EQUILIBRIUM ASSET AND OPTION PRICING UNDER JUMP DIFFUSION

This paper develops an equilibrium asset and option pricing model in a production economy under jump diffusion. The model provides analytical formulas for an equity premium and a more general pricing kernel that links the physical and risk‐neutral densities. The model explains the two empirical phenomena of the negative variance risk premium and implied volatility smirk if market crashes are expected. Model estimation with the S&P 500 index from 1985 to 2005 shows that jump size is indeed negative and the risk aversion coefficient has a reasonable value when taking the jump into account.     

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.     

Endogenous Random Asset Prices in Overlapping Generations Economies

This paper derives a general explicit sequential asset price process for an economy with overlapping generations of consumers. They maximize expected utility with respect to subjective transition probabilities given by Markov kernels. The process is determined primarily by the interaction of exogenous random dividends and the characteristics of consumers, given by arbitrary preferences and expectations, yielding an explicit random dynamical system with expectations feedback. The paper studies asset prices and equity premia for a parametrized class of examples with CARA utilities and exponential distributions. It provides a complete analysis of the role of risk aversion and of subjective as well as rational beliefs.     

Matching the BRIC equity premium: A structural approach

The equity risk premium (ERP) in BRIC markets is, on average, significantly higher than that in the US market. This paper employs an endowment economy with recursive preferences and long-run risk to explain the ERP generated by a portfolio of BRIC equity indices. The combination of recursive preferences and long-run risk partially explains the BRIC ERP. It turns out that there is a puzzle with respect to BRIC data as well. This holds even if we account for high levels of aversion to consumption and utility risk and for the empirically observed autoregressive structure of US consumption and BRIC dividend growth.     

Dissecting interbank risk using basis swap spreads

This paper analyses interbank risk using the information content of basis swap (BS) spreads, floating-to-floating interest rate swaps whose payments are associated with euro deposit rates for alternative tenors. To identify the impact of shocks affecting interbank risk, we propose an empirical model that decomposes BS quotes into their expected and unexpected components. These unobservable constituents of BS spreads are estimated by solving a signal extraction problem using a particle filter. We find that expected components covariate with aggregate liquidity and risk aversion while systemic risk arises as the main driver behind unexpected fluctuations. Our empirical findings suggest that macroprudential analysis emerges as a key device to ease asset pricing in a new multi-curve scenario.     

Estimation Uncertainty and the Equity Premium*

This paper studies a dynamic equilibrium model of asset prices in a partially observable exchange economy. It shows that the precautionary savings motive in response to estimation uncertainty can dominate the risk aversion effect, resulting in the reduction of the equity premium over short horizons. This exacerbates the equity premium puzzle. Over longer holding horizons, however, estimation uncertainty does induce higher risk premiums on equity over risk‐free coupon bonds of matching maturities, as long‐term bond yields are lowered due to the precautionary savings effect.     

ARROW–DEBREU EQUILIBRIA FOR RANK‐DEPENDENT UTILITIES

We provide conditions on a one‐period‐two‐date pure exchange economy with rank‐dependent utility agents under which Arrow–Debreu equilibria exist. When such an equilibrium exists, we show that the state‐price density is a weighted marginal rate of intertemporal substitution of a representative agent, where the weight depends on the differential of the probability weighting function. Based on the result, we find that asset prices depend upon agents' subjective beliefs regarding overall consumption growth, and we offer a direction for possible resolution of the equity premium puzzle.     

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.     

PRICING CORPORATE SECURITIES UNDER NOISY ASSET INFORMATION

This paper considers the pricing of corporate securities of a given firm, in particular equity, when investors do not have full information on the firm's asset value. We show that under noisy asset information, the pricing of corporate securities leads to a nonlinear filtering problem. This problem is solved by a Markov chain approximation, leading to an efficient finite-dimensional approximative filter for the asset value. We discuss several applications and illustrate our results with a simulation study.     

A BENCHMARK APPROACH TO FINANCE

This paper derives a unified framework for portfolio optimization, derivative pricing, financial modeling, and risk measurement. It is based on the natural assumption that investors prefer more rather than less, in the sense that given two portfolios with the same diffusion coefficient value, the one with the higher drift is preferred. Each such investor is shown to hold an efficient portfolio in the sense of Markowitz with units in the market portfolio and the savings account. The market portfolio of investable wealth is shown to equal a combination of the growth optimal portfolio (GOP) and the savings account. In this setup the capital asset pricing model follows without the use of expected utility functions, Markovianity, or equilibrium assumptions. The expected increase of the discounted value of the GOP is shown to coincide with the expected increase of its discounted underlying value. The discounted GOP has the dynamics of a time transformed squared Bessel process of dimension four. The time transformation is given by the discounted underlying value of the GOP. The squared volatility of the GOP equals the discounted GOP drift, when expressed in units of the discounted GOP. Risk-neutral derivative pricing and actuarial pricing are generalized by the fair pricing concept, which uses the GOP as numeraire and the real-world probability measure as pricing measure. An equivalent risk-neutral martingale measure does not exist under the derived minimal market model.     

