PDE Methods for Option Pricing under Jump Diffusion Processes

Transcription

1 PDE Methods for Option Pricing under Jump Diffusion Processes Prof Kevin Parrott University of Greenwich November 2009 Typeset by FoilTEX

2 Summary Merton jump diffusion American options Levy Processes - Variance Gamma and CGMY Asian Options under Jump Diffusions Conclusions 1

3 Merton s Jump-Diffusion model Merton s model consists of a jump term added to geometric Brownian motion i.e. ds t /S t = (µ λκ)dt + σdw t + dj t where J t is a compound Poisson process with rate λ i.e. J t = N t j=1 (Y j 1), where E[N t ] = λt and P {N t = n} = e λt(λt)n n! If the j th jump occurs at time t and S t is the asset price immediately before the jump, then S t = Y j S t. The compensation constant κ = E[Y 1]. 2

4 Merton derives an analytic solution for a call option when the Y are lognormally distributed ln Y N(a, b 2 ), i.e. f(y) = } 1 { y (2π)b exp (ln y a)2 2b 2 κ = e (a+0.5b2) 1 The solution to the jump-diffusion sde is S t = S 0 e ((µ λκ 0.5σ2 )t+σw t ) so multiplicative jumps can be seen to be a natural extension of GBM. N t j=1 Y j 3

5 Levy Processes This jump-diffusion model is an example of a more general class of exponential Levy models S t = S 0 e X t where X t is a Levy process consisting of a drift term, a Brownian motion term and a superposition of Poisson processes with various jump sizes and allowing for infinite jump rates or activities (of vanishingly small jumps). The Merton model has a Levy density ν(x) = } λ (x a)2 exp { (2π)b 2b 2, where x = ln y and is a finite activity process ( ν(x)dx = λ). 4

6 Variance Gamma Variance Gamma is a Levy process and is widely used as a share price model; it has infinite but relatively low activity of small jumps ν(x) = exp { λ px} cx 1 x>0 + exp { λ n x } 1 x<0, where x = ln y c x 5

7 CGMY CGMY is another Levy process and is widely used as a share price model; it is identical to Variance Gamma if Y = 0 ν(x) = C exp { Mx} x 1+Y 1 x>0 + C exp { G x } x 1+Y 1 x<0, where x = ln y 6

8 Pricing PIDE Under certain assumptions, the Option price V, when the underlying follows an exponential Levy process, can be shown to obey the following partial-integrodifferential equation V t = 1 2 σ2 S 2 2 V S 2 +rs V S rv + (V (Se x ) V (S) S(e x 1) V S )ν(x)dx or V t = LV + I(V ) or inequality in the case of American Options. A flexible approach to pricing is to solve a finite difference approximation to this PIDE/PIDI. 7

9 The PIDE can be integrated between two time levels at t m t m 1 V t dt = V (S j, t m 1 ) V (S j, t m ) = t m t m 1 (L V (S j, t) + I(V (S, t)) dt and using θ method time quadrature leads to the finite difference approximation V m 1 j V m j = t [ θ(l S V m 1 j + I j (V m 1 )) + (1 θ)(l S V m j + I j (V m )) ]. where L S and I j (V m ) are 2 nd order approximations to the continuous operators, and Vj m are discrete prices. A uniform spatial mesh in S, or in u = ln S, would not distribute the error efficiently; a better choice is either to use a non-uniform meshes (Forsyth, Toivanen) or uniform meshes with coordinate stretching (this work). 8

10 Co-ordinate stretching The region of asset price interest is determined by the payoff; S = k for a vanilla call or put. In this case a simple analytic transformation S = a sinh(ξ L) + c can be used to refine the mesh around S = K Choose the curve to pass through (0, 0) c = a sinh L and the maximum stretch to be at S = K c = K. i.e. S = K sinh(l) sinh(ξ L) + K. The parameter L controls the stretch. The transformed equations are more complex, however implementing the finite difference equations is straightforward (on a unifrom mesh in the transformed variable). 9

