Finite Difference Methods

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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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