ASSET PRICE BUBBLES IN INCOMPLETE MARKETS*

This paper studies asset price bubbles in a continuous time model using the local martingale framework. Providing careful definitions of the asset's market and fundamental price, we characterize all possible price bubbles in an incomplete market satisfying the “no free lunch with vanishing risk (NFLVR)” and “no dominance” assumptions. We show that the two leading models for bubbles as either charges or as strict local martingales, respectively, are equivalent. We propose a new theory for bubble birth that involves a nontrivial modification of the classical martingale pricing framework. This modification involves the market exhibiting different local martingale measures across time—a possibility not previously explored within the classical theory. Finally, we investigate the pricing of derivative securities in the presence of asset price bubbles, and we show that: (i) European put options can have no bubbles; (ii) European call options and discounted forward prices have bubbles whose magnitudes are related to the asset's price bubble; (iii) with no dividends, American call options are not exercised early; (iv) European put‐call parity in market prices must always hold, regardless of bubbles; and (v) futures price bubbles can exist and they are independent of the underlying asset's price bubble. Many of these results stand in contrast to those of the classical theory. We propose, but do not implement, some new tests for the existence of asset price bubbles using derivative securities.     

Convergence of utility indifference prices to the superreplication price in a multiple‐priors framework

This paper formulates a utility indifference pricing model for investors trading in a discrete time financial market under nondominated model uncertainty. Investor preferences are described by possibly random utility functions defined on the positive axis. We prove that when the investors's absolute risk aversion tends to infinity, the multiple‐priors utility indifference prices of a contingent claim converge to its multiple‐priors superreplication price. We also revisit the notion of certainty equivalent for multiple‐priors and establish its relation with risk aversion.     

OPTIMAL RISK SHARING FOR LAW INVARIANT MONETARY UTILITY FUNCTIONS

We consider the problem of optimal risk sharing of some given total risk between two economic agents characterized by law-invariant monetary utility functions or equivalently, law-invariant risk measures. We first prove existence of an optimal risk sharing allocation which is in addition increasing in terms of the total risk. We next provide an explicit characterization in the case where both agents' utility functions are comonotone. The general form of the optimal contracts turns out to be given by a sum of options (stop-loss contracts, in the language of insurance) on the total risk. In order to show the robustness of this type of contracts to more general utility functions, we introduce a new notion of strict risk aversion conditionally on lower tail events, which is typically satisfied by the semi-deviation and the entropic risk measures. Then, in the context of an AV@R-agent facing an agent with strict monotone preferences and exhibiting strict risk aversion conditional on lower tail events, we prove that optimal contracts again are European options on the total risk.     

A martingale representation theorem and valuation of defaultable securities

We consider a financial framework with two levels of information: the public information generated by the financial assets, and a larger flow of information that contains additional knowledge about a random time. This random time can represent many economic and financial settings, such as the default time of a firm for credit risk, and the death time of an insured for life insurance. As the random time cannot be seen before its occurrence, the progressive enlargement of filtration seems tailor‐fit to model the larger flow of information that incorporates both the public flow and the information about the random time. In this context, our interest focuses on the following challenges: (a) How to single out the various risks coming from the financial assets, the random time, and their correlations? (b) How these risks interplay and lead to the formation of any risk in the larger flow of information? It is clear that understanding how risks build‐up and interact, when one enlarges the flow of information, is vital for an efficient risk management and derivatives' evaluation in those informational markets. Our answers to these challenges are full and complete no matter what the model for the random time is and no matter how the random time is related to the public flow. In fact, we introduce “pure default” risks, and quantify and classify these risks afterward. Then we elaborate our martingale representation results, which state that any martingale in the large filtration stopped at the random time can be decomposed into orthogonal local martingales (i.e., local martingales whose product remains a local martingale). This constitutes our first principal contribution, while our second contribution consists in evaluating various defaultable securities according to the recovery policy, within our financial setting that encompasses any default model, using a martingale “basis.” Our pricing formulas explain the impact of various recovery policies on securities and determine the types of pure default risk they entail.     

On a Mean—Generalized Semivariance Approach to Determining the Hedge Ratio

A new mean‐risk hedge ratio based on the concept of generalized semivariance (GSV) is proposed. The proposed mean‐GSV (M‐GSV) hedge ratio is consistent with the GSV‐based risk–return model developed by Fishburn (1977), Bawa (1975, 1978), and Harlow and Rao (1989). The M‐GSV hedge ratio can also be considered an extension of the GSV‐minimizing hedge ratio considered by De Jong, De Roon, and Veld (1997) and Lien and Tse (1998, 2000). The M‐GSV hedge ratio is estimated for Standard & Poor's (S&P) 500 futures and compared to six other widely used hedge ratios. Because all the hedge ratios considered are known to converge to the minimum‐variance (Johnson) hedge ratio under joint normality and martingale conditions, tests for normality and martingale conditions are carried out. The empirical results indicate that the joint normality and martingale hypotheses do not hold for the S&P 500 futures. The M‐GSV hedge ratio varies less than the GSV hedge ratio for low and relevant levels of risk aversion. Furthermore, the M‐GSV hedge ratio converges to a value different from the values of the other hedge ratios for higher values of risk aversion. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21: 581–598, 2001     

