Horizon-unbiased Investment with Ambiguity

In the presence of ambiguity on the driving force of market randomness, we consider the dynamic portfolio choice without any predetermined investment horizon. The investment criteria is formulated as a robust forward performance process, reflecting an investor's dynamic preference. We show that the market risk premium and the utility risk premium jointly determine the investors' trading direction and the worst-case scenarios of the risky asset's mean return and volatility. The closed-form formulas for the optimal investment strategies are given in the special settings of the CRRA preference

Representation of homothetic forward performance processes in stochastic factor models via ergodic and infinite horizon BSDE

In an incomplete market, with incompleteness stemming from stochastic factors imperfectly correlated with the underlying stocks, we derive representations of homothetic (power, exponential and logarithmic) forward performance processes in factor-form using ergodic BSDE. We also develop a connection between the forward processes and infinite horizon BSDE, and, moreover, with risk-sensitive optimization. In addition, we develop a connection, for large time horizons, with a family of classical homothetic value function processes with random endowments.Comment: 34 page

Forward Exponential Performances: Pricing and Optimal Risk Sharing

In a Markovian stochastic volatility model, we consider financial agents whose investment criteria are modelled by forward exponential performance processes. The problem of contingent claim indifference valuation is first addressed and a number of properties are proved and discussed. Special attention is given to the comparison between the forward exponential and the backward exponential utility indifference valuation. In addition, we construct the problem of optimal risk sharing in this forward setting and solve it when the agents' forward performance criteria are exponential.Comment: 29 page

Coherent Pricing

Recent literature proved the existence of an unbounded market price of risk (MPR) or maximum generalized Sharpe ratio (GSR) if one combines the most important Brownian-motion-linked arbitrage free pricing models with a coherent and expectation bounded risk measure. Furthermore, explicit sequences of portfolios with a theoretical (risk, return) diverging to (��1;+1) were constructed and their performance tested. The empirical evidence revealed that the divergence to (��1;+1) is only theoretical (not real), but the MPR is much larger than the GSR of the most important international stock indices. The natural question is how to modify the available pricing models so as to prevent the caveat above. The theoretical MPR cannot equal inf nity but must be large enough (consistent with the empirical findings) and this will be the focus of this paper. It will be shown that every arbitrage free pricing model can be improved in such a manner that the new stochastic discount factor (SDF) satisfie the two requirements above, and the newMPR becomes bounded but large enough. This is important for several reasons; Firstly, if the existent models predict unrealistic price evolutions then these mistakes may imply important capital losses to practitioners and theoretical errors to researchers. Secondly, the lack of an unbounded MPR is much more coherent and consistent with equilibrium. Finally, the major discrepancies between the initial pricing model and the modifie one will affect the tails of their SDF, which seems to justify several empirical caveats of previous literature. For instance, it has been pointed out that it is not easy to explain the real quotes of many deeply OTM options with the existing pricing models

Explicit Description of HARA Forward Utilities and Their Optimal Portfolios

This paper deals with forward performances of HARA type. Precisely, for a market model in which stock price processes are modeled by a locally bounded $d$-dimensional semimartingale, we elaborate a complete and explicit characterization for this type of forward utilities. Furthermore, the optimal portfolios for each of these forward utilities are explicitly described. Our approach is based on the minimal Hellinger martingale densities that are obtained from the important statistical concept of Hellinger process. These martingale densities were introduced recently, and appeared herein tailor-made for these forward utilities. After outlining our parametrization method for the HARA forward, we provide illustrations on discrete-time market models. Finally, we conclude our paper by pointing out a number of related open questions.Comment: 39 page

Time--consistent investment under model uncertainty: the robust forward criteria

We combine forward investment performance processes and ambiguity averse portfolio selection. We introduce the notion of robust forward criteria which addresses the issues of ambiguity in model specification and in preferences and investment horizon specification. It describes the evolution of time-consistent ambiguity averse preferences. We first focus on establishing dual characterizations of the robust forward criteria. This offers various advantages as the dual problem amounts to a search for an infimum whereas the primal problem features a saddle-point. Our approach is based on ideas developed in Schied (2007) and Zitkovic (2009). We then study in detail non-volatile criteria. In particular, we solve explicitly the example of an investor who starts with a logarithmic utility and applies a quadratic penalty function. The investor builds a dynamical estimate of the market price of risk $\hat \lambda$ and updates her stochastic utility in accordance with the so-perceived elapsed market opportunities. We show that this leads to a time-consistent optimal investment policy given by a fractional Kelly strategy associated with $\hat \lambda$. The leverage is proportional to the investor's confidence in her estimate $\hat \lambda$

Stochastic Analysis in Finance and Insurance

[no abstract available