school of mathematicspjohnson/resources/math60082/lecture... · review improved finite difference...

ReviewImproved Finite Difference Methods

Exotic optionsSummary

FINITE DIFFERENCE - CRANK NICOLSON

Dr P. V. Johnson

School of Mathematics

2013

Dr P. V. Johnson MATH60082



OUTLINE

1 REVIEWLast time...Today’s lecture

2 IMPROVED FINITE DIFFERENCE METHODSThe Crank-Nicolson MethodSOR method

3 EXOTIC OPTIONSAmerican optionsConvergence and accuracy

4 SUMMARYOverview




Last time...Today’s lecture

Introduced the finite-difference method to solve PDEsDiscetise the original PDE to obtain a linear system ofequations to solve.This scheme was explained for the Black Scholes PDE andin particular we derived the explicit finite differencescheme to solve the European call and put optionproblems.

The convergence of the method is similar to the binomialtree and, in fact, the method can be considered a trinomialtree.Explicit method can be unstable - constraints on our gridsize.





Introduced the finite-difference method to solve PDEsDiscetise the original PDE to obtain a linear system ofequations to solve.This scheme was explained for the Black Scholes PDE andin particular we derived the explicit finite differencescheme to solve the European call and put optionproblems.The convergence of the method is similar to the binomialtree and, in fact, the method can be considered a trinomialtree.Explicit method can be unstable - constraints on our gridsize.





Here we will introduce the Crank-Nicolson methodThe method has two advantages over the explicit method:

stability;improved convergence.

Here we will need to solve a matrix equation.

In addition we will discuss how to price American optionsand how to remove nonlinearity error in a variety of cases.





Here we will introduce the Crank-Nicolson methodThe method has two advantages over the explicit method:

stability;improved convergence.

Here we will need to solve a matrix equation.In addition we will discuss how to price American optionsand how to remove nonlinearity error in a variety of cases.




The Crank-Nicolson MethodSOR method

CRANK NICOLSON METHOD

The Crank-Nicolson scheme works by evaluating thederivatives at V(S, t + ∆t/2).The main advantages of this are:

error in the time now (∆t)2

no stability constraints

Crank-Nicolson method is implicit, we will need to usethree option values in the future (t + ∆t)to calculate three option values at (t).





Crank-Nicolson grid

0 T∆t 2∆t (i+1)∆t... ...

0

∆S

2∆S

.

.

j∆S

.

.

SU

Vji

upper boundary

lower boundary

pde holds in this region

Vj-1i

Vj+1i

Vji+1

Vj-1i+1

Vj+1i+1

i∆t

Focus attention on i, j-th value Vji, and a little

piece of the grid around that point





APPROXIMATING AT THE HALF STEP

Now take approximations to the derivatives at the halfstep t + 1/2∆tThey are in terms of Vi

j , as follows:

∂V∂t≈

Vi+1j −Vi

j

∆t

∂V∂S≈ 1

4∆S(Vi

j+1 −Vij−1 + Vi+1

j+1 −Vi+1j−1)

∂2V∂S2 ≈

12∆S2 (V

ij+1 − 2Vi

j + Vij−1 + Vi+1

j+1 − 2Vi+1j + Vi+1

j−1)





DERIVING THE EQUATION

Here the Vi values are all unknown, so...rearrange our equations to have the known values on onesidethe unknown values on the other.

14 (σ

2j2 − rj)Vij−1 + (−σ2j2

2− r

2− 1

∆t)Vi

j +14 (σ

2j2 + rj)Vij+1 =

− 14 (σ

2j2 − rj)Vi+1j−1 − (−σ2j2

2− r

2+

1∆t

)Vi+1j − 1

4(σ2j2 + rj)Vi+1

j+1

There is one of these equations for each point in the grid





MATRIX EQUATIONS

•We can rewrite the valuation problem in terms of a matrix asfollows:

b0 c0 0 0 . . . . 0a1 b1 c1 0 . . . . .0 a2 b2 c2 0 . . . .. 0 a3 b3 c3 0 . . .. . . . . . . . .. . . 0 aj bj cj 0 .. . . . . . . . .0 . . . . . 0 ajmax bjmax

Vi0

Vi1

Vi2

Vi3..

