Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-16T07:28:12.186Z Has data issue: false hasContentIssue false
  • English
  • Français

Theory of Option Strategy Under Risk Aversion

Published online by Cambridge University Press:  19 October 2009

W. H. Hausman  and
W. L. White
Get access Rights & Permissions [Opens in a new window]

Extract

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.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 3 , Issue 3 , September 1968 , pp. 343 - 358
Copyright
Copyright © School of Business Administration, University of Washington 1968

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ayres, Herbert F., “Risk Aversion in the Warrant Markets,” Chapter 21 in Cootner (see Reference 8 below).Google Scholar
[2]Baumol, W. J., Malkiel, B. A., and Quandt, R. E., “The Valuation of Convertible Securities,” Quarterly Journal of Economics, February 1966.CrossRefGoogle Scholar
[3]Bellman, R. E., Dynamic Programming (Princeton, N. J.: Princeton University Press, 1957).Google ScholarPubMed
[4]Bierman, H. Jr., “The Valuation of Stock Options,” Journal of Financial and Quantitative Analysis, September 1967.CrossRefGoogle Scholar
[5]Boness, A. James, “Elements of a Theory of Stock-option Value,” Journal of Political Economy, Volume 72, No. 2 (April 1964), pp. 163–175.CrossRefGoogle Scholar
[6]Boness, A. James, “Some Evidence of the Profitability of Trading in Put and Call Options,” Chapter 20 in Cootner, op. cit.Google Scholar
[7]Brada, J. C., Ernst, H., and Van Tassel, J., “The Distribution of Stock Price Differences: Gaussian After All?” Operations Research, April 1966.CrossRefGoogle Scholar
[8]Cootner, P. A., The Random Character of Stock Market Prices (Cambridge, Mass.: M.I.T. Press, 1964).Google Scholar
[9] Division of Trading and Exchanges, Securities and Exchange Commission, “Report on Put and Call Options,” August 1961.Google Scholar
[10]Fama, G., “The Behavior of Stock-Market Prices,” Journal of Business, April 1965.CrossRefGoogle Scholar
[11]Holland, D. M., and Lewellen, W. G., “Probing the Record of Stock Options,” Harvard Business Review, March–April 1962.Google Scholar
[12]Karlin, S., “Stochastic Models and Optimal Policy for Selling an Asset,” Chapter 9 in Studies in Applied Probability and Management Science, Arrow, , Karlin, , and Scarf, , eds. (Stanford, California: Stanford University Press, 1962).Google Scholar
[13]Kruizenga, Richard J., “Profit Returns from Purchasing Puts and Calls,” Chapter 18 of Paul H. Cootner, The Random Character of Stock Market Prices (Cambridge, Mass.: M.I.T. Press, 1964).Google Scholar
[14]Osborne, M. F. M., “Brownian Motion in the Stock Market,” Chapter 4 in Cootner, op. cit.Google Scholar
[15]Poensgen, Otto H. “The Valuation of Convertible Bonds,” Part I, Industrial Management Review, Volume 7, No. 1 (Fall 1965), pp. 77–92; Part II, Industrial Management Review, Volume 7, No. 2 (Spring 1966), pp. 83–98.Google Scholar
[16]Pratt, John W., “Risk Aversion in the Small and in the Large,” Econometrica, January 1964.CrossRefGoogle Scholar
[17]Pye, G., “The Value of the Call Option on a Bond,” Journal of Political Economy, April 1966.CrossRefGoogle Scholar
[18]Samuelson, Paul A., “Rational Theory of Warrant Pricing,” Industrial Management Review, Volume 6, No. 2 (Spring 1965), pp. 13–31.Google Scholar
[19]Schlaifer, R., Introduction to Statistics for Business Decisions (New York, N. Y.: McGraw-Hill, 1961).Google Scholar
[20]Sprenkle, Case M., “Warrant Prices as Indicators of Expectations and Preferences,” Chapter 19 of Cootner, op. cit.Google Scholar
[21]Taylor, H. M., “Evaluating a Call Option and Optimal Timing in the Stock Market,” Management Science, September 1967.CrossRefGoogle Scholar