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12 Comments on the convolution integral The integrand is localised by the exponential tapering of a Levy density and moreover is identically zero when the prices are locally linear. The main accuracy requirement is consequently around the strike region (or between the strike and the optimal exercise boundary). Local mesh refinement of this region will control errors both in the Black-Scholes operator and convolution quadrature. Using V SS = 0 at S = S max as a mesh truncation condition enforces linearity in the prices beyond S max. The coordinate stretching is also an efficient way to provide a locally linear region to the right of the strike (mesh size increases exponentially). 11

13 Evaluating the discrete convolution integral ( [ ] ) V I j (V ) = V (S j e x ) V j S j (e x 1) ν(x)dx S S j Approximating this integral using quadrature appears to require interpolation for V (S j e x ), however this can be avoided. This is easier to follow with a transformation back to the asset price jump y = e x i.e. I j (V ) = 0 ( V (S j y) V j S j (y 1) [ ] ) V S S j ν(y)dy = 0 F j (y) ν(y)dy 12

14 Localisation Let Ω = {S 0, S 1,... S N = S max } be the finite difference mesh in S. Then the following localisation I j I loc j = y max j 0 F j (y) ν(y)dy where y max j = S N /S j is accurate providing that mesh truncation is sufficiently far from the strike. Note that the effect of asset price jumps outside the mesh are ignored e.g. for j = N, I N (V ) 1 0 F N(y) ν(y)dy; this is consistent with imposing the truncation condition V SS = 0, at S = S max since the integrand will then be zero for S > S max. 13

15 Mesh based quadrature For a given localisation yj max = y N = S N /S j, construct a partition of all possible jumps {[y 0, y 1 ],..., [y p, y p+1 ],..., [y N 1, y N ]}. Then I loc j = p=n 1 p=0 yp+1 y p F j (y) ν(y)dy Applying the trapezoidal rule to each interval, I loc j Îloc j = p=n 1 p=0 ( ) 1 2 (y p+1 y p )[F j (y p+1 ) ν(y p+1 ) + F j (y p ) ν(y p )] 14

16 Discrete densities The localised integral term can be rewritten as Î loc j = (y N y N 1 ) F j (y N ) ν(y N ) + 2 p=n 2 p=1 + (y 1 y 0 ) F j (y 0 ) ν(y 0 ) 2 p=n = ν jp F j (y p ) p=0 (y p+1 y p 1 ) F j (y p ) ν(y p ) 2 where F j (y p ) = V p V j (y p 1) [ ] V S and the ν S jp are discrete densities for a j jump from S j to S p. 15

17 Quadrature with coordinate transformations 1. The non-uniform mesh in S creates a non-uniform set of quadrature intervals in y = S/S j. 2. The quadrature sampling is most accurate in the region of the greatest transformation stretch (e.g. the strike S = K) i.e. where the integrand has its maximum variation. 3. Simpson s rule (note that the quadrature rule is uniformly spaced in ξ) improves the discrete quadrature measures but computational tests show little effect on the option prices. 16

18 European Price Convergence - Merton stretch = 1 stretch = 10 N M value error ratio Rich. value error ratio Rich n/a n/a Table 1: Numerical convergence comparison of uniform and stretched grids for an European Put with Merton jump diffusion. Prices at S=100, where K = 100, T = 0.25, r = 0.05, σ = 0.15, λ = 0.1, a = 0.9, b = Exact value =

19 American Options The pricing PIDE is now replaced by an inequality since the hedged portfolio value can only have a return bounded above by the risk-free return i.e. V t LV + 0 (V (Sy) V (S) (y 1)S V S ) ν(y)dy = LV + I(V ) An American option cannot fall beneath its immediate payoff g(s, t), thus V g must also hold. problem These two conditions combine into a linear complementarity (V g)( V t + LV + I(V )) = 0 18

20 Discrete Complementarity The discrete linear complementarity problem is ( V m 1 j g(s j, t m 1 ) ) ( V m 1 j Vj m t [ θ(l S V m 1 j + I j (V m 1 )) + (1 θ)(l S V m j + I j (V m )) ]) = 0 Rannacher timestepping is effective in avoiding oscillations in Gamma (cf Giles et al). 19

