Chapter 4: Ten Different Ways to Derive Black–Scholes 263
This partial differential equation can now be solved for
the Black–Scholes formulæ.
This method is not used in practice for finding these
formulæ, but rather, knowing the traded prices of van-
illas as a function ofKandTwe can turn this equation
around to findσ, since the above analysis is still valid
even if volatility is stock and time dependent.
Continuous-time Limit of the
Binomial Model
Some of our ten derivations lead to the Black–Scholes
partial differential equation, and some to the idea of the
option value as the present value of the option payoff
under a risk-neutral random walk. The following simple
model does both.
In the binomial model the asset starts atSand over a
time stepδteither rises to a valueu×Sor falls to a
valuev×S,with0<v< 1 <u. The probability of a rise
ispand so the probability of a fall is 1−p.
We choose the three constantsu,vandpto give the
binomial walk the same drift,μ, and volatility,σ,asthe
asset we are modelling. This choice is far from unique
and here we use the choices that result in the simplest
formulæ:
u= 1 +σ
√
δt,
v= 1 −σ
√
δt
and
p=
1
2
+
μ
√
δt
2 σ
.
Having defined the behaviour of the asset we are ready
to price options.
