Chapter 5 Finite Difference Methods. Math6911 W07, HM Zhu

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1 Chapter 5 Finite Difference Methods Math69 W07, HM Zhu

2 References. Chapters 5 and 9, Brandimarte. Section 7.8, Hull 3. Chapter 7, Numerical analysis, Burden and Faires

3 Outline Finite difference (FD) approximation to the derivatives Explicit FD method Numerical issues Implicit FD method Crank-Nicolson method Dealing with American options Further comments 3

4 Chapter 5 Finite Difference Methods 5. Finite difference approximations Math69 W07, HM Zhu

5 Finite-difference mesh Aim to approximate the values of the continuous function f(t, S) on a set of discrete points in (t, S) plane Divide the S-axis into equally spaced nodes at distance S apart, and, the t-axis into equally spaced nodes a distance t apart (t, S) plane becomes a mesh with mesh points on (i t, j S) We are interested in the values of f(t, S) at mesh points (i t, j S), denoted as fi,j = f i t,j S ( ) 5

6 The mesh for finite-difference approximation S S max =M S? j S fi,j = f i t,j S ( ) i t T=N t t 6

7 Black-Scholes Equation for a European option with value V(S,t) V + t σ S V V + rs rv = 0 S S where 0 < S < + and 0 t < T with proper final and boundary conditions (5.) Notes: This is a second-order hyperbolic, elliptic, or parabolic, forward or backward partial differential equation Its solution is sufficiently well behaved,i.e. well-posed

8 Finite difference approximations The basic idea of FDM is to replace the partial derivatives by approximations obtained by Taylor expansions near the point of interests For example, ( ) ( ) ( ) ( ) ( ) f S,t f S,t+ t f S,t f S,t+ t f S,t = lim t t 0 t t for small t, using Taylor expansion at point S,t ( S,t) f f ( S,t+ t) = f ( S,t) + t+ O t t ( ( ) ) ( ) 8

9 Forward-, Backward-, and Centraldifference approximation to st order derivatives backward central forward t t ( ) ( ) ( ) f t,s f t+ t,s f t,s Forward: + O( t) t t f ( t,s) f ( t,s) f ( t t,s) Backward: + O( t) t t f ( t,s) f ( t + t,s) f ( t t,s) Central: + O t t t t + t ( ( t) )

10 Symmetric Central-difference approximations to nd order derivatives ( ) ( ) ( ) ( ) ( S ) + + f t,s f t,s S f t,s f t,s S + O S ( ( S) ) ( ) ( ) ( ) ( + ) ( ) Use Taylor's expansions for f t,s S and f t,s S around point f t,s+ S =? + f t,s S =? t,s : 0

11 Finite difference approximations f f f f f f Forward Difference:, t t S S f f f f f f Backward Difference:, t t S S f f f f f f Central Difference:, t t S S As to the second derivative, we have: f i +,j i,j i,j+ i,j i,j i,j i,j i,j i +,j i,j i,j+ i,j i,j+ i,j i,j i,j S S S = f f f f f f + f i,j+ i,j i,j ( S ) S

12 Finite difference approximations Depending on which combination of schemes we use in discretizing the equation, we will have explicit, implicit, or Crank-Nicolson methods We also need to discretize the boundary and final conditions accordingly. For example, for European Call, Final Condition: f = max j S K, 0, for j = 0,,...,M N,j ( ) Boundary Conditions: fi, 0 = 0 r( N i) fi,m = Smax Ke where S = M S. max t, for i = 0,,...,N

13 Chapter 5 Finite Difference Methods 5.. Explicit Finite-Difference Method Math69 W07, HM Zhu

14 Explicit Finite Difference Methods f f f In + rs + σ S = rf, at point ( i t, j S ), set t S S f fi,j fi,j backward difference: t t f fi,j+ fi,j central difference:, S S and f S f + f f, r f = rf i,j, S = j S S i,j+ i,j i,j

15 Explicit Finite Difference Methods Rewriting the equation, we get an explicit scheme: where f = a f + b f + c f * * * i,j j i,j j i,j j i,j+ * a j = t j rj ( σ ) ( σ ) * b = t j + r j (5.) * ( c ) j = t σ j + rj for i = N -, N -,...,, 0 and j =,,..., M -.

