108156.pdf

Solutions 291

4) Finally, ifX′<X<X′′ <S(T), then (7.9) becomes an equality,

S(T)−X=α(S(T)−X′)+(1−α)(S(T)−X′′) becauseX=αX′+

(1−α)X′′.

7.16Suppose thatPE(S′)≤PE(S′′)forsomeS′<S′′, whereS′=x′S(0) and

S′′=x′′S(0). Write and sell a put option on a portfolio withx′′shares and

buy a put option on a portfolio withx′shares, investing the balancePE(S′′)−

PE(S′) without risk. If the written option is exercised at timeT, then the

liability can be met by exercising the other option. Sincex′<x′′, the payoffs

satisfy (X−x′S(T))+≥(X−x′′S(T))+with strict inequality wheneverX≥

x′S(T). It follows that this is an arbitrage strategy.

7.17Suppose thatX′<X′′, butCA(X′)≤CA(X′′). We can write and sell the

call with strike priceX′′and purchase the call with strike priceX′, investing

the balanceCA(X′′)−CA(X′) without risk. If the written option is exercised

at timet≤T, we will have to pay (S(t)−X′′)+. This liability can be met

by exercising the other option immediately, receiving the payoff (S(t)−X′)+.

Since (S(t)−X′′)+≤(S(t)−X′)+with strict inequality wheneverX′<S(t),

this strategy leads to arbitrage.

The inequality for put options can be proved in a similar manner.

7.18We shall prove Proposition 7.19 for American put options. The argument for

European puts is similar. By Proposition 7.15PA(S) is a decreasing function

ofS.WhenS≥X, the intrinsic value of a put option is zero, and so the time

value, being equal toPA(S), is also a decreasing function ofS. On the other

hand,PA(S′)−PA(S′′)≤S′′−S′for anyS′<S′′by Proposition 7.16. This

implies thatPA(S′)−(X−S′)+≤PA(S′′)−(X−S′′)+ifS′<S′′≤X,so

the time value is an increasing function ofSforS≤X. As a result, the time

value has a maximum atS=X.

Chapter 8

8.1Let us compute the derivative of the priceCE(0) of a call option with respect

tou. The formula for the price, assuming thatSd<X<Su,is

CE(0) =1+^1 rru−−dd[S(0)(1 +u)−X].

The derivative with respect touis equal to

(r−d)[X−S(0)(1 +d)]

(1 +r)(u−d)^2 =

(r−d)[X−Sd]

(1 +r)(u−d)^2.

This is positive in our situation, sincer>dandX>Sd,soCE(0) increases

asuincreases.

The derivative ofCE(0) with respect todis equal to

−(u−(1 +r)[S(0)(1 +r)(u−du)) 2 −X]=−(u−r)[S

u−X]

(1 +r)(u−d)^2 ,

which is negative, sincer<uandX<Su. The option price decreases asd

increases.