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Value at Risk for Linear and Non-Linear Derivatives

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Value at Risk for Linear and Non-Linear Derivatives Value at Risk for Linear and Non-Linear Derivatives by Clemens U. Frei (Under the direction of John T. Scruggs) Abstract This paper examines the question whether standard parametric Value at Risk approaches, such as the delta-normal and the delta-gamma approach and their assumptions are appropriate for derivative positions such as forward and option contracts. The delta-normal and delta-gamma approaches are both methods based on a first-order or on a second-order Taylor series expansion. We will see that the delta-normal method is reliable for linear derivatives although it can lead to signif- icant approximation errors in the case of non-linearity. The delta-gamma method provides a better approximation because this method includes second order-effects. The main problem with delta-gamma methods is the estimation of the quantile of the profit and loss distribution. Empiric results by other authors suggest the use of a Cornish-Fisher expansion. The delta-gamma method when using a Cornish-Fisher expansion provides an approximation which is close to the results calculated by Monte Carlo Simulation methods but is computationally more efficient. Index words: Value at Risk, Variance-Covariance approach, Delta-Normal approach, Delta-Gamma approach, Market Risk, Derivatives, Taylor series expansion. Value at Risk for Linear and Non-Linear Derivatives by Clemens U. Frei Vordiplom, University of Bielefeld, Germany, 2000 A Thesis Submitted to the Graduate Faculty of The University of Georgia in Partial Fulfillment of the Requirements for the Degree Master of Arts Athens, Georgia 2003 °c 2003 Clemens U. Frei All Rights Reserved Value at Risk for Linear and Non-Linear Derivatives by Clemens U. Frei Approved: Major Professor: John T. Scruggs Committee: James S. Linck Chris T. Stivers Electronic Version Approved: Maureen Grasso Dean of the Graduate School The University of Georgia August 2003 Acknowledgments I would like to thank my major professor, Dr. John T. Scruggs, for his patience, guidance, and extraordinary support he provided during the process of writing this thesis. Furthermore, I would like to thank Dr. James S. Linck for his advice throughout the whole year at the University of Georgia as well as for being on my committee. I would also like to thank Dr. Chris T. Stivers for being on my committee. I would also like to express my gratitude to the DAAD (German Academic Exchange Service) for supporting my year at the University of Georgia through a generous scholarship. Mein besonderer Dank gilt meiner Familie, die immer f¨urmich da ist, meine Entscheidungen unterst¨utztund mir meine Ausbildung erm¨oglicht hat. iv Table of Contents Page Acknowledgments . iv List of Figures . vii List of Tables . viii Chapter 1 Introduction . 1 2 Background . 4 2.1 Definition of Risk . 4 2.2 The Risk Model . 5 2.3 Derivative Pricing Models . 8 2.4 The Greeks . 11 2.5 Synthetic Options . 19 2.6 Payoff Functions . 22 3 The Variance-Covariance Value at Risk Approach . 26 3.1 Definition of Value at Risk . 26 3.2 Implicit Value at Risk Assumptions . 28 3.3 Value at Risk for Single and Multi Assets . 29 4 Parametric Linear Approximation Value at Risk Models . 35 4.1 The Delta-Normal Approach . 35 4.2 Risk Factor Coverage . 40 v vi 4.3 Advantages and Shortcomings . 42 4.4 Examples of the Delta-Normal Approach . 45 5 Parametric Non-Linear Approximation Value at Risk Models . 52 5.1 Introduction to Second-Order Effects . 52 5.2 Quadratic Models . 54 5.3 The Cornish-Fisher Expansion . 61 5.4 Alternative Approaches . 62 5.5 Delta-Normal vs. Delta-Gamma Approaches . 65 6 Conclusion . 71 Glossary of Notation . 74 Bibliography . 77 List of Figures 2.1 The Delta of a Call Option vs. the Asset Price and the Passage of Time 15 2.2 The Gamma of an Option vs. the Asset Price and the Passage of Time. 18 2.3 Linear and Non-Linear Payoff Functions. 24 4.1 “Delta-only” Approximation vs. Black-Scholes Value of a Long Call Option Position. 44 5.1 Comparison between the Delta-Normal and the Delta-Gamma Approximation with the Black-Scholes Model. 69 5.2 Absolute Approximation Error of the Delta-Normal vs. the Delta- Gamma Approach. 70 vii List of Tables 2.1 Data for a Hypothetical Call Option . 14 2.2 Hypothetical Call and Forward Data. 23 2.3 Relationship between Instrument and Underlying Price or Rate. 25 5.1 Statistical Properties of an Option vs. its Underlying Asset . 56 5.2 Statistical Properties of the Delta-Normal vs. the Delta-Gamma Approach . 