Chebyshev polynomial approximation to approximate partial differential equations

achieve a comparable degree of accuracy even increasing the order of the polynomial to
30. Furthermore, Chebyshev polynomials exhibit their usual oscillation which appears
fairly evenly over the interval we have considered. On the other hand, spline polynomials
exhibit larger oscillations at the edges of the interval.

4. Option Pricing

To show how to empirically use the proposed method we provide now two examples. We
start with the first example, which has already described in section 2. In the first case, we
use the proposed methodology to value an European put option written on a stock . In this
case, there is the possibility to compare the empirical result given by our methodology
with the Black and Scholes closed form solution. We use the absolute error (ASE) as a
measure of accuracy.

Table 1 shows the results for the entire set of options considered. We also report
the results using the Black and Scholes method (1973 - B&S henceforth).

Insert Table 1 here

We fit (4) using the first twenty Chebyshev basis to estimate the parameters c_i in (4)⁷.
The basis number has been chosen using Theorem 6.4.2 in Judd (1998). Once the
coefficients have been estimated, we estimate the boundary condition in (3) by simulating
200,000 paths for the stock. We can see that regardless the option considered, our method
produces rather accurate option prices. The absolute error reported at the bottom of Table
1 confirms that.

7

Note that we estimate these coefficients using Chebyshev nodes and Chebyshev basis.

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