Lecture 4 - Finite differences methods for PDEs

Transcription

1 Finite diff. Lecture 4 - Finite differences methods for PDEs Lina von Sydow Finite differences, Lina von Sydow, (1 : 18)

2 Finite difference methods Finite diff s v rv = 0 v(s T ; T ) = Φ(S) Grid in space and time: (1) Finite differences, Lina von Sydow, (2 : 18)

3 Finite diff. We know v(t ; s) = Φ(s). Want to compute v(0; s) using (1). Equation (1) holds for all s 0, for practical reasons we have to decide on computational domain 0 s s max ; 0 t T : As a rule of thumb we use s max = 4K. Finite differences, Lina von Sydow, (3 : 18)

4 Finite diff. Introduce grid-points (s ; t n ) ) s = s ; = 0; : : : ; M; t n = n ; n = 0; : : : ; N; s = smax M ; = T N : Finite differences, Lina von Sydow, (4 : 18)

5 Approximate derivatives (v n v(s ; t n )). (s ; t n ) = v n v n+1 (s ; t n ) = v +1 n v n 1 2 s + O(( s) 2 ) = = D 0 v n + O(( s) (s ; t n ) = v n +1 v n s + O( s) = D + v n + (s ; t n ) = v n v n 1 s + O( s) = D v n + O( s) Finite differences, Lina von Sydow, (5 : 18)

6 Finite 2 2 (s ; t n ) = D + D v n + O(( s) 2 ) = v n = D + = v +1 n v n s 2 v n 1 s v n + O(( s) 2 ) = v n 1 s 2 + O(( s) 2 ) = = v n +1 2v n +v n 1 s 2 + O(( s) 2 ) Finite differences, Lina von Sydow, (6 : 18)

7 Finite diff. Call option: Put option: v(0; t) = 0; v(s max ; t) = s max Ke r(t t) : (2) v(s max ; t) = 0; v(0; t) = Ke r(t t) : Other possible boundary 2 s = 0 and s = s max. 0 at Finite differences, Lina von Sydow, (7 : 18)

8 Finite diff. Example: Consider European call option. Use boundary conditions (2): = 1: v+1 n v n 1 2 s v+1 n 2v n + v n 1 s 2 = M 1: = v n +1 2 s ; = v +1 n 2v n s 2 : v n +1 v n 1 2 s v+1 n 2v n + v n 1 s 2 = s max K e r(t tn) v n 2 s 1 ; = s max K e r(t tn) 2v n + v n 1 s 2 : Finite differences, Lina von Sydow, (8 : 18)

9 Finite diff. Finite differences, Lina von Sydow, (9 : 18)

10 Finite diff. Finite differences - (European options) for = 0; : : : ; M v N = Φ(s ) (final condition) end for [ General formula: v n v n 1 rv n = 0: v+1 + rs n v n 1 2 s s2 Multiply with, move v n 1 ) v n +1 2v n +v n 1 s 2 (3) v n 1 = v n s2 + rs 2 s (v +1 n v n 1 )+ (v n s v n Finite differences, Lina von Sydow, (10 : 18) + v n 1 ) rv n : ]

11 for n = N; : : : ; 1 Finite diff. v n 1 0 = 0 v n 1 M n = s max K e r(t t n 1) for = 1; : : : ; M 1 v n s2 = v n + rs 2 s (v n +1 v n (v n s v n 1 )+ + v n 1 ) rv n end for end for Finite differences, Lina von Sydow, (11 : 18)

12 Finite diff. Taylor expanding ) local truncation error '(; s). Introduce notation v = v(s ; t n ). v(s ; t n 1 ) = v v t v 3 tt 6 v ttt + + O( 4 ) v(s 1 ; t n ) = v s v s + s2 s 2 v 3 ss 6 v sss + + s4 24 v ssss + O( s 5 ) v(s +1 ; t n ) = v + s v s + s2 2 v ss + s3 6 v sss + + s4 24 v ssss + O( s 5 ) Finite differences, Lina von Sydow, (12 : 18)

13 Use in approximation of PDE (3): Finite diff. v (v v t+ 2 2 vtt 3 6 vttt+o(4 )) + 2 s v s+ +rs s 3 3 vsss+o( s5 ) 2 s s2 v t s 2 v ss++ s4 12 vssss+o( s6 ) s 2 rv = 2 v tt + O( 2 ) + rs v s + O( s 2 )+ 2 2 s 2 v ss + O( s 2 ) rv = [Use(1)] = (4) O() + O( s 2 ) = ': Finite differences, Lina von Sydow, (13 : 18)

14 Finite diff. From Fourier-analysis it is possible to derive stability condition for periodic problem. These conditions also say something about stability for problem with boundary conditions. Finite differences, Lina von Sydow, (14 : 18)

15 Finite diff. For parabolic problems explicit methods have conditions like s 2 : Implicit methods do not have same type of stability conditions. Finite differences, Lina von Sydow, (15 : 18)

16 Euler backwards in time: Finite diff. v n v n 1 + rs v n 1 rv n 1 Crank-Nicholson: +1 v n s s2 = v n v n 1 + Lv n 1 = 0; ' = O() + O( s 2 ): v n v n v n v n 1 +v n 1 1 s 2 Lv n + Lv n 1 = 0; ' = O( 2 ) + O( s 2 ): Finite differences, Lina von Sydow, (16 : 18)

17 Finite diff. Introduce notation v n = v1 n v2 n : : : vm n 1 Explicit method: v n 1 = Av n + b. Implicit method: Av n 1 = b. A matrix of order M 1 M 1. T : Finite differences, Lina von Sydow, (17 : 18)

18 Transformations Finite diff. Possible to transform Black-Scholes PDE to @x 2 : Finite differences, Lina von Sydow, (18 : 18)