Theory of Option Strategy Under Risk Aversion

Abstract

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.