16 Numerical Computation Dependency S S max =M S (j+) S x j S x x (j-) S x 0 0 (i-) t i t T=N t t

17 Implementation. Starting with the final values, we apply (5.) to solve f for N,j j M. We use the boundary condition to determine f and f. N 0, N-,M f N,j f N,j. Repeat the process to determine and so on 7

18 Example We compare explicit finite difference solution for a European put with the exact Black-Scholes formula, where T = 5/ yr, S 0 =$50, K = $50, σ=30%, r = 0%. Black-Scholes Price: $.8446 EFD Method with S max =$00, S=, t=5/00: $.888 EFD Method with S max =$00, S=, t=5/4800: $

19 Example (Stability) We compare explicit finite difference solution for a European put with the exact Black-Scholes formula, where T = 5/ yr, S 0 =$50, K = $50, σ=30%, r = 0%. Black-Scholes Price: $.8446 EFD Method with S max =$00, S=, t=5/00: $.888 EFD Method with S max =$00, S=.5, t=5/00: $3.44 EFD Method with S max =$00, S=, t=5/00: -$.87E 9

20 Chapter 5 Finite Difference Methods 5.. Numerical Stability Math69 W07, HM Zhu

21 Numerical Accuracy The problem itself The discretization scheme used The numerical algorithm used

22 Conditioning Issue Suppose we have mathematically posed problem: y = f x ( ) where y is to evaluated given an input x. * Let x = x+ δx for small change δx. ( * ) ( ) If hen f x is near f x, then we call the problem is well - conditioned. Otherwise, it is ill-posed/ill-conditioned.

23 Conditioning Issue Conditioning issue is related to the problem itself, not to the specific numerical algorithm; Stability issue is related to the numerical algorithm One can not expect a good numerical algorithm to solve an illconditioned problem any more accurately than the data warrant But a bad numerical algorithm can produce poor solutions even to well-conditioned problems 3

24 Conditional Issue The concept "near" can be measured by further information about the particular problem: f x f x δ x C f x f x x ( ) ( * ) ( ) ( ( ) 0) where C is called condition number of this problem. If C is large, the problem is ill-conditioned. 4

25 Floating Point Number & Error Let x be any real number. Infinite decimal expansion : x = ±.x Truncated floating point number : x where x 0, 0 x i 9, Floating point or roundoff error : fl x fl x ( x) ( x) x d = 0 ±.x e x x d : an integer, precision of the floating point system e : an bounded integer d 0 e 5

26 Error Propagation When additional calculations are done, there is an accumulation of these floating point errors. Example : Let x = and where d = 3. Floating point error : Error propagation : fl fl ( x) fl ( x) x = = ( ) x x =

27 Numerical Stability or Instability Stability ensures if the error between the numerical soltuion and the exact solution remains bounded as numerical computation progresses. That is, f ( * x ) (the solution of a slightly perturbed problem) is near * f x the computed solution. ( )( ) Stability concerns about the behavior of ( ) as numerical computation progresses for fixed discretization steps t and S. f i,j f i t,j S 7

28 Convergence issue Convergence of the numerical algorithm concerns about the behavior of f f i t,j S as t, S 0 for fixed values i t,j S. ( ) i,j ( ) For well-posed linear initial value problem, Stability Convergence (Lax's equivalence theorem, Richtmyer and Morton, "Difference Methods for Initial Value Problems" (nd) 967) 8

29 Numerical Accuracy These factors contribute the accuracy of a numerical solution. We can find a good estimate if our problem is wellconditioned and the algorithm is stable Stable: ( ) ( ) f x f x * * Well-conditioned: ( * ) ( ) f x f x ( ) ( ) * f x f x 9

30 Chapter 5 Finite Difference Methods 5..3 Financial Interpretation of Numerical Instability Math69 W07, HM Zhu

31 Financial Interpretation of instability (Hall, page 43-4) If f S and f S are assumed to be the same at ( i +, j) as they are at ( i, j), we obtain equations of the form: f = aˆ f + bˆ f + cˆ f (5.3) where i,j j i +,j j i +,j j i +,j+ â j = tσ j tr j = π d + r t + r t ( ) ˆbj = σ j t = π 0 + r t + r t ĉ j = tσ j + tr j = π u + r t + r t for i = N -, N -,...,, 0 and j =,,..., M -.

