Pricing Multi-Dimensional Options by Grid Stretching and High Order Finite Differences

Transcription

1 Pricing Multi-Dimensional Options by Gri Stretching an High Orer Finite Differences Kees Oosterlee Numerical Analysis Group, Delft University of Technology Joint work with Coen Leentvaar Southern Ontario Numerical Analysis Day, Talk at Numerical Analysis Day, /nr. 1

2 Research Concept Numerical Treatment of Equations for Option Pricing Accurate iscretization with only a few gri points Example I: Black-Scholes equation Gri stretching High orer iscretization European option Example II: Basket options - Increasing problem imensions sparse gris A coorinate transformation Gri stretching an iscretization as in the 1D case European basket options Talk at Numerical Analysis Day, /nr. 2

3 Application: Option pricing Basic options European Call option: at maturity time T, the holer may purchase an asset for the exercise price K. The writer must sell the asset, if the holer ecies to buy it. European Put option: The right to sell an asset on a certain ate at a prescribe amount. s s 0 K 0 0 t T Exotic options: options epening on other functions of the stock price (average stock price, minimum, maximum, a basket of stocks) Talk at Numerical Analysis Day, /nr. 3

4 Options on a Single Asset Point of Departure - The asset price follows the lognormal ranom walk, S t = µs t t + σs t W t, with W t a Wiener process, µ is rift, σ volatility. - Interest rate r, ivien yiel δ an σ are known, - Transaction costs for heging are not inclue, - There are no arbitrage possibilities. Black-Scholes partial ifferential equation: (for a European option) u t σ2 s 2 2 u + (r δ)s u s2 s ru = 0 Nobel prize in 1997 for Merton an Scholes (Black ie in 1995). Talk at Numerical Analysis Day, /nr. 4

5 Final/Bounary conitions Single Asset European Call option: Right to buy assets at maturity t = T for exercise price K. Call option: Final conition: u(s,t ) = max(s K,0) = (s K) + u K s Bounary conitions for s 0: u(s,t) = 0. Bounary conitions at s : u(s,t) = se δ(t t) Ke r(t t) or: u ss = 0. The stanar European option can be solve exactly (serves as a reference). Talk at Numerical Analysis Day, /nr. 5

6 Important Quantities Hege Parameters Delta: the rate of change of the option value with respect to s. Portfolio with u ± s is instantaneously risk neutral. = u s Gamma: inicates the change in Delta Γ = 2 u s 2 If Gamma is high, the portfolio results for a very short time in a risk-less scenario. There are several other important heging parameters. Talk at Numerical Analysis Day, /nr. 6

7 Increasing imensions Multi-Asset Options The problem imension increases if the price of an option epens on more than one asset s i (the so-calle multi-asset options). Each unerlying asset is assume to follow a geometric (lognormal) iffusion process. It is assume that the correlation of each asset to all other assets is constant. Each aitional asset is represente by an extra imension in the problem: Lu : = u t u [σ i σ j ρ i, j s i s j ] + i, j=1 s i s j i=1 u [(r δ i )s i ] ru = 0. s i The require information to value a basket option is the volatility of each asset σ i an the correlation between each pair of assets ρ i, j. Talk at Numerical Analysis Day, /nr. 7

8 Introuction: Several option-types Final conitions etermine the type of the option Basket Call, option on a basket of assets: u(s,t) = max{ i=1 Call option on the maximum of several assets Exchange option on two assets n i s i K,0} u(s,t) = max{max{s 1,s 2,,...,s } K,0}} u(s,t) = max{s 1 s 2,0} Talk at Numerical Analysis Day, /nr. 8

9 Basket options A basket option is an option whose payoff epens on the value of a portfolio (or basket) of assets. Basket options are growing in popularity as a means of heging the risk of a portfolio an are highly interesting for banks nowaays. They are attractive because an option on a basket is cheaper than buying options on the iniviual assets. Furthermore, their payoff profile replicates the changes in a portfolio s value more closely than any combination of options on the unerlying assets. Talk at Numerical Analysis Day, /nr. 9

10 1D Gri Stretching General Consier a general parabolic PDE with non-constant coefficients u t = α(s) 2 u s 2 + β(s) u s + γ(s)u(s,t) Coorinate transformation x = ξ(s) (one-to-one), inverse s = η(x) = ξ 1 (x) an û(x,t) := u(s,t). Chain rule, the first an secon erivative: u s = 1 û η (x) x, 2 u s = 1 2 û 2 (η (x)) 2 x η (x) û 2 (η (x)) 3 x. Application changes the factors α, β an γ into: α(x) = α(η(x)) (η (x)) 2, β(x) = β(η(x)) η (x) α(η(x)) η (x) (η (x)) 3, γ(x) = γ(η(x)). Talk at Numerical Analysis Day, /nr. 10

11 Gri Stretching T Black-Scholes t o K S A coorinate transformation that clusters points in the region of interest, aroun s = K, the nonifferentiability in the final conition. Spatial transformation use for Black-Scholes [Clarke-Parrott, Tavella-Ranall]: x = ξ(s) = sinh 1 (ζ(s K)) + sinh 1 (ζk). An equiistant gri iscretization (n x an n t cells) after the analytic transformation Talk at Numerical Analysis Day, /nr. 11

