Utility valuation of multi-name credit derivatives and application to CDOs

We study the impact of risk-aversion on the valuation of credit derivatives. Using the technology of utility-indifference pricing in intensity-based models of default risk, we analyse resulting yield spreads in multi-name credit derivatives, particularly CDOs. We study first the idealized problem with constant intensities where solutions are essentially explicit. We also give the large portfolio asymptotics for this problem. We then analyse the case where the firms have stochastic default intensities driven by a common factor, which can be viewed as another extreme from the independent case. This involves the numerical solution of a system of reaction-diffusion PDEs. We observe that the nonlinearity of the utility-indifference valuation mechanism enhances the effective correlation between the times of the credit events of the various firms leading to non-trivial senior tranche spreads, as often seen from market data.     

An example of indifference prices under exponential preferences

The aim herein is to analyze utility-based prices and hedging strategies. The analysis is based on an explicitly solved example of a European claim written on a nontraded asset, in a model where risk preferences are exponential, and the traded and nontraded asset are diffusion processes with, respectively, lognormal and arbitrary dynamics. Our results show that a nonlinear pricing rule emerges with certainty equivalent characteristics, yielding the price as a nonlinear expectation of the derivatives payoff under the appropriate pricing measure. The latter is a martingale measure that minimizes its relative to the historical measure entropy.Received: July 2003, Mathematics Subject Classification: 93E20, 60G40, 60J75JEL Classification: C61, G11, G13The second author acknowledges partial support from NSF Grants DMS-0102909 and DMS-0091946. We have received valuable comments from the participants at the Conferences in Paris IX, Dauphine (2000), ICBI Barcelona (2001) and 14th Annual Conference of FORC Warwick (2001). While revising this work, we came across the paper by Henderson (2002) in which a special case of our model is investigated     

Pricing Dynamic Insurance Risks Using the Principle of Equivalent Utility

We introduce an expected utility approach to price insurance risks in a dynamic financial market setting. The valuation method is based on comparing the maximal expected utility functions with and without incorporating the insurance product, as in the classical principle of equivalent utility. The pricing mechanism relies heavily on risk preferences and yields two reservation prices - one each for the underwriter and buyer of the contract. The framework is rather general and applies to a number of applications that we extensively analyze.     

Optimal Investment With Undiversifiable Income Risk

This paper treats the problem of consumption and portfolio choice in continuous time, with stochastic income that cannot be replicated by trading the available securities. the optimal controls and value functions are characterized in terms of the viscosity solution of the associated Hamilton-Jacobi-Bellman equation, which is shown to exist and is characterized. the optimal policy is then given from the first-order conditions of the Hamilton-Jacobi-Bellman equation.     

A valuation algorithm for indifference prices in incomplete markets

A probabilistic iterative algorithm is constructed for indifference prices of claims in a multiperiod incomplete model. At each time step, a nonlinear pricing functional is applied that isolates and prices separately the two types of risk. It is represented solely in terms of risk aversion and the pricing measure, a martingale measure that preserves the conditional distribution of unhedged risks, given the hedgeable ones, from their historical counterparts.Received: 1 September 2003, Mathematics Subject Classification: 93E20, 60G40, 60J75JEL Classification: C61, G11, G13The second author acknowledges partial support from NSF Grants DMS 0102909 and DMS 0091946.     

Mean field and n‐agent games for optimal investment under relative performance criteria

We analyze a family of portfolio management problems under relative performance criteria, for fund managers having CARA or CRRA utilities and trading in a common investment horizon in log‐normal markets. We construct explicit constant equilibrium strategies for both the finite population games and the corresponding mean field games, which we show are unique in the class of constant equilibria. In the CARA case, competition drives agents to invest more in the risky asset than they would otherwise, while in the CRRA case competitive agents may over‐ or underinvest, depending on their levels of risk tolerance.     

Bounds on Derivative Prices in an Intertemporal Setting with Proportional Transaction Costs and Multiple Securities

The observed discrepancies of derivative prices from their theoretical, arbitrage-free values are examined in the presence of transaction costs. Analytic upper and lower bounds on the reservation write and purchase prices, respectively, are obtained when an investor's preferences exhibit constant relative risk aversion between zero and one. The economy consists of multiple primary securities with stationary returns, a constant rate of interest, and any number of American or European derivatives with, possibly, path-dependent arbitrary payoffs.     

Dynamically consistent investment under model uncertainty: the robust forward criteria

We combine forward investment performance processes and ambiguity-averse portfolio selection. We introduce robust forward criteria which address ambiguity in the specification of the model, the risk preferences and the investment horizon. They encode the evolution of dynamically consistent ambiguity-averse preferences.We focus on establishing dual characterisations of the robust forward criteria, which is advantageous as the dual problem amounts to the search for an infimum whereas the primal problem features a saddle point. Our approach to duality builds on ideas developed in Schied (Finance Stoch. 11:107–129, 2007) and ?itkovi? (Ann. Appl. Probab. 19:2176–2210, 2009). We also study in detail the so-called time-monotone criteria. We solve explicitly the example of an investor who starts with logarithmic utility and applies a quadratic penalty function. Such an investor builds a dynamic 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 \(\hat{\lambda}\) and with the leverage being proportional to the investor’s confidence in her estimate.