Higher Order Finite Difference Scheme for solving 3D Black-Scholes equation based on Generic Factored Approximate Sparse Inverse Preconditioning using Reordering Schemes E-N.G. Grylonakis, C.K. Filelis-Papadopoulos, G.A. Gravvanis
Dec 15, 2015
Higher Order Finite Difference Scheme for solving 3D Black-Scholes equation based on Generic
Factored Approximate Sparse Inverse Preconditioning using Reordering Schemes
E-N.G. Grylonakis, C.K. Filelis-Papadopoulos, G.A. Gravvanis
Introduction
Financial Instruments
Derivatives
Futures Forwards Swaps Options
Financial contracts that give the holder the right but not the obligation to buy (call option) or sell (put option) an underlying asset
for a fixed price at a specific date.
Options
ProblemHow much money should one pay to buy a specific
option contract?
Topic of InterestAccurate option pricing for three
underlying assets, using the multi-asset Black-Scholes
equation.
Options Pricing Methods
Binomial Options Pricing Model
Lattice Methods
Monte Carlo Methods
Black-Scholes-Merton Model
Black-Scholes (BS) Equation
ru-S
uSr+
SS
uSSssp
2
1=Lu
n
1=i ii
n
1=ji, ji
2
jijiij ∂
∂
∂∂
∂
Time Dependent Convection-Diffusion-ReactionPartial Differential Equation
Pricing with the BS equation
Single-Asset Option
1D BS PDE Closed-Form Solutions
Multi-Asset Option
N-D BS PDE ApproximateSolutions
Pricing Methodology
Option Contract
① Number of underlying assets② Parameters (Strike Price, Expiration date, etc.)③ Payoff Function (Initial Condition)④ Boundary Conditions
Numerical solution of the corresponding BS Partial Differential Equation
Three-Asset Basket Option
Payoff Function
Max { w[I(T)-K], 0 }
n
1=jjj tIw=tI
where wj is the total investment in asset j ( as a percentage) and Ij(t)
is the price of j-th asset.
Linear Boundary Conditions
Commonly used in practical pricing problems, providing stability when used with the Finite Difference Method
Spatial Discretization
Finite Difference Schemes(4rth order accuracy)
12
1
3
2 0
3
2
12
1
12
1
3
4
2
5
3
4
12
1
or
or
computationaldomain
boundary
ghostvalues
Ghost Values Treatment
Richardson’s extrapolation method (4rth order accuracy)
Modified Stencils
6
1 1
2
1
3
1First Derivatives:
Second Derivatives:
The imposition of linear boundary conditions forces the second derivatives to vanish on the
boundary. The first order derivatives were discretized by a fourth order one-sided
approximation:
4
1
3
4 3 4
12
25Leftmost boundary:
We denote by
1
kx
the discretized first order derivative
for coordinate xk . Then, the stencil of the derivative with respect
to coordinate k can be formed in a d-dimensional way:
mk
1k
1m
1
kmkd
1d
km
d
k
Ix
Ix
The cross-derivative can be approximated by the following expression:
m
1
1m
1
mk
1
1m
1
md
1d
m
d
k
Ix
Ix
Ixx
k
kk
2
The coefficient matrix is then formed by the following tensor product:
mkmkd x
1-k
1mkx
1-d
kmk exeX
where:
The above schemes reduce the programming effort substantiallywhile providing a compact method to discretize PDE’s in higherdimensions.
Numerical Time Integration
After the spatial discretization, a system of Ordinary Differential Equations of the following form, occurs:
This system can be solved by the implicit fourth order backward difference scheme (BDF4):
3211
4
1
3
434
12
25
nnnnn uuuuutAI
It can be observed that the BDF4 scheme requires the discretesolution in three previous time steps. These values can be obtained by the Implicit Runge-Kutta method (4rth order accurate):
,kbtyys
1iiin1n
,)ka+y,c+(x f=s
1j=jijninik tt
The coefficients of the 2-stage method are:
Solving the Linear System
The arising large, sparse,linear system was solved by thePreconditioned BiConjugate Gradient Stabilized (PBiCG-STAB) method, in conjunction with the Modified Generic Factored Approximate Sparse Inverse (MGenFAspI) scheme.
M=GH MGenFAspI matrix:
The MGenFAspI matrix is computed by solving the following systems:
The modified approach minimizes the searches for elements and enhances the performance of the method.
Approximate Minimum Degree(AMD) Reordering
When attempting to solve large sparse linear systems, reordering schemes can be used in order to minimize the fill-in during the factorization process.
