This paper was prepared for John B. Taylor and Michael Woodford, Editors

2

There are other potential explanations of the options smile in terms of nonnormality or jump

processes for returns, and these have received the attention in the options literature. Such explanations

might even provide a complete rational basis for the smile, though it is hard to know for sure.  Since

the 1987 stock market crash, the options smile has usually appeared distorted into an options “leer,”

with the left side of the mouth higher (e.g., the deep out-of-the-money puts are especially overpriced),

see Bates (1995), Jackwerth and Rubinstein (1995) and Bates (1991). Public memories of the 1987

crash are apparently at work in producing this “leer.”

5

options than it is for middle-range strike prices.  This options smile might possibly be

explained in terms of the distortion in probabilities represented by the Kahneman–Tversky

weighting function, since the theory would suggest that people act as if they overestimate

the small probability that the price of the underlying crosses the strike price and

underestimate the high probability that the price remains on the same side of the strike price.

The Kahneman–Tversky weighting function might even explain the down-turned corners

of the mouth that some smiles exhibit (see Fortune, 1996) if at these extremes the

discontinuities at the extremes of the weighting function become relevant.

2

  

We now turn to the other foundation of prospect theory, the Kahneman and Tversky

(1979) value function.  The value function differs from the utility function in expected

utility theory in a very critical respect:  the function (of wealth or payout) has a kink in it at

a point, the “reference point,” the location of which is determined by the subjective

impressions of the individual.  The reference point is the individual’s point of comparison,

the “status quo” against which alternative scenarios are contrasted.  Taking value as a

function of wealth, the Kahneman–Tversky (1979) value function is upward sloping

everywhere, but with an abrupt decline in slope at the reference point (today’s wealth or

whatever measure of wealth that is psychologically important to the subject).  For wealth

levels above the reference point, the value function is concave downward, just as are

conventional utility functions.  At the reference point, the value function may be regarded,

from the fact that its slope changes abruptly there, as infinitely concave downward.  For

wealth levels below the reference point, Kahneman and Tversky found evidence that the

value function is concave upward, not downward.  People are risk lovers for losses, they

asserted.

Perhaps the most significant thing to notice about the Kahneman–Tversky value

function is just the discontinuity in slope at the reference value, the abrupt downward

change in slope as one moves upward past the reference value.  Prospect theory does not nail

down accurately what determines the location of the reference point, just as it does not nail

down accurately, for the weighting function, what is the difference between very high

probabilities and extremely high probabilities.  The theory does not specify these matters

because experimental evidence has not produced any systematic patterns of behavior that

can be codified in a general theory.  However, the reference point is thought to be

determined by some point of comparison that the subject finds convenient, something

readily visible or suggested by the wording of a question.

This discontinuity means that, in making choices between risky outcomes, people will

behave in a risk averse manner, no matter how small the amounts at stake are.  This is a

contrast to the prediction of expected utility theory with a utility function of wealth without

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3

Mehra and Prescott did not discover the equity premium. Perhaps that honor should go to Smith

(1925), although there must be even earlier antecedents in some forms.  Mehra and Prescott’s original

contribution seems to have been, in the context of present-value investor intertemporal optimizing

models, to stress that the amount of risk aversion that would justify the equity premium, given the

observed correlation of stocks with consumption, would imply much higher riskless interest rates than

we in fact see.

6

kinks, for which, since the utility function is approximately linear for small wealth changes,

people should behave as if they are risk neutral for small bets.  That people would usually

be risk neutral for small bets would be the prediction of expected utility theory even if the

utility function has such a slope discontinuity, since the probability that wealth is currently

at the kink is generally zero.  With prospect theory, in contrast, the kink always moves with

wealth to stay at the perceived current level of wealth (or the current point of reference); the

kink is always relevant.

Samuelson (1963) told a story which he perceived as demonstrating a violation of

expected utility theory, and, although it came before Kahneman and Tversky’s prospect

theory, it illustrates the importance of the kink in the value function.  Samuelson reported

that he asked a lunch colleague whether he would accept a bet that paid him $200 with a

probability of .5 and lost him $100 with a probability of .5.  The colleague said he would not

take the bet, but that he would take a hundred of them.  With 100 such bets, his expected

total winnings are $5,000 and he has virtually no chance of losing any money.  It seems

intuitively compelling to many people that one would readily take the complete set of bets,

even if any element of the set is unattractive.  Samuelson proved that if his colleague would

answer the same way at any wealth level, then he necessarily violates expected utility

theory.

Samuelson’s colleague is not, however, in violation of prospect theory.  When viewing

a single bet, the kink in the value function is the dominant consideration.  If he were to judge

100 bets sequentially, the kink would always be relevant (the reference point would move

with each successive bet) and he would reject all of them.  But if he were to judge 100 bets

together, the collective outcomes would be far above today’s value function kink, and the

bet is, by prospect theory, clearly desirable.

The failures to accept many such bets when one considers them individually has been

called “myopic loss aversion” by Benartzi and Thaler (1995).  They argue that, under

estimated values for the magnitude of the kink in the Kahneman–Tversky value function,

the “equity premium puzzle” of Mehra and Prescott (1985) can be resolved; see also Siegel

and Thaler (1997).

 

Today, the term “equity premium puzzle,” coined by Mehra and Prescott (1985), is

widely used to refer to the puzzlingly high historical average returns of stocks relative to

bonds.

3

  The equity premium is the difference between the historical average return in the

stock market and the historical average return on investments in bonds or treasury bills.

According to Siegel (1994), the equity premium of U.S. stocks over short-term government

bonds has averaged 6.1% a year for the United States for 1926 to 1992, and so one naturally

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