Multi-Factor Finite Differences

Transcription

1 February 17, 2017

2 Aims and outline Finite differences for more than one direction The θ-method, explicit, implicit, Crank-Nicolson Iterative solution of discretised equations Alternating directions implicit (ADI) methods Analysis of accuracy and stability, Greeks High-dimensional problems, dimension reduction

3 Example: 2-factor basket For simplicity, we will focus on options on (baskets of) traded assets. Consider two stocks 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, where the two Wiener processes W 1,t, W 2,t have correlation ρ. An example of a payoff is a put on with expiry T and strike K. S 1,T + S 2,T

4 Black-Scholes PDE A standard hedging argument gives the pricing PDE V t σ2 1S1 2 2 V S1 2 The terminal condition is Boundary conditions are 2 V + ρσ 1 σ 2 S 1 S S 1 S 2 2 σ2 2S2 2 2 V S2 2 V V +rs 1 + rs 2 rv = 0. S 1 S 2 V (S 1, S 2, T ) = max(k S 1 S 2, 0). lim 1, S 2, t) S 1 = 0, lim 1, S 2, t) S 2 = 0.

5 Spatial discretisation Solve on [0, S max ] [0, S max ] [0, T ]. Use time steps t m = m t, m = 0,..., M, t = T /M. Introduce a grid {(j S, k S) : 0 j N, 0 k N}. Denote by V m the approximation to the exact solution V (j S, k S, t m ) at a point (j S, k S), and time t m Vj 1,k+1 m V+1 m V m j+1,k+1 V m j 1,k V m V m j+1,k Vj 1,k 1 m V 1 m V m j+1,k 1

6 Discretisation If we write A for the discrete operator, the backward finite difference equations for the θ-scheme have a similar structure to the 1-D case, V m V m 1 + t [ θa V m 1 ] + (1 θ)a V m = 0. The most obvious choice for the difference approximation is where A V m = A 0V m + A 1V m + A 2V m, A 1 V m = A 2 V m = A 0 V m = 1 [ 2 σ2 1 j2 V m j+1,k 2V m + V m ] 1 [ j 1,k + 2 r j V m j+1,k V m ] m j 1,k rv 1 [ 2 σ2 2 k2 V m +1 2V m + V m ] 1 [ r k V m +1 V m ] 1 1 [ 4 ρσ 1σ 2 jk V m j+1,k+1 V m j 1,k+1 V m j+1,k 1 + V m ] j 1,k 1 or (better) a 7-point-stencil (see later) for A 0.

7 Stencil notation The discretised equations can be written in stencil form as [a] [ V m 1] = [A] [V m ], where [a] = a NW a N a NE a W a C a E a SW a S a SE V j 1,k+1 V +1 V j+1,k+1, [V ] = V j 1,k V V j+1,k V j 1,k 1 V 1 V j+1,k 1 and stencil multiplication produces a scalar by multiplying corresponding entries and summing them up.,

8 Left-hand side a C = 1 + θ t [ σ1 2j 2 + σ2 2k2 + r ] a N = 1 2 θ t [ σ2 2k2 + rk ] a S = 1 2 θ t [ σ2 2k2 rk ] a E = 1 2 θ t [ σ1 2j 2 + rj ] a W = 1 2 θ t [ σ1 2j 2 rj ] a NE = 1 4 θ tρσ 1σ 2 jk a NW = 1 4 θ tρσ 1σ 2 jk a SW = 1 4 θ tρσ 1σ 2 jk a SE = 1 4 θ tρσ 1σ 2 jk

9 Right-hand side A C = 1 (1 θ) t [ σ 2 1 j 2 + σ 2 2 k2 + r ] A N = 1 2 (1 θ) t [ σ 2 2 k2 + rk ] A S = 1 2 (1 θ) t [ σ 2 2 k2 rk ] A E = 1 2 (1 θ) t [ σ 2 1 j 2 + rj ] A W = 1 2 (1 θ) t [ σ 2 1 j 2 rj ] A NE = 1 4 (1 θ) tρσ 1σ 2 jk A NW = 1 4 (1 θ) tρσ 1σ 2 jk A SW = 1 4 (1 θ) tρσ 1σ 2 jk A SE = 1 4 (1 θ) tρσ 1σ 2 jk

10 Intermezzo: 1 factor revisited In the special case of 1 stock, V t σ2 S 2 2 V V + rs rv = 0 S [0, ), t (0, T ). S 2 S Recall the θ-timestepping central difference scheme for the Black-Scholes equation in the form V m 1 V m j j t [ 1 V m 1 j+1 2V m 1 + V m 1 j j 1 S 2 = θ 2 σ2 S 2 j [ 1 + (1 θ) 2 σ2 S 2 Vj+1 m 2V j m j S 2 = θav m 1 + (1 θ)av m j j + V m j 1 m 1 V j+1 + rs j 2 S V m 1 j 1 rv m 1 j m Vj+1 + rs V j 1 m j rv m j 2 S on a grid S 0 = 0, S 1 = S, S 2 = 2 S,..., S N = S max, time steps t 0 = 0, t 1 = t, t 2 = 2 t,..., t M = M t = T, at which we introduce approximations V m n ] ] to V (n S, m t).

