66 Mathematics for Finance
3.3.2 Continuous-Time Limit
Discrete-time and discrete-price models have apparent disadvantages. They
clearly restrict the range of asset price movements as well as the set of time
instants at which these movements may occur. In this section we shall outline
an approach free from such restrictions. It will be obtained by passing to the
continuous-time limit from the binomial tree model.
We shall consider a sequence of binomial tree models with time stepτ=N^1 ,
lettingN→∞. For all binomial tree models in the approximating sequence it
will be assumed that the probability of upward and downward price movements
is^12 in each step.
In this context it proves convenient to use the logarithmic return
k(n) = ln(1 +K(n)) ={
ln(1 +u) with probability 1/2,
ln(1 +d) with probability 1/2.In place of the risk-free rate of return over one time step, we shall use the
equivalent continuous compounding rater, so that the return over a time step
of lengthτwill be eτr.
We denote bymthe expectation and byσthe standard deviation of the
logarithmic returnk(1)+k(2)+···+k(N) over the unit time interval from 0 to 1,
consisting ofNsteps of τ. The logarithmic returnsk(1),k(2),...,k(N)
are identically distributed and independent, just asK(1),K(2),...,K(N)are.
It follows that
m=E(k(1) +k(2) +···+k(N))
=E(k(1)) +E(k(2)) +···+E(k(N)) =NE(k(n)),
σ^2 =Var(k(1) +k(2) +···+k(N))
=Var(k(1)) + Var (k(2)) +···+Var(k(N)) =NVar (k(n))for eachn=1, 2 ,...,N. This means that eachk(n) has expectationmN=mτ
and standard deviation
√
σ^2
N =σ√
τ, so the two possible values of eachk(n)
must be
ln(1 +u)=mτ+σ
√
τ,
ln(1 +d)=mτ−σ√
τ.(3.7)
Exercise 3.22
Findmandσforu=0.02,d=− 0 .01 andτ=1/12.length