STOCHASTIC

MANAGEMENT SCIENCE
Vol. 14, No. 1, September, 1967
Printed in U.S.A.
EVALUATING A CALL OPTION AND OPTIMAL TIMING
STRATEGY IN THE STOCK MARKET*
HOWARD M. TAYLORf
Cornell University
The optimal strategy for the holder of a "put" or "call" option contract im
the stock market is studied under the' random walk model for stock prices.
Some results are distribution-free in that they depend only on the mean price
change. Other results are derived under the assumption that price changes have
a normal or Gaussian distribution. Under this assumption explicit results are
obtained for the limiting case where the expiration date of the contract is indefinitely
far in the future. Knowing the optimal strategy it is possible to
evaluate whether the purchase of a given option can be expected to be profitable.
1. Introduction and Summary
A call is an option entitling the holder to buy a block of shares in a given
company at a stated price at any time during a stated interval. Thus the call
listed in the financial section of the newspaper as:
U. S. Steel 49| 6 mos. $565.50
means that for a price of $565.50 one may purchase the privilege (option) of
buying 100 shares of the stock of U. S. Steel at a price of $49,375 per share at
any time during the next six months. (Call prices listed in the newspaper are
"asking" prices, and bargaining is often possible.)
Of the many reasons for purchasing a call, two are studied here, referred to as
the investor's problem (insurance problem [2]) and the speculator's problem. In
the first case an investor may have decided to add a block of the stock of a particular
company to his portfolio, leaving only the question of the timing of the
purchase. He feels that in the near future the price of the stock may drop, enabling
a more advantageous buy. However, not being sure and to guard against
a near future price rise he purchases a call which guarantees that he, at least,
need never pay more than the price stated in the call. Of course, should the price
of the stock drop in the near future, he will ignore the call option and make his
purchase at the lower price on the open market. As a current example (1966)
consider the stock of American Telephone and Telegraph, one of the bluest of
the blue chips, which is now selling at or near its lowest price in two years. The
price is depressed because of a pending investigation of the company by the
Federal Communications Commission, according to the financial newspapers.
There is no doubt that the company is sound and soundly managed, and it is
* Received May 1966 and revised January 1967.
t In refereeing an earlier draft of this paper, Arthur Veinott made many detailed and
helpful suggestions which resulted in simpler proofs under weaker assumptions with stronger
conclusions. I thank him for his conscientious help.
MODELS OF OPTION STRATEGY