Radial Basis Functions generated Finite Differences (RBF-FD) for Solving High-Dimensional PDEs in Finance

Transcription

1 1 / 27 Radial Basis Functions generated Finite Differences (RBF-FD) for Solving High-Dimensional PDEs in Finance S. Milovanović, L. von Sydow Uppsala University Department of Information Technology Division of Scientific Computing SIAM Student Computational Finance Day TU Delft, Netherlands, 216-May-23

2 2 / 27 Overview Introduction Option Pricing Methods Radial Basis Functions generated Finite Difference methods Results Numerical Experiments Conclusion Further Research

3 3 / 27 Option Pricing Black-Scholes-Merton model in higher dimensions db(t) = rb(t)dt, ds 1 (t) = µ 1 S 1 (t)dt + σ 1 S 1 (t)dw 1 (t), ds 2 (t) = µ 2 S 2 (t)dt + σ 2 S 2 (t)dw 2 (t),. ds D (t) = µ D S D (t)dt + σ D S D (t)dw D (t). Option price (1) u(s 1 (t),..., S D (t), t) = e r(t t) E Q t [g(s 1(T ),..., S D (T ))]. The Black-Scholes-Merton equation u t + r D u s i + 1 D 2 u ρ ij σ i σ j s i s j ru =, s i i 2 u i,j si s j u(s 1, s 2,..., s D, T ) = g(s 1, s 2,..., s D ). (2)

4 4 / 27 BENCHOP The BENCHmarking project in Option Pricing Monte Carlo methods converge slowly but are increasingly competitive in higher dimensions. Fourier-methods are fast, especially the COS-method. Finite difference methods are reasonably fast in lower dimensions but suffer from the curse of dimensionality. Methods based on approximations with Radial Basis Functions have the potential to be competitive in higher dimensions. L. von Sydow, L.J. Höök, E. Larsson, E. Lindström, S. Milovanovic, J. Persson, V. Shcherbakov, Y. Shpolyanskiy, S. Sirén, J. Toivanen, J. Waldén, M. Wiktorsson, J. Levesley, J. Li, C.W. Oosterlee, M.J. Ruijter, A. Toropov, and Y. Zhao. In International Journal of Computer Mathematics, volume 92, pp , 215.

5 5 / 27 Radial Basis Functions Method Discretize space using N nodes. Approximate solution N u(s, t) λ k (t)φ(ε s s k ), k = 1, 2,..., N, (3) k=1 where φ is a radial basis function and ε is a shape parameter. The linear combination coefficients λ k are found by enforcing interpolation conditions. Figure 1: Gaussian RBFs.

6 6 / 27 Local Radial Basis Functions Methods The global approximation leads to a dense linear system of equations which tends to be ill-conditioned when ε is small Local RBF approximations might be a better approach!

7 6 / 27 Local Radial Basis Functions Methods The global approximation leads to a dense linear system of equations which tends to be ill-conditioned when ε is small Local RBF approximations might be a better approach! Ideas Come up with a localization which leads to a sparse linear system of equations. Radial Basis Functions Partition of Unity (RBF-PU) Radial Basis Functions generated Finite Differences (RBF-FD) Come up with basis transformation which will heal the ill-conditioning. RBF-QR RBF-GA

8 7 / 27 Radial Basis Functions generated Finite Differences Try to exploit the best properties from both FD and RBF with minimal loss. For each point s i in space, define its neighborhood of M 1 points and observe it as a stencil. Approximate the differential operator at every point [Lu(s)] i M k=1 w (i) k u(i) k. (4)

9 7 / 27 Radial Basis Functions generated Finite Differences Try to exploit the best properties from both FD and RBF with minimal loss. For each point s i in space, define its neighborhood of M 1 points and observe it as a stencil. Approximate the differential operator at every point [Lu(s)] i M k=1 w (i) k u(i) k. (4) Compute the RBF-FD weights and place them in matrix W φ( s (i) 1 s(i) 1 )... φ( s(i) 1 s(i) M ) w 1 [Lφ( s s (i) )] s=s i.. =.. φ( s (i) M s(i) 1 )... φ( s(i) M s(i) M ) w M [Lφ( s s (i) M )] s=s i

