I use a new technique to derive a closed-form solution for the price of a European call option on an asset with stochastic volatility. The model allows arbitrary correlation between volatility and spotasset returns. I introduce stochastic interest rates and show how to apply the model to bond options and foreign currency options. Simulations show that correlation between volatility and the spot asset’s price is important for explaining return skewness and strike-price biases in the Black-Scholes (1973) model. The solution technique is based on characteristic functions and can be applied to other problems. Many plaudits have been aptly used to describe Black and Scholes ’ (1973) contribution to option pricing theory. Despite subsequent development of option theory, the original Black-Scholes formula for a European call option remains the most successful and widely used application. This formula is particularly useful because it relates the distribution of spot returns I thank Hans Knoch for computational assistance. I am grateful for the suggestions of Hyeng Keun (the referee) and for comments by participants

Using dealer’s quotes and transactions prices on straight industrial bonds, we investigate the determinants of credit spread changes. Variables that should in theory determine credit spread changes have rather limited explanatory power. Further, the residuals from this regression are highly cross-correlated, and principal components analysis implies they are mostly driven by a single common factor. Although we consider several macroeconomic and financial variables as candidate proxies, we cannot explain this common systematic component. Our results suggest that monthly credit spread changes are principally driven by local supply0 demand shocks that are independent of both credit-risk factors and standard proxies for liquidity.

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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.

: A three parameter stochastic process, termed the variance gamma process, that generalizes Brownian motion is developed as a model for the dynamics of log stock prices. The process is obtained by evaluating Brownian motion with drift at a random time given by a gamma process. The two additional parameters are the drift of the Brownian motion and the volatility of the time change. These additional parameters provide control over the skewness and kurtosis of the return distribution. Closed forms are obtained for the return density and the prices of European options. The statistical and risk neutral densities are estimated for data on the S&P500 Index and the prices of options on this Index. It is observed that the statistical density is symmetric with some kurtosis, while the risk neutral density is negatively skewed with a larger kurtosis. The additional parameters also correct for pricing biases of the Black Scholes model that is a parametric special case of the option pricing model d...

We investigate the relative importance of diffusion and jumps in a new jump diffusion model for asset returns. In contrast to the standard modelling of jumps for asset returns, the jump component of our process can display finite or infinite activity, and finite or infinite variation. Empirical investigations of time series indicate that index dynamics are essentially devoid of a diffusion component, while this component may be present in the dynamics of individual stocks. This result leads to the conjecture that the risk-neutral process should be free of a diffusion component for both indices and individual stocks. Empirical investigation of options data tends to confirm this conjecture. We conclude that the statistical and risk-neutral processes for indices and stocks tend to be pure jump processes of innite activity and finite variation.

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

This paper surveys the field of asset pricing. The emphasis is on the interplay between theory and empirical work and on the trade-off between risk and return. Modern research seeks to understand the behavior of the stochastic discount factor ~SDF! that prices all assets in the economy. The behavior of the term structure of real interest rates restricts the conditional mean of the SDF, whereas patterns of risk premia restrict its conditional volatility and factor structure. Stylized facts about interest rates, aggregate stock prices, and cross-sectional patterns in stock returns have stimulated new research on optimal portfolio choice, intertemporal equilibrium models, and behavioral finance. This paper surveys the field of asset pricing. The emphasis is on the interplay between theory and empirical work. Theorists develop models with testable predictions; empirical researchers document “puzzles”—stylized facts that fail to fit established theories—and this stimulates the development of new theories. Such a process is part of the normal development of any science. Asset pricing, like the rest of economics, faces the special challenge that data are generated naturally rather than experimentally, and so researchers cannot control the quantity of data or the random shocks that affect the data. A particularly interesting characteristic of the asset pricing field is that these random shocks are also the subject matter of the theory. As Campbell, Lo, and MacKinlay ~1997, Chap. 1, p. 3! put it: What distinguishes financial economics is the central role that uncertainty plays in both financial theory and its empirical implementation. The starting point for every financial model is the uncertainty facing investors, and the substance of every financial model involves the impact of uncertainty on the behavior of investors and, ultimately, on mar-* Department of Economics, Harvard University, Cambridge, Massachusetts

In this article we construct a model in which a consumer’s utility depends on the consumption history We describe a general equilibrium framework similar to Cox, Ingersoll, and Ross (1985a). A simple example is then solved in closedform in this general equilibrium setting to rationalize the observed stickiness of the consumption series relative to the fluctuations in stock market wealth. The sample paths of consumption generated from this model imply lower variability in consumption growth rates compared to those generated by models with separable utilizations. We then present a partial equilibrium model similar to Merton (1969, 1971) and extend Merton’s results on optimal consumption and portfolio rules to accommodate nonseparability in preferences. Asset pricing implications of our framework are briefly explored. The idea that a given bundle of consumption goods will provide the same level of satisfaction at any date regardless of one’s past consumption experience is implicit in models that use time-separable utility functions to represent preferences. Separable utility functions have been the mainstay in much of the literature on asset pricing and optimal consumption and portfolio The results reported in this article were first presented at the EFA meetings in Bern, Switzerland, in 1985 [see Sundaresan (1984)]. Subsequently the article was presented at a number of universities and conferences. I thank the participants at those presentations for their feedback. I am especially thankful to Doug Breeden, Michael Brennan, John Cox, Chi-fu Huang, and Krishna Ramaswamy for their thoughtful comments and criticisms. I also thank Tong-sheng Sun for explaining the simulation procedure for stochastic differential equations and for his comments and suggestions. I am responsible for any remaining errors. Correspondence should be sent to Suresh M. Sundaresan, Graduate

This paper solves the dynamic investment problem of a risk averse manager compensated with a call option on the assets he controls. Under the manager’s optimal policy, the option ends up either deep in or deep out of the money. As the asset value goes to zero, volatility goes to infinity. However, the option compensation does not strictly lead to greater risk seeking. Sometimes, the manager’s optimal volatility is less with the option than it would be if he were trading his own account. Furthermore, giving the manager more options causes him to reduce volatility. MANAGERS WITH CONVEX COMPENSATION SCHEMES play an important role in financial markets. This paper solves for the optimal dynamic investment policy for a risk averse manager paid with a call option on the assets he controls. The paper focuses on how the option compensation impacts the manager’s appetite for risk when he cannot hedge the option position. On one hand, the convexity of the option makes the manager shun payoffs that are likely to be near the money. Under the optimal policy, the manager