Using radial basis functions for option pricing

Transcription

1 Using radial basis functions for option pricing Elisabeth Larsson Division of Scientific Computing Department of Information Technology Uppsala University Actuarial Mathematics Workshop, March 19, 2013, University of Leicester E. Larsson, Leicester, March 19, 2013 (1 : 23)

2 The option pricing test problem Basic RBF approximation method Variations of the theme E. Larsson, Leicester, March 19, 2013 (2 : 23)

3 Introduction and motivation Basis functions: φ j (x) = φ( x x j ). Translates of one single function rotated around a center point. Example: Gaussians φ(εr) = exp( ε 2 r 2 ) Approximation: s ε (x ) = N j=1 λ jφ j (x ) Solution: Collocation with data yields {λ j } N j=1. E. Larsson, Leicester, March 19, 2013 (3 : 23)

4 Why use RBFs for option pricing? Advantages Flexibility wrt to the computational domain. Allows adaptive node placement As easy in d dimensions. Spectral accuracy / exponential convergence. Allows direct calculation of and Γ. Challenges Parameter selection strategies Ill-conditioning Computational cost E. Larsson, Leicester, March 19, 2013 (4 : 23)

5 The basic option pricing test problem Purpose: To determine the current value of an option. Example: European basket call option. Expiration date: Strike price: Dimensions: T =Dec 30, 2013 K=200 SEK d = 3 the number of underlying assets The multi-dimensional Black-Scholes equation: u d t = r u x i + 1 d [ ] σσ T x i 2 x 2 u ix j ru. ij x i x j i=1 i,j=1 Variables: x R d asset prices, t time left to expiration. Parameters: σ volatility matrix, r risk free interest rate. E. Larsson, Leicester, March 19, 2013 (5 : 23)

6 Initial and boundary conditions Contract function Boundary conditions ( Φ(x) = max 0, 1 ) d d i=1 x i K. E. Larsson, Leicester, March 19, 2013 (6 : 23) u(x, t) 1 d d x i Ke rt i=1 as x. S. Jansson and J. Tysk, 2006: Feynman-Kac formulas for Black-Scholes type operators

7 Discretization in time and space Solution form: u(x, t) Black-Scholes: N λ j (t)φ j (x) j=1 N λ j(t)φ j (x) = j=1 Time: 1 N (λ n+1 j λ n j )φ j (x) = k j=1 Boundary: Interior: Initially: N j=1 N j=1 N j=1 N λ j (t)lφ j (x) j=1 (αλ n+1 j + (1 α)λ n j )Lφ j (x) λ n+1 j φ j (x i ) = g(x i, t n+1 ), x i at Ω λ n+1 j a j (x i ) = N λ n j b j (x i ), j=1 N λ 0 j φ j (x i ) = Φ(x i ) j=1 E. Larsson, Leicester, March 19, 2013 (7 : 23) x i in Ω

8 How can we play with the nodes? I Square domain, N = 1165 E. Larsson, Leicester, March 19, 2013 (8 : 23)

9 How can we play with the nodes? Square domain, N = 1165 Triangular domain, N = 603 E. Larsson, Leicester, March 19, 2013 (8 : 23)

10 How can we play with the nodes? Square domain, N = 1165 Triangular domain, N = 603 E. Larsson, Leicester, March 19, 2013 (8 : 23)

11 How can we play with the nodes? Square domain, N = 1165 Triangular domain, N = 603 Adapted nodes, N = 599 E. Larsson, Leicester, March 19, 2013 (8 : 23)

12 How can we play with the nodes? Square domain, N = 1165 Triangular domain, N = 603 Adapted nodes, N = 599 Change of domain N N/d! Redistribution improves local accuracy E. Larsson, Leicester, March 19, 2013 (8 : 23)

13 Numerical results for different node sets Square Triangle Adaptive Error measured in the region of interest. Triangle and square same accuracy. Adaptive an order of magnitude better. Pettersson, Larsson, Marcusson, Persson, 2008: Improved radial basis function methods for multi-dimensional option pricing. E. Larsson, Leicester, March 19, 2013 (9 : 23)

14 Another game of nodes Domain and interior nodes are invariant with respect to 90 degree rotations, reflections in diagonals and axes. Employ the generalized Fourier transform to reduce memory and computational cost. However, operator must have same invariance. E. Larsson, Leicester, March 19, 2013 (10 : 23)

15 Transforming the Black-Scholes equation into the heat equation u t = r d i=1 x i u x i d [ ] σσ T x 2 u ix j ru. ij x i x j i,j=1 Change variables: x = exp(a T (Q T y + b)) Change function: u(t, y) = e γt+ξt y p(t, y) p t = y p A is computed from A T A = 1 2 d k=1 (σ kσ T k ). Q and b are arbitrary. γ and ξ depend on σ, r, A and Q. E. Larsson, Leicester, March 19, 2013 (11 : 23)