11 Practicalities Rewrite as a n V m 1 n 1 + b nv m 1 n + c n V m 1 n+1 = A nv m n 1 + B n V m n + C n V m n+1. In matrix notation, with boundary conditions included, b0 m c m V m 1 a 1 m b1 m c1 m 0 B... 0 V m 1 0 m C m A m 1 B1 m C1 m... 0 = an 1 m bn 1 m cn 1 m V m A m an m bn m N 1 V m 1 N 1 BN 1 m CN 1 m A m N N BN m This is (generally) a tridiagonal linear system, which can be solved directly in O(N) operations in each timestep. V m 0 V m 1... V m N 1 V m N

12 Analysis θ-method The scheme satisfies a discrete maximum principle, if A n, B n, C n 0 A n, B n, C n 1 a n, c n 0 a n + b n + c n 1 ( b n > 0) Setting θ = 0 (explicit scheme), from B n 0 we get stability only for small timesteps. θ = 1 gives the fully implicit method, which is unconditionally stable. Both schemes of first order accurate, Crank-Nicolson second order.

13 von Neumann analysis Since instabilities often appear highly oscillatory, we study the propagation of Fourier modes of the form sin(kx ), cos(kx ). We do this for the constant coefficient equation U t σ2 2 U X 2 + ( r 1 2 σ2) U X ru = 0 on (, ), i. e. X = log S. The θ-scheme for this equation reads ( U m 1 σ 2 ( ) n θ t 2 X 2 U m 1 n+1 2U m 1 n + U m 1 + r σ2 /2 ( ) ) U m 1 n 1 n+1 U m 1 ru m 1 n 1 n = 2 X ( U m σ 2 ( n + (1 θ) t m ) U 2 X 2 n+1 2Um n + r σ 2 ) /2 ( Um m ) n 1 + U n+1 2 X Um m n 1 ru n.

14 von Neumann analysis For U M n = e ikn, first show for m = M that 1 ( U m 2 X n+1 Un 1) m i = X sin k Um n, 1 ( U m X 2 n+1 2Un m + Un 1 m ) 4 = X 2 sin2 (k/2) Un m ; then deduce by induction where U m n = R θ ( X, t; k) M m e ink, R θ ( X, t; k) = 1 (1 θ) t(2σ2 / X 2 sin 2 (k/2) + r i(r 1 2 σ2 )/ X sin k) 1 + θ t(2σ 2 / X 2 sin 2 (k/2) + r i(r 1. 2 σ2 )/ X sin k)

15 von Neumann analysis The scheme is stable, if R θ ( X, t; k) 1 k This is true for all t if θ 0.5 unconditionally stable For θ = 0, need σ 2 t X 2 For θ = 1, For θ = 0.5, lim R 1 = lim R 1 = 0. X 0 t lim X 0 R 1 2 = lim t R 1 2 = 1. High-frequency modes lead to unstable Greeks (use Rannacher start-up).

16 Maximum principles in 2D The 2D scheme satisfies a discrete maximum principle, if A x 0 A x 1 x a x 0, x C a x 1 ( ac > 0) For θ = 1, no constraint on timestep. x For θ < 1, maximum principle only if t = O( S 2 ), but unconditionally stable for θ 1/2. Analysis of drift as for single factor (use upwinding where necessary).

17 7-point stencil Another observation is that for say ρ > 0 the terms a NW a SE, as well as ANW and A SE have the wrong sign. Instead of S for the cross-derivative, we can take the 7-point-stencil 1 2 S 2 [ ] = S 2 [ ] S 2 [ and This is an average of two shifted difference stencils along the direction of the principal component of the diffusion matrix. ].

18 7-point stencil We replace A 0 V m by A 0 V m = 1 [ 2 ρσ 1σ 2 kj V m j+1,k+1 V m +1 V m j+1,k + 2V m V m j 1,k V m 1 + V m ] j 1,k 1 This gives the correct sign for a NE, asw, ANW, ASE. It can be shown that the modified A N, AE etc also have, for constant coefficients and suitable equidistant meshes. For Black-Scholes, can use log-coordinates (or exponentially stretched meshes) to achieve this. The general situation is subject of current research. For ρ [ 1, 0], use points along the diagonal running from NW to SE.

