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Puzzles of Finance Six Practical Problems and Their Remarkable Solutions

Mark P. Kritzman

posted on 15 January 2002

reviewed by Joe McCauley

This small, entertaining monograph can be read superficially in a sitting and provides food for thought, as I will point out. It would be interesting and useful for econophysicists to try to find alternative solutions to several of the first five puzzles not based on either expected utility or equilibrium arguments. An alternative solution to the sixth puzzle has recently been worked out {1}. Kritzman prefaces the book by stating that he will not address either the equity premium or dividend puzzle because these two problems depend on agents? psychology. He addresses instead 6 puzzles that he calls purely logical and mathematical. However, he contradicts himself in that several of the solutions depend on utility functions and therfore on agents? psychology. His background is that of money manager.

Siegel?s Paradox is based on the incontrovertible fact that the average of a random variable differs from the inverse of the average of the same variable. Kritzman applies this to exchange rates with the question whether the difference is economically relevant.

In Liklihood of Loss the nonuniqueness of liklihood of loss is discussed, pointing out the connection to a first passage problem. The result is used to criticize the idea of using social security funds to place bets in the stock market. Lognormal returns are assumed throughout the book.

In Time Diversification he addresses the interesting question whether agents should be more risk-tolerant with long rather than short time horizons. The conventional wisdom assumes the former, but he discusses a solution by Samuelson that contradicts this viewpoint. Samuelson?s solution is, of course, based on an expected utility so (like Yi-Cheng?s beloved Lemon Problem) it would be interesting to analyse this question correctly without any reference to utility. Can one replace utility, which is arbitrary and unconvincing, with entropy of the distribution, which is unique and beyond fudge-work, and arrive at interesting results?

Why the Expected Return is Not to be Expected. Kritzman argues that the expected return has less than 50% probability of occuring. Here, one should use the cumulative distribution.

Half the Stocks All the Time or All the Stocks Half the Time? Should an agent switch or balance, try to time the market or buy and hold? The balanced strategy has lower risk. Again, expected utility is referred to, but after the fact.

The Irrelevance of the Expected Return for Option Valuation. This chapter extolls the use of the riskless return in the delta hedge strategy, an idea much beloved of theorists and ignored by traders. But, as Gemunu Gunaratne has pointed out, traders are more intelligent than theorists {1}. In this chapter, I am irritated that everyone under the sun (including Einstein and Wiener) gets credit for the background necessary for the Black-Scholes equation while Osborne, who introduced the lognormal distribution into finance in 1958, is completely ignored. As it has been said many times, it?s the victors who write history (Smoluchowski, Fokker, Planck and Kolmogorov might also have been included in Kritzman?s list?). The main point, however, is that the famous (anti-) arbitrage argument leading to a riskless hedge is wrong on two counts {1}, another example of how the economists? ,equilibrium? idea does not apply to reality. As we point out in our new stochastic theory of returns, volatility and option pricing, why would a trader go to the trouble to construct a complicated hedge that must be updated continually only to get the same return he?d get by letting his money rest in a CD or money market fund? Clearly, such a trader would not be intelligent.

Credit is given incorrectly to Einstein for his solution of a heat transfer problem whereas in reality all that was/is needed in order to solve the Black-Scholes equation (after a simple transformation) is the Green function for the diffusion equaion written down by Einstein, Bachelier and others. Also repeated is the irritating claim that CAPM is an ,equilibrium? model, which it patently is not {1}.}.


1. J. L. McCauley and Gemunu H. Gunaratne, An Alternative Option Pricing Model, preprint and submitted (2001).