68 viii Chapter 1 Introduction Financial disasters with derivatives in the early 1990’s such as Barings, Metallge- sellschaft, Orange County, and Daiwa have led to considerable concern among reg- ulators, senior management, and financial analysts about the trading activities of companies, financial institutions, and banks. The causes for those debacles, among others, range from rogue traders for Barings and Daiwa to misunderstood market risks for Metallgesellschaft and Orange County. Jorion [12] observes increased volatility in financial markets since the early 1970’s. Among many other factors the fact that more and more assets, such as bank loans, became liquid and tradable due to securitization led to more volatile markets. There- fore, controlling market risk became very important for financial and non-financial institutions, as well as regulators. It can be argued (see [12]) that the development of Value at Risk (VaR) is a direct response to the disasters in the early 1990’s. Eventually, the VaR approach became an industry-wide standard for measuring market risk. To determine how the VaR approach is different from other risk measures, Dennis Weatherstone, former chairman of J.P.Morgan, made the comment, “at close of each business day tell me what the market risks are across all businesses and locations.” Saunders and Cornett [19, p.235] describe the above comment in the following manner, “what he wants is a single dollar number which tells him J.P. Morgan’s market exposure the next business day especially if that day turns out to be a bad day.” VaR provides a consistent 1 2 and comparable measure of risk denominated in currency units (given a confidence- level and time horizon) across all instruments, business desks, and business lines. Managers think of risk in terms of dollars of loss. That makes a VaR measure more effective, such as communicating the risk exposure to senior management. Most VaR implementations consider linear exposure to market risks. As we will see later, linear approximations can be inaccurate for non-linear positions such as options. However, the rapid growth of derivative markets reveals that derivatives are widely used for hedging and speculation and non-linear exposures to risks are common characteristics in portfolios. Therefore, it is important to be aware of the implication of non-linear risk exposure. This paper attempts to answer the question of whether standard parametric VaR approaches, such as the delta-normal and the delta-gamma approach are appropriate for derivative positions such as forwards and options. The outline of the paper is as follows: Chapter 2 presents the underlying risk model which illustrates the set of assumptions about the risk factors as well as derivative pricing models for forwards, futures, and European-style options. Chapter 3 provides an introduction to the VaR methodology. I will also discuss the implicit assumptions of standard VaR models. Chapter 4 discusses the delta-normal method which is based on a first-order Taylor series expansion. I will conclude that the delta-normal method is reliable for portfolios with linear risk exposures but can lead to significant approximation errors for portfolios with non-linear risk exposures. Non-linear positions such as options will be discussed in chapter 5. The delta-gamma method provides a better approx- imation because it models second-order effects. Since the relationship between the value of the position and normally distributed risk factors is non-linear, the distri- bution of the position is non-normal. I will show that this distribution exhibits high skewness and excess kurtosis which makes a VaR calculation using the delta-normal 3 method inaccurate. The main difficulty with delta-gamma methods is the estimation of the quantile of the profit and loss distribution. Different methods for estimating this quantile are presented including the Cornish-Fisher expansion. In addition, a method which matches the known moments of the distribution to one of a family of distributions known as Johnson distributions and using a central Â2-distribution to match the first three moments is presented. Empirical results by Pichler and Selitsch [17] suggest the use of a higher moment Cornish-Fisher expansion. Delta- normal approaches are appropriate for linear derivatives and for positions that reveal only weak curvature. The delta-gamma approach offers a significant improvement over the delta-normal method in the case of non-linear derivatives. In particular, the delta-gamma method, when using a Cornish-Fisher expansion, provides an approxi- mation which is close to results calculated by Monte Carlo Simulation methods but is computationally more efficient. Chapter 2 Background 2.1 Definition of Risk Jorion [12] defines risk as the volatility of unexpected outcomes. Of the many sources of risk a financial institution is exposed to, I will focus on financial risk and, in particular, market risk. According to J.P. Morgan [13, p.2], market risk can be defined as a “risk related to the uncertainty of a financial institution’s earnings on its trading portfolio caused by changes in market conditions such as the price of an asset, interest rates, market volatility, and market liquidity.” Market risk can occur in terms of interest rate, foreign exchange, and equity price risk.
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