Lecture 19 - Purdue University · 2014-04-03 · Outline 1 Finite Di erence Method 2 Explicit Method Convergence, Stability and Consistency (STAT 598W) Lecture 19 2 / 22

Lecture 19

Xiaoguang Wang

STAT 598W

April 3rd, 2014

(STAT 598W) Lecture 19 1 / 22

Outline

1 Finite Difference Method

2 Explicit MethodConvergence, Stability and Consistency


Outline




Interested PDE

Wilmott et al (1994)

Recall: we are interested in the numerical solution of the PDE:

∂V

∂t+

1

2σ2S2∂

2V

∂S2+ rS

∂V

∂S− rV = 0

V (T ,S) = Φ(S)

where V (t, s) is the price at time t of an European Call option.(stock price=s).

By using the change of variables:

S = K · ex t = T − τ/1

2σ2

V (S , t) = Ke−12(k2−1)x−( 14 (k2−1)

2+k1)τu(x , τ)

where k1 = k2 = r/12σ2.


PDE: transformations

We can transform the PDE into a difussion equation:

∂u

∂τ=∂2u

∂x2

with initial and boundary conditions:

u(x , 0) = max(e12(k2+1)x − e

12(k2−1)x , 0)

limx→−∞

u(x , τ) = 0

limx→∞

u(x , τ) ∼ e12(k2+1)x+ 1

4(k2+1)2τ


Simplified format of the problem

In general, suppose we have the following PDE:

∂u

∂τ=∂2u

∂x2

with conditions:

u(x , 0) = u0(x) limx→−∞

u(x , τ) = f (x , τ)

limx→∞

u(x , τ) = g(x , τ)

Then we use finite difference method to get the numeric solution ofthe PDE.


Types of differences

The basic idea of finite-difference methods is to approximate the differentderivatives of the PDE by finite-differences. The derivative of a function fat a point x can be approximated by any of the following types ofdifferences:

Forward difference: ∆h,1f (x) = f (x+h)−f (x)h ;

Backward difference: ∆h,0f (x) = f (x)−f (x−h)h ;

Centered difference: ∆h,1/2f (x) = f (x+h)−f (x−h)2h ;

General difference:∆h,θf (x) = θ∆h,1f (x) + (1− θ)∆h,0f (x), 0 ≤ θ ≤ 1.

In terms of accuracy, both f ′(x)−∆h,1f (x) and f ′(x)−∆h,0f (x) areO(h) as h→ 0. However, the centered differences are more accurate since

f ′(x)−∆h,1/2f (x) = O(h2)


Approximation for higher orders of derivatives

Note that, by iterating the finite-difference operators ∆, one can easilyconstruct approximations for higher order derivatives. For instance, thefinite-difference approximation of f ′′(x) using centered differences (withmesh h/2) will be:

f ′′(x) = ∆h/2,1/2(∆h/2,1/2f ) = ∆h/2,1/2

(f (x + h/2)− f (x − h/2)

h

)

=1

h

(f (x + h)− f (x)

h− f (x)− f (x − h)

h

)=

f (x + h)− 2f (x) + f (x − h)

h2

Question: what’s the accuracy of the approximation above?


Finite Difference Equations

Once the types of finite-difference approximations have been selected, thenext step consists of approximating the derivatives of the PDE at thepoints of a regular lattice of the solution’s domain. For instance, considerthe head equation on R+ × R and the grid pointsGδt,δx := {(tm, xn)}n∈Z ,m∈N given by

xn = nδx , tm = mδt

where δt and δx are certain mesh parameters determined by the user.Then applying the forward differences in t and centered differences in x ,the solution u can be approximated by a function u : Gδt,δx → R on thelattice that is a solution of the finite-difference equations:

u(tm+1, xn)− u(tm, xn)

δt− u(tm, xn+1)− 2u(tm, xn) + u(tm, xn−1)

δx2= 0

u(0, xn) = Φ(xn)


Finite Difference Equations

For simplicity, we write umn := u(tm, xn), then we get

um+1n = αum

n+1 + (1− 2α)umn + αum

n−1

for any m ≤ 1 and n ∈ Z , where α = δtδx2

. This is the so-called explicitmethod (forward method).In practice, we need to restrict our lattice Gδt,δx and maybe impose someboundary conditions. For instance, if we are only interested in finding thesolution u for t = T and x = x0, we can set δt = T/M, for some large M,and take a triangle-shaped lattice

tm = mδt, xn = x0 + nδx ,m = 0, · · · ,M, n = −(N −m), · · · , (N −m),

with a small mesh δx and a large enough N. In fact, taking N ≥ M willsuffice to determine u(T , x0) uniquely from the values of u at t = 0.


Boundary Conditions

In most cases for derivative pricing, we will be interested in approximatingthe solution in a finite domain

Gδt,δx0 := {(tm, xn) : m = 0, · · · ,M, n = −N, · · · ,N},

and, thus, we will have to impose some conditions on the upper{tm, xN}Mm=0 and on the lower boundaries {tm, x−N}Mm=0. Such conditionsshould be consistent with the behavior of u(t, x) when x →∞ andx → −∞, respectively. There are two common types of boundaryconditions:

Dirichlet conditions: e.g.u(0, x) = Φ(x), u(t,∞) = β(t), u(t,−∞) = α(t), for some knownfunctions Φ, α, β.

Neumann conditions: e.g. ∂xu(t,∞) = β(t), ∂xu(t,−∞) = α(t), forsome known functions α and β.