16 What happens with the domain and the nodes? New variables Original variables The transformation leads to automatically adapted node placement. E. Larsson, Leicester, March 19, 2013 (12 : 23)

17 Numerical results with the generalized Fourier transform Error-Work Gain with GFT Red Uni square, Black GFT, Blue Adapted tri 2D * 3D Given N Same accuracy for square and RBF-GFT. Lower cost for RBF-GFT. 2D 48, 3D 864. Adapted is more efficient at least in 2D. Larsson, Åhlander, Hall, 2008: Multi-dimensional option pricing using radial basis functions and the generalized Fourier transform E. Larsson, Leicester, March 19, 2013 (13 : 23)

18 Exploiting the spectral accuracy Parabolic PDE. We need to get low frequencies right. Result sensitive to node placement. We use more nodes than we would like. E. Larsson, Leicester, March 19, 2013 (14 : 23)

19 Exploiting the spectral accuracy (cont.) Collocation Least squares E. Larsson, Leicester, March 19, 2013 (15 : 23)

20 A multi-level least-squares RBF approach Coarse grid with small ε for smooth part. Fine grid with larger ε for initial non-smoothness. Boundary conditions are collocated (necessary). Computational cost turns out to be less than for a collocation aproach. Larsson, Gomes, 2013: A least squares multi-level radial basis method with applications in finance E. Larsson, Leicester, March 19, 2013 (16 : 23)

21 Errors for the two-grid least-squares method Error evolution Final error blue: N = 40, ε = 10 red: N = 21, ε = 0.8 black: Two level 150 least-squares evaluation points. E. Larsson, Leicester, March 19, 2013 (17 : 23)

22 Two-dimensional problem N f = 320, N c = 76, N e = 932, ε f = 4, ε c = 1 Ref. sol.: Adaptive finite differences: Persson, von Sydow, E. Larsson, Leicester, March 19, 2013 (18 : 23)

23 Flexibility wrt the computational domain Easy way to save computations by going to simplex. Adaptivity, least squares or a multi-level/multi-scale approach is needed. Any of the discussed aproaches can be used in higher dimensions, but cost becomes an issue. Future direction: Adaptive partition of unity RBF-methods. E. Larsson, Leicester, March 19, 2013 (19 : 23)

24 Currently: Partition of unity RBF-methods (RBF-PU for interpolation Wendland 2002) Global approximant: M ũ(x) = w i (x)u i (x), i=1 where w i (x) are weight functions. Local RBF approximants: u i (x) = N i j=1 λi j φ j(x). Applying operators: M ũ(x) = w i u i + 2 w i u i + w i u i i=1 Sparsity reduces memory and computational cost. Subdomain approach introduces parallelism. E. Larsson, Leicester, March 19, 2013 (20 : 23)

25 Practical challenges in RBF approximation Conditioning for small ε and large N Spectral convergence with N requires fixed ε. For smooth solutions, the best ε is small. We need to compute in the red region. log 10 (cond(a)) Computational cost Coefficient matrices are typically dense (for infinitely smooth RBFs). Direct methods are O(N 3 ) and known fast methods are most efficient for larger ε. E. Larsson, Leicester, March 19, 2013 (21 : 23)

26 The RBF-QR method: Idea The Gaussian RBFs are expanded in terms of { Tj,m c (x) = e ε2 r 2 r 2m T j 2m (r) cos((2m + p)θ), Tj,m s (x) = e ε2 r 2 r 2m T j 2m (r) sin((2m + p)θ), leading to Φ(x ) = C D T (x ), where c ij is O(1) and D = diag(o(ε 0, ε 2, ε 2, ε 4, ε 4, ε 4, ε 6,...)). Then C is QR-factorized so that Φ(x ) = Q [ ] [ ] D R 1 R T (x ) 0 D 2 Form a new basis (in the same space) Ψ(x ) = D 1 1 R 1 1 QH Φ(x ) = [ I R ] T (x ). E. Larsson, Leicester, March 19, 2013 (22 : 23)

27 Stable computation as ε 0 and N The RBF-QR method allows stable computations for small ε. (Fornberg, Larsson, Flyer 2011) Consider a finite non-periodic domain. Theorem (Platte, Trefethen, and Kuijlaars 2010): Exponential convergence on equispaced nodes exponential ill-conditioning. Solution #1: Cluster nodes towards the domain boundaries. E. Larsson, Leicester, March 19, 2013 (23 : 23)