10 Adding polynomial terms φ( s (i) 1 s (i) 1 )... φ( s(i) 1 s (i) M ) 1 w 1 [Lφ( s s (i) )]s=s i.. φ( s (i). M s(i) 1 )... φ( s(i) M s(i) M ) 1 w =.. M [Lφ( s s (i) w M )]s=s i M+1 L Absolute Error for European Call Option K=1 T=1 r=.3 σ=.15 n=5 regular grid, standard stencil, standard matrix regular grid, standard stencil, polynomial matrix regular grid, adapted stencil, polynomial matrix 2 u max s Figure 2: Error in the solution. 8 / 27

11 9 / 27 Disctretization Discretize the Black-Scholes-Merton equation operator in space using RBF-FD D u t = r s i u si + 1 D ρ ij σ i σ j s i s j u si s 2 j ru W u. i Integrate in time using the standard implicit schemes BDF-1, i,j BDF-2. Au n+1 = b n

12 Solution of linear systems n= Figure 3: Structure of A. 1 / 27

13 11 / 27 Grids Figure 4: The nearest-neighbor based stencils used for approximating the differential operator.

14 12 / 27 Error distribution 4 RBF FD absolute error s s1 Figure 5: The absolute error distribution computed using 41 point in each dimension and a 5-point stencil on a full regular grid.

15 Grids 8K s2 2K K K 2K 8K s 1 Figure 6: Square domain. 13 / 27

16 Grids 8K s2 2K K K 2K 8K s 1 Figure 7: Triangular grid. 14 / 27

17 Grids 8K s2 2K K K 2K 8K s 1 Figure 8: Adapted grid. 15 / 27

18 Grids 8K s2 2K K K 2K 8K s 1 Figure 9: Adapted grid with some possible stencils. 16 / 27

19 17 / 27 Experiment European-style basket option with T = 1, K = 1, r =.3, σ 1 = σ 2 =.15, ρ =.5. Payoff function [ s1 (T ) + s 2 (T ) + u(s 1, s 2, T ) = K]. (5) 2

20 18 / 27 Details Boundary conditions u(,, t) =, (6) u(s 1, s 2, t) = 1 2 [s 1(t) + s 2(t)] e r(t t) K. (7) RBF choice: Gaussian φ(r) = e (ε r)2. (8) Error measured on a subdomain: (s 1, s 2 ) [ 1 3 K, 5 3 K].

21 19 / 27 Solution V(S 1,S 2 ) S S Figure 1: The computed solution on a regular triangular grid.

22 2 / 27 Choice of ε, n = n= Δu max h=.1 h= h=.1 ε(h)=.2228(1/h) h=.3 h= ε Figure 11: Error as a function of ε.

23 21 / 27 Choice of ε, n = n= Δu max h=.1 h= h=.1 ε(h)=.5256(1/h) h=.3 h= ε Figure 12: Error as a function of ε.

24 Results umax equidistant grid, n=3 equidistant grid, n=5 equidistant grid, n=7 adapted grid, n=3 adapted grid, n=5 adapted grid, n= h Figure 13: Spatial Convergence. 22 / 27

25 Results umax FD, equidistant grid RBF-FD, equidistant grid, ε const, n = 5 RBF-FD, equidistant grid, ε opt, n = 5 RBF-FD, adapted grid, ε avg, n = t [s] Figure 14: Computational performance. 23 / 27

26 24 / 27 Conclusions Proposed method: Works equally well with American options using operator splitting. Allows for local grid refinement which may deliver high accuracy in desired regions of the computational domain. Performs well due to sparsity and reduction in the computational domain allowed by the mesh-free discretization. Promises extendability to higher dimensional problems as well as for other models in the field.

27 25 / 27 Future Research Finalize study of shape parameter behavior. Developing adaptivity both in node placement and stencil size (hp-adaptivity). Optimizing stencils close to the boundaries. Smoothing of the terminal condition. Least squares instead of interpolation. Solving advanced financial models.

28 Future Research Higher dimensional problems s s s s s 1 12 s European Call Basket Option s s s / 27

29 Thank you for your attention. 27 / 27