This article develops a direct filtration-based maximum likelihood methodology for estimating the parameters and realizations of latent affine processes. Filtration is conducted in the transform space of characteristic functions, using a version of Bayes ’ rule for recursively updating the joint characteristic function of latent variables and the data conditional upon past data. An application to daily stock market returns over 1953-96 reveals substantial divergences from EMM-based estimates; in particular, more substantial and time-varying jump risk. The implications for pricing stock index options are examined. 3 “The Lion in Affrik and the Bear in Sarmatia are Fierce, but Translated into a Contrary Heaven, are of less Strength and Courage.” Jacob Ziegler; translated by Richard Eden (1555) While models proposing time-varying volatility of asset returns have been around for thirty years, it has proven extraordinarily difficult to estimate the parameters of the underlying volatility process,

The central insight of asset pricing is that a security’s value depends on both its distribution of payo¤s across economic states and state prices. In …xed income markets, many investors focus exclusively on estimates of expected payo¤s, such as credit ratings, without considering the state of the economy in which default is likely to occur. Such investors are likely to be attracted to securities whose payo¤s resemble those of economic catastrophe bonds–bonds that default only under severe economic conditions. We show that many structured …nance instruments can be characterized as economic catastrophe bonds, but o¤er far less compensation than alternatives with comparable payo ¤ pro…les. We argue that this di¤erence arises from the willingness of rating agencies to certify structured products with a low default likelihood as “safe ” and from a large supply of investors who view them as such.

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This paper applies the Extreme-Value (EV) Generalised Pareto distribution to the extreme tails of the return distributions for the S&P500, FT100, DAX, Hang Seng, and Nikkei225 futures contracts. It then uses tail estimators from these contracts to estimate spectral risk measures, which are coherent risk measures that reflect a user’s risk-aversion function. It compares these to more familiar VaR and Expected Shortfall (ES) measures of risk, and also compares the precision and discusses the relative usefulness of each of these risk measures.

We propose a continuous time utility maximization model to value stock and option compensation from the executive’s perspective. We allow the executive to invest non-option wealth in the market and riskless asset but not in the company stock itself. This enables executives to adjust exposure to market risk, but they are subject to firm-specific risk for incentive purposes. Since the executive is risk averse, this unhedgeable firm risk leads them to place less value on the options than their cost to the company, given by their market or Black Scholes value. By distinguishing between these two types of risks, we are able to examine the effect of stock volatility, firm-specific risk, and market risk on the value to the executive. Executives do not necessarily want to increase stock volatility, as their risk aversion can outweigh the option’s convexity effect. Firm-specific risk generally reduces option value, although if market risk is held fixed and the options are out-of-the-money it is possible for the reverse to hold. Generally, market risk has a positive effect on option value. An implication of the model is that the Black Scholes formula exaggerates the incentives for the executive to increase the company stock price. We examine the relationship between risk and optimal incentives, and find firm-specific risk decreases optimal incentives (regardless

The focus of this work is on the problem of tracking parameters describing both the stochastic discount factor and the objective / real-world measure dynamically, with the aim of monitoring value at risk or other related diagnostics of interest. The methodology presented incorporates information from derivative prices as well as from the underlying instrument’s price over time in order to perform on-line parameter inference. We construct a parametric model of the stochastic discount factor which is introduced based on empirical results in the literature (Aït-Sahalia and Lo, 2000; Jackwerth, 2000; Rosenberg and Engle, 2002, for example). This is used in a sequential Monte Carlo algorithm for tracking the parameters of this and of an objective density over time. Further, two new techniques for pricing European options in the framework are discussed. In applying this approach to price data, Variance Gamma and Normal Inverse Gaussian models of the underlying price process have been discussed. These are for illustrative purposes and other models could easily be also considered. Both models appear to track realistically; detailed results are presented for the Variance Gamma model. These cover the value at risk estimates, expected price change estimates and parameter estimates.

The power family, also known as the family of constant relative risk aversion (CRRA), is the most widely used parametric family for fitting utility functions to data. Its characteristics have, however, been little understood, and have led to numerous misunderstandings. This paper explains these characteristics in a manner accessible to a wide

We use a sample of option prices, and the method of Bakshi, Kapadia and Madan (2003), to estimate the ex ante higher moments of the underlying individual securities ’ risk-neutral returns distribution. We find that individual securities ’ volatility, skewness and kurtosis are strongly related to subsequent returns. Specifically, we find a negative relation between volatility and returns in the cross-section. We also find a significant relation between skewness and returns, with more negatively (positively) skewed returns associated with subsequent higher (lower) returns, while kurtosis is positively related to subsequent returns. To analyze the extent to which these returns relations represent compensation for risk, we use data on index options and the underlying index to estimate the stochastic discount factor over the 1996-2005 sample period, and allow the stochastic discount factor to include higher moments. We find evidence that, even after controlling for differences in co-moments, individual securities ’ skewness matters. However, when we combine information in the risk-neutral distribution and the stochastic discount factor to estimate the implied physical distribution of industry returns, we find little evidence that the distribution of technology stocks was positively skewed during the bubble period–in fact, these stocks have the lowest skew, and the highest estimated Sharpe ratio, of all stocks in our sample. All errors are the responsibility of the authors. We thank Robert Battalio, Patrick Dennis, and Stewart Mayhew for providing data and computational code. We thank Andrew Ang, Leonce Bargeron, and Paul Pfleiderer

Crisis events such as the 1987 stock market crash, the Asian Crisis and the bursting of the Dot-Com bubble have radically changed the view that extreme events in financial markets have negligible probability. This paper argues that the use of the Generalized Extreme Value (GEV) distribution to model the Risk Neutral Density (RND) function provides a flexible framework that captures the negative skewness and excess kurtosis of returns, and also delivers the market implied tail index of asset returns. We obtain an original analytical closed form solution for the Harrison and Pliska (1981) no arbitrage equilibrium price for the European option in the case of GEV asset returns. The GEV based option prices successfully remove the well known pricing bias of the Black-Scholes model. We explain how the implied tail index is efficacious at identifying the fat tailed behaviour of losses and hence the left skewness of the price RND functions, particularly around crisis events.