Time-Consistent Conditional Expectation Under Probability Distortion

We introduce a new notion of conditional nonlinear expectation under probability distortion. Such a distorted nonlinear expectation is not subadditive in general, so it is beyond the scope of Peng’s framework of nonlinear expectations. A more fundamental problem when extending the distorted expectation to a dynamic setting is time inconsistency, that is, the usual “tower property” fails. By localizing the probability distortion and restricting to a smaller class of random variables, we introduce a so-called distorted probability and construct a conditional expectation in such a way that it coincides with the original nonlinear expectation at time zero, but has a time-consistent dynamics in the sense that the tower property remains valid. Furthermore, we show that in the continuous time model this conditional expectation corresponds to a parabolic differential equation whose coefficient involves the law of the underlying diffusion. This work is the first step toward a new understanding of nonlinear expectations under probability distortion and will potentially be a helpful tool for solving time-inconsistent stochastic optimization problems.

CONIC PORTFOLIO THEORY

Portfolios are designed to maximize a conservative market value or bid price for the portfolio. Theoretically this bid price is modeled as reflecting a convex cone of acceptable risks supporting an arbitrage free equilibrium of a two price economy. When risk acceptability is completely defined by the risk distribution function and bid prices are additive for comonotone risks, then these prices may be evaluated by a distorted expectation. The concavity of the distortion calibrates market risk attitudes. Procedures are outlined for observing the economic magnitudes for diversification benefits reflected in conservative valuation schemes. Optimal portfolios are formed for long only, long short and volatility constrained portfolios. Comparison with mean variance portfolios reflects lower concentration in conic portfolios that have comparable out of sample upside performance coupled with higher downside outcomes. Additionally the optimization problems are robust, employing directionally sensitive risk measures that are in the same units as the rewards. A further contribution is the ability to construct volatility constrained portfolios that attractively combine other dimensions of risk with rewards.