This is a continuation of Black-Scholes-Merton Derivation (Part 2: Option Value Evolution).
A hedging portfolio starts with initial capital and invests in the stock and money market account so that the portfolio value
at each time
is equal to
. This is equivalent to the condition that
for all
.
One way to impose this equality is to ensure that
for all
and
.
Comparing the expressions and
calculated previously in Part 1 and Part 2 respectively, we see that we require the following to hold:
Comparing the terms, we get
. This is known as the delta-hedging rule.
Next, comparing the terms, we have
Simplifying and rearranging, we get
Let , then we have derived the Black-Scholes-Merton partial differential equation:
for all
.
The terminal condition to be satisfied is , where
is the strike price of the call option.
Reference
Shreve, Steven E. Stochastic calculus for finance II: Continuous-time models. Vol. 11. New York: Springer, 2004.