21 American Price Convergence - Merton stretch = 1 stretch = 10 N M value Rich error ratio value Rich error ratio Table 2: Numerical convergence comparison of uniform and stretched grids for an American Put with Merton jump diffusion. Prices at S=100, where K = 100, T = 0.25, r = 0.05, σ = 0.15, λ = 0.1, a = 0.9, b = Exact value

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23 Figure 1: Stretched grid Gamma for Merton jump diffusion, American Put 22

24 a j V m 1 j 1 A j Vj 1 m + B j Vj m In matrix notation Algebraic structure + b j V m 1 j p=n + c j V m 1 j+1 θ t p=n + C j Vj+1 m + (1 θ) t p=0 (M + M c )V m 1 = f p=0 ν jp F p = ν jp F p, j = 0,... N See approaches in Forsyth et al and Cont and Voltchkova for fast techniques for dealing with the computational cost of the full matrix M c. PSOR has been found to work satisfactorily for the implicit case and early exercise. Note that M c entries will decay exponentially fast away from the diagonal. 23

25 Infinite activity Levy densities This approach needs to be modified for Variance Gamma and CGMY since ν(y p ) = ν(y p ) is singular for p = j i.e. y = 1. The terms that have to be treated separately in I j are p=j p=j 1 yp+1 y p F (S j y) ν(y)dy = yj+1 y j 1 ( V (S j y) V (S j ) S j (y 1) V ) S ν(y)dy The approach described here follows Rama Cont (and also Peter Forsyth). y j 1 = S j 1 S j y j = 1 y j+1 = S j+1 S j S j 1 S j S j+1 24

26 I sing j = yj+1 y j 1 Effective volatility σ ϵ ( V (S j y) V (S j ) S j (y 1) V ) S ν(y)dy Expand V (S j y) in a Taylor series about S j for small jumps y 1 = ϵ 1 V (S j y) = V (S j ) + S j (y 1) V S S2 j (y 1) 2 2 V S Then I sing where σ 2 ϵ = j yj+1 y j 1 = 1 2 σ2 ϵ S 2 j 2 V S 2 (y 1) 2 ν(y)dy. 25

27 σ 2 ϵ = xj+1 x j 1 (e x 1) 2 C Evaluating σ 2 ϵ [ ] e Mx x 1+Y 1 x>0 + e G x x 1+Y 1 x<0 dx, where x = ln(y). x j 1 = ln( S j 1 S j ) x j = 0 x j+1 = ln( S j+1 S j ) S j 1 S j 1 S j S j S j+1 Define x D j = x j x j 1 = 0 ln( S j 1 ) = ln(1 + S j 1 ) S j 1 /S j S j S j 1 and x U j = x j+1 x j = ln( S j+1 ) 0 = ln(1 + S j ) S j /S j S j S j 26

28 Then σ 2 ϵ = x U j 0 e Mx x 1+Y (ex 1) 2 dx + x D j 0 e Gx x 1+Y (ex 1) 2 dx = x U j 0 x 1 Y e Mx (1+ x x D j 2! +x2 3! +...)2 dx+ 0 x 1 Y e Gx (1+ x 2! +x2 3!!+...)2 dx which for Y < 2, can be expressed in terms of incomplete Gamma functions, and converges quickly (3 terms needed). 27

29 European Price Convergence - Variance Gamma stretch = 1 stretch = 10 stretch = 20 N value error value error value error Table 3: Numerical convergence of European Call Prices with Variance Gamma comparing different stretched grids with Rannacher timestepping. Prices linearly interpolated to S=90 where K = 98, r = 0.0, T = 0.5, ν = , λ n = , λ p = Convergence rates not quoted as they were affected by the interpolation. Exact

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31 30 Figure 2: VG American Put showing lack of smooth pasting

32 American Price Convergence - Variance Gamma stretch = 1 stretch = 10 stretch = 20 N value error r tio value error r tio value error r tio Table 4: Numerical convergence comparison of uniform and stretched grids for an American Put under VG. Prices at S=100, where K = 100, T = 0.5, r = 0.05, σ = 0.0, ν = , λ n = , λ p = Limiting value

33 European Price Convergence - CGMY Wang et al this work N Value Error Value Error Table 5: Comparison of stretched grid (10 : 1) with Wang, Wan and Forsyth (2006) for CGMY. European Put prices linearly interpolated to S = 90 where K = 98, r = 0.06, σ = 0.0, C = 16.97, G = 7.08, M = 29.97, Y = Exact value is

