risk mgmt for derivatives

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UVA-F-1432Risk Management for Derivatives

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UVA-F-1432Oct. 3, 2008

This technical note was prepared by Professor Robert M. Conroy. Copyright 2007 by the University of VirginiaDarden School Foundation, Charlottesville, VA. All rights reserved. To order copies, send an e-mail [email protected] part of this publication may be reproduced, stored in a retrieval system,used in a spreadsheet, or transmitted in any form or by any meanselectronic, mechanical, photocopying,

recording, or otherwisewithout the permission of the Darden School Foundation.

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 riskreturn 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 themost 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 therelative 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 volatilethan 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 riskreturn 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 lowerrisk, they adjust the Beta of their portfolio accordingly. In a capital asset pricing model (CAPM

framework), Betaor market risk is the only relevant risk for portfolios.For bonds, the most important source of risk is changes in interest rates. Interest rate

changes directly affect bond prices. Modified duration1 is the most frequently used measure of

how bond prices change relative to a change in interest rates. Relatively higher modified durationmeans more price volatility for a given change in interest rates. For both bonds and equities, risk

can be distilled down to a single risk factor. For bonds, it is modified duration, and for equities, it

1For a complete discussion of duration, see Duration and Convexity, (UVA-F-1238); or R. Brealy and S.Myers,Principles of Corporate Finance, 7thed., (McGraw-Hill: Boston, 2003): 6748.


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UVA-F-1432-2-

is Beta. In each case, the risk can be measured, and then adjusted or managed to suit your risk

tolerance.

In each of those cases, financial theory provides a measure of risk. Using these riskmeasures, holders of either bonds or equities can adjust or manage the risk level of the securities

that they hold. What is nice about these asset categories is that they have a single measure of

risk. Derivative securities are more challenging.

Risk Measures for Derivatives

In the discussion that follows we will define risk as the sensitivity of price to changes in

factors that affect an assets value. More price sensitivity will be interpreted as more risk.

The basic option pricing model

A simple European call option can be valued using the Black-Scholes Model2:

European call option value = ( ) ( )21 dNeXdNUAVTrf

where

( )N = Cumulative standard normal function

( )T

TrXUAVd

f

++=

2

1

2

1

ln,

Tdd = 12 ,

UAV =Underlying asset value

T = Time to maturity = UAVvolatility

X = Exercise (strike) price

rf = Risk-free rate

This very basic option pricing model demonstrates that the value of a European call option

depends on the value of five factors: underlying asset value, risk-free rate, volatility, time to

maturity, and exercise price. If the value of any one of those factors should change, then thevalue of the option would change. Of the five inputs or factors, one is fixed (exercise price),

another is deterministic (time to maturity), and the other three change randomly over time

2For the Black-Scholes Model, see Brealy Myers,Principles of Corporate Finance, 6027.


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