148 Frequently Asked Questions In Quantitative Finance
hedge the option with the stock you don’t care whether
the stock rises or falls, and so you don’t care what the
probabilities are. People can therefore disagree on the
probability of a stock rising or falling but still agree on
the value of an option, as long as they share the same
view on the stock’s volatility.
In probabilistic terms we say that in a complete mar-
ket there exists a unique martingale measure, but for
an incomplete market there is no unique martingale
measure. The interpretation of this is that even though
options are risky instruments we don’t have to specify
our own degree of risk aversion in order to price them.
Enough of complete markets, where can we find in-
complete markets? The answer is ‘everywhere.’ In
practice, all markets are incomplete because of real-world
effects that violate the assumptions of the simple models.
Take volatility as an example. As long as we have a
lognormal equity random walk, no transaction costs,
continuous hedging, perfectly divisible assets,...,and
constant volatility then we have a complete market.
If that volatility is a known time-dependent function
then the market is still complete. It is even still com-
plete if the volatility is a known function of stock price
and time. But as soon as that volatility becomes ran-
dom then the market is no longer complete. This is
because there are now more states of the world than
there are linearly independent securities. In reality,
we don’t know what volatility will be in the future so
markets are incomplete.
We also get incomplete markets if the underlying follows
a jump-diffusion process. Again more possible states
than there are underlying securities.
Another common reason for getting incompleteness
is if the underlying or one of the variables governing
