We shall investigate the problem of optimal exercising strategy for option holders for the case in which option holders are averse to risk. A model of stock price changes incorporating the Lognormal random walk assumption will be combined with a class of utility functions containing diminishing marginal utility of money. In general, the strategy of waiting until the last possible day to exercise an option, which maximizes expected value, will not maximize expected utility. The strategy which maximizes expected utility is obtained by a dynamic programming formulation of the decision problem. At each day (or decision stage), the option holder may choose to act (exercise) or wait until the next day. Working backwards from the last day, a series of critical prices are obtained, with the optimal strategy being as follows: act if the stock price on any day is greater than the critical price for that day; otherwise, wait. Using the concept of proportional risk aversion developed by Pratt, we will demonstrate that, under certain conditions, a utility function which exhibits increasing proportional risk aversion is sufficient to create a series of finite critical prices. Moreover, once an option is exercised, the option holder continually faces a tactical decision to hold the stock and wait for capital gains or sell and take profits as ordinary income, thereby avoiding further risk. This decision may also be optimized by a dynamic programming scheme similar to the approach used above.
[1]
M. Osborne.
Brownian Motion in the Stock Market
,
1959
.
[2]
R. Quandt,et al.
The Valuation of Convertible Securities
,
1966
.
[3]
E. Fama.
The Behavior of Stock-Market Prices
,
1965
.
[4]
Herbert E. Scarf,et al.
Studies in applied probability and management science
,
1962
.
[5]
J. Pratt.
RISK AVERSION IN THE SMALL AND IN THE LARGE11This research was supported by the National Science Foundation (grant NSF-G24035). Reproduction in whole or in part is permitted for any purpose of the United States Government.
,
1964
.
[6]
C. Granger,et al.
The Random Character of Stock Market Prices.
,
1965
.
[7]
H. F. Ayres.
Risk aversion in the warrant markets
,
1963
.
[8]
A. Boness.
Elements of a Theory of Stock-Option Value
,
1964,
Journal of Political Economy.
[9]
Gordon Pye.
The Value of the Call Option on a Bond
,
1966,
Journal of Political Economy.
[10]
R. Bellman.
Dynamic programming.
,
1957,
Science.
[11]
Robert Schlaifer.
Introduction to statistics for business decisions
,
1963
.