12 Discretization Fourth Orer Discretization Finite ifferences, base on Taylor s expansion O(h 2 +k 2 ) is easily achieve by central ifferencing an Crank-Nicolson iscretization Our aim: High accuracy with only a few gri points 4th orer long stencil iscretizations in space an in time O(h 4 + k 4 ) The 4th orer implicit Backwar Differentiation Formula, BDF4, time integration is use. Talk at Numerical Analysis Day, /nr. 12

13 Discretization Fourth orer in space (long stencils): û i t = 1 12h 2 α i ( û i û i+1 30û i + 16û i 1 û i 2 ) h β i ( û i+2 + 8û i+1 8û i 1 + û i 2 ) + γ i û i + O(h 4 ), 2 i N 2. Fourth orer in time: BDF4 scheme (precee by CN, BDF3). BDF4 reas ( ) I kl No stability complications observe (1) û j+1 = 4û j 3û j û j 2 1 4û j 3, (2) Well-suite for linear complementarity problems (for American options) Talk at Numerical Analysis Day, /nr. 13

14 Accuracy European option pricing experiment, ivien yiel Error in u h an hege parameters h,γ h (comparison with analytic solution u ex ). K = 15, σ = 0.3, r = 0.05, δ = 0.03, T = 0.5. Scheme Gri u u ex rate ex rate Γ Γ ex rate O(h 2 + k 2 ) no stretching O(h 4 + k 4 ) gri stretching Talk at Numerical Analysis Day, /nr. 14

15 European pricing experiment on stretche gri u h, h, Γ h S S S Talk at Numerical Analysis Day, /nr. 15

16 Higher Dimensions: Sparse Gris Combination Technique S 2 S 1 Talk at Numerical Analysis Day, /nr. 16

17 Sparse Gris, Combination of solutions Zenger, Griebel (1990/1991) The combination equation for the sparse gri solutions in 2D reas: u comb n = u sparse u sparse I =n+1 I =n where I correspons to the number of gri points in each irection. If n = 4 then, all combinations of I = 5 are: (16,2),(8,4),(4,8),(2,16). The combination equation for the sparse gri solutions in in general imensions reas: u comb n = 1 k=0 ( 1) k ( 1 k ) u sparse I =n+ 1 k Sparse Gri Techniques converge nicely if mixe secon erivatives in the problem are boune. Talk at Numerical Analysis Day, /nr. 17

18 Higher Dimensions: Sparse Gris Overall Gri Number of points processe O(N(logN) 1 ) versus O(N ) (full gri) Accuracy of solutions O(N 2 (logn) 1 ) versus O(N 2 ) (full gri, h = 1/N) Talk at Numerical Analysis Day, /nr. 18

19 Secon orer Accurate Discretization of 2 u i=1 xi 2 + i=1 Full Gri N max error Conv Sparse Gri Test Case 5D Reference Equation u x i 5u = 0, with solution: u(x 1,...,x ) = e i=1 ( 1)i+1 x i Sparse Gri N max error Conv Asymptotic convergence factors sparse gri: 2D: 3.61, 3D: 3.13, 4D: 2.86, 5D: 2.61 Talk at Numerical Analysis Day, /nr. 19

20 Higher Dimensional B-S Transformation, Stretching an Sparse Gris! See also Reisinger (2003/2004) Coorinate transformation with Linear transformation: u t = i=1 j=1 α i j 2 u u + s i s j β i ru i=1 s i α i j = 1 2 ρ i jσ i σ j s i s j, β i = (r δ i )s i (3) X = ΓS, S = Γ 1 X = ZX (4) x i = γ im s m, s i = m=1 z im x m, m=1 x i s k = γ ik (5) Talk at Numerical Analysis Day, /nr. 20

21 Transforme equation: an Final Result u t = i=1 i=1 k=1 u t = i=1 ˆα i j = j=1 m=1 k=1 k=1 l=1 j=1 l=1 m=1 ˆα i j 2 u u + x i x j ˆβ i ru i=1 x i α kl x i s k x j s l, ˆβi = n=1 x i β k (6) k=1 s k 2 u ρ kl σ k σ l γ ik γ jl z km z ln x m x n + x i x j (r δ k )γ ik z km x m u x i ru (7) Talk at Numerical Analysis Day, /nr. 21

22 Actual Coorinate Transformation Transformation use (Tavella, Ranall): 2D : x 1 = n 1 s 1 + n 2 s 2 x 2 = n 1 s 1 + n 2 s 2 3D : x 1 = n 1 s 1 + n 2 s 2 + n 3 s 3 x 2 = n 1 s 1 + n 2 s 2 + n 3 s 3 x 3 = n 1 s 1 n 2 s 2 + n 3 s 3 Talk at Numerical Analysis Day, /nr. 22

