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1

Finite Difference Methods

A finite difference method obtains a price for a derivative by solving the partial differential equation numerically

Example: • An American put option on a stock that pays a continuous dividend yield q.

ƒ ƒ 2ƒ 2 2 (r q) S ½ 2 S rf t S S

• Finite difference methods aim to represent the differential equation in the form of a difference equation • We form a grid by considering equally spaced time values and stock price values • Define ƒi,j as the value of ƒ at time iDt when the stock price is jDS

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Implicit Finite Difference Method

In

ƒ ƒ 1 2 2 2ƒ (r q) S S rƒ 2 t S 2 S

we set at the node (i,j):

ƒ ƒ i ,j 1 ƒ i ,j 1 S 2DS and: 2 ƒ ƒ i ,j 1 ƒ i ,j ƒ i ,j ƒ i ,j 1 DS 2 DS DS S 2 ƒ ƒ i ,j 1 ƒ i ,j 1 2ƒ i ,j 2 S D S2

or

If we also set

ƒ ƒ i 1,j ƒi ,j t Dt

we obtain the implicit finite difference method.

Re-arranging we get:

a j ƒi ,j 1 b j ƒi ,j c j ƒi ,j 1 ƒi 1,j where: The Boundary Conditions

• Example: American put fN,j=0 fN,j=0 N,j

a j ƒi,j 1 b j ƒi,j c j ƒi ,j 1 ƒi 1,j

• N-1 equations • N-1 unkowns • Exactly 1 solution fN,j=max(K-jΔS,0) f0,j=K

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Explicit Finite Difference Method

If f S and 2 f S 2 are assumed to be the same at the (i 1,j ) point as they are at the (i,j ) point we obtain the explicit finite difference method f i 1, j 1 f i 1, j 1 f S 2 DS and : f i 1, j 1 f i 1, j 1 2 f i 1, j f 2 S DS 2

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With the explicit method we do not need to solve any equations. We can immediatel y calculate the value for f i , j from the values one time - step ahead : ƒi , j a* ƒ i 1, j 1 b* ƒ i 1, j c* ƒ i 1, j 1 j j j

With:

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Implicit vs Explicit Finite Difference Methods ƒi , j +1 ƒi , j ƒi , j –1

Implicit Method

ƒi +1, j +1 ƒi +1, j ƒi , j ƒi +1, j ƒi +1, j –1

Explicit Method

Implicit

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Explicit

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Finite Difference Methods versus Trees

• The explicit finite difference method is equivalent to the trinomial tree approach

Remember:

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Explicit FD method as a Tree pu pm

pd

We need: pu>0,pm>0 and pu>0

This not always true!

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• One can show that the implicit finite difference method is equivalent to a multinomial tree approach

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Improving Efficiency

• It is computationally more to use a FD method with a grid for lnS than S itself. • For Z=lnS we have:

• For the implicit FD method we get: where: 15

• The step-size is now proportional with S • For the implicit method we get: with: 16

• The explicit equation now becomes:

• with:

• If we set DZ 3Dt , then the three probabilities are positive pu pm pd

• The explicit FD method is now identical to the trinomial tree.

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Crank-Nicolson method

• Rolling back with the implicit method: ƒ i , j a j ƒ i 1, j 1 b j ƒ i 1, j c j ƒ i 1, j 1 • Rolling back with the implicit method: ƒ i 1, j a * ƒ i , j 1 b* ƒ i , j c * ƒ i , j 1 j j j

• If we sum the two, we get: ƒ i , j ƒ i , j 1 a j ƒ i 1, j 1 b j ƒ i 1, j c j ƒ i 1, j 1 a * ƒ i , j 1 b* ƒ i , j c* ƒ i , j 1 j j j

• Defining gi,j as, g i , j f i , j a * ƒ i , j 1 b* ƒ i , j c * ƒ i , j 1 j j j

• we obtain:

a j ƒ i 1, j 1 1 b j ƒ i 1, j c j ƒ i 1, j 1 g i , j

• This is the Crank-Nicolson scheme

• Why do we care? • The Crank-Nicolson scheme converges faster then either the implicit or the explicit method.

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