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We investigate the informational content of options-implied probability density functions (PDFs) for the future price of oil. Using a semiparametric variant of the methodology in Breeden and Litzenberger (1978), we investigate the fit and smoothness of distributions derived from alternative PDF estimation methods, and develop a set of robust summary statistics. Using PDFs estimated around episodes of high geopolitical tensions, oil supply disruptions, and macroeconomic data releases, we explore the extent to which oil price movements are expected or unexpected, and whether agents believe these movements to be persistent or temporary.
Recently a market in options based on CPI inflation (inflation caps and floors) has emerged in the US. This paper uses quotes on these derivatives to construct probability densities for inflation. We study how these pdfs respond to news announcements, and find that the implied odds of deflation are sensitive to certain macroeconomic news releases. We compare the option-implied probability densities with those obtained by time series methods, and use this information to construct empirical pricing kernels. The options-implied densities assign considerably more mass to extreme inflation outcomes (either deflation or high inflation) than do their time series counterparts. This yields a U-shaped empirical pricing kernel, with investors having high marginal utility in states of the world characterized by either deflation or high inflation.
This paper applies classic linear inverse theory to the estimation of the implied risk neutral probability density function (PDF) from option prices. To overcome non-uniqueness and instability inherent in the option inverse problem, smoothness requirement for the shape of a PDF and an initial model are introduced by a penalty function. Positivity constraints are included as a hard bond on the PDF values. Then the option inverse problem becomes a non-negative least-squares problem which can be solved by the classic methods such as the non-negative least squares program of Lawson and Hanson (1974). The best solution is not the solution that gives best fit, but the solution that gives the optimal trade-off between the goodness of fit and smoothness of the estimated risk natural PDF. The proposed inversion technique is compared to the models of Black-Scholes (BS), a mixture of two lognormals (MLN), Jarrow and Rudd's Edgeworth expansion (JR), and jump diffusion (JD) for the estimation of the PDF from the option prices associated with the September 2007 NYMEX natural gas futures. It is found that the inversion technique not only gives best goodness of fit, but also the significantly better model resolution. BS, JD and MLN models basically cannot resolve the densities far away from the strikes where option prices are observed and can resolve long wavelength features of the densities inside the strikes where option prices are observed. On the other hand, the inversion model can resolve not only the significant details of the densities inside the strikes where option prices are observed, but also the long wavelength features of the densities away from the strikes where option prices are observed. The empirical study for the last three months of the September 2007 futures contract shows that the shapes of the estimated PDFs become more symmetric as the futures contract is closer to the expiration date. The dispersion of the estimated PDFs decreases with decreasing the time to expire, indicating the resolution of uncertainty with time.
We develop a general method for estimating the implied, martingale equivalent, probability density function (PDF) for futures prices from American options prices. The early exercise feature of American options precludes expressing the price of the option in terms of the PDF. There exist tight bounds for the price of American options in terms of the PDF. We demonstrate how these bounds, together with observed option prices, can be used to estimate the parameters of the PDF. We estimate the distribution for crude oil during the Persian Gulf crisis and find the distribution differs significantly from that recovered using standard techniques.
An overview of methods used for estimation of option-implied risk-neutral probability density functions (PDFs) is presented in the study, and one of such methods, double lognormal approach, is used for the analysis of the information content of the EUR/PLN currency options on the Polish market. Estimated PDFs have proven to provide superior information concerning future volatility than historical volatility, yet their forecasting power is comparable to that of the Black-Scholes model. There are no strong grounds for using PDFs as a predictor of the future EUR/PLN exchange rate. Low informative content does not directly follow, as PDFs can be used as an indicator of markets conditions. The issues that could be addressed more thoroughly in the future studies concern the assumption of risk neutrality and the impact of the estimation method on the higher moments of the distribution.
In The Medium of Contingency Elie Ayache builds upon his ground-breaking book The Blank Swan, in exploring the intersection of philosophy and finance, introducing new notions of price and market. Inverting the received view, he now sees a creation of matter in both the market and its metaphysics, rather than pure speculation. Once recognized as the proper medium of contingency and disassociated from the probabilistic and statistical tools traditionally used to model it, the market can be thought as 'real', in a new sense of reality corresponding to the new sense of matter. To bring this new and original perspective, The Medium of Contingency builds on probability theory as first formalized by von Mises and Kolmogorov, and later revisited by Shafer and Vovk. It utilises the author's extensive experience in derivatives pricing technology and software, as well as his work in the philosophy of contingency and contingent claims, to propose a new philosophical interpretation of Brownian motion and of the Black-Scholes-Merton formula. Then it completes the overturning of the traditional view of the market by arguing that there should be no difference, ultimately, between an underlying asset and the derivative written on it. This book does not aim to change the market but the way we must think of it. It is the author's conviction that there can be no philosophy of the market, and consequently no thinking of it, without a philosophy of contingent claims and of derivative pricing. The book provides the missing piece, which the philosophy of probability cannot provide alone. Its scope, however, extends beyond the strict critique of financial mathematics, as it also, and perhaps most importantly, delivers the author's definitive treatment of the philosophically prominent and recently much discussed notion of contingency.
Commodities have become an important component of many investors' portfolios and the focus of much political controversy over the past decade. This book utilizes structural models to provide a better understanding of how commodities' prices behave and what drives them. It exploits differences across commodities and examines a variety of predictions of the models to identify where they work and where they fail. The findings of the analysis are useful to scholars, traders and policy makers who want to better understand often puzzling - and extreme - movements in the prices of commodities from aluminium to oil to soybeans to zinc.
Author : American Bar Association. House of Delegates Publisher : American Bar Association Page : 216 pages File Size : 31,20 MB Release : 2007 Category : Law ISBN : 9781590318737
The Model Rules of Professional Conduct provides an up-to-date resource for information on legal ethics. Federal, state and local courts in all jurisdictions look to the Rules for guidance in solving lawyer malpractice cases, disciplinary actions, disqualification issues, sanctions questions and much more. In this volume, black-letter Rules of Professional Conduct are followed by numbered Comments that explain each Rule's purpose and provide suggestions for its practical application. The Rules will help you identify proper conduct in a variety of given situations, review those instances where discretionary action is possible, and define the nature of the relationship between you and your clients, colleagues and the courts.