190 Mathematics for Finance
Rather than looking at the details of this limit, we just refer to Figure 8.2
for illustration. It shows the priceCEof a European call with strikeX= 100
on a stock withS(0) = 100,σ=0.3andm=0.2. (Thoughmis irrelevant for
the Black–Scholes formula, it still features in the discrete time approximation.)
The continuous compounding interest rate is taken to ber=0.2. The option
price is computed in two ways, as a function of the timeTremaining before
the option is exercised:
a) (solid line) from the Black–Scholes formula forTbetween 0 and 1;b) (dots) using the Cox–Ross–Rubinstein formula withT increasing from 0
to1inN= 10 steps of durationτ=0.1 each; the discrete growth rates for
each step are computed using formulae (3.7).
Even with as few as 10 steps there is remarkably good agreement between the
discrete and continuous time formulae.