Inserting or Stretching Points in Finite Difference Discretizations

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Inserting or Stretching Points in Finite Diﬀerence

Discretizations

Jherek Healy

*Correspondence: jherekhealy@protonmail.com

Abstract:

Partial diﬀerential equations sometimes have critical points where the solution or some of its

derivatives are discontinuous. The simplest example is a discontinuity in the initial condition. It is well known

that those decrease the accuracy of ﬁnite diﬀerence methods. A common remedy is to stretch the grid, such

that many more grid points are present near the critical points, and fewer where the solution is deemed smooth.

An alternative solution is to insert points such that the discontinuities fall in the middle of two grid points.

This paper compares the accuracy of both approaches in the context of the pricing of ﬁnancial derivative

contracts in the Black-Scholes model and proposes a new fast and simple stretching function.

Keywords: ﬁnite diﬀerence method, grid stretching, Black-Scholes

1. Introduction

Partial diﬀerential equations (PDEs) sometimes have critical points where the solution or some of its

derivatives are discontinuous. The simplest example is a discontinuity in the initial condition. This situation

arises in the pricing of nearly all ﬁnancial derivative contracts. The vanilla European option of given maturity

and strike price, the simplest non-linear contract, has indeed a discontinuous ﬁrst derivative at the strike price.

It is well known that such critical points decrease the accuracy of ﬁnite diﬀerence methods. A common

remedy, detailed in [Tavella and Randall 2000, p. 167], is to stretch the grid such that many more grid points are

present near the critical points, and fewer where the solution is deemed smooth. The stretching transformation

for a single point reads

S(u)=B+αsinh(c2u+c1(1 −u)),(1)

where

c1=asinh Smin−B

c2=asinh Smax−B

, and

controls the density of points near the critical point

. For

u∈[0,1], we have S(u)∈[Smin,Smax].

Independently of such a stretching, Giles and Carter [2005], Tavella and Randall [2000] also show that the

error in the solution is signiﬁcantly decreased when the critical points are located in the middle of two grid

points. There are several ways to place the critical points in such manner. A ﬁrst approach is to move the grid.

This is applicable only for a single critical point, and if the boundaries can be moved. A second approach is to

simply insert a point in the grid, around the critical point such that the critical point is exactly in the middle of

two grid points. A third approach is to use a smooth deformation, typically a monotonic cubic spline, to place

the critical point approximately (but not exactly) in the middle of two grid points [Tavella and Randall 2000, p.

171].

The advantage of the cubic spline smooth deformation is to preserve the second-order convergence. A

robust implementation is however more involved than the insertion approach. The insertion approach, due to

its lack of smoothness, will a priori not preserve the second-order convergence, but this does not mean that its

accuracy is worse.

arXiv:2210.02541v4 [math.NA] 20 Dec 2022

2 of 13

In this paper, we compare the accuracy of the two approaches, using concrete examples of options in the

Black-Scholes model, on nearly uniform grids, as well as on stretched grids. We also propose a faster stretching

transformation, similar to the sinh transformation and give a simple extension to multiple critical points.

2. Cubic stretching

2.1. Single critical point

According to Noye [1983, p. 307], a stretching function should have the following properties:

(i) dS/du

should be ﬁnite over the whole interval - if it becomes inﬁnite at some point, then there is poor

resolution near that point;

(ii) dS/du

must be smaller near at critical point than elsewhere in the interval, which ensures high resolution

near the critical point, but dS/ dushould be non zero at the critical point.

An intuitive candidate would be a function based on a probability density function. A mixture distribution

makes it easy to ensure a higher density around the critical points. A numerical inversion of the mixture

distribution, for example via a monotonic interpolation scheme, leads to the desired stretching function.

Unfortunately, such a stretching will typically have very large derivatives near the boundaries (corresponding to

the inverse of the cumulative density tails) and thus does not obey property (i).

For a single critical point, an interesting stretching function candidate is the cubic based on the Taylor

series of the sinh function:

S(u)=B+α·1

χ(c2u+c1(1 −u))3+c2u+c1(1 −u)¸,(2)

where

is the solution of the depressed cubic equation

χc3

1+c1+B−Smin

α=0

and

is the solution of

χc3

2+c2+

B−Smax

α=0. The value χ=6matches the sinh expansion, other positive values are also possible.

Figure 1shows the cubic transformation to be close to the sinh transformation in practice. As expected, it

is not exponential and thus closer to linear, far away from the critical point. For the same value of

, the slope

is slightly diﬀerent at the critical point. The slope is matched using a lower

α=0.9

for the cubic stretching.

One main advantage of the cubic stretching is performance, as the transformation doesn’t involve any costly

function at all. In practice, the cubic stretching is around ﬁve times faster.

2.2. Many critical points

Tavella and Randall [2000] propose to use the following jacobian for multiple critical points (Bk):

J(u,S)=∂S

∂u=AÃX

α2

k+(S(u)−Bk)2!−1

,(3)

where

is a normalizing constant used to ensure that

S(1) =Smax

with initial condition

S(0) =Smin

. The Jacobian

is nearly constant for

S≈Bk

which corresponds to a uniform discretization and is nearly linear for

SÀBk

S¿Bkwhich corresponds to a exponential grid.

Equation 3is an ordinary diﬀerential equation (ODE) whose initial condition consists in the function

values at two end-points: it is a two-points boundary problem. A standard method to solve this kind of problem

is the shooting method: we are shooting a projectile from point

S(0) =Smin

so that it lands at point

S(1) =Smax

Any solver can be used so solve for

. The ODE can be solved with the fourth-order Runge-Kutta method for a

given guess A.

3 of 13

Figure 1. Stretching around the point B=125 using 63 points in the interval [0,150] with α=1.50.

Similarly, the derivative of Equation 2provides a candidate stretching for multiple points:

du=αA

i=1

(u−bi)2+α.

The solution

(A,b1,...,bn)

such that

S(0) =Smin

S(1) =Smax

and

S(bi)=Bi

involves a

-dimensional non-linear

optimization and may not be practical for large n.

Solving such non-linear problems makes the overall technique much slower than the single critical point

case, and more challenging to implement in a robust fashion. We thus describe below simpler, better performing

and more robust techniques below.

2.2.1. Direct piecewise-cubic representation

Based on Equation 2, we consider a piecewise-cubic representation of class

. Let

(Bi)i=1,...,m

be the

ordered

critical points in the interval

(Smin,Smax)

. Let

Di=Bi+Bi+1

be the corresponding mid-points for

i=1,...,m−1

, and

D0=Smin,Dm=Smax

for notation convenience. The piecewise cubic interpolant on the

interval [di−1,di)reads

pi(u)=Bi+αi·1

χ(c2i(u−di−1)+c2i−1(di−u))3+c2i(u−di−1)+c2i−1(di−u)¸,(4a)

where diis such that

pi(di)=Di,pi(di−1)=Di−1.(4b)

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摘要：

ArticleInsertingorStretchingPointsinFiniteDierenceDiscretizationsJherekHealy*Correspondence:jherekhealy@protonmail.comAbstract:Partialdierentialequationssometimeshavecriticalpointswherethesolutionorsomeofitsderivativesarediscontinuous.Thesimplestexampleisadiscontinuityintheinitialcondition.Itiswel...

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Inserting or Stretching Points in Finite Difference Discretizations

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