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Exotic Options with Stochastic Volatilities
Abstract
In 1973, Fischer Black and Myron Scholes made a major breakthrough by developing a model for pricing stock options. In 1997, they were awarded the Nobel Prize in Economics for their outstanding work. The BlackScholes model and its extensions are very popular for pricing many types of options. Individuals, corporations, and many financial institutions use derivatives like options and futures to reduce risk exposures. The BlackScholes model is based on the assumption that the asset volatility is either constant or a known function of time over the life of the option. In 1987, J. Hull and A. White examined the problem of pricing of a European call option on a stock that has a stochastic volatility. They proved that when there is a positive correlation between the stock price and its volatility, outofthe money options are under priced by the BlackScholes formula, while inthemoney options are overpriced. When the correlation is negative, the effect is reversed. Derivatives with more complicated payoffs than the standard European or American call options and put options are sometimes referred to as exotic options. Most exotic options trade in overthe counter market and are designed by financial institutions to meet the requirements of their clients. In Vaidya [6], some computational techniques for pricing standard options and certain exotic options, Lookback options are discussed. The payoff from the Lookback options depend upon the maximum or minimum stock price reached during the life of the option. In 1979, M. Goldman, H. Sosin, and M. Gatto found valuation formulas for European Lookback Call and put options on a stock when the volatility is constant or a known function of time. In Vaidya [7], the problem of pricing European Lookback call options on a nondividend paying stock with stochastic volatility is examined. It turned out that the price of a European lookback call on a stock that has a stochastic volatility has a bias relative to the price of the European lookback call on the stock with constant volatility. When the volatility is positively correlated with the stock price, the price of the European lookback call options is below the price of European lookback call options on the stock with constant volatility. When the volatility is negatively correlated with the stock price, the reverse is true. Another important type of Exotic options is Asian options. The payoff from an Asian option depends upon the average price of the underlying asset during the life of the option. There are no exact formulas for pricing Asian options. In 1993, J. Hull and A. White found efficient procedures for pricing these average options on stocks when the volatility is constant. So this leads to the following question. Research Question: What happens to pricing of European average call options on assets when the volatility is stochastic? Research Method and Conclusion: We will use Monte Carlo Simulation to analyze the problem. We will use the antithetic variable technique and the control variable technique as described in Hammersley and Handscomb [3] and J. Hull and A. White [4]. It turned out that the price of a European average call option on a stock that has a stochastic volatility has a bias relative to the price of the European average call on the stock with constant volatility. When the volatility is positively correlated with the stock price the price of European average call options is below the price of the options on the stock with constant volatility. When the volatility is negatively correlated with the stock price the price of European average call options is above the price of the options on the stock with constant volatility.

