20 Mathematics for Finance
- at time 0 borrow money to buy a call option with strike price $100; then, at
time 1 repay the loan with interest and purchase the stock, exercising the
option if the stock price goes up.
The investor will be open to considerable risk if she chooses to follow the first
strategy. On the other hand, following the second strategy, she will need to
borrowC(0)∼= 31 .8182 dollars to pay for the option. At time 1 she will have
to repay $35 to clear the loan and may use the option to purchase the stock,
hence the cost of purchasing one share will be
S(1)−C(1) + 35 =
{
135 if stock goes up,
75 if stock goes down.Clearly, the risk is reduced, the spread between these two figures being narrower
than before.
Exercise 1.11
Compute the risk (as measured by the standard deviation of the return)
involved in purchasing one share with and without the option if a)p=
0 .25, b)p=0.5, c)p=0.75.Exercise 1.12
Show that the risk (as measured by the standard deviation) of the above
strategy involving an option is a half of that when no option is purchased,
no matter what the probability 0<p<1is.If two options are bought, then the risk will be reduced to nil:S(1)− 2 ×C(1) + 70 = 110 with probability 1.This strategy turns out to be equivalent to a long forward contract, since the
forward price of the stock is exactly $110 (see Section 1.5). It is also equivalent
to borrowing money to purchase a share for $100 today and repaying $110 to
clear the loan at time 1.
Chapter 9 on financial engineering will discuss various ways of managing
risk with options: magnifying or reducing risk, dealing with complicated risk
exposure, and constructing payoff profiles tailor made to meet the specific needs
of an investor.