Sparse Wavelet Methods for Option Pricing under Lévy Stochastic Volatility models

Transcription

1 Sparse Wavelet Methods for Option Pricing under Lévy Stochastic Volatility models Norbert Hilber Seminar of Applied Mathematics ETH Zürich Workshop on Financial Modeling with Jump Processes p. 1/18

2 Outline BNS model PIDE for pricing European contingent claims FEM formulation and stabilization Sparse grid Numerical examples Conclusions and outlook Workshop on Financial Modeling with Jump Processes p. 2/18

3 References Barndorff-Nielsen, O. E. and Shepard, N.: Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics (with discussion). J. R. Stat. Soc. B, 63, (2001). Benth, F. E. and Groth, M.:The minimal entropy martingale measure and numerical option pricing for the Barndorff-Nielsen-Shepard stochastic volatility model. Burman, E. and Ern, A.: Continuous interior penalty hp-finite Element methods for transport operators. EPFL-IACS report , (2005). Nicolato, E. and Venardos, E.: Option pricing in stochastic volatility models of the Ornstein-Uhlenbeck type. Mathematical Finance, 13(4), (2003). von Petersdorff, T. and Schwab, C.: Numerical solution of parabolic equations in high dimensions. ESAIM: Mathematical Modelling and Numerical Analysis 38 N 1 (2004) Workshop on Financial Modeling with Jump Processes p. 3/18

4 The model Let (Ω, F, (F t ) 0 t T, P) be a filtered probability space. dx t = (µ + βσt 2 )dt + σ t dw t + ρ dz λt dσt 2 = λσt 2 dt + dz λt, σ0 2 > 0 where price process of stock S = (S t ) is S t = e X t W = (W t ) a Brownian motion Z = (Z λt ) a subordinator (background driving Lévy process (BDLP)) W and Z are independent β, µ, ρ, λ R, λ > 0, ρ 0 Workshop on Financial Modeling with Jump Processes p. 4/18

5 The model Extension: superposition of m independent OU-processes Z k dx t = (µ + βσ 2 t )dt + σ t dw t + σ 2 t = mx w k σk,t 2 k=1 with P m k=1 w k = 1, w k > 0, k. mx ρ k dz k,λk t k=1 dσ 2 k,t = λ k σ 2 k,tdt + dz k,λk t, σ 2 k,0 > 0 Equivalent martingale measures preserving the BNS structure (for m = 1). Let and M := {Q P : e rt+xt is a Q-martingale} M := {Q M : Q is structure preserving} Workshop on Financial Modeling with Jump Processes p. 5/18

6 The model Theorem. (E. Nicolato, E. Venardos (2003)) M. The dynamics of X under Q M are given by dx t = r λκ 1 2 σ2 t dt + σ t dw t + ρ dz λt dσ 2 t = λσ 2 t dt + dz λt, where (W t ) is a Q-Brownian motion and (Z λt ) is a Q-Lévy process with Lévy density k w (z) := w(z)k(z) with w : R + R +, Z R + p w(z) 1 2k(z)dz < The cumulant transform κ is given by κ := κ(ρ) = Z R + e ρz 1 k w (z)dz. Workshop on Financial Modeling with Jump Processes p. 6/18

7 Option pricing model problem: European put option with payoff h(x T ). Its arbitrage free price P t, t T is given by P t = E Qh i e r(t t) h(x T ) F t the process (X, σ 2 ) is Markovian P t = P (t, X t, Y t ), where Y = σ 2 and u(t, x, y) := P (t, x, y) is u(t, x, y) = E Qh e r(t t) h(x T ) i X t = x, Y t = y if u is sufficiently smooth, it satisfies the following partial integro-differential equation t u Au + ru = 0 on (0, T ] R R + u(t, ) = h(x) on R R + where A is the infinitesimal generator of the Markov process (X, σ 2 ): A := A W + A δ + A J. Workshop on Financial Modeling with Jump Processes p. 7/18

8 Option pricing A := A W + A δ + A J with A W ϕ := 1 Q(x, 2 y) ϕ, Q(x, y) := y A δ ϕ := γ(x, y) ϕ, γ(x, y) := y λκ + r 2 λy Z h i A J ϕ(x, y) := λ ϕ(x + ρz, y + z) ϕ(x, y) k w (z)dz R + ««since z Qz 0, z := (x, y) R R + the equation is degenerate since the order β of A J is β < 1 and there is no diffusion in y-direction (Q 22 = 0), the equation is hyperbolic in y in case of m BDLPs, the PIDE has m + 1 space variables (x, y 1,..., y m ) Workshop on Financial Modeling with Jump Processes p. 8/18

