Numerical Derivative Pricing in Non-BS Markets

Transcription

1 Numerical Derivative Pricing in Non-BS Markets C. Schwab joint work with N. Hilber, N. Reich, C. Winter ETH Zurich, Seminar for Applied Mathematics Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007

2 Goal: Unified methodology for pricing & hedging for all market models and contracts X = log S R d, strong Markov process. Find arbitrage-free price u of contract on X, Solve generalized BS-equation, u(t, x) = E[g(X T ) X t = x]. u t + Au = 0, u t=t = g, where A is the infinitesimal generator of X. C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 2

3 Numerical pricing: MCM, FFT, FDM, FEM FEM Find a solution u(t, ) V such that u, v + Au, v = 0 for all v V Domain(E). t }{{} E(u,v) Finite dimensional subspace 1 V L V Matrix Problem: MU L (t) + AU L (t) = 0, t (0, T ), U L (0) = U 0. This setting applies to any underlying modelled by strong Markov process. 1 L indicates number of refinement steps or Level. C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 3

4 Quadratic Hedging and Greeks Calculate derivative price u. Solve same PIDE for hedging 2 error h or sensitivity s = η u to a model parameter η, e.g. vega ( σ u) ) Γ(u, id)2 h(t, x) Ah(t, x) = (Γ(u, u) (t, x). t Γ(id, id) t s(t, x) As(t, x) = ηa[u](t, x). Remaining Greeks, e.g. delta, theta, gamma,... can be obtained directly via postprocessing 2 cf. Föllmer, Sondermann, 1986; Bouleau, Lamberton, 1989; Kallsen, Vesenmayer, 2007; Γ = carré-du-champ operator. C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 4

5 Dirichlet form of multivariate Lévy process X determined by characteristic triplet (γ, Q, ν). Then E(u, v) = ψ X (ξ)û(ξ) v(ξ)dξ = E γ (u, v) + E Q (u, v) + E ν (u, v), R d with E γ (u, v) = γ u, v, E Q (u, v) = 1 Q u, v, 2 ( ) E ν (u, v) = u(x + z) u(x) z u(x)1 z 1 v(x)ν(dz)dx. R d R d Semigroup: T t u(x) = R d e i ξ,x e tψ X (ξ)û(ξ)dξ. C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 5

6 Contracts on baskets multivariate processes Modelling issue: Parametrization of dependence Drift part deterministic no dependence, Diffusion part Pure jump part Characteristic exponent: ψ X (ξ) = i γ, ξ 1 2 ξ, Qξ + covariance matrix, Lévy copula. R d (e i ξ,x 1 i ξ, x 1 x 1 ) ν(dx), with drift vector γ R d ; volatility correlation matrix Q R d d ; Lévy measure ν(dz), R d (1 z 2 )ν(dz) <. C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 6

7 Sklar s theorem for Lévy copulas The tail integral U : R d \{0} R d d { U(x 1,..., x d ) = sgn(x j )ν (xj, ) for x j > 0 (, x j ] for x j < 0 i=1 Theorem. Lévy Process X Lévy Copula F such that Conversely, j=1 }. U I ((x i ) i I ) = F I ((U i (x i )) i I ), I {1,..., d}. (1) Lévy Copula F and given U i : R R + Lévy Process X such that (1) holds for marginal tail integrals of X. Density of X: k(x 1,..., x d ) = [ 1... d F] (U 1 (x 1 ),..., U d (x d )) k 1 (x 1 )... k d (x d ). C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 7

8 Well-posedness requires certain assumptions Recall: Find a solution u(t, ) Domain(E) such that Assumptions: u, v + E(u, v) = 0 for all v Domain(E). (2) t F is 1-homogeneous, i.e. F(tx 1,..., tx d ) = tf(x 1,..., x d ), t > 0 Marginal measures ν i (dx i ) = k i (x i )dx i with k i (z) 1 z 1+Y, z 1. i Admissible margins: CGMY, NIG, Meixner, spectrally negative,... C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 8

