8. Option Pricing..............................................
By aEuropean derivative securityorcontingent claim with stockSas the
underlying asset we mean a random variable of the formD(T)=f(S(T)),
wherefis a given function, called thepayoff. This is a direct generalisation of
a call option withf(S)=(S−X)+, a put option withf(S)=(X−S)+,ora
forward contract withf(S)=S−X(for the long position).
We have already learnt the basic method of pricing options in the one-step
model (see Section 1.6) based on replicating the option payoff. Not surprisingly,
this idea extends to a general binomial tree model constructed out of such one-
step two-state building blocks. Developing this extension will be our primary
task in this chapter.
Theorem 8.1
Suppose that for any contingent claimD(T) there exists a replication strategy,
that is, an admissible strategyx(t),y(t) with final valueV(T)=D(T). Then
the priceD(0) of the contingent claim at time 0 must be equal to that of the
replicating strategy,V(0) =D(0).
Proof
The proof is just a modification of that of Proposition 1.3. IfD(0)>V(0),
then we write the derivative security and take a long position in the strategy.
Our obligation will be covered by the strategy, the differenceD(0)−V(0) being
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