Pricing Algorithms for financial derivatives on baskets modeled by Lévy copulas

Transcription

1 Pricing Algorithms for financial derivatives on baskets modeled by Lévy copulas Christoph Winter, ETH Zurich, Seminar for Applied Mathematics École Polytechnique, Paris, September 6 8, 26

2 Introduction Option pricing Discretization Numerical results Conclusion Introduction Option pricing Partial integrodifferential equation Variational formulation Discretization Sparse tensor product finite element space Galerkin discretization Numerical quadrature of the Lévy copula kernel Numerical results Conclusion C. Winter École Polytechnique, Paris, September 6 8, 26 p. 2

3 Literature Introduction Option pricing Discretization Numerical results Conclusion W. Farkas, N. Reich, C. Schwab, Anisotropic stable Lévy copula processes Analytical and numerical aspects., Research report No. 26-8, SAM, ETH Zurich, 26. J. Kallsen and P. Tankov, Characterization of dependence of multidimensional Lévy processes using Lévy copulas., Journal of Multivariate Analysis, Vol. 97, pp , (26). A.-M. Matache, T. von Petersdorff, C. Schwab, Fast deterministic pricing of options on Lévy driven assets., M2AN Math. Model. Numer. Anal., Vol. 38, pp. 37 7, (24). T. von Petersdorff and C. Schwab, Numerical solution of parabolic equations in high dimensions., M2AN Math. Model. Numer. Anal., Vol. 38, pp , (24). C. Winter École Polytechnique, Paris, September 6 8, 26 p. 3

4 Introduction Option pricing Discretization Numerical results Conclusion Lévy copula and tail integral A function F : R d R is called Lévy copula if F(u,..., u d ) for (u,..., u d ) (,..., ), F(u,..., u d ) = if u i = for at least one i {,..., d}, F is d-increasing, F {i} (u) = u for any i {,..., d}, u R. The tail integral U : R d \{} R d d U(x,..., x 2 ) = sgn(x j )ν I(x j ). i= j= C. Winter École Polytechnique, Paris, September 6 8, 26 p. 4

5 Introduction Option pricing Discretization Numerical results Conclusion Theorem (Sklar s theorem for Lévy copulas) For any Lévy process X R d exists a Lévy copula F such that the tail integrals of X satisfy U I ((x i ) i I ) = F I ((U i (x i )) i I ), for any nonempty I {,..., d} and any (x i ) i I R I \{}. The Lévy copula F is unique on d i= RanU i. Lévy density k with marginal Lévy densities k,..., k d, k(x,..., x d ) =... d F ξ =U (x ),...,ξ d =U d (x d )k (x)... k d (x). C. Winter École Polytechnique, Paris, September 6 8, 26 p. 5

6 Clayton Lévy copula In two dimensions (d=2) F(u, v) = Introduction Option pricing Discretization Numerical results Conclusion ( u θ + v θ) θ ( η{uv } ( η) {uv } ), (α, α 2 )-stable marginal densities k(x, x 2 ) = ( + θ) α θ+ α θ+ 2 x αθ x 2 α 2θ ( ) α θ x αθ + α θ 2 x 2 α 2θ θ 2 ( η {x x 2 } + ( η) {x x 2 <}). C. Winter École Polytechnique, Paris, September 6 8, 26 p. 6

7 Introduction Option pricing Discretization Numerical results Conclusion Clayton Lévy copula with marginal of CGMY type C i =, G i = M i = 4, Y i = for i =, 2 and η = 2 Independent θ =.5 (left) and dependent θ = (right) tails. C. Winter École Polytechnique, Paris, September 6 8, 26 p. 7

8 Introduction Option pricing Discretization Numerical results Conclusion Tempered stable Lévy copula processes Let densities k,..., k d be tempered stable. With Sklar s theorem for Lévy copulas there exist a Lévy process X t R d with marginal densities k,..., k d. Log prices are solution of the generalized BS equation u t + Au =, u t=t = g, where A is the infinitesimal generator of the process X t with domain D(A). C. Winter École Polytechnique, Paris, September 6 8, 26 p. 8

9 Partial integrodifferential equation Assume S i t = S i ert+x i t, i d. The price is the solution of Introduction Option pricing Discretization Numerical results Conclusion ( ) V (t, S) = E e r(t t) g(s T ) S t = S, V t (t, S) + d 2 V S i S j A ij + r 2 S i S j i,j= ( d + V (t, Se z ) V (t, S) R d i= Terminal condition V (T, S) = g(s). d i= S i V S i (t, S) rv (t, S) S i (e z i ) V S i (t, S) ) ν(dz) =. C. Winter École Polytechnique, Paris, September 6 8, 26 p. 9

10 Transformation to log price Let x i = log S i, τ = T t. with A BS [ϕ] = 2 A J [ϕ] = R d Initial condition i,j= Introduction Option pricing Discretization Numerical results Conclusion u τ + A BS[u] + A J [u] =, d 2 ϕ d ( ) ϕ A ij + x i x j 2 A ii r + rϕ, x i i= ( ) d ϕ(x + z) ϕ(x) (e z i ) ϕ (x) ν(dz). x i i= u(, x) := u = g(e x,..., e x d ). C. Winter École Polytechnique, Paris, September 6 8, 26 p.