9 Well-posedness + explicit domain characterization Lemma. Multivariate process X satisfies Iψ X (ξ) Rψ X (ξ), ξ R d. Theorem. Domain(E) = H (Y 1/2,...,Y d /2) (R d ) and there exist γ, c > 0, C 0 such that and E(u, u) γ u 2 H (Y 1 /2,...,Y d /2) (R d ) C u 2 L 2 (R d ), E(u, v) c u H (Y 1 /2,...,Y d /2) (R d ) v H (Y 1 /2,...,Y d /2) (R d ). The associated semigroup is therefore analytic and the variational problem is well-posed. C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 9

10 Implementation in three steps 1. Localization: Truncation to bounded log price domain 2. Space discretization: Matrix problem MU L (t) + A(t)U L (t) = 0, t (0, T ), U L (0) = U Time discretization: E.g. backward Euler 3 (M + ta(t m )) U(t m ) = MU(t m 1 ), 1 m T / t, U(0) = U 0. Not restricted to Lévy processes; e.g. additive processes. 3 Other schemes: Crank-Nicolson, hp discontinuous Galerkin. C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 10

11 Localization based on marginal tail decay For implementation, a bounded spatial domain := [ R, R] d is required. Let H (s 1,...,s d ) ( ) := {ū u C 0 ( )}Hs 1,...s d (R d ) where ū is zero-extension of u to R d. Find u R (t, ) H (Y 1/2,...,Y d /2) ( ) such that u R, v R + E(u R, v R ) = 0 for all v R t H (Y 1/2,...,Y d /2) ( ). Proposition. This is still well-posed and u R u L e cr, provided marginal Lévy densities have semi-heavy tails. C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 11

12 Space discretization: Number of matrix entries needs to be reduced Issue 1: Curse of dimension O ( h 2d ) matrix entries 4. Issue 2: Non-local generator A matrix not sparse. Remedies: Sparse Grids: O ( h 2 log h 2(d 1)) entries, Wavelet compression : O ( h d) entries. Combination 5 : Asymptotically optimal complexity, O ( h 1 log h 2(d 1)) non-zero matrix entries. 4 Meshwidth h = 2 L. 5 Provided mixed Sobolev smoothness of solution u. C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 12

13 Curse 2 of dimension d = 1 d = 2 2 log S PSfrag replacements log S 2 d = 3 log S 3 h h -1 0 h 1 log S log S 1 0 log S 2 1 h h N = O(h 1 ) N = O(h 2 ) N = O(h 3 ) C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 13

14 ... overcome by sparse grid d = 2 d = 3 N = O(h 1 log h ) N = O(h 1 log h 2 ) Sparse grid preserves stability and convergence rates. C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 14

15 Advantages of wavelet bases Norm equivalences Sharp Numerical Analysis Break curse of dimension Multiscale compression of jump measure of X Efficient preconditioning Wavelets allow for efficient treatment of (moderate) multidimensional problems. C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 15

16 Sparse tensor product wavelets V L = span { ψ l 1 j 1 (x 1 ) ψ l d j d (x d ) 1 j i M l i, 0 l i L }, V L = span { ψ l 1 j 1 (x 1 ) ψ l d j d (x d ) 1 j i M l i, l l d L }. Left: V L. Right: V L. C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 16

17 Sparse tensor product spaces reduce complexity N u u L L V L V L V L V L L L Left: Dimension of V L, V L Right: L 2 -Convergence. C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 17

18 Wavelet basis reduces Lévy to BS complexity Density pattern for FEM matrix A for L = 8 refinement steps (512 mesh points) Left: Matrix for Black-Scholes process Right: Matrix for tempered stable process A-priori wavelet compression preserves convergence rate. C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 18

19 Numerical compression of jump measure nz = Left: Actual stiffness matrix of A for L = 5, Clayton Lévy copula with CGMY margins Right: Prediction by the compression scheme. 6 Reich, C. Schwab Diss ETH 2008 Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 19

