Solutions 289
7.6IfCE−PE<S(0)e−rdivT−Xe−rT, then at time 0 sell short e−rdivT of a
share, write and sell a put, and buy a call option, investing the balance at
the rater. Between time 0 andTpay dividends to the stock owner, raising
cash by shorting the stock. This will lead to one shorted share held at timeT.
If the put option is exercised at timeT, you will have to buy a share forX.
Use this share to close the short position in stock. You will be left with a call
option and a positive amount (−CE+PE+S(0)e−rdivT−Xe−rT)erT>0. If
the put option is not exercised at all, then you can use the call to buy a share
forXat timeT, closing the short position in stock. You will also be left with
a positive final balance (−CE+PE+S(0)e−rdivT−Xe−rT)erT>0.
On the other hand, ifCE−PE>S(0)e−rdivT−Xe−rT, then the opposite
strategy will also lead to arbitrage.
7.7The strike price is equal to the forward price (more precisely, the exchange
rate) of 0.9883 euros to a dollar computed in Solution 6.5.
7.8If S(0)−Xe−rT<CA−PA, then write and sell a call, buy a put, and buy
a share, investing (or borrowing, if negative) the balance at the rater.Now
the same argument as in the first part of the proof of Theorem 7.2 applies,
except that the arbitrage profit may also include the dividend if the call is
exercised after the dividend becomes due. (Nevertheless, the dividend cannot
be included in this inequality because the option may be exercised and the
share sold before the dividend is due.)
IfCA−PA<S(0)−div 0 −X, then at time 0 sell short a share, write and
sell a put, and buy a call option, investing the balance at the rater. If the put
is exercised at timet<T, you will have to buy a share forX,borrowing the
amount at the rater. As the dividend becomes due, borrow the amount at the
raterand pay it to the owner of the share. At timeTreturn the share to the
owner, closing the short sale. You will be left with the call option and a positive
amount (S(0) +PA−CA−div 0 )erT−Xer(T−t)>XerT−Xer(T−t)≥0. (If
the put is exercised before the dividend becomes due, you can increase your
arbitrage profit by closing out the short position in stock immediately, in which
case you would not have to pay the dividend.) If the put is not exercised before
expiryT, then the second part of Solution 7.5 applies.
7.9IfS(0)−Xe−rT<CA−PA, then use the same strategy as in the first part
of the proof of Theorem 7.2. The resulting arbitrage profit will in fact be
increased by the dividends accumulated up to the time when the call option
is exercised.
IfCA−PA<S(0)e−rdivT−X, then at time 0 sell short e−rdivT of a
share, write and sell a put, and buy a call option, investing the balance at the
rater. Between time 0 andTpay dividends to the stock owner, raising cash
by shorting the stock. This will lead to one shorted share held at timeT.If
the put option is exercised at timet≤T, you will have to buy a share forX,
borrowing this amount at the rater. At timeTuse this share to close the short
position in stock. You will be left with a call option and a positive amount
(−CA+PA+S(0)e−rdivT)erT−Xer(T−t)≥(−CA+PA+S(0)e−rdivT−
X)erT>0 plus any dividends accumulated since the share was bought at
timet. If the put option is not exercised at all, then you can use the call to
buy a share forXat timeT, closing the short position in stock. You will also
be left with a positive final balance (−CA+PA+S(0)e−rdivT)erT−X>
(−CA+PA+S(0)e−rdivT−X)erT>0.
7.10IfCE>CA, then write and sell a European call and purchase an American
call. The differenceCE−CAwill be your arbitrage income. To keep it, do not