18 Mathematics for Finance
a call option with strike price $100 and exercise time 1 if a)A(1) = 105
dollars, b)A(1) = 115 dollars.Aputoption with strike price $100 and exercise time 1 gives the right to
sellone share of stock for $100 at time 1. This kind of option is worthless if
the stock goes up, but it brings a profit otherwise, the payoff being
P(1) =
{
0 if stock goes up,
20 if stock goes down,given that the pricesA(0),A(1),S(0),S(1) are the same as above. The notion
of a portfolio may be extended to allow positions in put options, denoted byz,
as before.
The replicating and pricing procedure for puts follows the same pattern as
for call options. In particular, the priceP(0) of the put option is equal to the
time 0 value of a replicating investment in stock and bonds.
Remark 1.4
There is some similarity between a put option and a short forward position:
both involve selling an asset for a fixed price at a certain time in the future.
However, an essential difference is that the holder of a short forward contract
is committed to selling the asset for the fixed price, whereas the owner of a put
option has the right but no obligation to sell. Moreover, an investor who wants
to buy a put option will have to pay for it, whereas no payment is involved
when a forward contract is exchanged.
Exercise 1.9
Once again, let the bond and stock pricesA(0),A(1),S(0),S(1) be as
above. Compute the priceP(0) of a put option with strike price $100.An investor may wish to trade simultaneously in both kinds of options and,
in addition, to take a forward position. In such cases new symbolsz 1 ,z 2 ,z 3 ,...
will need to be reserved for all additional securities to describe the positions
in a portfolio. A common feature of these new securities is that their payoffs
depend on the stock prices. Because of this they are calledderivative securities
orcontingent claims. The general properties of derivative securities will be
discussed in Chapter 7. In Chapter 8 the pricing and replicating schemes will
be extended to more complicated (and more realistic) market models, as well
as to other financial instruments.