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,Vt(St,t).
Suppose also that the process for S is described by the following geometric
Brownian motion:
dSt = Strdt + StσdZt (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 = rSVs + Vt +1S 2σ 2 Vss (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)