20 Matrix estimates based on distance of supports Example: Dimension d = 2. Spline-wavelets { ψ l,k, l = (l 1, l 2 ) : l 1 L } of degree p N, and p p vanishing moments. Theorem. Under the above assumptions, Aψ l,k, ψ l,k ( l 1+ l 1 ) 2 ep(l(1) +l (2)) dist (2+max{Y i }+2ep) xy, where l (1) l (2) may be any two of the four level indices. eplacements dist x > 0 C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 20

21 Matrix estimates based on distance of sing-supp ts Theorem. Assume l 1 > l 1, l 2 > l 2. If d sing x 2 l 1, If d sing y 2 l 2, Aψ l,k, ψ l,k 2 l 1 l epl 1 d (1+Y 1 +ep) sing x. Aψ l,k, ψ l,k 2 l 2 l epl (1+Y 2 d 2 +ep) sing y. eplacements d sing x > 0 C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 21

22 Implementation Quadrature w.r.t. jump kernel k 7 Need to find u V L such that u L t, v + Au L, v = 0, with ( ) d u A ν (u)(x) = u(x + z) u(x) z i 1 { z <1} (x) ν(dz). R x d i i= Left: Kernel k, weak dependence θ = 0.5. Right: Quadrature points, N = 6 refinement levels. 7 Winter, Diss ETH 2008 C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 22

23 Results: Time value for basket options Let T = 1.0, r = 0 and payoff g = ( 1 2 (S 1 + S 2 ) 1 ) theta = 0.01 theta = 0.5 theta = rho = 0 rho = 0.25 rho = S S2 S S2 Left: Lévy model with Clayton copula, Y = (0.5, 1.5) and η = 1. Right: Black-Scholes model with same correlation ρ. C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 23

24 American style contracts Optimal stopping, free boundary problem u(t, x) = sup E[g(X τ ) X t = x]. t τ T Solve generalized BS-inequality (in viscosity sense), Variational inequality: u t + Au 0, u(t, ) g, (u g)(u t + Au) = 0. Find u(t, ) K g := {v V v g a.e.} such that u, v u + Au, v u 0 for all v K g. t }{{} E(u,v u) C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 24

25 American contracts Sequence of matrix LCPs Find U(t m ) K such that for all v K ( v U(tm ) ) (M + ta)u(tm ) ( v U(t m ) ) MU(tm 1 ), U(0) = U 0, with K := {v R dim V L v g}. Equivalent U(t m ) g, (M + ta)u(t m MU(t m 1 ), (U(t m ) g) ( (M + ta)u(t m ) MU(t m 1 ) ) = 0. C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 25

26 Results: European and American option, CGMY 8 Let C = 1, G = 12, M = 14, Y = 1 and refinement level L = American European Payoff r=1e 10 r=1e 08 r=1e 06 r= Option Price Stock Price Exercise boundary Time to maturity Left: European and American option price for r = 0.2. Right: Free boundary for r 0. 8 Matache, Nitsche, Schwab, 2005 C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 26

27 American swing option in BS market 9 [ p ] u(t, s) := sup E e r(τi t) g(τ i, S τi ) S t = s τ T t i=1 with refraction period δ, p exercise rights and stopping times T t := {τ = (τ 1,..., τ p ) τ i+1 τ i δ} p = 1 p = 2 p = 3 p = 4 p = p = 1 p = 2 p = 3 p = 4 p = swing option price spot price spot price Left: Swing option price. Right: free boundary time to maturity 9 Wilhelm, Winter, 2006 C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 27

28 Stochastic Volatility: Pure diffusion model 10 Mean reverting OU process: ds t = µs t dt + σ t S t dŵt σ t = f (Y t ) where dy t = α(m Y t )dt + βd W t, W t, Ŵt are Brownian motions, W t = ρŵt + 1 ρ 2 W t, ρ [ 1, 1]. f : R R + (Stein-Stein: f (y) = y for ρ = 0). α > 0, m > 0, β, µ R. 10 Fouque, Papanicolaou, Sircar, 2000 C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 28

