This paper describes the application of the Principle of Maximum Entropy to the estimation of the distribution of an underlying asset from a set of option prices. The resulting distribution is least committal with respect to unknown or missing information and is hence the least prejudiced. The maximum entropy distribution is the only information about the asset that can be inferred from the price data alone. An extension to the Principle of Minimum Cross-Entropy allows the inclusion of prior knowledge of the asset distribution. We show that the maximum entropy distribution is able to accurately fit a known density, given simulated option prices at different strikes.

We obtain the maximum entropy distribution for an asset from call and digital option prices. A rigorous mathematical proof of its existence and exponential form is given, which can also be applied to legitimise a formal derivation by Buchen and Kelly (JFQA 31:143-159, 1996). We give a simple and robust algorithm for our method and compare our results to theirs. We present numerical results which show that our approach implies very realistic volatility surfaces even when calibrating only to at-the-money options. Finally, we apply our approach to options on the S&P 500 index.

Option price contains information on the distribution of the underlying asset. Under insufficient condition we employ the maximum entropy principle to estimate the probability density of the asset price. The problem is equivalent to finding the Lagrange multipliers of a linear functional defined by entropy and payoff functions. Buchen and Kelly proved that the maximum entropy distribution recovered from observed option prices is quite similar with the original asset distribution. In this article we apply a similar method to recover the probability density function of an asset from given option prices for binary options and European options.

It is widely acknowledged that many financial modelling techniques failed during the financial crisis, and in our post-crisis environment many techniques are being reconsidered. This single volume provides a guide to lessons learned for practitioners and a reference for academics. Including reviews of traditional approaches, real examples, and case studies, contributors consider portfolio theory; methods for valuing equities and equity derivatives, interest rate derivatives, and hybrid products; and techniques for calculating risks and implementing investment strategies. Describing new approaches without losing sight of their classical antecedents, this collection of original articles presents a timely perspective on our post-crisis paradigm. Highlights pre-crisis best classical practices, identifies post-crisis key issues, and examines emerging approaches to solving those issues Singles out key factors one must consider when valuing or calculating risks in the post-crisis environment Presents material in a homogenous, practical, clear, and not overly technical manner

The focus of this volume is on the development of new approaches for the market-conform valuation of newly issued derivatives. The first chapter presents a flexible approach to construct the binomial process of the underlying asset price by using a simultaneously backward and forward induction algorithm. This framework can be used to price and hedge a wide range of plain-vanilla and exotic options. In the second chapter this new approach is compared to existing models using a sample of plain-vanilla options, American call options and European Barrier options from two competing markets. In the third chapter new methods to value American-style options via Monte Carlo simulations in accordance with given market prices are discussed. After a short introduction to Monte Carlo methods, two new approaches are proposed. These new frameworks are illustrated via pricing examples for standard American put options.

This book shows how current and recent market prices convey information about the probability distributions that govern future prices. Moving beyond purely theoretical models, Stephen Taylor applies methods supported by empirical research of equity and foreign exchange markets to show how daily and more frequent asset prices, and the prices of option contracts, can be used to construct and assess predictions about future prices, their volatility, and their probability distributions. Stephen Taylor provides a comprehensive introduction to the dynamic behavior of asset prices, relying on finance theory and statistical evidence. He uses stochastic processes to define mathematical models for price dynamics, but with less mathematics than in alternative texts. The key topics covered include random walk tests, trading rules, ARCH models, stochastic volatility models, high-frequency datasets, and the information that option prices imply about volatility and distributions. Asset Price Dynamics, Volatility, and Prediction is ideal for students of economics, finance, and mathematics who are studying financial econometrics, and will enable researchers to identify and apply appropriate models and methods. It will likewise be a valuable resource for quantitative analysts, fund managers, risk managers, and investors who seek realistic expectations about future asset prices and the risks to which they are exposed.