23 Gri Stretching Stretching coorinate x i with function x i = x i (y i ), gives a solution u = u(y 1,y 2,...,y ). The equation then reas u t = i=1 i=1 i=1 k=1 j=1 k=1 k=1 l=1 m=1 l=1 m=1 m=1 n=1 n=1 ρ kl σ k σ l γ ik γ jl z km z ln x m x n 1 J i J j ρ kl σ k σ l γ ik γ il z km z ln x m x n H i J 3 i (r δ k )z km x m γ ik J i u x i ru u y i + 2 u y i y j + (8) With J i the first, H i the secon erivative of the stretching function: J i = x i /y i, H i = 2 x i /y 2 i. In practice: Stretching only in x 1 Talk at Numerical Analysis Day, /nr. 23

24 Discretization an Kronecker proucts Each erivative must be taken apart an summe up. To simplify this, we use Kronecker proucts, which combine the 1D iscretization stencils to the imensional case. The secon erivative of the assets is in the 2D case: [ 2 ] u 2 s 2 [ 2 ] u 2 s [ 2 ] 1 u = I s [ 2 ] u = I 2 [ ] 2 s 1 1 Talk at Numerical Analysis Day, /nr. 24

25 Kronecker proucts General Let A be a matrix of size k l an B a matrix of size m n. Then the Kronecker prouct A B is a matrix of size k m n l with a 11 B a 12 B... a 1l B A B = a 21 B a 22 B... a 2l B a k1 B a k2 B... a kl B Assume that the matrices are of appropriate size to calculate the proucts. The multiple Kronecker prouct is given by an the following expression: NO B A 1 A 2... A N = i=1 NO i=1 A i A i = B (A 1 A 2... A N ) Talk at Numerical Analysis Day, /nr. 25

26 Discretization an Kronecker proucts The secon erivative of the i-th asset in -imensions can be written in the same way as the 2D case: [ 2 ] u 2 s i = i 1 O n=0 [ 2 ] u I n 2 s i Also non-constant coefficients can easily be implemente an it also usable for the first erivative. 1 i 1 O n=1 I i n Talk at Numerical Analysis Day, /nr. 26

27 Kronecker proucts: Correlation To use Kronecker proucts with the correlation term, we rewrite this term as: 2 u s 1 s 2 = s 2 ( ) u s 1 It follows that: [ 2 ] u s 1 s 2 2 = [ ] u s 2 1 [ ] u s 1 1 Talk at Numerical Analysis Day, /nr. 27

28 Kronecker proucts: Correlation In general, for imensions, the correlation matrix reas: [ 2 ] i 1 [ ] u O j 1 [ ] u O u = I n I n s i s j s j s i n=0 1 n=i O n= j+1 This can also be use with non-constant coefficients an therefore it is usable for the high-d Black-Scholes equation. I n Talk at Numerical Analysis Day, /nr. 28

29 Results: 2D Two-asset European basket call option (from Tavella s book). K = 100 r = 4.5% δ 1 = 5%, δ 2 = 7% T = 1 σ 1 = 0.25, σ 2 = 0.35 an ρ 12 = 0.63 Payoff: max{0.58s S 2 K,0} Talk at Numerical Analysis Day, /nr. 29

30 Coorinate Transformation + Stretching + Sparse Gris Payoff: max(n 1 s 1 + n 2 s 2 K,0) (left: no transformation, no stretching, right: stretching an transformation). Payoff Payoff U0 100 U S S X X Talk at Numerical Analysis Day, /nr. 30

31 Coorinate Transformation + Sparse Gris Solution u at t = 0 : (left: no transformation, no stretching, right: stretching an transformation). Solution Solution U U S S X X Talk at Numerical Analysis Day, /nr. 31

32 Numerical Results 2D Black-Scholes Full Eq Eq Trafo-Stretch Trafo-Stretch Gri Secon Fourth Secon Fourth (16 16) (32 32) (64 64) Value (Tavella) Sparse Eq Eq Trafo-Stretch Trafo-Stretch Gri Secon Fourth Secon Fourth Value (Tavella) Talk at Numerical Analysis Day, /nr. 32

33 Basket Option: 3D Three-asset European basket call option (from Tavella s book). K = 100 r = 4.5% δ 1 = 5%, δ 2 = 7%, δ 3 = 4% T = 1 σ 1 = 0.25, σ 2 = 0.35, σ 3 = 0.20 ρ 12 = 0.63, ρ 13 = 0.25, ρ 23 = 0.5 Payoff: max{0.38s S S 3 K,0} Talk at Numerical Analysis Day, /nr. 33

34 Numerical Results Full gri computations: Eq Eq Stretch Stretch Secon Fourth Secon Fourth Tavella 3D Sparse gri with transformation, stretching an 4th orer iscretization: Talk at Numerical Analysis Day, /nr. 34

35 Conclusions Options on Single Asset: Accurate option values with gri stretching in space an 4th orer iscretization in space an time The sparse gri metho (recombination technique) is an interesting choice if the problem imension increases Basket options: Accurate option values with coorinate transformation, gri stretching, 4th orer iscretization in space an time an the sparse gri metho. Talk at Numerical Analysis Day, /nr. 35