D.4 Derivation
of the PDE

e
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.
n the standard
BlackScholes option pricing analysis, the underlying asset pays zero dividends
over the life of the option. Here we consider
relaxing this assumption to allow for the effects of a constant continuous
dividend yield over the life of the option.
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 riskfree return. In
other words,
Substituting
for dV and V, we get
or
This
is called the BlackScholes (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,SX}.
The
solution of the differential equation for the call option is given by the BlackScholes
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.