34 American Price Convergence - CGMY Wang et al this work N Value Error Value Error Table 6: Comparison of stretched grid (10 : 1) with Wang, Wan and Forsyth (2006) for CGMY. American Put prices linearly interpolated to S = 90 where K = 98, r = 0.06, σ = 0.0, C = 0.42, G = 4, 37, M = 191.2, Y = Exact value

35 American Price-Convergence - CGMY stretch = 1 stretch = 10 N value error r tio value error r tio na na na Table 7: Convergence comparison of uniform and stretched grids for an American Put under CGMY. Prices at S=98, where K = 98, T = 0.25, r = 0.06, σ = 0.0, C = 0.42, G = 4.37, M = 191.2, Y = Limiting value

36 Asian Options Define the continuously sampled average A t price as A t = 1 t t 0 S τ dτ where A 0 = S 0, and which evolves according to da t = 1 t (S t A t )dt where as before ds t /S t = (µ λκ)dt + σdw t + dj t 35

37 Asian Pricing PDE With no jumps, the PDE for the price V (A, S, t) follows an ultra-parabolic equation V t = 1 2 σ2 S 2 2 V S 2 + rs V S + 1 t with a final condition (payoff) at time t = T, V (S, A, T ) = g(s, A, T ) (S A) V A rv There are transformations that reduce the dimensionality but this is not always possible. In this case the pde can be solved efficiently using finite differences combined with a semi-lagrange integration method. 36

38 The Lagrangian derivative dv dt The Asian pde then simplifies to Semi-Lagrangian approach along any path in the A t plane is given by dv dt = V t + V da A dt. along paths P(A, t; S) such that dv dt = 1 2 σ2 S 2 2 V S 2 + rs V S rv = LV da dt = 1 (S A). t 37

39 Discretisation Take the integral of the PDE along the path P m k (A, t; S j) (Ãk, t m ) (A k, t m 1 ), i.e. V (S j, A k, t m 1 ) V (S j, Ãk, t m ) = P m k (A,t;S j) L S V (S, A, t)dt. Finite difference prices {V m j,k } on the mesh {S 0,...S N } {A 0,...A N } can be derived using theta method quadrature and L S, the O(h 2 ) approximation to L V m 1 jk ( = Ṽ jk m + t θl S (V m 1 jk ) + (1 θ) L ) S (Vjk m ) Ṽjk m is the mesh price interpolated to Ãk. Vj,k m V (S j, A k, t m ) to O( t 2 )+O(h 2 ) for θ = 0.5 and the equations are unconditionally stable for θ

40 Determining Ãk Integrating () with respect to time along the path P m k (A, t; S) gives Ãk A k 1 S j A da = t m t m 1 1 t dt thus = ln[(s j A k )/(S j Ãk)] = ln[(t m /t m 1 )] Ã k = S j ( t m 1 t m ) (S j A k ) = αa k + (1 α)s j, where α = tm 1 t m. 39

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42 Boundary condition at S = 0 If at some time t the asset price S t = 0 then it remains zero; hence the final average price is known A T = 1 T T 0 S t dt = 1 T t 0 S t dt = t T A t and hence the payoff, A 0 is also known, e.g. for an average rate put V (0, A, t) = e r(t T ) max(k t T A, 0) Alternatively use the exact solution of the pde for S = 0 (it becomes linear hyperbolic). 41

43 Procedure for S-L Integration The solution is carried out backwards in time, so beginning with values at T, values at t m 1 are obtained, and so on. A set of tri-diagonal equations are solved at each timestep (no early exercise) Prices and difference approximations at trajectory endpoints are obtained by cubic-spline interpolation in the A-direction. If the option has early exercise then the A-parameterised problems becomes parameterised LCP s and can be solved using PSOR. 42

44 Price Evolution of a Fixed-strike Asian call 43

45 44 Figure 3: At t = expiry (6 months).

46 45 Figure 4: At t = 4 months.

47 46 Figure 5: At t = 2 months.

48 47 Figure 6: At t = 0.

