European Option And American Option Pricing With Stochastic Interest Rate And Their Applications

In this paper, first, we consider the Exotic options—compound options and reset options— pricing. A generalization of the Geske formula for compound call options is derived in the case of time-dependent volatility and time-dependent interest rate by applying martingale method and change of numeraire or change of probability measure. A more generalization of compound option is also obtained when the interest rate is stochastic. A closed-form pricing formula of compound call option is given in an extended Vasicek's interest rate framework. Either when the interest rate and volatility of the the underlying are all time-varying or when the interest rate is stochastic, the pricing formula for reset call options with predetermined dates is also derived. Numerical research shows that the correlated coefficient between the stock price and interest rate is almost unacted on the price of reset call option with short maturity and Monte Carlo method is inefficient. Monte Carlo method should be only used if there is no closed-formed solution for option pricing;Second, we discuss four cases of the quanto options: an option of floating exchange rate, written on foreign stock in terms of foreign currency, whose value is transformed to domestic currency value by exchange rate at maturity time; an option of pre-agreed exchange rate, whose value is transformed to domestic currency value by pre-specified exchange rate at maturity time; an option written on foreign stock, which strike price is domestic currency value and payoffs at maturity time is by means of domestic currency value of the foreign stock value transformed by temporal exchange rate; an exchange rate option with the change of foreign stock price.