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What is the Finite-Difference Method?

Short answer

The finite-difference method is a way of approximating differential equations, in continuous variables, into difference equations, in discrete variables, so that they may be solved numerically. It is a method particularly useful when the problem has a small number of dimensions, that is, independent variables.

Example

Many financial problems can be cast as partial differential equations. Usually these cannot be solved analytically and so they must be solved numerically.

Long answer

Financial problems starting from stochastic differential equations as models for quantities evolving randomly, such as equity prices or interest rates, are using the language of calculus. In calculus we refer to gradients, rates of change, slopes, sensitivities. These mathematical 'derivatives' describe how fast a dependent variable, such as an option value, changes as one of the independent variables, such as an equity price, changes. These sensitivities are technically defined as the ratio of the infinitesimal change in the dependent variable to the infinitesimal change in the independent. And we need an infinite number of such infinitesimals to describe an entire curve. However, when trying to calculate these slopes numerically, on a computer, for example, we cannot deal with infinites and infinitesimals, and have to resort to approximations.

Technically, a definition of the delta of an option is

where V(S, t) is the option value as a function of stock price, S, and time, t. Of course, there may be other independent variables. The limiting procedure in the above is the clue to how to approximate such derivatives based on continuous variables by differences based on discrete variables.

The first step in the finite-difference methods is to lay down a grid, such as the one shown in Figure 2.8.

The grid typically has equally spaced asset points, and equally spaced time steps, although in more sophisticated

The finite-difference grid.

Figure 2.8: The finite-difference grid.

schemes these can vary. Our task will be to find numerically an approximation to the option values at each of the nodes on this grid.

The classical option pricing differential equations are written in terms of the option function, V(S, t), say, a single derivative with respect to time, the option's theta, the first derivative with respect to the underlying, |y, the option's delta, and the second derivative with respect to the underlying, j^-, the option's gamma. I am explicitly assuming we have an equity or exchange rate as the underlying in these examples. In the world of fixed income we might have similar equations but just read interest rate, r, for underlying, S, and the ideas carry over.

A simple discrete approximation to the partial derivative for theta is

where St is the time step between grid points. Similarly,

where SS is the asset step between grid points. There is a subtle difference between these two expressions. Note how the time derivative has been discretized by evaluating the function V at the 'current' S and t, and also one time step before. But the asset derivative uses an approximation that straddles the point S, using S + SS and S — SS. The first type of approximation is called a one-sided difference, the second is a central difference. The reasons for choosing one type of approximation over another are to do with stability and accuracy. The central difference is more accurate than a one-sided difference and tends to be preferred for the delta approximation, but when used for the time derivative it can lead to instabilities in the numerical scheme. (Here I am going to describe the explicit finite-difference scheme, which is the

easiest such scheme, but is one which suffers from being unstable if the wrong time discretization is used.)

The central difference for the gamma is

Slightly changing the notation so that V k is the option value approximation at the th asset step and kth time step, we can write

Finally, plugging the above, together with S = i SS, into the Black-Scholes equation gives the following discretized version of the equation:

This can easily be rearranged to give Vik 1 in terms of Vik 1, Vk and Vk_ 1, as shown schematically in Figure 2.9.

In practice we know what the option value is as a function of S, and hence , at expiration. And this allows us to work backwards from expiry to calculate the option value today, one time step at a time.

The above is the most elementary form of the finite-difference methods, there are many other more sophisticated versions.

The advantages of the finite-difference methods are in their speed for low-dimensional problems, those with up to three sources of randomness. They are also particularly good when the problem has decision features such as early exercise

The relationship between option values in the explicit method.

Figure 2.9: The relationship between option values in the explicit method.

because at each node we can easily check whether the option price violates arbitrage constraints.

References and Further Reading

Wilmott, P 2006 Paul Wilmott on Quantitative Finance, second edition. John Wiley & Sons Ltd

 
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