Risk Management for Derivatives

Abstract

This technical note addresses the basics of risk measure for options. It introduces the different risk measures for options: Delta, Gamma, Vega, Rho, and Theta. Although the note focuses primarily on price risk (Delta) and the Delta risk (Gamma), it does address volatility risk (Vega), interest-rate risk (Rho), and time decay (Theta). In addition to providing derivations and basic calculations, the note provides a full description of Delta hedging.

Excerpt

UVA-F-1432

Oct. 3, 2008

Risk Management for Derivatives

“The Greeks are coming, the Greeks are coming!”

Managing risk is important to a large number of individuals and institutions. The most fundamental aspect of business is a process where we invest, take on risk, and in exchange earn a compensatory return. The key to success in this process is to manage your risk–return tradeoff. Managing risk is a nice concept, but often difficulty arises when measuring risk. There is a saying: “What gets measured, gets managed.” To alter that slightly, “What cannot be measured, cannot be managed.” Hence, risk management always requires some measure of risk. Risk, in the most general context, refers to how much the price of a security changes for a given change in some factor.

In the context of equities, Beta is a frequently used measure of risk. Beta measures the relative risk of an asset. High-Beta stocks, or portfolios, have more variable returns relative to the overall market than low-Beta assets. If a Beta of 1.00 means the asset has the same risk characteristics as the market, then a portfolio with a Beta greater than 1.00 will be more volatile than the market portfolio and consequently more risky with higher expected returns. Conversely, assets with a Beta less than 1.00 are less risky than average and have lower expected returns. Portfolio managers use Beta to measure their risk–return tradeoff. If they are willing to take on more risk (and return), they increase the Beta of their portfolio, and if they are looking for lower risk, they adjust the Beta of their portfolio accordingly. In a capital asset pricing model (CAPM framework), Beta or market risk is the only relevant risk for portfolios.

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Keywords: Derivatives, Options, Delta, Hedging