I use a new technique to derive a closed-form solution for the price of a European call option on an asset with stochastic volatility. The model allows arbitrary correlation between volatility and spotasset returns. I introduce stochastic interest rates and show how to apply the model to bond options and foreign currency options. Simulations show that correlation between volatility and the spot asset’s price is important for explaining return skewness and strike-price biases in the BlackScholes (1973) model. The solution technique is based on characteristi c functions and can be applied to other problems.

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A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options

This paper summarizes and explains the main events of the liquidity and credit crunch in 2007-08. Starting with the trends leading up to the crisis, I explain how these events unfolded and how four different amplification mechanisms magnified losses in the mortgage market into large dislocations and turmoil in financial markets.

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Deciphering the Liquidity and Credit Crunch 2007-08

(2002). Modelling Extremal Events for Insurance and Finance. Journal of the American Statistical Association: Vol. 97, No. 457, pp. 360-360.

Modelling Extremal Events for Insurance and Finance

This book provides the most comprehensive treatment of the theoretical concepts and modelling techniques of quantitative risk management. Whether you are a financial risk analyst, actuary, regulator or student of quantitative finance, Quantitative Risk Management gives you the practical tools you need to solve real-world problems. Describing the latest advances in the field, Quantitative Risk Management covers the methods for market, credit and operational risk modelling. It places standard industry approaches on a more formal footing and explores key concepts such as loss distributions, risk measures and risk aggregation and allocation principles. The book's methodology draws on diverse quantitative disciplines, from mathematical finance and statistics to econometrics and actuarial mathematics. A primary theme throughout is the need to satisfactorily address extreme outcomes and the dependence of key risk drivers. Proven in the classroom, the book also covers advanced topics like credit derivatives. Fully revised and expanded to reflect developments in the field since the financial crisis Features shorter chapters to facilitate teaching and learning Provides enhanced coverage of Solvency II and insurance risk management and extended treatment of credit risk, including counterparty credit risk and CDO pricing Includes a new chapter on market risk and new material on risk measures and risk aggregation

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Quantitative Risk Management: Concepts, Techniques, and Tools

The financial market turmoil in 2007 and 2008 has led to the most severe financial crisis since the Great Depression and threatens to have large repercussions on the real economy The bursting of the housing bubble forced banks to write down several hundred billion dollars in bad loans caused by mortgage delinquencies At the same time, the stock market capitalization of the major banks declined by more than twice as much While the overall mortgage losses are large on an absolute scale, they are still relatively modest compared to the $8 trillion of US stock market wealth lost between October 2007, when the stock market reached an all-time high, and October 2008 This paper attempts to explain the economic mechanisms that caused losses in the mortgage market to amplify into such large dislocations and turmoil in the financial markets, and describes common economic threads that explain the plethora of market declines, liquidity dry-ups, defaults, and bailouts that occurred after the crisis broke in summer 2007 To understand these threads, it is useful to recall some key factors leading up to the housing bubble The US economy was experiencing a low interest rate environment, both because of large capital inflows from abroad, especially from Asian countries, and because the Federal Reserve had adopted a lax interest rate policy Asian countries bought US securities both to peg the exchange rates at an export-friendly level and to hedge against a depreciation of their own currencies against the dollar, a lesson learned from the Southeast Asian crisis of the late 1990s The Federal Reserve Bank feared a deflationary period after the bursting of the Internet bubble and thus did not counteract the buildup of the housing bubble At the same time, the banking system underwent an important transformation The

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Deciphering the Liquidity and Credit Crunch 2007-2008

We consider default by firms that are part of a single clearing mechanism. The obligations of all firms within the system are determined simultaneously in a fashion consistent with the priority of debt claims and the limited liability of equity. We first show, via a fixed-point argument, that there always exists a "clearing payment vector" that clears the obligations of the members of the clearing system; under mild regularity conditions, this clearing vector is unique. Next, we develop an algorithm that both clears the financial system in a computationally efficient fashion and provides information on the systemic risk faced by the individual system firms. Finally, we produce qualitative comparative statics for financial systems. These comparative statics imply that, in contrast to single-firm results, even unsystematic, nondissipative shocks to the system will lower the total value of the system and may lower the value of the equity of some of the individual system firms.

Systemic Risk in Financial Systems

A revised dispatch procedure for power is described that meets the demand for energy while accounting for both cost and emission considerations. First, a tradeoff curve, representing all alternative dispatch policies, is derived for each demand level. Since air pollution damage is a function of the spatial distribution of pollutants in the atmosphere, an air pollution diffusion model is utilized in order to evaluate each dispatch policy on the tradeoff curve from an air impact point of view. Under the postulate of rational behavior, the utility function of the decision maker can be described by a quasi-concave function whose maximum point is found among points on the tradeoff curve. An interactive search method, based on the golden section search technique, is then devised in order to guide the decision maker toward finding the dispatch policy that maximizes his utility function.

Economic-Enviromental Power Dispatch

This paper presents a Dynamic Emissions Management System for the management of sulfur oxide emissions from fossil fuel-fired generating stations. Based on a multiple-strategy approach to emissions control via generation shifting, this system can be used for contingency evaluation, during a shortage of low sulfur fuel, for supplemental control or during air pollution emergencies. Simulation results of emissions management of an actual power system are presented.

Sulfur oxide emissions management for electric power systems

An application of a decision approach for controlling air pollution emission from power generation, presented by the authors in an earlier paper, is described. Trade-off curves between pollution emission and total operating cost are first computed for a portion of a realistic system containing coal, oil, and gas turbine generation. Points on the trade-off curve are then mapped into an exposure measure by using an air pollution diffusion model for the region being analyzed. The computations were carried out for several demand levels and various weather conditions. A comparative analysis of the results and their implications to power dispatch decisions are discussed.

An Application of the Economic-Environmental Power Dispatch

This article presents the theory of option pricing with random volatilities in complete markets As such, it makes two contributions First, the newly developed martingale measure technique is used to synthesize results dating from Merton (1973) through Eisenberg, (1985, 1987) This synthesis illustrates how Merton's formula, the CEV formula, and the Black-Scholes formula are special cases of the random volatility model derived herein The impossibility of obtaining a self-financing trading strategy to duplicate an option in incomplete markets is demonstrated This omission is important because option pricing models are often used for risk management, which requires the construction of synthetic options

Larry Eisenberg

Papers

Systemic Risk in Financial Systems

Economic-Enviromental Power Dispatch

Sulfur oxide emissions management for electric power systems

An Application of the Economic-Environmental Power Dispatch

Option pricing with random volatilities in complete markets