Portfolio Optimization and Martingale Measures

The paper studies connections between risk aversion and martingale measures in a discrete-time incomplete financial market. An investor is considered whose attitude toward risk is specified in terms of the index b of constant proportional risk aversion. Then dynamic portfolios are admissible if the terminal wealth is positive. It is assumed that the return (risk) processes are bounded. Sufficient (and nearly necessary) conditions are given for the existence of an optimal dynamic portfolio which chooses portfolios from the interior of the set of admissible portfolios. This property leads to an equivalent martingale measure defined through the optimal dynamic portfolio and the index 0 < b ≤ 1. Moreover, the option pricing formula of Davis is given by this martingale measure. In the case of b = 1; that is, in the case of the log-utility, the optimal dynamic portfolio defines the numéraire portfolio.     

MINIMAL ENTROPY–HELLINGER MARTINGALE MEASURE IN INCOMPLETE MARKETS

This paper defines an optimization criterion for the set of all martingale measures for an incomplete market model when the discounted price process is bounded and quasi-left continuous. This criterion is based on the entropy–Hellinger process for a nonnegative Doléans–Dade exponential local martingale. We develop properties of this process and establish its relationship to the relative entropy "distance." We prove that the martingale measure, minimizing this entropy–Hellinger process, is unique. Furthermore, it exists and is explicitly determined under some mild conditions of integrability and no arbitrage. Different characterizations for this extremal risk-neutral measure as well as immediate application to the exponential hedging are given. If the discounted price process is continuous, the minimal entropy–Hellinger martingale measure simply is the minimal martingale measure of Föllmer and Schweizer. Finally, the relationship between the minimal entropy–Hellinger martingale measure (MHM) and the minimal entropy martingale measure (MEM) is provided. We also give an example showing that in contrast to the MHM measure, the MEM measure is not robust with respect to stopping.     

EQUILIBRIUM STATE PRICES IN A STOCHASTIC VOLATILITY MODEL1

In a stochastic volatility model, the no-free-lunch assumption does not induce a unique arbitrage price because of market incompleteness. In this paper, we consider a contingent claim on the primitive asset, traded in zero net supply. Given a system of Arrow-Debreu state prices, we provide necessary and sufficient conditions for consistency with an intertemporal additive equilibrium model that we fully characterize. We show that the risk premia corresponding to the minimal martingale of Föllmer and Schweizer (1991) are consistent with logarithmic preferences, while the Hull and White model (1987) (volatility risk premium independent of the asset price) is consistent with a class of utility functions including constant relative risk aversion (CRRA) ones.     

RISK-AVERSION BEHAVIOR IN CONSUMPTION/INVESTMENT PROBLEMS1

In this paper, we study the risk-aversion behavior of an agent in the dynamic framework of consumption/investment decision making that allows the possibility of bankruptcy. Agent's consumption utility is assumed to be represented by a strictly increasing, strictly concave, continuously differentiable function in the general case and by a HARA-type function in the special case treated in the paper. Coefficients of absolute and relative risk aversion are defined to be the well-known curvature measures associated with the derived utility of wealth obtained as the value function of the agent's optimization problem. Through an analysis of these coefficients, we show how the change in agent's risk aversion as his wealth changes depends on his consumption utility and the other problem parameters, including the payment at bankruptcy. Moreover, in the HARA case, we can conclude that the agent's relative risk aversion is nondecreasing with wealth, while his absolute risk aversion is decreasing with wealth only if he is sufficiently wealthy. At lower wealth levels, however, the agent's absolute risk aversion may increase with wealth in some cases.     

CLOSED-FORM SOLUTIONS FOR OPTIMAL PORTFOLIO SELECTION WITH STOCHASTIC INTEREST RATE AND INVESTMENT CONSTRAINTS

We examine the portfolio choice problem of an investor with constant relative risk aversion in a financial market with partially hedgeable interest rate risk. The individual shadow price of the portfolio constraint is characterized as the solution of a new backward equation involving Malliavin derivatives. A generalization of this equation is studied and solved in explicit form. This result, applied to our financial model, yields closed-form solutions for the shadow price and the optimal portfolio. The effects of parameters such as risk aversion, interest rate volatility, investment horizon, and tightness of the constraint are examined. Applications of our method to a monetary economy with inflation risk and to an international setting with currency risk are also provided.     

