Chapter 4: Ten Different Ways to Derive Black–Scholes 261
Now we go over to the risk-neutral world to value
the local-time term, ending up, eventually, with the
Black–Scholes formula.
It is well worth simulating this strategy on a spread-
sheet, using a finite time step and let this time step get
smaller and smaller.
Parameters as Variables
The next derivation is rather novel in that it involves
differentiating the option value with respect to the
parameters strike,K, and expiration,T, instead of the
more usual differentiation with respect to the variables
Sandt. This will lead to a partial differential equation
that can be solved for the Black–Scholes formulæ. But
more importantly, this technique can be used to deduce
the dependence of volatility on stock price and time,
given the market prices of options as functions of strike
and expiration. This is an idea due to Dupire (1993)
(also see Derman & Kani, 1993, and Rubinstein, 1993, for
related work done in a discrete setting) and is the basis
for deterministic volatility models and calibration.
We begin with the call option result from above
V=e−r(T−t)EtQ[max(ST−K, 0)],
that the option value is the present value of the risk-
neutral expected payoff. This can be written as
V(K,T)=e−r(T−t
∗)
∫∞
0
max(S−K,0)p(S∗,t∗;S,T)dS
=e−r(T−t
∗)
∫∞
K
(S−K)p(S∗,t∗;S,T)dS,
wherep(S∗,t∗;S,T) is the transition probability density
function for the risk-neutral random walk withS∗being
