Chapter 4: Ten Different Ways to Derive Black–Scholes 253
Hedging and the Partial Differential
Equation
The original derivation of the Black–Scholes partial
differential equation was via stochastic calculus, Itˆo’s
lemma and a simple hedging argument (Black & Scholes,
1973).
Assume that the underlying follows a lognormal ran-
dom walk
dS=μSdt+σSdX.
Useto denote the value of a portfolio of one long
option position and a short position in some quantity
of the underlying:
=V(S,t)−S. (4.1)
The first term on the right is the option and the second
term is the short asset position.
Ask how the value of the portfolio changes from timet
tot+dt. The change in the portfolio value is due partly
to the change in the option value and partly to the
change in the underlying:
d=dV−dS.
From Itˆo’s lemma we have
d=
∂V
∂t
dt+
∂V
∂S
dS+^12 σ^2 S^2
∂^2 V
∂S^2
dt−dS.
The right-hand side of this contains two types of terms,
the deterministic and the random. The deterministic
terms are those with thedt, and the random terms are
those with thedS. Pretending for the moment that we
