The validity of the classic Black-Scholes option pricing formula dcpcnds on the capability of investors to follow a dynamic portfolio strategy in the stock that replicates the payoff structure to the option. The critical assumption required for such a strategy to be feasible, is that the underlying stock return dynamics can be described by a stochastic process with a continuous sample path. In this paper, an option pricing formula is derived for the more-general cast when the underlying stock returns are gcncrated by a mixture of both continuous and jump processes. The derived formula has most of the attractive features of the original Black&holes formula in that it does not dcpcnd on investor prcfcrenccs or knowledge of the expcctsd return on the underlying stock. Morcovcr, the same analysis applied to the options can bc extcndcd to the pricingofcorporatc liabilities. 1. Intruduction In their classic paper on the theory of option pricing, Black and Scholcs (1973) prcscnt a mode of an:llysis that has rcvolutionizcd the theory of corporate liability pricing. In part, their approach was a breakthrough because it leads to pricing formulas using. for the most part, only obscrvablc variables. In particular,

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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

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Economics is primarily a non-experimental science. Typically, we cannot generate new data sets on which to test hypotheses independently of the data that may have led to a particular theory. The common practice of using the same data set to formulate and test hypotheses introduces data-snooping biases that, if not accounted for, invalidate the assumptions underlying classical statistical inference. A striking example of a datadriven discovery is the presence of calendar effects in stock returns. There appears to be very substantial evidence of systematic abnormal stock returns related to the day of the week, the week of the month, the month of the year, the turn of the month, holidays, and so forth. However, this evidence has largely been considered without accounting for the intensive search preceding it. In this paper we use 100 years of daily data and a new bootstrap procedure that allows us to explicitly measure the distortions in statistical inference induced by data-snooping. We find that although nominal P-values of individual calendar rules are extremely significant, once evaluated in the context of the full universe from which such rules were drawn, calendar effects no longer remain significant.

Markets have internal dynamics leading to excess volatility and other phenomena that are difficult to explain using rational expectations models. This paper studies these using a nonequilibrium price formation rule, developed in the context of trading with market orders. Because this is so much simpler than a standard inter-temporal equilibrium model, it is possible to study multi-period markets analytically. There price dynamics have second order oscillatory terms. Value investing does not necessarily cause prices to track values. Trend following causes short term trends in prices, but also causes longer-term oscillations. When value investing and trend following are combined, even though there is little linear structure, there can be boom-bust cycles, excess and temporally correlated volatility, and fat tails in price fluctuations. The long term evolution of markets can be studied in terms of flows of money. Profits can be decomposed in terms of aggregate pairwise correlations. Under reinvestment of profits this leads to a capital allocation model that is equivalent to a standard model in population

A longstanding controversy in economics and finance is whether financial markets are governed by rational forces or by emotional responses. We study the importance of emotion in the decisionmaking process of professional securities traders by measuring their physiological characteristics, e.g., skin conductance, blood volume pulse, etc., during live trading sessions while simultaneously capturing real-time prices from which market events can be detected. In a sample of 10 traders, we find statistically significant differences in mean electrodermal responses during transient market events relative to no-event control periods, and statistically significant mean changes in cardiovascular variables during periods of heightened market volatility relative to normal-volatility control periods. We also observe significant differences in these physiological response across the 10 traders which may be systematically related to the traders’ levels of experience.

The menu-cost interpretation of sticky prices implies that the probability of a price change should depend on the past history of prices and fundamentals only through the gap between the current price and the frictionless price. We find that this prediction is broadly consistent with the behavior of 9 Philadelphia gasoline wholesalers. We nevertheless reject the menu-cost model as a literal description of these firms' behavior, arguing instead that price stickiness arises from strategic considerations of how customers and competitors will react to price changes.

We develop an equilibrium model of the term structure of forward prices for storable commodities. As a consequence of a nonnegativity constraint on inventory, the spot commodity has an embedded timing option that is absent in forward contracts. This option’s value changes over time due to both endogenous inventory and exogenous transitory shocks to supply and demand. Our model makes predictions about volatilities of forward prices at different horizons and shows how conditional violations of the “Samuelson effect ” occur. We extend the model to incorporate a permanent second factor and calibrate the model to crude oil futures data. COMMODITY MARKETS IN RECENT YEARS have experienced dramatic growth in trading volume, the variety of contracts, and the range of underlying commodities. Market participants are also increasingly sophisticated about recognizing and exercising operational contingencies embedded in delivery contracts. 1 For all of these reasons, there is a widespread interest in models for pricing and hedging commodity-linked contingent claims. In this paper we present an equilibrium model of commodity spot and forward prices. By explicitly incorporating the microeconomics of supply, demand, and storage, our model captures some fundamental differences between commodities and financial assets. Empirically, commodities are strikingly different from stocks, bonds and other conventional financial assets. Among these differences are:

A model of an incomplete market with the incorporation of a new notion of "inside information" is posed. The usual assumption that the stock price is Markovian is modified by adjoining a hidden Markov process to the Black-Scholes exponential Brownian motion model for stock fluctuations. The drift and volatility parameters take different values when the hidden Markov process is in different states. For example, it is 0 when there is no subset of the market which has or which believes it has, extra information. However, the hidden process is in state 1 when information is not equally shared by all, and then the behavior of the members in the subset causes increased fluctuations in the stock price. This model