This article presents convenient reduced-form models of the valuation of contingent claims subject to default risk, focusing on applications to the term structure of interest rates for corporate or sovereign bonds. Examples include the valuation of a credit-spread option

In the setting of ‘‘affine’ ’ jump-diffusion state processes, this paper provides an analytical treatment of a class of transforms, including various Laplace and Fourier transforms as special cases, that allow an analytical treatment of a range of valuation and econometric problems. Example applications include fixed-income pricing models, with a role for intensity-based models of default, as well as a wide range of option-pricing applications. An illustrative example examines the implications of stochastic volatility and jumps for option valuation. This example highlights the impact on option ‘smirks ’ of the joint distribution of jumps in volatility and jumps in the underlying asset price, through both jump amplitude as well as jump timing.

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Implicit in the prices of traded financial assets are Arrow-Debreu prices or, with continuous states, the state-price density (SPD). We construct a nonparametric estimator for the SPD implicit in option prices and derive its asymptotic sampling theory. This estimator provides an arbitrage-free method of pricing new, complex, or illiquid securities while capturing those features of the data that are most relevant from an asset-pricing perspective, e.g., negative skewness and excess kurtosis for asset returns, volatility "smiles" for option prices. We perform Monte Carlo experiments and extract the SPD from actual S&P 500 option prices.

If asset returns have systematic skewness, expected returns should include rewards for accepting this risk. We formalize this intuition with an asset pricing model that incorporates conditional skewness. Our results show that conditional skewness helps explain the cross-sectional variation of expected returns across assets and is significant even when factors based on size and book-to-market are included. Systematic skewness is economically important and commands a risk premium, on average, of 3.60 percent per year. Our results suggest that the momentum effect is related to systematic skewness. The low expected return momentum portfolios have higher skewness than high expected return portfolios. THE SINGLE FACTOR CAPITAL ASSET PRICING MODEL ~CAPM! of Sharpe ~1964! and Lintner ~1965! has come under recent scrutiny. Tests indicate that the crossasset variation in expected returns cannot be explained by the market beta alone. For example, a growing number of studies show that “fundamental” variables such as size, book-to-market value, and price to earnings ratios

This article studies the design and valuation of debt contracts in a general dynamic setting under uncertainty. We incorporate some insights of the recent corporate finance literature into a valuation framework. The basic framework is an extensive form game determined by the terms of a debt contract and applicable bankruptcy laws. Debtholders and equityholders behave noncooperatively. The firm’s reorganization boundary is determined endogenously. Strategic debt service results in significantly higher default premia at even small liquidation costs. Deviations from absolute priority and forced liquidations occur along the equilibrium path. The design tends to stress higher coupons and sinking funds when firms have a higher cash payout ratio. This article studies the design and valuation of debt contracts in a general dynamic setting under uncertainty. In doing so we draw together two strands of the finance literature that have developed significantly in recent years, but have done so in large part indepen-We have benefited from the comments of seminar participants at the

Motivated by recent financial crises in East Asia and the United States where the downfall of a small number of firms had an economy-wide impact, this paper generalizes existing reduced-form models to include default intensities dependent on the default of a counterparty. In this model, firms have correlated defaults due not only to an exposure to common risk factors, but also to firm-specific risks that are termed “counterparty risks.” Numerical examples illustrate the effect of counterparty risk on the pricing of defaultable bonds and credit derivatives such as default swaps.

We consider the problem of pricing an American contingent claim whose payoff depends on several sources of uncertainty. Using classical assumptions from the Arbitrage Pricing Theory, the theoretical price can be computed as the maximum over all possible early exercise strategies of the discounted expected cash flows under the modified risk-neutral information process. Several efficient numerical techniques exist for pricing American securities depending on one or few (up to 3) risk sources. They are either lattice-based techniques or finite difference approximations of the Black-Scholes diffusion equation. However, these methods cannot be used for high-dimensional problems, since their memory requirement is exponential in the

This paper explores the ability of conditional versions of the CAPM and the consumption CAPM—jointly the (C)CAPM—to explain the cross section of average stock returns. Central to our approach is the use of the log consumption–wealth ratio as a conditioning variable. We demonstrate that such conditional models perform far better than unconditional specifications and about as well as the Fama-French three-factor model on portfolios sorted by size and book-to-market characteristics. The conditional consumption CAPM can account for the difference in returns between low-book-to-market and high-bookto-market portfolios and exhibits little evidence of residual size or book-to-market effects. We are grateful to Eugene Fama and Kenneth French for graciously providing the

Central banks have several reasons for extracting information from asset prices. Asset prices may embody more accurate and more up-to-date macroeconomic data than what is currently published or directly available to policy makers. Aberrations in some asset prices may indicate