Chebyshev polynomial approximation to approximate partial differential equations

We fit the functional at Chebyshev nodes to estimate the coefficients¹. The advantage of
our approach is its flexibility, and the fact that it is easily implementable and since the
functional, at least in the first empirical example, is approximated using deterministic
nodes, we obtain less disperse estimates of the coefficients. In addition, beside our
specific applications, our method it is applicable in other fields, providing efficient
solutions to complex systems of partial differential equations. These features make our
approach very attractive. One reason why polynomial approximations of this type are
underutilised (in comparison to direct ad hoc approximation methods) by applied
researchers might be lack of familiarity. Therefore, in Section 3, we provide some
guidance on how to use them to solve systems of differential equations.

The layout of the paper is the following. Section 2 describes the option pricing
valuation model, which is our application. Section 3 outlines the approximation method
we advocate to obtain the solution to the option pricing problem. Section 4 evaluates its
empirical performance. Section 5 summarises the main findings of this study and offers
some concluding remarks.

¹ Tzavalis and Wang (2003) use a similar approach based on Chebyshev approximation to approximate the
optimal exercise boundary in the context of a stochastic volatility model. Their method also relies on
extrapolation procedures.

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