An extensive empirical literature in finance has documented not only the presence of anamolies in the Black-Scholes model, but also the "term-structures" of these anamolies (for instance, the behavior of the volatility smile or of unconditional returns at different maturities). Theoretical efforts in the literature at addressing these anamolies have largely focussed on two extensions of the Black-Scholes model: introducing jumps into the return process, and allowing volatility to be stochastic. This paper employs commonly-used versions of these two classes of models to examine the extent to which the models are theoretically capable of resolving the observed anamolies. We find that each model exhibits some "term-structure" patterns that are fundamentally inconsistent with those observed in the data. As a consequence, neither class of models constitutes an adequate explanation of the empirical evidence, although stochastic volatility models fare better than jumps in this regard.

This article provides several new insights into the economic sources of skewness. First, we document the differential pricing of individual equity options versus the market index, and relate it to variations in return skewness. Second, we show how risk aversion introduces skewness in the risk-neutral density. Third, we derive laws that decompose individual return skewness into a systematic component and an idiosyncratic component. Empirical analysis of OEX options and 30 stocks demonstrates that individual risk-neutral distributions differ from that of the market index by being far less negatively skewed. This paper explains the presence and evolution of risk-neutral skewness over time and in the cross-section of individual stocks.

i Abstract: The volatility smile changed drastically around the crash of 1987 and new option pricing models have been proposed in order to accommodate that change. Deterministic volatility models allow for more °exible volatility surfaces but refrain from introducing additional risk-factors. Thus, options are still redundant securities. Alternatively, stochastic models introduce additional risk-factors and options are then needed for spanning of the pricing kernel. We develop a statistical test based on this di®erence in spanning. Using daily S&P500 index options data from 1986-1995, our tests suggest that both in- and out-of-the-money options are needed for spanning. The ¯ndings are inconsistent with deterministic volatility models but are consistent with stochastic models which incorporate additional priced risk-factors such as stochastic volatility, interest rates, or jumps. What is a good model to price equity derivatives and to manage risk? Starting from Black and Scholes (1973), a common approach in the derivative pricing literature has been to model the underlying asset as a geometric Brownian motion with constant volatility. Early tests of options on stocks such as Rubinstein (1985) more or less supported the empirical implications of a geometric

One of the central questions in financial economics is the determination of asset prices, such as the value of a stock. Over the past three decades, research on this topic has converged on a concept called the “state-price density”. However, a puzzle has arisen. On the one hand, Cox, Ingersoll, and Ross (1985) and others argue that the ratio of the state-price density to the statistical probability density, which is commonly known as the pricing kernel, should decrease monotonically as the aggregate wealth of an economy rises. On the other hand, recent empirical work on options on the S&P 500 index suggests that, for a sizable range of index levels, the pricing kernel is increasing instead of decreasing. We investigate theoretical explanations to this puzzle. Our existing work has ruled out some alternative hypotheses, such as data imperfections and methodological problems. The current paper focuses on the hypothesis of state-dependent utility. State-dependent utility can arise from diverse causes such as habit persistence, stochastic index volatility, or dependence on interest rates or other asset prices. In this paper we characterize a general relation between the index and the state variables that is required to explain the puzzle. However, it remains for us to provide economic reasoning to justify this relation. The existing literature is largely unsatisfactory with respect to this puzzle, and most research does not touch on the puzzle altogether.

We study the cross-sectional performance of option pricing models in which the volatility of the underlying stock is a deterministic function of the stock price and time. For each date in our sample of FTSE 100 index option prices we t an implied binomial tree to the panel of all European style options with dierent strike prices and maturities and then examine how well this model prices a corresponding panel of American style options. We nd that the implied binomial tree model performs no better than an ad-hoc procedure of smoothing Black-Scholes implied volatilities across strike prices and maturities. Our cross-sectional results complement the time-series ndings of Dumas, Fleming, and Whaley (1998). Financial Support from the Rodney White Center at the Wharton School is gratefully acknowledged. y Philadelphia, PA 19104-6367. Phone: (215) 898-3609. E-mail: brandtm@wharton.upenn.edu. z Philadelphia, PA 19104-6367. Phone: (215) 898-5071. E-mail: taow@wharton.upenn.e...

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Traditional parametric methods have limited success in estimating and forecasting the volatility of financial securities. Recent advance in evolutionary computation has provided additional tools to conduct data mining effectively. The current work applies the genetic programming in a Time Series Data Mining framework to characterize the S&P100 high frequency data in order to forecast the one step ahead integrated volatility. Results of the experiment have shown to be superior to those derived by the traditional methods.