108156.pdf

8. Option Pricing 175

The initial value of the replicating portfolio isx(1)S(0) +y(1). By Theorem 8.1

D(0) =x(1)S(0) +y(1)

=

f(Su)−f(Sd)

u−d

−

(1 +d)f(Su)−(1 +u)f(Sd)

(u−d)(1 +r)

. (8.1)

Exercise 8.1

Show that the price of a call option grows withu, the other variables

being kept constant. Analyse the impact of a change ofdon the option

price.

Exercise 8.2

Find a formula for the priceCE(0) of a call option ifr=0andS(0) =

X= 1 dollar. Compute the price foru=0.05 andd=− 0 .05, and also

foru=0.01 andd=− 0. 19 .Draw a conclusion about the relationship

between the variance of the return on stock and that on the option.

Recall the notion of the risk-neutral probability, given by

p∗=

r−d

u−d

, (8.2)

which turns the discounted stock price process (1+r)−nS(n) into a martingale,
see Chapter 3.

Theorem 8.2

The expectation of the discounted payoff computed with respect to the risk-
neutral probability is equal to the present value of the contingent claim,

D(0) =E∗

(

(1 +r)−^1 f(S(1))

)

. (8.3)