9 Truncation change to time-to-maturity τ := T t define excess to payoff U := u h (assuming w.l.o.g r = 0) truncate problem to a bounded domain Ω := [ a, b] [0, c] with a, b, c > 0 sufficiently large impose boundary conditions on Ω τ U + AU = f := Ah on (0, T ] Ω u(τ, ) = 0 on (0, T ] Γ D u(0, ) = 0 in Ω y Ω c Γ D Γ D a b x Workshop on Financial Modeling with Jump Processes p. 9/18

10 Stabilized FEM E. Burmann, A. Ern (and others): the addition of p(u h, v h ) = X F F x Z F F F y h 2 F γ y L (F )[ y u h ][ y v h ]ds {z } =:p y (u h,v h ) + X Z h 2 F γ x L (F )[ x u h ][ x v h ]ds, u h, v h V h F {z } =:p x (u h,v h ) to the standard continuous Galerkin formulation is used to stabilize transport operators F x (F y ) is the set of all horizontal (vertical) faces of the quadrangle mesh on Ω γ = (γ x, γ y ) Workshop on Financial Modeling with Jump Processes p. 10/18

11 Stabilized FEM let V h be the space of continuous piecewise bilinear functions (tensor product of Finite Element space of continuous piecewise linear functions on mesh of width h) let M N. Define the time step τ := T/M. the stabilized FEM formulation reads: Given u 0 h, for m = 0,..., M 1, find V h such that v V h u m+1 h 1 τ u m+1 h u m h, v L 2 (Ω) where the bilinear form a(, ) is defined as a(ϕ, ψ) := + a(u m+1 h, v) + p y (u m+1 h, v) = f, v, A[ϕ], ψ. L 2 (Ω) Workshop on Financial Modeling with Jump Processes p. 11/18

12 Stabilized FEM equivalent to: Given u 0 R dim V h, for m = 0,..., M 1, find u m+1 R dim V h such that M + τ(a + P y ) u m+1 = Mu m + τf where u m+1 is the coefficient vector of u m+1 h w.r.t basis {ϕ j } dim V h j=1 of V h A ij = a(ϕ j, ϕ i ), M ij = (ϕ j, ϕ i ) L 2 (Ω), P y ij = p y (ϕ j, ϕ i ) f j = f, ϕ j curse of dimension: for m BDLPs the space V h has dim V h = O(h (m+1) ) use sparse grid to reduce the dimension of approximation space b V h with dim b V h = O(h 1 log h m ) Workshop on Financial Modeling with Jump Processes p. 12/18

13 Sparse grid Let I = [0, 1]. Define the increment spaces W 1 = span{x, 1 x} W l = span{ψ l j : 1 j 2 l }, l 0, where ψ l j(x) := 1 2 l+1 x (2j 1). + The sparse grid space on level L 0 is then defined as bv L := M 0 l 1 +l 2 L W l 1 W l 2 M 0 l 1 L W l 1 W 1, 0 l 1, l 2 L. Workshop on Financial Modeling with Jump Processes p. 13/18

14 Sparse grids Example: L = 3 l 1 l 2 Workshop on Financial Modeling with Jump Processes p. 14/18

15 Example model parameters: Ω = [ 7, 5] [0, 4], T = 0.5, K = 1, r = 0, λ = 2.5, IG(γ, δ)-lévy kernel k(z) = δ 2 3 2π z 2 (1 + γz)e 1 2 γz with γ = 2, δ = discretization parameters: L = 7, implicit Euler with time step t = two values for ρ: left: ρ = 0.01, right: ρ = 4 Workshop on Financial Modeling with Jump Processes p. 15/18

16 Matrix compression? due to non-locality of Z ea J [ϕ](x, y) := R 2 h ϕ(x + z 1, y + z 2 ) ϕ(x, y)i ek(z1, z 2 )dz 1 dz 2 the corresponding operator matrix e A J in sparse wavelet basis is fully populated hence, complexity is O( b N 2 L), b N L = dim b V L in the BNS model, the non-local operator is A J [ϕ](x, y) = Z R + h i ϕ(x + ρz, y + z) ϕ(x, y) k(z)dz, because the kernel is one dimensional, the matrix A J is not fully populated Workshop on Financial Modeling with Jump Processes p. 16/18

17 Matrix compression? example: IG(γ, δ)-lévy kernel with density δ 3 γ 2 k(z) = z (1 + γz)e 2 z, 2 2π δ = , γ = 10, ρ = 4 53% of entries are zero Workshop on Financial Modeling with Jump Processes p. 17/18

18 Conclusions parabolic-hyperbolic PIDE to price European contingent claim under the BSN model stabilized sparse grid FEM to approximate the value function not discussed: more sophisticated time-stepping (e.g. hp-dg) multilevel wavelet preconditioning matrix compression Workshop on Financial Modeling with Jump Processes p. 18/18