| Summary: | 碩士 === 東吳大學 === 財務工程與精算數學系 === 103 === The market information of foreign exchange option often presents by implied volatility, and there is often a volatility smile phenomenon on implied volatility. In other words, the volatility of the underlying asset is not constant. Heston (1993) proposed stochastic volatility models to explain this phenomenon. The stochastic volatility indicates that the volatility of the underlying asset is also a stochastic process. In this paper, using the implied volatility data of the European foreign exchange option in actual market with 3 types of numeric integration which is Gauss-Laguerre quadrature, Gauss-Legendre quadrature, and Gauss-Lobatto quadrature and 25 situations of initial parameters calibrate parameters in Heston’s stochastic volatility model, and make the implied volatility of European foreign exchange option under Heston’s stochastic volatility model close to the market actual implied volatility. The implied volatility calibrated by Heston’s stochastic volatility model is called model implied volatility. Besides, comparing model implied volatility to market actual implied volatility, which is called validation. The results show that model implied volatility calibrated by Gauss-Lobatto quadrature is the nearest to approach market actual implied volatility. Furthermore, except short expiry dates, Heston’s stochastic volatility model can describe that the implied volatility of the market has volatility smile phenomenon.
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