and x are sufficiently small, then there exists a positive constant K independent of τ and x such that for 1 n N ξ n l 2 K( τ 2 + x ɛ). (11) x 3/2 Corollary Suppose that all hypotheses in Theorem above are satisfied. If the conditions of ɛ = O( x 7/2 ) and x = O( τ) hold, then ξ n l 2 K( τ 2 ). (12)

21 Numerical results Merton model Example Under Merton model, parameters used in the simulation were σ = 0.15, r = 0.05, σ J = 0.45, µ J = 0.90, λ = 0.10, T = 0.25, K = 100. The order q of convergence rate was computed by U( τ, x) U( τ/2, x/2) q = log l 2 2. (13) U( τ/2, x/2) U( τ/4, x/4) l 2

22 Numerical results Merton model Table: Values of European call options obtained by the implicit method with three time levels under the Merton model. The reference values are at S = 90, at S = 100, and at S = 110. The truncated domain is [ 1.5, 1.5]. N is the number of time steps and M is the number of space steps. S = 90 S = 100 S = 110 N M Value Error Value Error Value Error

23 Numerical results Merton model Table: The rate of l 2 -errors obtained by the implicit method with three time levels under the Merton model. The truncated domain is [ 1.5, 1.5]. N is the number of time steps and M is the number of space steps. q is the rate of convergence defined by (13). N M U( τ, x) U( τ/2, x/2) l 2 q

24 Numerical results Kou model Example Under Kou model, parameters used in the simulation were σ = 0.15, r = 0.05, λ + = , λ = , p = , λ = 0.10, T = 0.25, K = 100.

25 Numerical results Kou model Table: Values of European call options obtained by the implicit method with three time levels under the Kou model. The reference values are at S = 90, at S = 100, and at S = 110. The truncated domain is [ 1.5, 1.5]. N is the number of time steps and M is the number of space steps. S = 90 S = 100 S = 110 N M Value Error Value Error Value Error

26 Numerical results Kou model Table: The rate of l 2 -errors obtained by the implicit method with three time levels under the Kou model. The truncated domain is [ 1.5, 1.5]. N is the number of time steps and M is the number of space steps. q is the rate of convergence defined by (13). N M U( τ, x) U( τ/2, x/2) l 2 q

27 Conclusions 1 The finite difference method with three time levels to solve the PIDE. 2 Consistency and stability. 3 The second-order convergence in the discrete l 2 -norm with a constant ratio τ/ x.

28 Thank you for your attention.