290 Mathematics for Finance
exercise the American option until expiry, when either both options will turn
out worthless, or the loss from settling the European call will be recovered by
exercising the American call. The argument for put options is similar.
7.11Suppose thatCE≥S(0)−div 0. In this case write and sell a call option and
buy stock, investing the balance on the money market. As soon as you receive
the dividends, also invest them on the money market. On the exercise date
you can sell the stock for at least min(S(T),X), settling the call option. Your
final wealth will be positive, (CE−S(0) + div 0 )erT+ min(S(T),X)>0. This
proves thatCE<S(0)−div 0.
The remaining bounds follow by the put-call parity relation (7.5) for
dividend-paying stock:S(0)−div 0 −Xe−rT≤CE,sincePE≥0;−S(0) +
div 0 +Xe−rT≤PE,sinceCE≥0; andPE<Xe−rT,sinceCE<S(0)−div 0.
7.12For dividend-paying stock the regions determined by the bounds on call and
put prices in Proposition 7.3 are shown as shaded areas in Figure S.9.Figure S.9 Bounds on European call and put prices for dividend-paying stock
7.13IfCA≥S(0), then buy a share, write and sell an American call and invest
the balanceCA−S(0) without risk. If the option is exercised before or at
expiry, then settle your obligation by selling the share forX. If the option
is not exercised at all, you will end up with the share worthS(T)atexpiry.
In either case the final cash value of this strategy will be positive. The final
balance will in fact also include the dividend collected, unless the option is
exercised before the dividend becomes due.
7.14Suppose thatX′ <X′′, butCE(X′)≤CE(X′′). We can write and sell a
call with strike priceX′′and buy a call with strike priceX′, investing the
differenceCE(X′′)−CE(X′) without risk. If the option with strike priceX′′
is exercised at expiry, we will need to pay (S(T)−X′′)+. This amount can
be raised by exercising the option with strike priceX′and cashing the payoff
(S(T)−X′)+. BecauseX′<X′′and (S(T)−X′)+≥(S(T)−X′′)+with
strict inequality wheneverX′<S(t), it follows that an arbitrage profit will
be realised.
The inequality for puts follows by a similar arbitrage argument.
7.15Consider four cases:
1) IfS(T)≤X′<X<X′′, then (7.9) reduces to 0≤0.
2) IfX′<S(T)≤X<X′′, then (7.9) becomes 0≤α(S(T)−X′),
obviously satisfied forX′<S(T).
3) IfX′<X<S(T)≤X′′,then(7.9)canbewrittenasS(T)−X≤
α(S(T)−X′). This holds becauseX=αX′+(1−α)X′′andS(T)≤
X′′.