This article examines the hedging positions derived from the Black-Scholes(B-S) model
and the GARCH(1,1) models, respectively, when the log returns of underlying asset exhibits
GARCH(1,1) process.
The result shows that Black-Scholes and GARCH options deltas, one of the hedging
parameters, are similar for near-the-money options, and Black-Scholes options delta is
higher then GARCH delta in absolute terms when the options are deep out-of-money, and
Black-Scholes options delta is lower then GARCH delta in absolute terms when the options
are deep in-the-money.
Simulation study of hedging procedure of GARCH(1,1) and B-S models are performed,
which also support the above findings.
1 Introduction..................1
2 Preliminaries.................4
3 Literature Review............11
4 The GARCH(1,1) Model.........15
5 Main Result..................21
6 Simulation Study.............29
7 Conclusion...................36
References.....................37
Figure 1~3.....................39
Table 1........................40
Table 2........................41
Table 3........................42
Table 4........................43
Table 5........................44
Table 6........................45
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4. Chang, Bin [2002], ”Evaluating the Black-Scholes Model and the GARCH Option
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8. Engle, Robert F., Joshua Rosenberg [1995], ”GARCH Gamma”, Journal of Derivatives, Summer 1995, 47-59.
9. Hagerud, Gustaf E. [1996], ”Discrete Time Hedging of OTC Options in a GARCH
Environment: A Simulation Experiment”, Working Paper Series in Economics and
Finance No. 165.
10. Huang, Liu-Yuen [2002], ”Fitting Financial Time Series Data to Heavy Tailed Distribution”,
Department of Applied Mathematics, National Sun Yat-Sen University, Taiwan.
11. Hull and White [1987], ”The Pricing of Options on Assets with Stochastic Volatilities”, Journal of Finance, 42, 281-300.
12. Hull, J.C. [1997], Options, Futures, & Other Derivatives, 4th ed. Prentice-Hall International, Inc.
13. Michael Sabbatini, Oliver Linton [1998], ”A GARCH Model of the Implied Volatility of the Swiss Market Index from Option Prices”, International Journal of Forecasting, 14, 199-213.
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15. Tsay, Ruey S. [2002], Analysis of Financial Time Series, John Wiley & Sons, Inc.
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