266 Frequently Asked Questions In Quantitative Finance
future option value using the probabilitiesp′for an up
move and 1−p′for a down.
Again this is the idea of the option value as the present
value of the expected payoff under a risk-neutral random
walk. The quantityp′is the risk-neutral probability, and
it is this that determines the value of the option not
the real probability. By comparing the expressions for
pandp′we see that this is equivalent to replacing the
real asset driftμwith the risk-free rate of returnr.
We can examine the equation forVin the limit asδt→0.
We write
V=V(S,t), V+=V(uS,t+δt)andV−=V(vS,t+δt).
Expanding these expressions in Taylor series for small
δtwe find that
∼
∂V
∂S
as δt→0,
and the binomial pricing equation forVbecomes
∂V
∂t
+^12 σ^2 S^2
∂^2 V
∂S^2
+rS
∂V
∂S
−rV= 0.
This is the Black–Scholes equation.
CAPM
This derivation, originally due to Cox & Rubinstein
(1985) starts from theCapital Asset Pricing Modelin
continuous time. In particular it uses the result that
there is a linear relationship between the expected
return on a financial instrument and the covariance
of the asset with the market. The latter term can be
thought of as compensation for taking risk. But the
asset and its option are perfectly correlated, so the
