STOCHASTIC

112 HOWARD M. TAYLOB
easy to picture an investor having decided to add AT and T to his portfolio as a
long run commitment. However, more news of the pending investigation may
depress the price further and thus perhaps a call should be considered, rather
than immediate purchase.
A speculator, on the other hand, may purchase a call if he feels a sharp price
rise in the offing. Again he's not sure and rather than make an outright purchase,
for much less money (or more stock controlled) he purchases a call. Should the
price rise during the period of the call he will exercise the option to buy at the
stated price and immediately resell in the open market at a profit. Should the
price fall, he will not exercise his option.
A put is a stock option contract which obligates the writer, or seller, to accept
delivery of 100 shares of a particular stock at a set price within a specified period
of time. Puts are bought and sold for reasons similar to those underlying calls,
and the analysis of puts is similar to the subsequent analysis of calls. Both puts
and calls are often purchased for other reasons, usually of a tax planning nature,
but our analyses are confined to the investor's and the speculator's problem as
outlined above.
The central problem studied is the call owner's optimal strategy in deciding
between purchasing on the market or waiting one more day, based on the current
market price of the stock and the number of days left in the option contract.
Once the optimal strategy is known, it is relatively easy to decide whether or not
the purchase of a call could be expected to yield a profit.
Before summarizing the results the fundamental assumptions will be presented
so that a reader not willing to make these assumptions need not read further.
Most important is the model chosen to represent short term fluctuations in the
market price of a stock. We have assumed these prices to be described by a
random walk, and examined two models which, following Samuelson [20], we
term the absolute random walk and the geometric random walk. Let Yr be
today's market price, F„_i tomorrow's price, and so on, down to Y0, the price
on the last day the call is effective. Most of our results are for the absolute
random walk model where we assume F