288 Mathematics for Finance
6.11The return on the index will be 3.37%. ForrF= 1% this gives the futures prices
f(0,3)∼= 916 .97 andf(1,3)∼= 938 .49. If the beta coefficient for a portfolio is
βV =1.5, then the expected return on this portfolio will beμV ∼= 4 .56%.
To construct a portfolio withβV ̃ = 0 and initial valueV(0) = 100 dollars,
supplement the original portfolio withN ∼= 0 .1652 short futures contracts
(observe thatNis the same as in Example 6.4).
If the actual return on the original portfolio during the first time step
happens to be equal to the expected return, then its value after one step will
beV(1)∼= 104 .56 dollars. Marking to market requires the holder ofN∼= 0. 1652
short forward contracts to pay $3. 56 .As a result, after one step the value of
the portfolio with forward contracts will beV ̃(1)∼= 104. 56 − 3 .56 = 101. 00
dollars, once again matching the risk-free growth.Chapter 7
7.1The investment will bring a profit of
(36−S(T))+− 4 .50e^0.^12 ×^123 =3,
whereS(T) is the stock price on the exercise date. This givesS(T)∼= 28. 36
dollars.
7.2E((S(T)−90)+−8e^0.^09 ×^126 )∼=− 5 .37 dollars.
7.3By put-call parity 2. 83 −PE=15. 60 − 15 .00e−^123 ×^0.^0672 ,soPE∼= 1 .98 dollars.
7.4Put-call parity is violated, 5. 09 − 7. 78 > 20. 37 − 24 e−^0.^0748 ×^126. Arbitrage can
be realised as in the first part of the proof of Theorem 7.1:- Buy a share for $20.37;
- Buy a put option for $7.78;
- Write and sell a call option for $5.09;
- Borrow $23.06 at the interest rate of 7.48%.
The balance of these transactions is zero. After six months - Sell one share for $24 by exercising the put option or settling the short
position in calls, depending on whether the share price turns out to be
below or above the strike price; - Repay the loan with interest, amounting to 23. 06 e^12 ×^0.^0748 ∼= 23 .94 dollars
in total.
The balance of 24− 23 .96 = 0.06 dollars will be your arbitrage profit.
7.5IfCE−PE>S(0)−div 0 −Xe−rT, then at time 0 buy a share and a put
option, write and sell a call option, and invest (or borrow, if negative) the
balance on the money market at the interest rater. As soon as you receive
the dividend, invest it at the rater. At the exercise timeTclose the money
market investment and sell the share forX, either by exercising the put if
S(T)<X,or by settling the call ifS(T)≥X.The final balance (CE−PE−
S(0) + div 0 )erT+X>0 will be your arbitrage profit.
On the other hand, ifCE−PE<S(0)−div 0 −Xe−rT, then at time 0 sell
short a share, write and sell a put, and buy a call option, investing the balance
on the money market. When a dividend becomes due on the shorted share,
borrow the amount and pay it to the owner of the stock. At timeTclose the
money market position, buy a share forXby exercising the call ifS(T)>X
or settling the put ifS(T)≤X, and close the short position in stock. Your
arbitrage profit will be (−CE+PE+S(0)−div 0 )erT−X>0.