OPTIMAL INVESTMENT WITH AN UNBOUNDED RANDOM ENDOWMENT AND UTILITY-BASED PRICING

This paper studies the problem of maximizing the expected utility of terminal wealth for a financial agent with an unbounded random endowment, and with a utility function which supports both positive and negative wealth. We prove the existence of an optimal trading strategy within a class of permissible strategies—those strategies whose wealth process is a super-martingale under all pricing measures with finite relative entropy. We give necessary and sufficient conditions for the absence of utility-based arbitrage, and for the existence of a solution to the primal problem. We consider two utility-based methods which can be used to price contingent claims. Firstly we investigate marginal utility-based price processes (MUBPP's). We show that such processes can be characterized as local martingales under the normalized optimal dual measure for the utility maximizing investor. Finally, we present some new results on utility indifference prices, including continuity properties and volume asymptotics for the case of a general utility function, unbounded endowment and unbounded contingent claims.     

DO ARBITRAGE‐FREE PRICES COME FROM UTILITY MAXIMIZATION?

In this paper we ask whether, given a stock market and an illiquid derivative, there exists arbitrage‐free prices at which a utility‐maximizing agent would always want to buy the derivative, irrespectively of his own initial endowment of derivatives and cash. We prove that this is false for any given investor if one considers all initial endowments with finite utility, and that it can instead be true if one restricts to the endowments in the interior. We show, however, how the endowments on the boundary can give rise to very odd phenomena; for example, an investor with such an endowment would choose not to trade in the derivative even at prices arbitrarily close to some arbitrage price.     

ON AGENT’S AGREEMENT AND PARTIAL‐EQUILIBRIUM PRICING IN INCOMPLETE MARKETS

We consider two risk‐averse financial agents who negotiate the price of an illiquid indivisible contingent claim in an incomplete semimartingale market environment. Under the assumption that the agents are exponential utility maximizers with nontraded random endowments, we provide necessary and sufficient conditions for negotiation to be successful, i.e., for the trade to occur. We also study the asymptotic case where the size of the claim is small compared to the random endowments and we give a full characterization in this case. Finally, we study a partial‐equilibrium problem for a bundle of divisible claims and establish existence and uniqueness. A number of technical results on conditional indifference prices is provided.     

DYNAMIC INDIFFERENCE VALUATION VIA CONVEX RISK MEASURES

The (subjective) indifference value of a payoff in an incomplete financial market is that monetary amount which leaves an agent indifferent between buying or not buying the payoff when she always optimally exploits her trading opportunities. We study these values over time when they are defined with respect to a dynamic monetary concave utility functional, that is, minus a dynamic convex risk measure. For that purpose, we prove some new results about families of conditional convex risk measures. We study the convolution of abstract conditional convex risk measures and show that it preserves the dynamic property of time-consistency. Moreover, we construct a dynamic risk measure (or utility functional) associated to superreplication in a market with trading constraints and prove that it is time-consistent. By combining these results, we deduce that the corresponding indifference valuation functional is again time-consistent. As an auxiliary tool, we establish a variant of the representation theorem for conditional convex risk measures in terms of equivalent probability measures.     

On the Existence of Minimax Martingale Measures

Embedding asset pricing in a utility maximization framework leads naturally to the concept of minimax martingale measures. We consider a market model where the price process is assumed to be an ^d‐semimartingale X and the set of trading strategies consists of all predictable, X‐integrable, ^d‐valued processes H for which the stochastic integral (H.X) is uniformly bounded from below. When the market is free of arbitrage, we show that a sufficient condition for the existence of the minimax measure is that the utility function u : → is concave and nondecreasing. We also show the equivalence between the no free lunch with vanishing risk condition, the existence of a separating measure, and a properly defined notion of viability.     

Bounds on European Option Prices under Stochastic Volatility

In this paper we consider the range of prices consistent with no arbitrage for European options in a general stochastic volatility model. We give conditions under which the infimum and the supremum of the possible option prices are equal to the intrinsic value of the option and to the current price of the stock, respectively, and show that these conditions are satisfied in most of the stochastic volatility models from the financial literature. We also discuss properties of Black–Scholes hedging strategies in stochastic volatility models where the volatility is bounded.     

ANALYTICAL COMPARISONS OF OPTION PRICES IN STOCHASTIC VOLATILITY MODELS

This paper gives an ordering on option prices under various well-known martingale measures in an incomplete stochastic volatility model. Our central result is a comparison theorem that proves convex option prices are decreasing in the market price of volatility risk, the parameter governing the choice of pricing measure. The theorem is applied to order option prices under q -optimal pricing measures. In doing so, we correct orderings demonstrated numerically in Heath, Platen, and Schweizer ( Mathematical Finance , 11(4), 2001) in the special case of the Heston model.     

ON UTILITY-BASED PRICING OF CONTINGENT CLAIMS IN INCOMPLETE MARKETS

We study the uniqueness of the marginal utility-based price of contingent claims in a semimartingale model of an incomplete financial market. In particular, we obtain that a necessary and sufficient condition for all bounded contingent claims to admit a unique marginal utility-based price is that the solution to the dual problem defines an equivalent local martingale measure.     

On the Pricing of Contingent Claims with Frictions

This paper studies the problem of pricing contingent claims in a market which has frictions in the form of costs, such as penalty functions corresponding to constraints. An arbitrage-free interval is identified, and a fair price based upon utility functions is proposed. It provides a framework to study incomplete markets that is simplier than the one related to constraints on portfolios introduced by Karatzas and Kou.     

MEAN–VARIANCE PORTFOLIO CHOICE: QUADRATIC PARTIAL HEDGING

In this paper we investigate the problem of mean–variance portfolio choice with bankruptcy prohibition. For incomplete markets with continuous assets' price processes and for complete markets, it is shown that the mean–variance efficient portfolios can be expressed as the optimal strategies of partial hedging for quadratic loss function. Thus, mean–variance portfolio choice, in these cases, can be viewed as expected utility maximization with non-negative marginal utility.     

The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets

Let χ be a family of stochastic processes on a given filtered probability space (Ω, F, (F_t)_t∈T, P) with T?R₊. Under the assumption that the set M_e of equivalent martingale measures for χ is not empty, we give sufficient conditions for the existence of a unique equivalent martingale measure that minimizes the relative entropy, with respect to P, in the class of martingale measures. We then provide the characterization of the density of the minimal entropy martingale measure, which suggests the equivalence between the maximization of expected exponential utility and the minimization of the relative entropy.     

Option Pricing in Stochastic Volatility Models of the Ornstein-Uhlenbeck type

Stochastic volatility models of the Ornstein-Uhlenbeck type possess authentic capability of capturing some stylized features of financial time series. In this work we investigate this class of models from the viewpoint of derivative asset analysis. We discuss topics related to the incompleteness of this type of markets. In particular, for structure preserving martingale measures, we derive the price of simple European-style contracts in closed form. Furthermore, the range of viable prices is determined and an empirical application is presented.