Exchange risk and asset returns: A theoretical and empirical study of an open economy asset pricing model

This study develops a consumption-based asset pricing model in which domestic consumers can buy goods from domestic and foreign markets but can only invest in domestic markets. In this model, the exchange rate influences asset prices through the marginal utility of consumption and increases the risks investors face. We find that our model can successfully price the 25 Fama–French portfolios and industry portfolios in the Chinese market, and the exchange rate is an important pricing factor in the unconditional linear model. We also find that the exchange risk is time-varying and countercyclical, which can help to explain the countercyclicality in equity premium.     

Optimal Consumption-Portfolio Policies With Habit Formation1

This is a companion paper to the authors ‘Asset Prices in an Exchange Economy with Habit Formation” in Econometrica which focuses on consumption demand and asset pricing when preferences are habit forming. Here we prove existence of optimal consumption-portfolio policies for (i) utility functions for which the marginal cost of consumption (MCC) interacts with the habit formation process and satisfies a recursive integral equation with forward functional Lipschitz integrand and (ii) utilities for which the MCC is independent of the standard of living and satisfies a recursive integral equation with locally Lipschitz integrand. Result (i) is demonstrated here for the first time. Result (ii) is novel and enables us to consider Cobb-Douglas utilities without placing lower bounds on the system of Arrow-Debreu prices. We also review and extend our earlier results in the linear case; in particular, we provide new insights about the structure of optimal portfolios. Additional new features of the model include the possibility of finite marginal utility of consumption at zero and habit formation mechanisms with stochastic coefficients. an extension to a financial market model with general processes is outlined. A byproduct of the analysis is a set of fixed-point theorems for recursive integral equations with forward functional Lipschitz or locally Lipschitz integrands.     

Treasury Bond risk and return, the implications for the hedging of consumption and lessons for asset pricing

All consumption-based models of asset pricing imply that the relation between the conditional mean and conditional volatility of any asset reflects the effectiveness of holding that asset as a hedge against intertemporal variation in the marginal utility of consumption. For Treasury Bonds of various maturities, we find significant positive relations. Our empirical findings support the conclusion that investors must sell bonds short to hedge shocks to marginal utility, because realized bond returns tend to be high (low) when investors least (most) desire an additional dollar of consumption. Implications for special cases of the general consumption-based model are also discussed.     

Equilibrium Pricing in the Presence of Cumulative Dividends Following a Diffusion

The paper presents some security market pricing results in the setting of a security market equilibrium in continuous time. The theme of the paper is financial valuation theory when the primitive assets pay out real dividends represented by processes of unbounded variation. In continuous time, when the models are also continuous, this is the most general representation of real dividends, and it can be of practical interest to analyze such models.
Taking as the starting point an extension to continuous time of the Lucas consumption-based model, we derive the equilibrium short-term interest rate, present a new derivation of the consumption-based capital asset pricing model, demonstrate how equilibrium forward and futures prices can be derived, including several examples, and finally we derive the equilibrium price of a European call option in a situation where the underlying asset pays dividends according to an Itô process of unbounded variation. In the latter case we demonstrate how this pricing formula simplifies to known results in special cases, among them the famous Black–Scholes formula and the Merton formula for a special dividend rate process.     

Step Options

Motivated by risk management problems with barrier options, we propose a flexible modification of the standard knock‐out and knock‐in provisions and introduce a family of path‐dependent options: step options . They are parametrized by a finite knock‐out (knock‐in) rate , ρ. For a down‐and‐out step option, its payoff at expiration is defined as the payoff of an otherwise identical vanilla option discounted by the knock‐out factor exp(-ρτ_B^‐) or max(1‐ρτ^-B,0), where &\tau;_B^‐ is the total time during the contract life that the underlying price was lower than a prespecified barrier level ( occupation time ). We derive closed‐form pricing formulas for step options with any knock‐out rate in the range $[0,∞). For any finite knock‐out rate both the step option's value and delta are continuous functions of the underlying price at the barrier. As a result, they can be continuously hedged by trading the underlying asset and borrowing. Their risk management properties make step options attractive "no‐regrets" alternatives to standard barrier options. As a by‐product, we derive a dynamic almost‐replicating trading strategy for standard barrier options by considering a replicating strategy for a step option with high but finite knock‐out rate. Finally, a general class of derivatives contingent on occupation times is considered and closed‐form pricing formulas are derived.     