32 Explicit Finite Difference Methods π u ƒ i +, j + π 0 ƒ i, j ƒ i +, j π d ƒ i +, j These coefficients can be interpreted as probabilities times a discount factor. If one of these probability < 0, instability occurs.

33 Explicit Finite Difference Method as Trinomial Tree Check if the mean and variance of the Expected value of the increase in asset price during t: [ ] E = Sπ + 0π + Sπ = rj S t = rs t Variance of the increment: 0 = ( S) πd + π0 + ( S) π σ ( ) [ ] [ ] σ ( ) E 0 d u u = j S t = σ S t Var = E E = S t r S t σ S t which is coherent with geometric Brownian motion in a risk-neutral world

34 Change of Variable Define Z = lns. The B-S equation becomes f σ f σ f + r + = rf t Z Z The corresponding difference equation is fi +,j fi,j σ fi +,j+ fi +,j σ fi +,j+ fi +,j + fi +,j+ + r + = t Z Z or f * * * i,j = α j fi +,j + β j fi +,j + γ j f i +,j+ 54. ( ) rf i,j 34

35 Change of Variable where α β γ * j * j * j σ t σ t = r + + r t Z Z t = + r t Z σ t σ t = r + + r t Z Z σ It can be shown that it is numerically most efficient if Z = σ 3 t. 35

36 36 Reduced to Heat Equation Get rid of the varying coefficients S and S² by using change of variables: Equation (5.) becomes heat equation (5.5): ( ) ( ) ( ) ( ) = = = = + 4,,,, σ τ σ τ τ r k x u Ee t S V T t Ee S k x k x (5.5) for and 0 u u x x τ τ = < <+ >

37 Explicit Finite Difference Method With finite u u u m+ n m+ n m n Ignoring this by = u δτ u difference m n = α u where α = terms ( nδx,mδτ ) m n+ δτ + ( δx) + O of n+ n ( δτ ) = ( δx ) O( δτ ) and O ( δx ) ( α ), for, this equations -N u u m m n of involves + α u n the ( ) u N form : m m n + + solving u, and m n we m + = O a system of ( δx) ) can approximat 0,,...M = e σ T δτ

38 Stability and Convergence (P. Wilmott, et al, Option Pricing) Stability: The solution of Eqn (5.5) is δτ i) Stable if 0 < α = ; ii) Unstable if α > Convergence: ( δ x) If 0 < α, then the explicit finite-difference approximation converges to the exact solution as δτ, δ x 0 m n (, ) ( δτ ) (in the sense that u u nδx mδτ as δτ, δx 0) Rate of Convergence is O 38

39 Chapter 5 Finite Difference Methods 5.3. Implicit Finite-Difference Method Math69 W07, HM Zhu

40 Implicit Finite Difference Methods f f f In + rs + σ S = r f, we use t S S f fi+,j fi, j forward difference: t t f fij, + fij, central difference:, S S and f fij, + + fij, fij,, rf = rf i,j S S

41 Implicit Finite Difference Methods Rewriting the equation, we get an implicit scheme: where a f + b f + c f = f (5.6) j i, j j i, j j i, j+ i +,j a j = t j + rj ( σ ) ( σ ) b = + t j + r j ( c ) j = t σ j + rj for i = N-,N-,...,, 0 and j =,,...,M-.