Vijmax−1Vi

jmax

=

di

0di

1di

2di

3..

dijmax−1di

jmax





MATRIX EQUATIONS

where:

aj = 14 (σ

2j2 − rj)

bj = −σ2j2

2− r

2− 1

∆tcj = 1

4 (σ2j2 + rj)

dj = − 14 (σ

2j2 − rj)Vi+1j−1 − (−σ2j2

2− r

2+

1∆t

)Vi+1j

− 14 (σ

2j2 + rj)Vi+1j+1





WHAT TO DO ON THE BOUNDARIES?

Boundary conditions are an important part of solving anyPDEFor most PDEs we know the boundary conditions for largeand small SFor call options ajmax = 0, bjmax = 1,djmax = Sue−δ(T−i∆t) −Xe−r(T−i∆t), b0 = 1, c0 = 0, d0 = 0

For put options b0 = 1, c0 = 0, d0 = Xe−r(T−i∆t), ajmax = 0,bjmax = 1, djmax = 0In general we can always determine the values of b0, c0, d0,ajmax, bjmax and djmax from our boundary conditions.





THE CRANK-NICOLSON METHOD

At each point in time we need to solve the matrix equationin order to calculate the Vi

j values.

There are two approaches to doing this,solve the matrix equation directly (LU decomposition),solve the matrix equation via an iterative method (SOR).

If possible, the LU approach is the preferred approach as itgives you an exact value for Vi

j and is much faster.

However, not possible to use LU approach with Americanoptions.The SOR (Successive Over Relaxation) can be easilyadapted to value American options





MATRIX EQUATIONS

The SOR method is a simpler approach but can take a littlelonger as it relies upon iteration.If we consider each of the individual equations fromAV = d we have that

a1Vi0 + b1Vi

1 + c1Vi2 = di

1

a2Vi1 + b2Vi

2 + c2Vi3 = di

1

............ = ...ajVi

j−1 + bjVij + cjVi

j+1 = dij

............ = ...ajmax−1Vi

jmax−2 + bjmax−1Vijmax−1 + cjmax−1Vi

jmax = dijmax−1





JACOBI ITERATION

Rearrange these equations to get:

Vij =

1bj(di

j − ajVij−1 − cjVi

j+1)

The Jacobi method is an iterative one that relies upon theprevious equation.

Taking an initial guess for Vij , denoted as Vi,0

jiterate using the formula below for the (k + 1)th iteration:

Vi,k+1j =

1bj(di

j − ajVi,kj−1 − cjV

i,kj+1)

carry on until the difference between Vi,kj and Vi,k+1

j issufficiently small for all j.

For Gauss-Seidel use the most up-to-date guess wherepossible:

Vi,k+1j =

1bj(di

j − ajVi,k+1j−1 − cjV

i,kj+1)





SOR

The SOR method is another slight adjustment. It startsfrom the trivial observation that

Vi,k+1j = Vi,k

j + (Vi,k+1j −Vi,k

j )

and so (Vi,k+1j −Vi,k

j ) is a correction term.Now try to over correct value, should work faster.This is true if Vi,k

j → Vij monotonically in k.

So the SOR algorithm says that

yi,k+1j =

1bj(di


i,kj+1)

Vi,k+1j = Vi,k

j + ω(yi,k+1j −Vi,k

j )

where 1 < ω < 2 is called the over-relaxation parameter.




American optionsConvergence and accuracy

AMERICAN OPTIONS: EXPLICIT

American option pricing problem requires an optimalearly exercise strategy.To generate one, compare the continuation value with theearly exercise value - take the larger.With the explicit finite difference method is prettystraightforward

calculate the continuation value CoVij

CoVij =

11 + r∆t

(AVi+1j+1 + BVi+1

j + CVi+1j−1)

then compare this to the early exercise payoff.

Thus for a put:

Vij = max[X− j∆S,

11 + r∆t

(AVi+1j+1 + BVi+1

j + CVi+1j−1)]

This is similar to using the binomial tree





AMERICAN OPTIONS: EXPLICIT

American option pricing problem requires an optimalearly exercise strategy.To generate one, compare the continuation value with theearly exercise value - take the larger.With the explicit finite difference method is prettystraightforward

calculate the continuation value CoVij

CoVij =

11 + r∆t

(AVi+1j+1 + BVi+1

j + CVi+1j−1)

then compare this to the early exercise payoff.Thus for a put:

Vij = max[X− j∆S,

11 + r∆t

(AVi+1j+1 + BVi+1

j + CVi+1j−1)]

This is similar to using the binomial treeDr P. V. Johnson MATH60082




American put option: Explicit

...0

∆S

2∆S

.

