Risk-Neutral Densities & Risk Premia
Adjusting risk-neutral densities to account for the existence of risk premia


Market participants frequently use measures implicit in financial asset prices to obtain estimates of market expectations. Prices of financial contracts are commonly used to extract information about market expectations of future asset prices; information that is, in turn, often interpreted in terms of market participants' expectations of future underlying fundamentals, such as growth and profitability prospects, inflation, etc.
A large number of techniques aiming at
modeling the entire probability density of the price of a financial contract for a future date have consequentially been developed in order to measure uncertainty. Prices of options and other derivatives are usually utilized to derive such implied densities. Option-based techniques of inference have the advantage of  providing information not only about expectations of the level of future variables, but also about the degree of uncertainty associated with these expectations as well as any possible asymmetries in terms of the range of expected future outcomes.

A well known problem with deriving expectations of future asset price movements from market prices is the existence of risk premia, as they introduce significant differences between discounted expected future asset prices and currently observable asset prices. The current price of an equity index in a years time is in general not equal to the market's expectation of that index a year ahead because of the impact of risk premia. Despite the fact that those differences are well researched and significant, they are nonetheless often explicitly or implicitly disregarded when expectations about future asset prices are extracted from observed prices. It is common practice to assume that implied forward levels are a direct measure of an asset's expected future level. In reality, risk premia should be considered when examining and interpreting the information from financial indicators in the context of asset markets, as financial assets' implied probability densities are directly impacted by them.

Almost all models available to infer implied probability densities generate "risk-neutral" estimates of the underlying price at the time of expiration, rather than true "real-world" probability densities (risk-neutral and real-world densities are only equivalent under the hypothesis that investors are risk neutral i.e. absence of risk premia). The advantage of modeling "risk-neutral" as opposed to "real-world" densities is that it avoids having to make explicit assumptions about the dynamic evolution of the underlying assets. However, the rub is that there is little scope to assess risk premia within that framework. 
There are multiple possible solutions to address this issue, such as modeling the stochastic behavior of the underlying asset by linking it to a number of underlying factors with certain properties. In that setup, once the dynamics of the underlying factors are specified, the dynamics of the asset considered can be derived for any maturity using no-arbitrage principles, and the distributional properties can then be uniquely determined. Moreover, given an assumption about the functional form of the risk premium in terms of the factors, it is possible to transform risk-neutral densities into real-world densities, and vice-versa, once the parameters of the model have been estimated using historical data. This allows us to quantify the differences between these two types of probability densities. For example, we can explore the differences in the means and variances of the two densities for any maturity, and examine if and how these differences vary over time.

Empirical evidence suggests that there are non-negligible differences between risk-neutral and real-world densities, and that these differences change over time as a result of time-variation in risk premia. Comparing estimated real-world densities with respect to actual future outcomes reveals that correcting risk-neutral densities for the underlying asset's risk premia is a superior approach to converging towards the true real-world probability density of future outcomes of the underlying asset.

Cross-sections of option prices embed the risk-neutral probability densities functions (PDFs) for the future values of the underlying asset. We utilize utility functions to adjust the risk-neutral PDF to produce subjective PDFs, we can obtain measures of the risk aversion implied in option prices. Using FTSE 100 and S&P 500 options, and both power and exponential utility functions, we show that subjective PDFs accurately forecast the distribution of realizations, while risk-neutral PDFs do not. The estimated coefficients of relative risk aversion are all reasonable. The relative risk aversion estimates are remarkably consistent across utility functions and across markets for given horizons. The degree of relative risk aversion declines with the forecast horizon and is lower during periods of high market volatility.
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