19 von Neumann analysis Consider the model problem u t u 2 x ρ 2 u u x 1 x 2 2 x 2 1 = 0 and its finite difference discretisation. Try solutions of the form u m n,l = R θ ( x, t; k) M m e ink+ilj. If θ 1 2, the scheme is unconditionally stable. If θ < 1 2, stable if (1 2θ)r c, where r = t/ x 2, c depends on the model parameters (here ρ) but not t, x. Similar behaviour for general PDEs.

20 Explicit scheme In the case θ = 0, V m 1 is explicitly computable, = [A] [V m ] starting with the payoff and stepping backwards in time. The stability constraint necessitates a large number of timesteps.

21 Implicit methods The case θ > 0 requires the solution of a potentially large system of equations at each timestep, [a] [ V m 1] = [A] [V m ] = f, the solution of which can be ordered into a single vector [ V m 1 = V0,0 m 1, V1,0 m 1,..., V m 1 N,0, V m 1 0,1, V m 1 1,1,..., V m 1 N,1,. ] V m 1 0,N, V m 1 1,N,..., V m 1 N,N, and the finite difference equations including boundary conditions written as (shorthand V = V m 1 ) KV = f.

22 System matrix The matrix K R (N+1) (N+1) has the shape a0 C a0 E 0 a0 N a NE 0 0 a 1 W a1 C a1 E a1 SN a1 N a NE 1 0 a p S ap SE 0 ap C ap E 0 ap N a NE p K = a p+1 SW ap+1 S ap+1 SE ap+1 W ap+1 C ap+1 E ap+1 NW ap+1 N ap+1 NE 0 a SW p 2 a S 2 p 2 a SE 2 p 2 a W 2 p 2 a C 2 p 2 a E 2 p a SW p 2 a S 1 p 2 a W 1 p 2 a C 1 p 2 1 where al C a C etc, with l = j + kp, p = N + 1, j, k = 0, 1,..., N. This is known as lexicographic ordering.

23 Linear systems Instead of a tridiagonal structure, there are nine non-zero elements per line, not all adjacent ( block-tridiagonal ). N + 1 is the number of points in each direction Gaussian elimination and LU-decomposition lead to fill-in. The complexity increases from O(N) in 1D to O(N 4 ) in 2D! The optimum is O(N 2 ), i.e. proportional to problem size. Need to exploit sparsity pattern, which leads to iterative methods.

24 Relaxation methods To solve: [a] [V ] = f K V = f. Solve by iteration, and let V (n) be the n th iteration for V. Idea: Split K = L + D + U, where L is lower triangular, D diagonal, and U an upper triangular matrix. Define the Jacobi scheme and the Gauss-Seidel scheme D V (n+1) = (L + U)V (n) + f, (D + L)V (n+1) = U V (n) + f.

25 Classical iterations The Jacobi iteration in stencil notation is a NW a N a NE 0 a C [ 0 V (n+1)] = a W 0 a E a SW a S a SE [ V (n)] +f, Gauss-Seidel (GS) with lexicographic ordering a W a C 0 a SW a S a SE [ V (n+1)] = a NW a N a NE 0 0 a E [ V (n)] +f j,

26 Discussion Advantages: exploit sparse matrix structure; easy to implement; Gauss-Seidel proven to converge for positive-definite matrices; can speed up by successive over-relaxation (see lectures) Disadvantages: rate of convergence (good if 1) grows with system size and close to 1 for fine grids; number of iterations required for given accuracy increases rapidly with size. The key word is linear complexity: solution time proportional to number of unknowns. So-called multi-grid solvers can achieve this.

27 American options Projected Gauss-Seidel (PGS) extends as per a C V (n+1/2) a W 0 0 a SW a S a SE [V (n+1)] = a NW a N a NE 0 0 a E [V (n)] + f, followed by V (n+1) ( ) = max V (n+1/2), payoff. Alternatively: use penalty method and faster converging iteration for linear systems (i.e., faster than linear Gauss-Seidel).

28 Splitting schemes The θ-scheme (I θ ta)v m 1 = (1 + (1 θ) ta)v m can be seen as the sequence of an explicit Euler step of size (1 θ) t; an implicit Euler step of size θ t. Idea: Split A = A 0 + A 1 + A 2 into simple components Y 0 = (I + ta 0 )V m (I ta 1 )Y 1 = Y 0 (I ta 2 )Y 2 = Y 1 V m 1 = Y 2 will define a first order accurate, stable solution.

29 Douglas scheme Consider modified discretisations to restore tri-diagonal structure. Idea: Factorise I tθ(a 1 + A 2 + A 0 ) approximately into tri-diagonal factors. Useful to rewrite (I θ ta)v m 1 = (I + (1 θ) ta)v m as (I θ ta)(v m 1 V m ) = ta V m, then any reasonable factorisation is consistent; Approximate I θ t(a 1 + A 2 + A 0 ) = (I tθa 1 )(I θ ta 2 ) θ 2 t 2 A 1 A 2 θ ta 0. Dropping the last two terms, gives the new scheme (I θ ta 1 )(I θ ta 2 )(V m 1 V m ) = ta V m. }{{} =:B

30 Douglas scheme This can alternatively be written as, for general dimension d: Y 0 = V m + ta V m Y j = Y j 1 + θ ta j (Y j V m ) j = 1,..., d V m 1 = Y d The first step is explicit. This is followed by unidirectional implicit corrections. Second order accuracy in t if A 0 = 0 (ρ = 0) and θ = 1 2, first order accurate if ρ 0. For d = 1, the scheme reduces to the well-known θ-scheme.

