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.

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