98 Frequently Asked Questions In Quantitative Finance
could calculate the expected stock price at expiration
asE[ST], and then the payoff at that expected price
P(E[ST]). Alternatively we could look at the various
option payoffs and then calculate the expected payoff
asE[P(ST)]. The latter makes more sense, and is indeed
the correct way to value options, provided the expec-
tation is with respect to therisk-neutralstock price. If
the payoff is convex then
E[P(ST)]≥P(E[ST]).
We can get an idea of how much greater the left-hand
side is than the right-hand side by using a Taylor series
approximation around the mean ofS.Write
S=S+ ,
whereS=E[S], soE[ ]=0. Then
E[f(S)]=E
[
f(S+ )
]
=E
[
f(S)+ f′(S)+^12
2 f′′(S)+···
]
≈f(S)+^12 f′′(S)E
[
2
]
=f(E[S])+^12 f′′(E[S])E
[
2
]
.
Therefore the left-hand side is greater than the right by
approximately
1
2 f
′′(E[S])E
[
2
]
.
This shows the importance of two concepts
- f′′(E[S]): Theconvexityof an option. As a rule this
adds value to an option. It also means that any
intuition we may get from linear contracts (forwards
and futures) might not be helpful with non-linear
instruments such as options.
- E
[
2
]
: Randomness in the underlying, and its
variance. Modelling randomness is the key to
modelling options.
The lesson to learn from this is that whenever a con-
tract has convexity in a variable or parameter, and that
