254 Frequently Asked Questions In Quantitative Finance
knowVand its derivatives then we know everything
about the right-hand sideexcept for the value of dS,
because this is random.
These random terms can be eliminated by choosing
=
∂V
∂S
.
After choosing the quantity, we hold a portfolio
whose value changes by the amount
d=
(
∂V
∂t
+^12 σ^2 S^2
∂^2 V
∂S^2
)
dt.
This change is completelyriskless. If we have a com-
pletely risk-free changedin the portfolio value
then it must be the same as the growth we would get
if we put the equivalent amount of cash in a risk-free
interest-bearing account:
d=rdt.
This is an example of the no arbitrage principle.
Putting all of the above together to eliminateand
in favour of partial derivatives ofVgives
∂V
∂t
+^12 σ^2 S^2
∂^2 V
∂S^2
+rS
∂V
∂S
−rV=0,
the Black–Scholes equation.
Solve this quite simple linear diffusion equation with the
final condition
V(S,T)=max(S−K,0)
and you will get the Black–Scholes call option formula.
This derivation of the Black–Scholes equation is perhaps
the most useful since it is readily generalizable (if not
necessarily still analytically tractable) to different under-
lyings, more complicated models, and exotic contracts.
