Chapter 2: FAQs 147
Consider, for example, the binomial model in which
there are two states of the world at the next time step,
and there are also two securities, cash and the stock.
That is a complete market. Now, after two time steps
there will be three possible states of the world, assum-
ing the binomial model recombines so that an up-down
move gets you to the same place as down-up. You might
think that you therefore need three securities for a com-
plete market. This is not the case because after the
first time step you get to change the quantity of stock
you are holding, this is where the dynamic part of the
replication comes in.
In the equity world the two most popular models for
equity prices are the lognormal, with a constant volatil-
ity, and the binomial. Both of these result in complete
markets, you can replicate other contracts in these
worlds.
In a complete market you can replicate derivatives with
the simpler instruments. But you can also turn this on
its head so that you can hedge the derivative with the
underlying instruments to make a risk-free instrument.
In the binomial model you can replicate an option from
stock and cash, or you can hedge the option with the
stock to make cash. Same idea, same equations, just
move terms to be on different sides of the ‘equals’ sign.
As well as resulting in replication of derivatives, or
the ability to hedge them, complete markets also have
a nice mathematical property. Think of the binomial
model. In this model you specify the probability of the
stock rising (and hence falling because the probabili-
ties must add to one). It turns out that this probability
does not affect the price of the option. This is a sim-
ple consequence of complete markets, since you can