49 Results with early-exercise Table 8: Semi-Lagrangian (S-L) convergence for Fixed Strike Put Asians with early exercise, where r = 0.1, quoted at strike K = 100. nt = 40, for all maturities. λ = 5. σ T Calculated by S-L (BP, 1996) 20x20 40x40 80x80 160x

50 Fixed-strike Asian Put with early exercise 49

51 50 Figure 7: At t = expiry (3 months).

52 51 Figure 8: At t = 2 months.

53 52 Figure 9: At t = 1 month.

54 53 Figure 10: At t = 0.

55 Asian Pricing PIDE With jumps, the price V (A, S, t) now follows an ultra-parabolic PIDE V t = 1 2 σ2 S 2 2 V S 2 + rs V S + 1 (S A) V t A rv + (V (Se x ) V (S) S(e x 1) V S )ν(x)dx 54

56 This simplifies as before with semi-lagrange integration to dv dt = 1 2 σ2 S 2 2 V S 2 +rs V S rv + (V (Se x ) V (S) S(e x 1) V S )ν(x)dx along paths P(A, t; S) such that = LV + I(V ) da dt = 1 (S A). t This structure is again an A-parameterised version of the single factor jump diffusion problem, so the same finite difference approach can be used for a fixed set of average price values. 55

57 Discretisation Take the integral of the PIDE along the path P m k (A, t; S j) (Ãk, t m ) (A k, t m 1 ), i.e. V (S j, A k, t m 1 ) V (S j, Ãk, t m ) = P m k (A,t;S j) (L S V (S, A, t) + I(V (S, A, t))) dt. Finite difference prices {V m j,k } on the mesh {S 0,...S N } {A 0,...A N } can be derived using theta method quadrature and L S, the O(h 2 ) approximation to L V m 1 jk ( ) ( ) = Ṽ jk m +θ t L S (V m 1 jk ) + I j (V m 1 ) +(1 θ) t L S (Vjk m ) + Ĩj(V m 1 ) Ṽjk m is the mesh price interpolated to Ãk. Vj,k m V (S j, A k, t m ) to O( t 2 )+O(h 2 ) for θ = 0.5 and the equations are unconditionally stable for θ

58 Comments Mesh based quadrature with coordinate stretching leads to accurate finite difference approximations to the for jump diffusion option pricing PIDEs (Merton, VG, CGMY) Semi-Lagrange time integration simplifies an Asian pricing PIDE into a A- parameterised set of one-factor problems very similar to the above and permits the same approach to early exercise The S-L method is unconditionally stable when combined with implicit finite differences and can be applied to discrete averaging and to volatility surfaces. The combined use of S-L and mesh based quadrature with coordinate transformations can be expected to lead to flexible, efficient and accurate early-exercise Asian pricing for a jump diffusion models 57

59 For more complex Asian options e.g. assets with stochastic volatility, see Parrott & Clarke, Parallel Solution of American Asian Options, Procs. of the 11th Domain Decomposition Conference, 1999.) Y. d Halluin, P.A. Forsyth, G. Labahn, A semi-lagrangian approach for American Asian options under jump diffusion, SIAM Journal on Scientific Computing, 27 (2005)

60 Merton RC Option pricing when the underlying stock is discontinuous, Journal of Financial Economics, (1976). A Hirsa and D. B. Madan, Pricing American Options Under Variance Gamma, Journal of Computational Finance, (2004). Y D Halluin P A Forsyth, G Labahn, A Penalty method for American Options with Jump Diffusion Processes, Numerische Mathematik, (2004). R Cont and P Tankov, Financial Modelling for Jump Processes, Chapman and Hall,

61 R Cont and E Voltchkova, A Finite Difference scheme for Option Pricing in Jump Diffusion and Exponential Levy Models, SIAM J Num Anal, A Almandral and C W Oosterlee, Highly accurate evaluation of European and American Options under the Variance Gamma Process, Working Paper K Oosterlee, A. Almendral, Accurate Evaluation of European and American Options Under the CGMY Process. SIAM J. Sci Comput. 29: (2007) I R Wang, J W I Wan, P A Forsyth, Robust numerical valuation of European and American options under the CGMY process, Working Paper (August 23, 2006) 60