Stochastic Modeling of Risk Aversion and its Implications for Derivative Pricing and Risk Management

Columbia University, New York NY

Investigators

Abstract

In many financial applications, investors are inevitably exposed to unhedgeable risks due to, for example, hedging constraints or other market frictions. The value of derivative securities naturally depends on how risks are managed, as well as the investor?s preferences between risk and return. This project will systematically investigate some recent problems that involve incorporating the investor's risk preferences into the valuation of derivatives, with a focus on employee stock options (ESOs) and credit derivatives. The project objective is to design efficient and robust valuation models that can be calibrated given market data and applied to construct the optimal risk management strategies consistent with the investor?s risk preferences. The research involves developing applicable mathematical tools to model, estimate, and analyze the complex risk structures of these derivatives, which have a significant bearing on their values. Moreover, the project includes the analytical and numerical studies of a number of stochastic control and optimal stopping problems that naturally arise in these valuation problems. By quantifying their risks and returns, the research will also shed light on some optimal contract design problems for these derivatives. Employee stock options are an important component of compensation in the US. Concerned about their cost to shareholders, the US Financial Accounting Standards Board since 2004 has required firms to estimate and report the granting cost of ESOs. The central challenge of ESO valuation lies in the uncertain timing of option exercises, which depends on the employee's risk aversion and various contractual restrictions of the options. On the other hand, credit derivatives are financial instruments whose payoffs are contingent on the occurrence of default event(s), such as the bankruptcy of a firm. The trading volume of credit derivatives has grown dramatically in the past decade, but the valuation and risk management technologies have not kept up. The recent crisis in the credit markets reflects ineffective risk management of complex credit derivatives by major financial institutions. Hence, this research aims to develop a valuation framework that explicitly accounts for the investor's risk preferences, contractual features and market conditions, so as to accurately quantify the risks of these derivatives. These valuation problems are important not only for individual or institutional investors, but also for regulators.