On the role of the utility function in the estimation of preference parameters

Modern economic theory summarizes the main characteristics of individual preferences through a definite set of parameters: risk aversion, prudence, and the elasticity of intertemporal substitution. Despite their importance, the results of the literature devoted to the parameters’ estimation are controversial. This paper highlights the neglected role that may have been played by the constraints that the quantitative definition of the parameters and the utility functions employed impose on the estimation. A number of simulation exercises are presented, which show that the same saving behaviour can be associated with quite different values of the parameters depending on the utility function adopted.     

Currency Prices, the Nominal Exchange Rate, and Security Prices in a Two‐Country Dynamic Monetary Equilibrium

This paper examines a continuous‐time two‐country dynamic monetary equilibrium in which countries with possibly heterogeneous tastes and endowments hold their own money for the purpose of transaction services formulated via money in the utility function. Given a price system, no‐arbitrage pricing results are provided for the price of each money and the nominal exchange rate. Characterizations are provided for equilibrium prices for general time‐additive preferences and non‐Markovian exogenous processes. Under a Markovian structure of model primitives, the currency prices are shown to solve a bivariate system of partial differential equations. Assuming that each country is endowed with heterogeneous separable power utility and the exogenous quantities all follow geometric Brownian motions, an equilibrium is shown to exist and additional characterization is provided. A further example of nonseparable Cobb–Douglas preferences is investigated. The additional features over the customary environment of homogeneous logarithmic preferences are emphasized.     

Risk,external adjustment and capital flows

This paper studies the process of external adjustment. I develop an open economy model with endowment and preference shocks that can account for the empirical behavior of real exchange rates, interest rates and consumption in the U.S. and Europe. The model includes cross border holdings of bonds and equity, and financial frictions that impede international risk-sharing. I find that external adjustment following endowment shocks predominantly takes place via trade flows, consistent with the intertemporal approach to the current account. In contrast, preference shocks that change investors' risk aversion induce adjustment via the trade and valuation channels; where the latter includes the effects of unexpected capital gains and loss on existing cross border holdings and changes in the expected future return differentials between foreign assets and liabilities. The model estimates imply that the valuation channel of external adjustment is more important for the U.S. than the trade channel. Consistent with this implication, I show that forecasts of future return differentials contributed most to the volatility of the U.S. net foreign asset position in the post Bretton-Woods era.     

Valuation of housing index derivatives

This study analyzes the valuation of housing index derivatives traded on the Chicago Mercantile Exchange (CME). Specifically, to circumvent the nontradability of housing indices, we propose and implement an equilibrium valuation framework. Assuming a mean-reverting aggregate dividend process and a utility function characterized by constant relative risk aversion, we show that the value of a housing index derivative depends only on parameters characterizing the underlying housing index, the endogenized interest rate and their correlation. We also analytically and numerically examine risk premiums for the CME futures and options and obtain three important findings. First, risk premiums are significant for all contracts with maturities longer than one year. Second, the expected growth rate of the underlying index is the key determinant for risk premiums. Third, risk premiums can be positive or negative, depending on whether the expected growth rate of the underlying index is higher or lower than the risk-free yield-to-maturity. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:660–688, 2010     

EXPLICIT SOLUTIONS OF CONSUMPTION-INVESTMENT PROBLEMS IN FINANCIAL MARKETS WITH REGIME SWITCHING

We consider a consumption and investment problem where the market presents different regimes. An investor taking decisions continuously in time selects a consumption–investment policy to maximize his expected total discounted utility of consumption. The market coefficients and the investor's utility of consumption are dependent on the regime of the financial market, which is modeled by an observable finite-state continuous-time Markov chain. We obtain explicit optimal consumption and investment policies for specific HARA utility functions. We show that the optimal policy depends on the regime. We also make an economic analysis of the solutions, and show that for every investor the optimal proportion to allocate in the risky asset is greater in a "bull market" than in a "bear market." This behavior is not affected by the investor's risk preferences. On the other hand, the optimal consumption to wealth ratio depends not only on the regime, but also on the investor's risk tolerance: high risk-averse investors will consume relatively more in a "bull market" than in a "bear market," and the opposite is true for low risk-averse investors.