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pp.567-611. Contents

. Statistical analysis of data from the nancial markets shows that generalized hyperbolic (GH) distributions allow a more realistic description of asset returns than the classical normal distribution. GH distributions contain as subclasses hyperbolic as well as normal inverse Gaussian (NIG) distributions which have recently been proposed as basic ingredients to model price processes. GH distributions generate in a canonical way Levy processes, i.e. processes with stationary and independent increments. We introduce a model for price processes which is driven by generalized hyperbolic Levy motions. This GH model is a generalization of the hyperbolic model developed by Eberlein and Keller (1995). It is incomplete. We derive an option pricing formula for GH driven models using the Esscher transform as martingale measure and compare the prices with classical Black-Scholes prices. The objective of this study is to examine the consistency of our model assumptions with the empirically obser...

This paper provides an empirical investigation of the risk neutral variance process and the market price of variance risk implied in the foreign currency options market. There are three principal contributions. First, the parameters of Heston's (1993) meanreverting square root stochastic volatility model are estimated using dollar/mark option prices from 1987 to 1992. Second, it is shown that these implied parameters can be combined with historical moments of the dollar/mark exchange rate to deduce an estimate of the market price of variance risk. These estimates are found to be nonzero, time varying, and of sufficient magnitude to imply that the compensation for variance risk is a significant component of the risk premia in the currency market. Finally, the out-of-sample test suggests that the historical variance and the Hull and White implied variance contain no additional information than those imbedded in the Heston implied variance. KEY WORDS: Market price of variance ...

Weisbach are gratefully acknowledged. I bear full responsibility for all remaining errors. Forecasting Future Volatility from Option Prices Evidence exists that option prices produce biased forecasts of future volatility across a wide variety of options markets. This paper presents two main results. First, approximately half of the forecasting bias in the S&P 500 index (SPX) options market is eliminated by constructing measures of realized volatility from five minute observations on SPX futures rather than from daily closing SPX levels. Second, much of the remaining forecasting bias is eliminated by employing an option pricing model that permits a non-zero market price of volatility risk. It is widely believed that option prices provide the best forecasts of the future volatility of the assets which underlie them. One reason for this belief is that option prices have the ability to impound all publicly available information – including all information contained in the history of past prices – about the future volatility of the underlying assets. A second related reason is that option pricing theory maintains that if an option prices fails to embody optimal forecasts of the future volatility of the underlying asset, a profitable trading strategy should be available whose implementation would push the option price to the level that reflects the best possible forecast of future volatility.

The paper shows how to convert statistical prediction sets into worst case hedging strategies for derivative securities. The prediction sets can, in particular, be ones for volatilities and correlations of the underlying securities, and for interest rates. This permits a transfer of statistical conclusions into prices for options and similar financial instruments. A prime feature of our results is that one can construct the trading strategy as if the prediction set had a 100 % probability. If, in fact, the set has probability 1−α, the hedging strategy will work with at least the same probability. Different types of prediction regions are considered. The starting value A0 for the trading strategy corresponding to the 1 − α prediction region is a form of long term value at risk. At the same time, A0 is coherent.

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The paper studies inference for volatility type objects and its implications for the hedging of options. It considers the nonparametric estimation of volatilities and instantaneous covariations between diffusion type processes. This is then linked to options trading, where we show that our estimates can be used to trade options without reference to the specific model. The new options “delta ” becomes an additive modification of the (implied volatility) Black-Scholes delta. The modification, in our example, is both substantial and statistically significant. In the inference problem, explicit expressions are found for asymptotic error distributions, and it is explained why one does not in this case encounter a bias-variance tradeoff, but rather a variance-variance tradeoff. Observation times can be irregular. Some key words and phrases: volatility estimation, statistical uncertainty, small interval asymptotics, mixing convergence, option hedging

This paper investigates the predictive power of implied variances extracted from currency options. Under the assumption that the options market is informationally efficient and that options are priced according to the model of Hull and White (1987), implied variances are estimated from transaction prices of currency options traded on PHLX. If implied variances are efficient, variance forecasts from time series models should have no additional predictive power in predicting subsequently realized market variances. The in-sample tests suggest that implied variances contain incremental information in the GARCH models for conditional variance. In contrast with recent findings on stock and stock index options, the in-sample and out-of-sample tests indicate that the implied variance is a superior, but biased forecast of future variance; and that the variance forecasts from GARCH and MA(60) models do not contain significant incremental information in predicting future variance. Key Words: Impl...

Option prices can be used to infer the level of uncertainty about future asset prices. The first two parts of this article explain such measures (implied volatility) and how they can differ from the market’s true expectation of uncertainty. The third then estimates the implied volatility of threemonth eurodollar interest rates from 1985 to 2001 and evaluates its ability to predict realized volatility. Implied volatility shows that uncertainty about short-term interest rates has been falling for almost 20 years, as the levels of interest rates and inflation have fallen. And changes in implied volatility are usually coincident with major news about the stock market, the real economy, and monetary policy. Federal Reserve Bank of St. Louis Review, May/June 2005, 87(3), pp. 407-25. Economists often use asset prices along with models of their determination to derive financial markets ’ expectations of events. For example, monetary economists use federal funds futures prices to measure expectations of interest rates (Krueger and Kuttner, 1995; Pakko and Wheelock, 1996). Similarly, a large literature on fixed and target zone exchange rates has used forward exchange rates to measure the credibility of exchange rate regimes or to predict their collapse (Svensson,