Chebyshev polynomial approximation to approximate partial differential equations

2. The Valuation Model

In this paper we present a simple and flexible method to approximate partial differential
equations. We start showing how to use the method by pricing an European option.
Suppose that the price of a non-dividend-paying asset in period 0 is S₀ , and denote with
K the strike price of a put option written on that asset.

Assumption 1: We assume that the option value depends on the stock price at expiry of
the option and time,V_t(S_t,t).

Suppose also that the process for S is described by the following geometric
Brownian motion:

dS_t = S_trdt + S_tσdZ_t                      (1)

where dZ is a standard increment of a Wiener process, and σthe variance parameter.

We can expand E ( dV )ɪ, using Ito's Lemma and the stochastic process above to
dt

obtain:

rV = rSV_s + V_t +¹S ²σ ² V_ss             (2)

s                     ss

where V_(.) represents the derivative with respect to the argument in the subscript.

All European options, in absence of arbitrage, must satisfy Equation (2). A call option
will have at expiry a payoff given byS - K , if S > K , while for a put option we have at
expiry the payoff K - S , if K > S . Therefore, in our specific case, the boudary
condition is given by

V(S,t) = max(0, K -S)                    (3)

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