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Discrete time option pricing with flexible volatility estimation. (English) Zbl 0951.91027
It has long been recognized in the option pricing literature that the Black-Scholes prices reveal certain empirical anomalies, e.g. the well-known “smile” effect. In recent years, the most prominent explanation for these anomalies has been stochastic volatility of the underlying asset. Empirically less significant are the effects of trading in discrete time [see P. Bossaerts and P. Hillion, J. Econom. 81, 243-272 (1997; Zbl 0944.62095)] and feedback effects of hedging on the stock price process [see E. Platen and M. Schweizer, Math. Finance 8, 67-84 (1998; Zbl 0908.90026)]. Since their introduction by R. Engle [Econometrica 50, 987-1007 (1982; Zbl 0491.62099)] autoregressive conditional heteroskedaticity models (ARCH) have been successfully applied to financial time series. It is thus natural to consider pricing models for options on assets whose prices follow ARCH-type processes. To this end, J.-C. Duan [Math. Finance 5, 13-32 (1995; Zbl 0866.90031)] established a discrete-time option pricing model for the case of a GARCH volatility process.
The aim of this paper is to show that for a given preference structure the results of J.-C. Duan may be very sensitive to alternative specifications of the volatility process. This concerns the statistical properties of the asset price process under the equivalent martingale measure as well as the simulated prices. The authors extend the results of J.-C. Duan to the case of a threshold GARCH process and provide extensive Monte Carlo simulation results for three typical parameter constellations. In particular, the simulated GARCH option prices is compared with corresponding TGARCH and Black-Scholes prices. In an empirical analysis, it is shown that the observed call option market prices indeed reflect the asymmetry found for the news impact curve of a DAX series.

MSC:
91G20 Derivative securities (option pricing, hedging, etc.)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
62P05 Applications of statistics to actuarial sciences and financial mathematics
62P20 Applications of statistics to economics
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