42 Numerical Computation Dependency S S max =M S (j+) S j S (j-) S x x x x 0 0 (i-) t i t (i+) t T=N t t

43 Implementation Equation (5.6) can be rewritten in matrix form: Cf = f + b i i+ i 57. ( ) where f and b are ( M ) dimensional vectors i i T fi = f i,,f i,,f i, 3,f i,m, bi = af i, 0, 00,,, 0, cm fi,m and C is ( M ) ( M ) symmetric matrices C b c 0 0 a b c 0 = 0 a3 b3 cm 0 0 a b M M T

44 Implementation. Starting with the final values, we need to solve a linear system (5.7) to obtain fn,j for j M using LU factorization or iterative methods. We use the boundary condition to determine f and f. f N,j N 0, N-,M f N,j. Repeat the process to determine and so on 44

45 Example We compare implicit finite difference solution for a European put with the exact Black-Scholes formula, where T = 5/ yr, S 0 =$50, K = $50, σ=30%, r = 0%. Black-Scholes Price: $.8446 IFD Method with S max =$00, S=, t=5/00: $.894 IFD Method with S max =$00, S=, t=5/4800: $

46 Example (Stability) We compare implicit finite difference solution for a European put with the exact Black-Scholes formula, where T = 5/ yr, S 0 =$50, K = $50, σ=30%, r = 0%. Black-Scholes Price: $.8846 IFD Method with S max =$00, S=, t=5/00: $.888 IFD Method with S max =$00, S=.5, t=5/00: $3.35 IFD Method with S max =$00, S=, t=5/00: $

47 Implicit vs Explicit Finite Difference Methods ƒ i +, j + ƒ i, j + ƒ i, j ƒ i +, j ƒ i, j ƒ i +, j Explicit Method ƒ i +, j ƒ i, j Implicit Method (always stable)

48 Implicit vs Explicit Finite Difference Method The explicit finite difference method is equivalent to the trinomial tree approach: Truncation error: O( t) Stability: not always The implicit finite difference method is equivalent to a multinomial tree approach: Truncation error: O( t) Stability: always 48

49 Other Points on Finite Difference Methods It is better to have ln S rather than S as the underlying variable in general Improvements over the basic implicit and explicit methods: Crank-Nicolson method, average of explicit and implicit FD methods, trying to achieve Truncation error: O(( t) ) Stability: always

50 Appendix A. Matrix Norms Math69 W07, HM Zhu

51 Vector Norms - - A vector norm is - There are various Norms serve as a x > c x For example, 0 for any = x + y x x p c n i= max i n x x x i x way to measure the length of a function mapping for any + i ways p v = [ x 0; y p 4 for any to define a - x c R ( p = is the Euclidean norm) 3]. = x, 0 iff v y R norm = x?, x R = n 0 v n = to a real number?, a vector or a v =? matrix x s.t. 5

52 Matrix Norms - Similarly, a matrix norm is a function mapping A R m n to a real number A s.t. A > 0 for any A 0; A = 0 iff A = 0 c A = c A for any c R A + B A + B for any A, B R m n - Various commonly used matrix norms A p sup x 0 Ax x p p A F m n i= j= a ij A A ρ max j n ρ m i= a ij ( T A A), ( B) max{ λ : λ is an eigenvalue of B} k A the spectral k max norm, i m n j= where a ij

53 An Example A = A A =? =? A A F =? =? 53

54 Basic Properties of Norms Let A, B n R n and x,y R. Then x 0; and x = 0 x = 0 x + y x + y α x = α x whereα is a real number Ax A x AB A B n 54

55 Condition number of a square matrix ( m n) n All norms in R R are equivalent. That is, if and are norms α β, c,c x, n n on R then > 0 such that for all R we have c x x c x α β α Condition Number of A Matrix The C A A. n n :, where A R condition number gives a measure of how close a matrix is close to singular. The bigger the C, the harder it is to solve Ax = b. 55

56 Convergence - vectors x k converges to x x k x converges to 0 - matrix A k 0 A k

57 Appendix B. Basic Row Operations Math69 W07, HM Zhu

58 Basic row operations = * Three kinds of basic row operations: ) Interchange the order of two rows or (equations) 0 0 a a a a a a 3 = 0 0 a 3 a 3 a 33 a a a 3 a a a 3 a 3 a 3 a 33 58

59 Basic row operations = * ) Multiply a row by a nonzero constant c 0 0 a a a a a a a 3 a 3 a 33 = ca ca ca 3 a a a 3 a 3 a 3 a 33 3) Add or subtract rows 0 0 a a a 3 0 a a a a 3 a 3 a 33 = a a a 3 a a a a a 3 a 3 a 3 a 3 a 33 59