Vega Risk and the Smile - Columbia University

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Vega risk and the smile Allan M. Malz The RiskMetrics Group, 44 Wall Street, New York, New York 10005, USA Vega risk is analytically easy to ``nest'' into the standard risk management framework, but is complicated by the prevalence of volatility smiles and term structures in most option markets. Volatility smiles, in spite of their occasionally treacherous effects on option books, are often neglected by risk managers. This paper provides a guide to incorporating vega risk into a ``classical'' value-at-risk VaR) model. The paper includes a tractable approach to capturing the effects of the volatility smile and term structure on vega risk and their interaction with other risk factors. The author also presents a summary of the statistical behavior of implied volatility and relates it to observed differences in the pattern of implied volatilities from that predicted by the Black± Scholes model. The VaR computation strategy employs the smile to compensate for the shortcomings of the Black±Scholes model and provide improved VaR forecasts without a speci®c alternative model. 1. INTRODUCTION Vega risk, the risk arising from ¯uctuations in implied volatility, can be a large part of the risk of a portfolio containing options. Vega risk is analytically easy to ``nest'' into the standard risk management framework. The treatment of vega risk in portfolios is, however, complicated by the prevalence of volatility smiles and term structures in most option markets. These option market phenomena run counter to the benchmark Black±Scholes option pricing model, in which implied and historical volatilityare constant over time and invariant with respect to option maturity and likelihood of exercise. Researchers have developed models incorporating stochastic volatility, jump-diusions, and other alterna- tives to the pure constant-volatility diusion environment of the Black±Scholes model. Some option dealers and risk managers use these models to re®ne their option prices and return forecasts. However, volatility smiles, in spite of their occasionally treacherous eects on option books, are often neglected by risk managers. A common metric for market risk for a portfolio of ®nancial assets is value-at-risk VaR), a measure of the maximum loss to the portfolio that can occur over a speci®ed time horizon with a speci®ed con®dence level. This paper provides a guide to incorporating vega risk into a ``classical'' VaR model. The paper suggests a practical approach to capturing the eects of the volatility smile and term structure on vega risk and their interaction with other risk factors. In our discussion, we will present several examples using a high-quality database of foreign exchange implied volatilities. 1 1 The data, sourced from JP Morgan's foreign exchange desk and available from RiskMetrics, includes implied volatilities for a wide range of currency pairs and maturities, extensive coverage of volatility smiles, and term structures. 41

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