Frequently Asked Questions In Quantitative Finance

Chapter 4: Ten Different Ways to Derive Black–Scholes 267

compensation in excess of the risk-free rate for taking

unit amount of risk must be the same for each.

For the stock, the expected return (dividing bydt)isμ.

Its risk isσ.

From Itˆowehave

dV=

∂V

∂t

dt+^12 σ^2 S^2

∂^2 V

∂S^2

dt+

∂V

∂S

dS.

Therefore the expected return on the option is

1

V

(

∂V

∂t

+^12 σ^2 S^2

∂^2 V

∂S^2

+μS

∂V

∂S

)

and the risk is

1

V

σS

∂V

∂S

.

Since both the underlying and the option must have the

same compensation, in excess of the risk-free rate, for

unit risk

μ−r

σ

=

1

V

(

∂V

∂t+

1

2 σ

(^2) S 2 ∂^2 V
∂S^2
+μS∂∂VS
)
1
VσS
∂V
∂S
.
Now rearrange this. Theμdrops out and we are left
with the Black–Scholes equation.

Utility Theory

The utility theory approach is probably one of the least

useful of the ten derivation methods, requiring that we

value from the perspective of an investor with a utility

function that is a power law. This idea was introduced

by Rubinstein (1976).