finite diff

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Finite Differences (FD)

As with any class of option, the price of the derivative is governed by solving

the underlying partial differential equation. The use of finite differencemethods allows us to solve these PDEs by means of an iterative procedure.

We can start by looking at the Black-Scholes partial differential equation:

where dV is the change in the value of an option, dt is a small change in time.is the volatility of the underlying, S is the underlying price and is thecarry (r-q).

By specifying initial and boundary conditions, one can attain numericalsolutions to all the derivatives of the Black-Scholes PDE using a finitedifference grid. The grid is typically set up so that partitions in twodimensions - space and time (in our case, we would be looking at the asset

price and the change in time): Once the grid is set up, there are three methods to evaluate the PDE at each

time step. The difference between each of the three methods is contingenton the choice of differenceused for time (i.e. forward, backward or centraldifferences). Central differences is used for the space grid (S).


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Explicit Finite Differences


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Explicit FD uses forward differencesat each time node t. By splitting the differentialequation into the time element and space elements, we can apply forwarddifferences to time as follows:

if we substitute x = ln(S), the equation becomes:

Applying the finite differences method, the above equation can be broken down andapproximated:

becomes

For the space grid, we can apply central differencesfor all order of derivatives:

becomes

and

becomes

and

becomes


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Combining the terms gives:

Which is the same as:

where the probabilities of each of the nodes is:

This case is actually equivalent to the trinomial tree where probabilities canbe assigned to the likelihood of an up move, a down move as well as nomove. It can also be shown that the following approximation holds:


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Explicit Finite-Difference

For stability and convergence,

As we can see, the result from this method isexplicitly given because we know the value(the claim) at the boundary where the option

expires. Then, we perform the calculationbackwards in time until the valuation date.Since the time dependence (i) only dependson future dates (i+1) we can explicitlycalculate the change, node by node backwardin time.

.3 tx


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Implicit Finite Differences


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Implicit Finite Differences

The implicit method takes backward differencesfor the timederivativebut still using central differencesfor the space derivatives.

Although similar in nature to the explicit finite differences method, theimplicit FD method is typically more stable and convergent than theexplicit FD method - however, it is often more computationally intensive.

The approximation to the PDE under an implicit FD method is given bythe following:

Note that the main difference between the above equation and the onefor the explicit FD method is in the selection of time step i. The

subsequent simplification of the approximation and the associatedprobabilities is similar to that of the explicit FD method.


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Crank-Nicholson Scheme


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Crank-Nicholson Scheme

An improvement over the implicit FD method is the Crank-NicolsonScheme which uses central differences for both time and spacedimensions. The result is that over smaller time steps dt, the methodis more accurate, stable and convergent than both implicit andexplicit methods - however, like the implicit FD method (which

requires evaluating equations at each time step) it is morecomputationally intensive than the explicit FD method.

The approximation for the PDE is given as:

finite diff

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