Chapter 2: FAQs 149
the behaviour of the underlying is random. Options
on terrorist acts cannot be hedged since terrorist acts
aren’t traded (to my knowledge at least).
We still have to price contracts even in incomplete
markets, so what can we do? There are two main ideas
here. One is to price the actuarial way, the other is to
try to make all option prices consistent with each other.
The actuarial way is to look at pricing in some average
sense. Even if you can’t hedge the risk from each option
it doesn’t necessarily matter in the long run. Because
in that long run you will have made many hundreds
or thousands of option trades, so all that really mat-
ters is what the average price of each contract should
be, even if it is risky. To some extent this relies on
results from theCentral Limit Theorem. This is called
the actuarial approach because it is how the insurance
business works. You can’t hedge the lifespan of indi-
vidual policyholders but you can figure out what will
happen to hundreds of thousands of themon average
using actuarial tables.
The other way of pricing is to make options consis-
tent with each other. This is commonly used when we
have stochastic volatility models, for example, and is
also often seen in fixed-income derivatives pricing. Let’s
work with the stochastic volatility model to get inspira-
tion. Suppose we have a lognormal random walk with
stochastic volatility. This means we have two sources of
randomness (stock and volatility) but only one quan-
tity with which to hedge (stock). That’s like saying
that there are more states of the world than underly-
ing securities, hence incompleteness. Well, we know we
can hedge the stock price risk with the stock, leaving
us with only one source of risk that we can’t get rid
of. That’s like saying there is one extra degree of free-
dom in states of the world than there are securities.