178 Mathematics for Finance
Theorem 8.3
The expectation of the discounted payoff computed with respect to the risk-
neutral probability is equal to the present value of the derivative security,
D(0) =E∗(
(1 +r)−^2 f(S(2)))
Exercise 8.5
LetS(0) = 120 dollars,u=0.2,d=− 0 .1andr=0.1. Consider a call
option with strike priceX= 120 dollars and exercise timeT=2.Find
the option price and the replicating strategy.Exercise 8.6
Using the data in the previous exercise, find the price of a call and the
replicating strategy if a 15 dollar dividend is paid at time 1.8.1.3 GeneralN-Step Model ..............................
The extension of the results above to a multi-step model is straightforward.
Beginning with the payoff at the final step, we proceed backwards, solving the
one-step problem repeatedly. Here is the procedure for the three-step model:
D(3) =f(S(3)),D(2) =1
1+r[p∗f(S(2)(1 +u)) + (1−p∗)f(S(2)(1 +d))]
=g(S(2)),D(1) =1
1+r[p∗g(S(1)(1 +u)) + (1−p∗)g(S(1)(1 +d))]
=h(S(1)),D(0) =1
1+r[p∗h(S(0)(1 +u)) + (1−p∗)h(S(0)(1 +d))],where
g(x)=^1
1+r[p∗f(x(1 +u)) + (1−p∗)f(x(1 +d))],h(x)=1
1+r[p∗g(x(1 +u)) + (1−p∗)g(x(1 +d))].