29 Infinitesimal generator splits into two parts Need to solve u t + Au = 0. Under risk-neutral measure, where 11 A = A γ + A Q, A γ (u)(s, y) = β(s, y) u(s, y), ( ) rs β(s, y) = α(m y) β ( ρ µ r f (y) + δ(s, y) 1 ρ 2), A Q (u)(s, y) = 1 2 Q(S, y) : D2 u(s, y), ( ) f Q(S, y) = 2 (y)s 2 βρs βρs β δ(s, y) represents the risk premium factor. C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 29

30 Stochastic Volatility: Jump models The BNS model 12 : where dx t = (µ + βσ 2 t )dt + σ tdw t + ρ dz λt dσ 2 t = λσ 2 t dt + dz λt, σ 2 0 > 0, W t a Brownian motion. Z t a subordinator. W and Z are independent. β, µ, ρ, λ R, λ > 0, ρ Barndorff-Nielsen and Shepard 01 C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 30

31 Infinitesimal generator splits into three parts Generator of Markov process (X, σ 2 ), under risk-neutral measure: A = A γ + A Q + A ν, where A γ (u)(x, y) = β(x, y) u(x, y), ( 1 β(x, y) = 2 y + λκ r λy ( y A Q (u)(x, y) = 1 2 Q(x, y) : D2 u(x, y), Q(x, y) = [ ] A ν (u)(x, y) = λ u(x + ρz, y + z) u(x, y) ν(dz). R + ), ), C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 31

32 Stochastic Volatility: CoGARCH(p,q) Brockwell et al. (2005). For 1 p q: where dx t = σ t dl t σ t = α 0 + α Y t dy t = BY t dt + eσ t d[l, L] (d) t, L a R-valued Lévy process. α 0 > 0, α R q with α p 0, α p+1 =... = α q = 0. e R q the q-th standard basis vector ( ) 0 I R q q B = β q β, β q 0, β R q 1. C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 32

33 Infinitesimal generator of CoGARCH 14 A = A γ + A Q + A ν, where (assume R z2 ν L (dz) < ) 13 A γ (u)(x) = β(x) u(x), β(x) = ( γ L x2, α ), B x, (B x) 3,..., (B x) q+1, (B x) q A Q (u)(x) = 1 ( ) σ 2 Q(x) : 2 D2 u(x), Q = L x 2 0, 0 0 [ ( ) A ν (u)(x) = u x + δ(x, z) u(x) x2 z u (x) ] ν L (dz), x 1 R δ(x, z) = ( x 2 z, α q x 2 z 2, 0,..., 0, x 2 z 2). 13 (γ L, σ L, ν L ) characteristic triplet of L, x R q+2, ex = (x 3,..., x q+2 ) R q. 14 Kallsen, Vesenmayer 2005; Hilber 2006 C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 33

34 Results: Stylized facts can be recovered Example: American put option in mean-reverting OU model implied volatility PriceBs PriceSV moneyness correlation moneyness correlation Left: Implied volatility as function of moneyness S/K and correlation factor ρ. Right: Price difference P BS P SV. C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 34

35 Results: European put in the BNS model T = 0.5, K = 1, r = 0, λ = 2.5, IG(γ, δ)-lévy kernel k(z) = δ 2 2π z 3 2 (1 + γz)e 1 2 γz, γ = 2, δ = Left: Price for ρ = 4. Right: Difference of prices for ρ = 0.01 and ρ = 4. C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 35

36 Wrap up Unified numerics beyond Lévy (additive, affine processes,...): Derivative pricing Quadratic hedging and Greeks Exotic contracts (Asians, Bermudans,...) Rough payoffs (digitals, binaries,...) Multi period contracts (of swing type) Local & stochastic volatility, local jump intensity Baskets using sparse (log) price space Well conditioned matrices, singularity-free computation Discretization & modelling error control, adaptivity,... C. Schwab Workshop PDE+Finance: Marne-la-Vallée Oct15+16, 2007 p. 36