292 Mathematics for Finance
8.2Ifr=0andS(0) =X= 1, thenCE(0) =−u−udd.Foru=0.05 andd=− 0. 05
we haveu−d=0.1andCE(0) = 0.025 dollars. However, ifu=0.01 and
d=− 0 .19, thenu−d=0. 2 .The variance of the return on stock is equal to
(u−d)^2 p(1−p) and is higher in the latter case, but the option price is lower:
CE(0) = 0.0095 dollars.
8.3To replicate a call option the writer needs to buy stock initially and sell it
when the option is exercised. The following system of equations needs to be
solved to find the replicating portfolio:
{
110(1−c)x+1. 05 y=10,
90(1−c)x+1. 05 y=0.Forc= 2% we obtainx∼= 0 .5102 andy∼=− 42 .8471, so the initial value of
the replicating portfolio will be 100x+y∼= 8 .1633 dollars. Ifc= 0, then the
replicating portfolio will be worth 7.1429 dollars. Note that the money market
positionyis the same for eachc.
8.4The borrowing rate should be used to replicate a call, since the money market
position will be negative. This givesx(1)∼= 0 .6667 andy(1)∼=− 40 .1786, so the
replicating portfolio value is 9.8214 dollars. For a put we obtainx(1)∼=− 0. 3333
andy(1)∼= 27 .7778 using the rate for deposits, so the replicating portfolio will
be worth 2.7778 dollars initially.
The results are consistent with the put and call prices obtained from (8.3)
with the appropriate risk-neutral probability (computed using the correspond-
ing interest rate),p∗∼= 0 .7333 for a call andp∗∼= 0 .6foraput.
8.5The option price at time 0 is 22.92 dollars. In addition to this amount, the
option writer should borrow 74.05 dollars and buy 0.8081 of a share. At time 1,
ifS(1) = 144, then the amount of stock held should be increased to 1 share,
the purchase being financed by borrowing a further 27.64 dollars, increasing
the total amount of money owed to 109.09 dollars. If, on the other hand,
S(1) = 108 dollars at time 1, then some stock should be sold to reduce the
number of shares held to 0.2963, and 55.27 dollars should be repaid, reducing
the amount owed to 26.18 dollars. (In either case the amount owed at time 1
includes interest of 7.40 dollars on the amount borrowed at time 0.)
8.6At time 1 the stock pricesSu= 144 andSd= 108 dollars (the so-called cum-
dividend prices) drop by the amount of the dividend to 129 and 93 dollars
(the so-called ex-dividend prices). These prices generate the prices at time 2,
henceSuu= 154.80,Sud= 116.10,Sdu= 111.60 andSdd=83.70 dollars. The
option will be exercised with payoff 34.80 dollars if the stock goes up twice.
In the remaining scenarios the payoff will be zero. The option value at time 1
will be 21.09 dollars in the up state and zero dollars in the down state. The
replicating portfolio constructed at time 1 and based on ex-dividend prices
consists of 0.8992 shares and a loan of 94.91 dollars in the up state, and no
shares and no money market position in the down state. The option price
at time 0 is 12.78 dollars. Replication (based on the cum-dividend prices at
time 1, since the dividend is available to the owner of the stock purchased at
time 0) involves buying 0.5859 shares and borrowing 57.52 dollars.
8.7From the Cox–Ross–Rubinstein formulaCE(0)∼= 5 .93 dollars,PE(0)∼= 7. 76
dollars.
8.8The least integermsuch thatS(0)(1 +u)m(1 +d)N−m>Xism= 35.
From the Cox–Ross–Rubinstein formula we obtainCE(0)∼= 3 .4661 dollars
andx(1) = [1−Φ(m− 1 ,N,q)]∼= 0 .85196 shares.