RISK-NEUTRAL PROBABILITIES AGAIN

6.3 RISK-NEUTRAL PROBABILITIES AGAIN

Risk-neutral probabilities, conveniently, allow linear pricing measures. These probabilities are defined in terms of market parameters (although their existence

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hinges importantly on a risk-free rate, R f , and rational traders) and differ markedly from historical probabilities. This difference, contrasting two cultures, is due to economic assumptions that the market price of a traded asset ‘internalizes’ all the past, future states and information that such an asset can be subjected to. If this is the case, and it is so in markets we call complete markets, then the current price ought to be determined by its future values as we have shown here. In other words, the market determines the price and not historical (probability) uncertainty! If there is no unique set of risk-neutral pricing measures, then market prices are not unique and we are in a situation of market incompleteness, unable to value the asset price uniquely.

It is therefore important to establish conditions for market completeness. Our ability to construct a unique set of risk neutral probabilities for the valuation of the stock at period 1 or the value of buying an option depends on a number of assumptions that are of critical importance in finance and must be maintained theoretically and practically. Pliska (1997) for example, emphasizes the impor- tance of these assumptions and their implications for risk-neutral probabilities. Namely, there can be a linear pricing measure if and only if there are no dominat- ing trading strategies. Further, if there are no dominant trading strategies, then the law of the single price holds, albeit the converse need not necessarily be true. And finally, if there were a dominating trading strategy, then there exists an arbitrage opportunity, but the converse is not necessarily true. Thus, risk-neutral pricing requires, as stated earlier:

r No arbitrage opportunities. r No dominant trading strategies. r The law of the single price.

When the assumption of market completeness is violated, it is no longer possible to obtain a unique set of risk-neutral probabilities. This means that one cannot duplicate the option with a portfolio or price it uniquely. In this case, an appropriate portfolio is optimized for the purpose of selecting risk-neutral probabilities. Such an optimization problem can be based on the best mean forecast as we shall outline below. These probabilities will, however, be a function of a number of parameters implied by the portfolio and decision makers’ preferences and of course the information available to the decision maker. When this is not possible, we can, for a given set of parameters bound the relevant option prices.

6.3.1 Rational expectations and optimal forecasts

Rational expectations mean that economic agents can forecast the ‘mean’ price (since risk-neutral probabilities imply that an expected value is used to value the asset). In this case, a mean forecast can be selected by minimizing the forecast error

(in which case the mean error is null). Explicitly, say that {x} = {x 1 , x 2 ,..., x t } stands for an information set (a time series, a stock price record, financial variables

etc.). A forecast is thus an estimate based on the information set {x} written for convenience by the function f (.) such that ¯y = f (x) whose error forecast is

147 ε = y − ¯y where y is the actual record of the series investigated and its forecast

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is obtained by minimizing the least squares errors. Assume that the forecast is unbiased, that is, based on all the relevant information available, I; the forecast equals the conditional expectation, or ¯y = E(y |I ) whose error is ε = y − ¯y = y − E(y |I ) . In this case, rational expectations exist when the expected errors are both null and uncorrelated with its forecast as well as with any observation in the information set. This is summarized by the following three conditions of rational expectations:

E (ε) = 0; E(ε ¯y) = E(εE(y |I )) = 0 ; E(εx) = cov(ε, x) = 0, ∀x ∈ I Of course, there can be various information sets as well as various mechanisms

that can be used to generate rational expectations. However, it is essential to note that the behaviour of forecast residual errors determine whether these forecasts are rational expectations forecasts or not.