Summaries: general steps of finite-difference method

Domain discretization: Discretize time and space on a region ofinterest, say [t0, tN ]× [x−N , xN ], leading to a lattice

Gδt,δx0 = {(tm, xn)} determined by given mesh parameters δt and δxas follows:

xn = x0 + nδx , tm = t0 + mδt, n = −N, · · · ,N, m = 0, · · · ,M

Discretization of the differential equations: Approximate thederivatives of the PDE at each point of the lattice by some type finitedifference. The discretization process will result in a system offinite-difference equations that the approximating function u(tm, xn)have to satisfy.

Boundary conditions: Impose boundary conditions that, together withthe finite-difference equations of step 2, determines uniquely u(tm, xn)

in the lattice Gδt,δx0 .

Solving the finite-difference equations: Solve the system offinite-differences with the boundary conditions on the lattice points.


Outline




Explicit Method

We limit our discussion on the heat equation since most PDE we areinterested in can be further transformed into this simple format.

∂u

∂τ=∂2u

∂x2

with conditions:

u(0, x) = u0(x) limx→−∞

u(τ, x) = f (τ, x)

limx→∞

u(τ, x) = g(τ, x)

Then applying the forward differences in t and centered differences in x ,the solution u can be approximated by a function u : Gδt,δx → R on thelattice that is a solution of the finite-difference equations:

u(tm+1, xn)− u(tm, xn)


δx2= 0

u(0, xn) = Φ(xn)


Explicit Method

This method is named as ”explicit” because we can solve the function uexplicitly:

um+1n = puum

n+1 + ps umn + pd um

n−1,

where

pu = pu =δt

(δx)2, ps = 1− 2

δt

(δx)2

Question: What are the conditions needed to make this method holdingnice properties such as consistency, stability?


Outline




Basic Concepts

Let L be the differential operator associated with the heat equation:

(Lu)(t, x) =∂u

∂t(t, x)− ∂2u

∂x2(t, x)

We can write the operator L in the following shorthand notation:

L =∂

∂t− ∂2

∂x2

Let Lδt,δx be a ”finite difference” operator on the lattice Gδt,δx :

(Lδt,δx u)(tm, xn)

=u(tm+1, xn)− u(tm, xn)


δx2


Consistency

A finite-difference operator Lδt,δx is consistent for a differential operator Lif the finite difference approximation of a function u, which is Lδt,δxu,converges to Lu as the mesh parameters shrink to 0. Concretely, for anyfunction u in the domain of L and any bounded domainD = [0,T ]× [c , d ], we have

limδt,δx→0

sup(tm,xn)∈D

|(Lδt,δx u)(tm, xn)− (Lu)(tm, xn)| = 0,

where above u is the restriction of u on the lattice Gδt,δx . Note that theexplicit method is consistent since we have

(Lδt,δx u)(tm, xn)− (Lu)(tm, xn) = O(δt) + O((δx)2)

We say the that the discretization error of Lδt,δx is O(δt) + O((δx)2).


Convergence and Stability

Definition

We say the approximation method is convergent on a given boundeddomain D := [0,T ]× [c , d ] if

limδt,δx

sup(tm,xn)∈D

|uδt,δx(tm, xn)− u(tm, xn)| = 0

Definition

We say that the finite-difference approximation is stable on a givenbounded domain D = [0,T ]× [c , d ] and for any ”nice” bounded Φ on[c, d ] (that is, supx∈[c,d ] |Φ(x)| <∞) if there exists a constant K <∞such that the solution uδt,δx satisfy

max(tm,xn)∈D

|uδt,δx(tm, xn)| ≤ K

for any δt > 0 and δx > 0 small enough.


LAX Equivalence Theorem

Theorem

Let L be a differential operator defining a well-posed initial value problem

Lu(t, x) = 0, u(0, x) = Φ(x)

and let Lδt,δx be a linear and consistent finite-difference approximationoperator for L. Let uδt,δx be the solution of the finite difference equationsand let D = [0,T ]× [c , d ] be a given domain. Then it holds that

limδt,δx

sup(tm,xn)∈D

|uδt,δx(tm, xn)− u(tm, xn)| = 0

if and only if Lδt,δx is stable on the domain D.

Remark: The condition α := δt(δx)2

≤ 12 is a sufficient condition for the

finite-difference approximation in the explicit method to be stable.


Performance of explicit method under round-off errors

We assume that there is a small error in our initial conditions as a result offinite-precision computer errors. We assume that

|u0n − u(0, xn)| ≤ η

for some precision bound η. Define umn := u(tm, xn) and um

n := umn − um

n .Moreover, let’s assume some small roundoff errors εmn :

um+1n − um

n

δt−

umn+1 − 2um

n + umn−1

δx2= εmn + εmn , |εmn | ≤ η

where |εmn | ≤ K1δt + K2δx2. Then finally we can show that

|umn | ≤ (T + 1)η + T (K1δt + K2δx2)

which will converge to zero as η → 0 and δt, δx → 0.


Exercise

Implement the Finite-difference algorithm in order to price an EuropeanCall option with the following parameters:

S = 50

K = 50

r = 0.1

σ = 0.4

T = 0.4167


Documents

Lecture 19 - Purdue University · 2014-04-03 · Outline 1 Finite Di erence Method 2 Explicit Method Convergence, Stability and Consistency (STAT 598W) Lecture 19 2 / 22