On the exit value of a forward contract

Default risk associated with forward contracts can be substantial, yet these financial instruments are widely used to hedge price risk. An objectively priced exit option on the forward contract would help reduce the likelihood of litigation associated with contract default. A method is proposed to compute the exit option's value for an arbitrary forward contract, using Black's (1976) model and option premium data. The time series dynamics of the exit option value are confirmed to be, like its underlying, well described by a martingale with heavy‐tailed (Student) GARCH residuals. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 29: 179–196, 2009     

A GENERAL EQUILIBRIUM MODEL OF A MULTIFIRM MORAL‐HAZARD ECONOMY WITH FINANCIAL MARKETS

We present a general equilibrium model of a moral‐hazard economy with many firms and financial markets, where stocks and bonds are traded. Contrary to the principal‐agent literature, we argue that optimal contracting in an infinite economy is not about a tradeoff between risk sharing and incentives, but it is all about incentives. Even when the economy is finite, optimal contracts do not depend on principals’ risk aversion, but on market prices of risks. We also show that optimal contracting does not require relative performance evaluation, that the second best risk‐free interest rate is lower than that of the first best, and that the second‐best equity premium can be higher or lower than that of the first best. Moral hazard can contribute to the resolution of the risk‐free rate puzzle. Its potential to explain the equity premium puzzle is examined.     

On the optimal portfolio for the exponential utility maximization: remarks to the six-author paper

This note contains ramifications of results of Delbaen et al. (2002). Assuming that the price process is locally bounded and admits an equivalent local martingale measure with finite entropy, we show, without further assumption, that in the case of exponential utility the optimal portfolio process is a martingale with respect to each local martingale measure with finite entropy. Moreover, the optimal value always can be attained on a sequence of uniformly bounded portfolios.     

Revisiting the risk/return relations in the Asian Pacific markets: New evidence from alternative models

This paper examines the risk/return relations in eleven Asian Pacific stock markets and explores if the 1997 Asian financial crisis significantly influenced market behavior in the region. We use a plain vanilla time-series regression approach as well as various GARCH models. Although results significantly vary across model specifications, the overall evidence from GARCH models supports a significantly positive risk/return relation in several markets but only prior to the Asian financial crisis. These results accord with Glosten et al. (1993) and Harvey (2001) and suggest that the relative risk aversion is sensitive to both model specifications and structural breaks.     

On the market viability under proportional transaction costs

This paper studies the market viability with proportional transaction costs. Instead of requiring the existence of strictly consistent price systems as in the literature, we show that strictly consistent local martingale systems (SCLMS) can successfully serve as the dual elements such that the market viability can be verified. We introduce two weaker notions of no arbitrage conditions on market models named no unbounded profit with bounded risk (NUPBR) and no local arbitrage with bounded portfolios (NLABPs). In particular, we show that the NUPBR and NLABP conditions in the robust sense are equivalent to the existence of SCLMS for general market models. We also discuss the implications for the utility maximization problem.     

Financial performance comparison between international joint ventures and wholly foreign-owned enterprises in China

International joint ventures (IJVs) and wholly foreign-owned enterprises (WFOEs) are the two major competing modes of Chinese market entry for foreign investors. However, the financial performance of and relative financial efficacy between the two modes have not been researched. The present study utilizes firmspecific financial data and examines financial ratios and characteristics of IJVs and WFOEs in the current Chinese financial environment. Results of this study reveal that the WFOE is superior to the IJV in terms of asset efficiency, financial risk aversion, and export growth whereas the IJV outperforms the WFOE with respect to local market growth and environment adaptation. © 1995 John Wiley & Sons, Inc.     

When in peril, retrench: Testing the portfolio channel of contagion

The impact of past gains and losses on international investors' risk aversion is an important factor in the propagation of financial shocks across countries. We first present a stylized model illustrating how changes in investors' risk aversion affect portfolio decisions and stock prices. We then examine empirically the behavior of international mutual funds. When funds' returns are below average, they reduce their exposure to countries in which they were overweight and vice versa. An index of “financial interdependence” that reflects the extent to which countries share overexposed funds helps explain the pattern of stock market comovement across countries and the pattern of contagion during crises.     

MAXIMIZING THE GROWTH RATE UNDER RISK CONSTRAINTS

We investigate the ergodic problem of growth-rate maximization under a class of risk constraints in the context of incomplete, Itô-process models of financial markets with random ergodic coefficients. Including value-at-risk , tail-value-at-risk , and limited expected loss , these constraints can be both wealth-dependent (relative) and wealth-independent (absolute). The optimal policy is shown to exist in an appropriate admissibility class, and can be obtained explicitly by uniform, state-dependent scaling down of the unconstrained (Merton) optimal portfolio. This implies that the risk-constrained wealth-growth optimizer locally behaves like a constant relative risk aversion (CRRA) investor, with the relative risk-aversion coefficient depending on the current values of the market coefficients.     

Exponential Hedging and Entropic Penalties

We solve the problem of hedging a contingent claim B by maximizing the expected exponential utility of terminal net wealth for a locally bounded semimartingale X . We prove a duality relation between this problem and a dual problem for local martingale measures Q for X where we either minimize relative entropy minus a correction term involving B or maximize the Q -price of B subject to an entropic penalty term. Our result is robust in the sense that it holds for several choices of the space of hedging strategies. Applications include a new characterization of the minimal martingale measure and risk-averse asymptotics.