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252 Forecasting Expected Returns in the Financial Markets

Asset pricing theorists have explained that risk-neutral probabilities are not probabilities

in a natural sense (e.g. Boyle 1992: 156; Stoll and Whaley, 1993: 203–204). Instead,

they are ‘pseudo-probabilities’ or weights used to find arbitrage-free asset prices via an

ingenious shortcut that avoids explicit reference to actual probabilities.

The argument presented below suggests that risk-neutral probabilities are interpretable

in the same way as natural (‘physical’ or ‘subjective’) probabilities in at least one concrete

sense. By buying or selling an asset, the investor makes a bet on the asset going up or

down at effective betting odds derived from risk-neutral probabilities. These odds, or

their corresponding risk-neutral probabilities, do not represent the market’s aggregated

subjective beliefs, but they do indicate actual monetary payoffs, the same as betting odds

at the racetrack.

Extending the analogy between betting and investment, it is shown that a ‘Kelly bet’,

otherwise known as a ‘log optimal’ or ‘growth optimal’ investment, is a simple derivative

of the underlying asset and can be replicated with a weighted portfolio of call options, put

options and risk-free bonds. The payoff from Kelly betting is a function of the

market-

implied (risk-neutral) probability and the gambler’s subjective probability of the event

observed. The only other factor in the payoff function is the risk-free interest rate. Bets

with this payoff function can be made in whatever dollar amount the gambler’s utility

function demands.

To formalize ‘Kelly betting’ and allow for the possibility that investors might like

to gamble just a fraction of their wealth growth-optimally, we introduce the notion of a

‘$1 Kelly bet’. This is the bet – in effect, the portfolio of bets – that a true Kelly (log utility)

gambler would make if his wealth was just $1. Described another way, it is a derivative

of the underlying asset that replicates the portfolio that an investor would hold if he had

set aside $1 and wanted to invest this amount growth-optimally in the underlying asset.

An investor could possess wealth of $100 and elect to buy twenty ‘$1 Kelly bets’, so as

to invest 20% of his wealth in a fund that will grow as quickly as possible, subject to the

accuracy of his probability assessments, over the long run. The remainder could be invested

more risk aversely, to allow perhaps for the possibility that he is not very good at judging

probabilities and hence is likely to lose some or most of the 20% set aside for Kelly betting.

As with other binary assets, it is possible to write options on Kelly bets. These can be

valued using conventional binomial pricing models. Depending on the gambler’s

ratio-

nality or ready cash, he might prefer to buy or sell options on Kelly bets rather than the

underlying asset (the bets themselves). From the perspective of a true Kelly gambler, there

is no difference between investing in options and investing in the underlying asset.

Ter-

minal utility is the same either way, and depends simply on the accuracy of the gambler’s

probability forecasts relative to those of the market.

12.2 Actual versus risk-neutral probabilities

Risk-neutral probabilities, regarded correctly as not ‘real’ probabilities in the way of

either relative frequencies or personal degrees of belief, are often mistaken as the market’s

beliefs. To see the different roles of actual and risk-neutral probabilities in asset valuation,

consider the following non-standard ‘decision theoretic’ derivation of the price of a call

option in a binomial tree (the more standard derivation is shown in Cox and Rubinstein

1985: 172–173; Ingersoll, 1987: 308–309; and Luenberger 1998: 328–329, for example).

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