calibration_final - The Dangers of Calibration and Hedging...

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Unformatted text preview: The Dangers of Calibration and Hedging the Greeks in Option Pricing Arkadev Chatterjea Robert A. Jarrow Kenan-Flagler Business School Johnson Graduate School of Management The University of North Carolina Cornell University Chapel Hill, NC 27599 Ithaca, NY 14853 email: chatterjea@cornell.edu email: raj15@cornell.edu Abstract This paper shows the dangers of calibration and hedging the greeks in option pricing theory. We focus on two problems that aren&t adequately addressed in existing textbooks. First, the inappropriate use of vega hedging. And second, the errors that a misspecied but calibrated model introduces into delta hedging. A simple example illustrates our insights in a transparent manner. KEY WORDS: option pricing, calibration, vega hedging, delta hedging, gamma hedging, model error. 1 1 INTRODUCTION When teaching the Black Scholes Merton model, the standard textbook pre- sentation (see [Chance, 1998], [Hull, 2006], [Jarrow and Turnbull, 2000], [Kolb, 2003], [Whaley, 2006]) includes calibration - implicit volatility esti- mation - and the procedure for hedging the "greeks" - delta, gamma, and vega. In light of the controversy surrounding the use of models and their role in the current credit crisis, it is important to teach both the advantages and disadvantages of these modeling procedures. Although the advantages are always discussed, none of these textbooks adequately discuss the disad- vantages. This is the purpose of this paper. Using the Black Scholes Merton model as the vehicle for our presenta- tion, we show that the danger of calibration is that it can lead to the false impression that a model which correctly matches market prices is useful for hedging. This is not true in general. Second, we show that hedging a pa- rameter of the model - the greek vega - is nonsensical. Consequently, vega hedging should not be taught even in introductory courses. An outline for this paper is as follows. Section 2 presents the Black Scholes Merton model. Sections 3 and 4 discuss calibration and hedging the greeks, respectively. Section 5 presents an example, while section 6 concludes 2 the paper. 2 THE BLACK SCHOLES MERTON MODEL To facilitate presentation, we concentrate on the Black Scholes Merton for- mula for pricing European call options. As will become clear, however, our arguments are more general and they apply to other derivatives and deriv- ative pricing models. To start, we need some notation. Let us consider a stock with price S t at time t . For simplicity, we assume that the stock pays no dividends. We let the stock price follow a geometric Brownian motion with constant volatility & . Last, we assume that the default free spot rate is the constant r ....
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calibration_final - The Dangers of Calibration and Hedging...

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