174 Mathematics for Finance
our arbitrage profit. IfD(0)<V(0),then we take the opposite positions, with
V(0)−D(0) the resulting arbitrage profit.
Replication also solves the problem of hedging the position of the option
writer. If the cash received for the option is invested in the replicating strategy,
then all the risk involved in writing the option will be eliminated.
In this chapter we shall gradually develop such pricing methods for options,
starting with a comprehensive analysis of the one-step binomial model, which
will then be extended to a multi-step model. Finally, the Black–Scholes formula
in continuous time will be introduced.
8.1 European Options in the Binomial Tree Model ...............
8.1.1 One Step..........................................
This simple case was discussed in Chapter 1. Here we shall reiterate the ideas
in a more general setting: We shall be pricing general derivative securities and
not just call or put options. This will enable us to extend the approach to the
multi-step model.
We assume that the random stock priceS(1) at time 1 may take two values
denoted by {
Su=S(0)(1 +u),
Sd=S(0)(1 +d),
with probabilitiespand 1−p, respectively. To replicate a general derivative
security with payofffwe need to solve the system of equations
{
x(1)Su+y(1)(1 +r)=f(Su),
x(1)Sd+y(1)(1 +r)=f(Sd),
forx(1) andy(1). This gives
x(1) =f(Su)−f(Sd)
Su−Sd,
which is the replicating position in stock, called thedeltaof the option. We
also find the money market position
y(1) =−(1 +d)f(Su)−(1 +u)f(Sd)
(u−d)(1 +r)