Chebyshev polynomial approximation to approximate partial differential equations

If we set the value function above V(s,t) ≈ φ(s)c(t), where φ is a suitable basis for an n-
dimensional family of approximating functions and c (t) is an n-vector of time-varying
coefficients, equation (2) can be re-written as follows:

φ(s)c'(t) ≈ [rsφ(s) + 2δ²s²φ"(s) -rφ(s)]c(t) ≈ ψ(s)c(t)                  (4)

To determine c(t), we select n-values (nodes) of s , s_i, and solve (4) for that particular
set of values. Given the n-dimensional family of basis functions chosen, (4) can now be
re-written in the form of a system as follows:

Φc'(t)=Ψc(t)                                  (5)

where Φ and Ψ are two n × n matrices.

Once the coefficients have been obtained as in (5), to price the financial option, we, first,
use the process in (1) to obtain estimates of (3). Finally, we multiply this by the estimated
coefficients. Averaging gives the price of the option.

3. Polynomial Approximation

In this section we describe in greater detail the approximation method adopted in this
paper. Let V ∈ Kⁿ⁺¹ be a function defined on the interval [a, b], the latter may well not be
tractable analytically, and assume that P is a polynomial that interpolates V at the
n

distinct n + 1 points s_i ∈ [a,b], with P(s) = ∑ c_iφ_i(s) . In order to solve the problem in
i=0

Section 2 by approximation we need to define: (a) the family of basis functions to
approximate the function V , (b) the interpolation nodes, s_i . In this section we show that
Chebyshev polynomials in conjunction with Chebyshev nodes offer the best solution to
our problem.

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