D.4  Derivation of the PDE

We start with a more general argument.  Let f(S,t) denote the value of any derivative security (e.g., a call option) at time t when the stock price is S.  Note that, by assumption,  the derivative price does not depend on the path of S, but only on the current value of S.

By Ito's lemma,

and recall that for the stock price

Now, consider the riskless hedge portfolio in Chapter 2, topic 2.4.

-k   derivative security

1    share of stock.

The number of derivative securities, k, is

.

The value of this portfolio, V, is given by

and the change in the value is

Substituting the df equation and the dS equation into this portfolio and rearranging, we get

The important fact about this equation is that dV is independent of the stochastic term z.  This means that all the risk of the derivative can be hedged by holding

units of the derivative.

Since dV is instantaneously deterministic, the portfolio return must equal the instantaneous risk-free return.  In other words,

Substituting for dV and V, we get

or

This is called the Black-Scholes (partial) differential equation.  The solution to this equation, subject to the appropriate boundary condition for f, determines the value of the derivative security.  For a European call option, the boundary condition is

f(S,T) = max{0,S-X}.

The solution of the differential equation for the call option is given by the Black-Scholes formula:

where

We can also apply Ito’s Lemma to derive the implied distribution of stock prices from a geometric Brownian motion model of stock prices.  The next topic Distribution of Stock Prices provides a technical overview of this issue.

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