31 Analysis The main advantage is that the discretisation matrices are tridiagonal (after appropriate re-ordering). What is more, for each j, the system decouples along grid lines. The complexity is therefore O(N 2 ). The order of accuracy is determined by Taylor expansion. Stability analysis by von Neumann. Consider the model problem u t u 2 x ρ 2 u u x 1 x 2 2 x 2 2 = 0.

32 von Neumann stability, d = 2 For un,l M = e ink+ilj : Show, similar to 1D, Bu m = ˆBu m, t Au m = Âu m, where ˆB = (1 + 2rθsk 2 )(1 + 2rθs2 j )  = 2r(sk 2 + s2 j 2ρs k c k s j c j ) with r = t/ x 2, s k = sin(k/2), c k = cos(k/2), etc. Deduce  0 and  2 ˆB if θ 1/2, then for R = ( ˆB Â)/ ˆB, u m 1 = R u m u m. Use Fourier isometry to deduce stable in the l 2 norm.

33 Greeks The Douglas scheme produces stable prices. This is not necessarily true for the sensitivities. The reason is R 1 for t. High frequency errors are not dampened, and blow up when taking finite differences. Can be avoided by Rannacher-type start-up.

34 Rannacher start-up Recall the 1D setting: optimal to replace 2 Crank-Nicolson steps by 4 fully implicit steps of half the step size, Giles and Carter (2006). Idea: Keep A 0 terms explicit, other factors implicit, e.g. (I ta 1 )(I ta 2 )V m 1 = (I + ta 0 )V m. This scheme is strongly stable, and first order accuracy is sufficient for start-up.

35 Craig & Sneyd scheme The scheme corrects for the O( t) error due to the mixed term. Idea: Approximate A 0 V m 1 from earlier using Y d and repeat, (I tθa 1 )(I tθa 2 )(Y d V m ) = tav m (I tθa 1 )(I tθa 2 )(V m 1 V m ) = tav m + θ ta 0 (Y d V m ). This correction step makes the scheme second order accurate in t for θ = 1/2 also for non-zero correlation. Note we have not used any special form of A 0 for the definition of the scheme. The form of A 0 is relevant for stability though.

36 Craig & Sneyd scheme Rewrite as, and generalise to (d 2), Y 0 = V m + tav m Y j = Y j 1 + θ ta j (Y j V m ) j = 1,..., d Ỹ 0 = Y 0 + θ ta 0 (Y d V m ) Ỹ j = Ỹ j 1 + θ ta j (Ỹ j V m ) j = 1,..., d V m 1 = Ỹ d Can be shown to be unconditionally stable in 2D if θ 1 2.

37 Efficiency The complexity of an algorithm is a measure of its computational cost C (in algebraic operations). The ɛ-complexity C(ɛ) is defined as the cost required to achieve a certain accuracy ɛ. In two dimensions, with N mesh points in each direction and M time steps, C = O(MN 2 ). The error of, say Crank-Nicolson and central differences, is O(( S) 2 + ( t) 2 ) or O(M 2 + N 2 ). Therefore, C(ɛ) = O(ɛ 3/2 ). This compares to a Monte Carlo error of M 1 + N 0.5 for N paths and M timesteps, i.e. C(ɛ) = O(ɛ 3 ), or O(ɛ 2 ) if time-stepping can be avoided.

38 High dimensions? In d dimensions, with N mesh points in each direction and M time steps, C = O(MN d ). In terms of the error, C = O(ɛ (d+1)/2 ). This exponential increase of the computational complexity for increasing dimension is referred to as curse of dimensionality. FD asymptotically better than MC up to dimension 3 5. Then use PCA for dimension reduction, exploiting dependency.

39 Literature D. Tavella, C. Randall: Pricing Financial Instruments The Finite Difference Method. Wiley, M. B. Giles, R. Carter: Convergence analysis of Crank-Nicolson and Rannacher time-marching. Journal of Computational Finance, 9(4), I. J. D. Craig, A. D. Sneyd: An alternating-direction implicit scheme for parabolic equations with mixed derivatives, Comput. Math. Applic., 16(4), Karel in t Hout: ADI schemes in the Numerical Solution of the Heston PDE, Proceedings of the International Conference Numerical Analysis and Applied Mathematics, 2007.