.

j∆S

.

.

SUImpose upper boundary at SU

Impose lower boundary at 0

terminalboundary

0 T∆t 2∆t (i+1)∆t ...i∆t

Move through “interior” ofmesh/grid using this rule





AMERICAN PUT OPTION: C-N

The American option pricing problem is slightly morecomplex for the Crank-Nicolson method.Consider the process of calculating Vi

j ...

The value of the option Vij , for all values of j, depends also

upon the value of Vij−1 and Vi

j+1.

Optimally deciding when to early exercise requires that wealready know these values.If we early exercise at some point this could change Vi

j forall j.





PSOR

A simple solution to this problem is to project our SORmethod (Projected SOR)In order to project, check whether or not it would beoptimal to exercise at each iteration.

This changes

yi,k+1j =

1bj(di


i,kj+1)

Vi,k+1j = Vi,k


j )

to (in the case of the American put option)

yi,k+1j =

1bj(di


i,kj+1)

Vi,k+1j = max(Vi,k


j ), X− j∆S)





PSOR

A simple solution to this problem is to project our SORmethod (Projected SOR)In order to project, check whether or not it would beoptimal to exercise at each iteration.This changes

yi,k+1j =

1bj(di


i,kj+1)

Vi,k+1j = Vi,k


j )

to (in the case of the American put option)

yi,k+1j =

1bj(di


i,kj+1)

Vi,k+1j = max(Vi,k


j ), X− j∆S)





CONVERGENCE

If the option price and the derivatives are well behavedthen the errors of the

Explicit method are O(∆t, (∆S)2)Crank-Nicolson method are O((∆t)2, (∆S)2).

These can be considered similar to the distribution errorfor the binomial tree.If convergence is smooth we can use extrapolation.

Finite-difference methods can suffer from non-linearityerror if the grid is not correctly aligned with respect to anydiscontinuities

in the option value,or in the derivatives of the option value.





CONVERGENCE

If the option price and the derivatives are well behavedthen the errors of the

Explicit method are O(∆t, (∆S)2)Crank-Nicolson method are O((∆t)2, (∆S)2).

These can be considered similar to the distribution errorfor the binomial tree.If convergence is smooth we can use extrapolation.Finite-difference methods can suffer from non-linearityerror if the grid is not correctly aligned with respect to anydiscontinuities

in the option value,or in the derivatives of the option value.





NON-LINEARITY ERROR

Now have the freedom to construct the grid as desired.Makes it is simple to construct the grid so that you have agrid point upon any discontinuities.For example, if we consider an European call or put optionthen the only source of non-linearity error is at S = X atexpiry.

Always choose ∆S so that X = j∆S for some integer valueof j.So if in this case S0 = 100 and X = 95, you need a suitablylarge SU and a ∆S which is a divisor of 95.





NON-LINEARITY ERROR

Now have the freedom to construct the grid as desired.Makes it is simple to construct the grid so that you have agrid point upon any discontinuities.For example, if we consider an European call or put optionthen the only source of non-linearity error is at S = X atexpiry.Always choose ∆S so that X = j∆S for some integer valueof j.So if in this case S0 = 100 and X = 95, you need a suitablylarge SU and a ∆S which is a divisor of 95.





BARRIER OPTIONS

When pricing barrier options, there is a large amount ofnon-linearity error that comes from not having the nodesin the tree aligned with the position of the barrier.Thus with barrier options we have two sources ofnon-linearity error

the error from the barrierthe error from the discontinuous payoff.

Simply match the grid to the barrier and the payoff.For a down and out barrier option choose SL (the lowervalue of S) to be on the barrier and then, as in the previousexample, choose ∆S so that the exercise price is also on anode.




Overview

OVERVIEW

We have introduced the Crank-Nicolson finite differencemethod.It is:

slightly harder to program;has faster convergence;better stability properties.

Applying the method to American options requires the useof PSORmore complex than the method for valuing Americanoptions using the explicit method.Can choose the dimensions of the grid so as to remove thenonlinearity error.


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