11 Variational formulation Basket option g(e x, e x 2,..., e x d ) = Weighted Sobolev space H η (Rd ) := Introduction Option pricing Discretization Numerical results Conclusion ( d i= ex i ) +. { } ϕ L loc (Rd ) e η(x) η(x) ϕ ϕ, e L 2 (R d ), i =,..., d x i Payoff g H η(r d ) where, η(x) = d i= ( µ + i {xi >} + µ i {xi <}) xi, with µ + i >, µ i >. C. Winter École Polytechnique, Paris, September 6 8, 26 p.

12 Bilinear forms Introduction Option pricing Discretization Numerical results Conclusion We associate with A BS the bilinear form a η BS R (u, v) = A BS [u](x)v(x)e 2η(x) dx, d and with A J a η J (u, v) = R d A J [u](x)v(x)e 2η(x) dx, and set a η (u, v) = a η BS (u, v) + aη J (u, v). C. Winter École Polytechnique, Paris, September 6 8, 26 p. 2

13 Continuity and Gårding inequality Assume A > and η L loc (Rd ) satisfies (i) η x i L (R d ) i d, Introduction Option pricing Discretization Numerical results Conclusion (ii) η(x + θz) η(x) η(z) x, z R d, θ [, ], (iii) e η(z) z { z >} ν(dz) <. R d Then, there exist constants C, C 2, C 3 > such that a η (u, v) C u H η (R d ) v H η (Rd ), a η (u, u) C2 u 2 H η (Rd ) C 3 u 2 L 2 η (Rd ). C. Winter École Polytechnique, Paris, September 6 8, 26 p. 3

14 Introduction Option pricing Discretization Numerical results Conclusion Sparse tensor product finite element space d=: Wavelet basis on [ R, R] { V L = span ψj l l L, j M l} = W W l, with increment spaces In [ R, R] d W := V, V L := V L V L = Sparse tensor product space V L := { W l := span ψj l : j M l}. l + +l d L l i L W l W l d. W l W l d. C. Winter École Polytechnique, Paris, September 6 8, 26 p. 4

15 Introduction Option pricing Discretization Numerical results Conclusion Sparse tensor product space (d = 2) Difference between V L and V L for level L = 3. C. Winter École Polytechnique, Paris, September 6 8, 26 p. 5

16 Introduction Option pricing Discretization Numerical results Conclusion Galerkin discretization Ansatz u L (τ, x) = l,j u l j (τ)ψl j (x). Linear system M d U (τ) + A d U(τ) =, τ (, T ), U() = U. Backward Euler time stepping ( M d + ta d) U(τ m ) = M d U(τ m ), m =,..., M, U() = U. C. Winter École Polytechnique, Paris, September 6 8, 26 p. 6

17 Discretized operator (d = 2) We need to compute a BS (ψ l j, ψl j ) = In matrix form d i,k= Introduction Option pricing Discretization Numerical results Conclusion 2 A ψl j ψj l ik dx + Ω R x i x k d i= A 2 BS := 2 A S M + 2 A 22M S + A 2 C ( C) + 2 A C M + 2 A 22M C. 2 A ψl j ii ψj l Ω R x dx. i C. Winter École Polytechnique, Paris, September 6 8, 26 p. 7

18 Discretized operator (d = 2) For the jump part a J (ψj l, ψl j ) = R d In matrix form A 2 J := R d d i= ( Introduction Option pricing Discretization Numerical results Conclusion Ω R ( ψ l j (x + z)ψl j (x) ψl j (x)ψl j (x) (e z i ) ψl j x i ψ l j dx )ν(dz). M,z M,z 2 M M (e z )C M (e z 2 )M C ) ν(dz). C. Winter École Polytechnique, Paris, September 6 8, 26 p. 8

19 Introduction Option pricing Discretization Numerical results Conclusion Numerical quadrature of the Lévy Copula kernel Quadrature points for N = 6 and θ =.5. Computation of Ω R z 2 k(z)dz with C i =, G i = M i = 8, Y i = for i =, 2 and η = theta =.5 theta = Error Number of Refinments C. Winter École Polytechnique, Paris, September 6 8, 26 p. 9

20 Introduction Option pricing Discretization Numerical results Conclusion Operator Matrix (in wavelet basis) For d = 2, L = 5 and R = 5 C = (, ), Y = (, ), G = (8, 8), M = (8, 8) and η = 2, θ =. Matrix on full grid (left) and on sparse grid (right) C. Winter 5 2 École Polytechnique, Paris, September 6 8, p. 2

21 Introduction Option pricing Discretization Numerical results Conclusion Comparison stable vs tempered stable processes For level L = 6. Matrix for stable processes (left) and tempered stable processes (right) C. Winter École Polytechnique, Paris, September 6 8, 26 p. 2

22 Multi-asset options Let T =.5 and r =. Introduction Option pricing Discretization Numerical results Conclusion Maximum put options (left) g = ( max(s, S 2,..., S d )) + ( Basket options (right) g = ) + d i= S i Sy Sx.2..5 Sy Sx C. Winter École Polytechnique, Paris, September 6 8, 26 p. 22

23 Introduction Option pricing Discretization Numerical results Conclusion Influence of the dependence structure Difference between strong and weak dependence for maximum put option (left) and basket option (right) 2 x 3 5 x Sy Sx.5 Sy Sx C. Winter École Polytechnique, Paris, September 6 8, 26 p. 23

24 Introduction Option pricing Discretization Numerical results Conclusion Conclusion Efficient quadrature rule Sparse grid technique Influence of exponential tails on matrix Influence of dependence structure on option prices C. Winter École Polytechnique, Paris, September 6 8, 26 p. 24