The AMD algorithm produces a reordering such that the vertices with minimum degree are to be eliminated first.
The degree of each vertex is approximated through an upper bound created by the sum of the weights of the neighboring vertices, increasing the performance of the resulting ordering scheme.
AMD
Implementation Issues
In order to compute the three initial solutions with the Runge-Kuttamethod, the solution of four linear systems at every time step is required.
Recalling the vectors, required by the R-K method:
,)ka+y,c+(x f=s
1j=jijninik tt
The 2-stage method requires the computation of vectors k1 and k2
which can be obtained by the following system:
The above system can be expressed in the following blockform:
n
n
2
1
11
11
Au
Au
k
k
DC
BA
tt
n1
11n
n
2
111
AuACAu
Au
k
k
S0
BA
ttt
where:
The computation of k1 and k2 is then performed by solving thefollowing linear systems at every time step:
The Schur complement is computed implicitly, since iterative methods do not require the coefficient matrix explicitly, because the product of a matrix by a vector is only needed. Thus:
Numerical Results
The estimated price of the basket option:
Performance (“seconds.hundreds”) of the PBiCG-STAB, based on the MGenFAspI in conjunction with AMD reordering scheme, for various values of N and droptol:
Convergence behavior of the PBiCG-STAB, based on the MGenFAspI in conjunction with AMD reordering scheme, for various values of N and droptol:
The number of nonzero elements in the G and H factors of the MGenFAspI for various values of N and droptol:
Conclusions1. The Black-Scholes PDE can be used to price options with many
underlying assets, without relying solely on Monte Carlo methods.
2. The high order schemes combined with a multi-dimensional PDE result in a large, sparse, linear system, thus, iterative methods are the best choice.
3. Preconditioners and reordering schemes can be used to enhance the performance of the chosen iterative method.
4. The MGenFAspI matrix has been proved to be an effective preconditioner and combined with various iterative methods has achieved better convergence behavior in comparison with other methods.
5. Moreover, the applicability of the MGenFAspI matrix in conjunction with the PBiCG-STAB method has been evaluated, for various model problems, derived from Computational Fluid Dynamics, Computational Structural Analysis and Plasma Physics.
ReferencesAchdou, Y., and Pironneau, O. 2005. Computational methods for option pricing. SIAM.
Amestoy, P., Davis, T.A., and Duff, I.S. 1996. An approximate minimum degree ordering algorithm. SIAM Journal on Matrix Analysis and Applications 17(4), 886-905.
Chow, E. 2000. A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM J. Sci. Comput. 21, 1804-1822.
Duffy, D.J. 2006. Finite difference methods in financial engineering: A partial differential equation approach. John Wiley and Sons.
Filelis-Papadopoulos, C.K., and Gravvanis, G.A. A class of generic factored and multilevel recursive approximate inverse techniques for solving general sparse systems (submitted).
Grylonakis E-N.G. 2014. On the study and numerical solution of the Black-Scholes equation. Dissertation Thesis, Department of Electrical and Computer Engineering, Democritus University of Thrace.
Gustafsson, B. 2008. High order difference methods for time dependent PDE. Springer Series in Computational Mathematics, Vol.38
Haug, E.G. 2007. The complete guide to option pricing formulas. McGraw-Hill.Hull, J. 2009. Options, futures and other derivatives. Prentice-Hall.
Jeong, D., Kim, J., and Wee, I-S. 2009. An accurate and efficient numerical method for Black-Scholes equations. Commun. Korean Math. Soc. 24(4), 617–628
Leentvaar, C.C.W, and Oosterlee, C.W. 2008. On coordinate transformation and grid stretching for sparse grid pricing of basket options. J. Comp. Appl. Math. 222, 193-209.
Persson, J., and von Sydow, L. 2007. Pricing European multi-asset options using a space-time adaptive FD-method. Comput. Vis. Sci. 10, 173–183.
Saad, Y., Schultz, M.H. 1986. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Comp. 7, 856-869.
Süli, E., and Mayers, D.F. 2003. An introduction to numerical analysis. Cambridge University Press.
Trottenberg, U., Osterlee, C.W., and Schuller, A. 2000. Multigrid. Academic Press.
Van der Vorst, H.A. 1992. Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. and Stat. Comput. 13(2), 631–644.
Wilmott, P., Dewynne, J., and Howison, S. 1994. Option pricing: Mathematical models and computation. Oxford Financial Press.
Zhang, P.G. 1998. Exotic Options: A guide to second-generation options. World Scientific, 2nd edition.
Thank you for your attention!