pp.567-611. Contents

As is well known, the classic Black-Scholes option pricing model assumes that returns follow Brownian motion. It is widely recognized that return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to non-normal return innovations. Second, return volatilities vary stochastically over time. Third, returns and their volatilities are correlated, often negatively for equities. We propose that time-changed Lévy processes be used to simultaneously address these three facets of the underlying asset return process. We show that our framework encompasses almost all of the models proposed in the option pricing literature. Despite the generality of our approach, we show that it is straightforward to select and test a particular option pricing model through the use of characteristic function technology.

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When the underlying price process is a one-dimensional diffusion, as well as in certain restricted stochastic volatility settings, a contingent claim's delta is always bounded by the infimum and supremum of its delta at maturity. Further, if the claim's payoff is convex (concave), then the claim's price is a convex (concave) function of the underlying asset's value. However when volatility is less specialized, or when the underlying price follows a discontinuous or non-Markovian process, then call prices can have properties very different from those of the Black-Scholes model: a call's price can be a decreasing, concave function of the underlying price over some range; increasing with the passage of time; and decreasing in the level of interest rates. Much of the financial options literature derives precise option prices, when the underlying asset price process is completely specified. Since it is empirically difficult to ascertain what the true underlying process is, another part of t...

Abstract. That information surprises result in discontinuous interest rates is no surprise to participants in the bond markets. We develop a class of Poisson-Gaussian models of the Fed Funds rate to capture surprise effects, and show that these models offer a good statistical description of short rate behavior, and are useful in understanding many empirical phenomena. Estimators are used based on analytical derivations of the characteristic functions and moments of jump-diffusion stochastic processes for a range of jump distributions, and are extended to discrete-time models. Jump (Poisson) processes capture empirical features of the data which would not be captured by Gaussian models, and there is strong evidence that existing models would be well-enhanced by jump and ARCH-type processes. The analytical and empirical methods in the paper support many applications, such as testing for Fed intervention effects, which are shown to be an important source of surprise jumps in interest rates. The jump model is shown to mitigate the non-linearity of interest rate drifts, so prevalent in pure-diffusion models. Day-of-week effects are modelled explicitly, and the jump model provides evidence of bond market overreaction, rejecting the martingale hypothesis for interest rates. Jump models mixed with Markov switching processes predicate that conditioning on regime is important in determining short rate behavior.

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A paradigm of statistical mechanics of financial markets (SMFM) using nonlinear nonequilibrium algorithms, first published in L. Ingber, Mathematical Modelling, 5, 343-361 (1984), is fit to multivariate financial markets using Adaptive Simulated Annealing (ASA), a global optimization algorithm, to perform maximum likelihood fits of Lagrangians defined by path integrals of multivariate conditional probabilities. Canonical momenta are thereby derived and used as technical indicators in a recursive ASA optimization process to tune trading rules. These trading rules are then used on out-ofsample data, to demonstrate that they can profit from the SMFM model, to illustrate that these markets are likely not efficient.

We document a surprising pattern in market prices of S&P 500 index options. When implied volatilities are graphed against a standard measure of moneyness, the implied volatility smirk does not flatten out as maturity increases up to the observable horizon of two years. This behavior contrasts sharply with the implications of many pricing models and with the asymptotic behavior implied by the central limit theorem (CLT). We develop a parsimonious model which deliberately violates the CLT assumptions and thus captures the observed behavior of the volatility smirk over the maturity horizon. Calibration exercises demonstrate its superior performance against several widely used alternatives.

A rapidly growing literature has documented important improvements in financial return volatility measurement and forecasting via use of realized variation measures constructed from high-frequency returns coupled with simple modeling procedures. Building on recent theoretical results in Barndorff-Nielsen and Shephard (2004a, 2005) for related bi-power variation measures, the present paper provides a practical and robust framework for non-parametrically measuring the jump component in asset return volatility. In an application to the DM/ $ exchange rate, the S&P500 market index, and the 30-year U.S. Treasury bond yield, we find that jumps are both highly prevalent and distinctly less persistent than the continuous sample path variation process. Moreover, many jumps appear directly associated with specific macroeconomic news announcements. Separating jump from non-jump movements in a simple but sophisticated volatility forecasting model, we find that almost all of the predictability in daily, weekly, and monthly return volatilities comes from the non-jump component. Our results thus set the stage for a number of interesting future econometric developments and important financial applications by separately modeling, forecasting, and pricing the continuous and jump components of the total return variation process.

The paper proposes an original class of models for the continuous time price process of a financial security with non-constant volatility. The idea is to define instantaneous volatility in terms of exponentially-weighted moments of historic log-price. The instantaneous volatility is therefore driven by the same stochastic factors as the price process, so that unlike many other models of non-constant volatility, it is not necessary to introduce additional sources of randomness. Thus the market is complete and there are unique, preference-independent options prices. We find a partial differential equation for the price of a European Call Option. Smiles and skews are found in the resulting plots of implied volatility. Keywords: Option pricing, stochastic volatility, complete markets, smiles. Acknowledgement. It is a pleasure to thank the referees of an earlier draft of this paper whose perceptive comments have resulted in many improvements. 1 Research supported in part by Record Treasu...