Generic pricing of foreign exchange, inflation and stock options under stochastic interest rates and stochastic volatility
In this paper we deal with the pricing of stock, foreign exchange and inflation options under stochastic interest rates and stochastic volatility. We consider a foreign exchange framework for the pricing inflation-indexed options in which the valuation of stock and foreign exchange options can be treated as a nested case. We assume multi-factor Gaussian rates for both the nominal (domestic) as the real (foreign) economy, which economies (currencies) can be exchanged against each other by means of the inflation index (exchange rate) which is driven by log-normal dynamics with a stochastic volatility component. Furthermore we allow for a general correlation structurebetween the drivers of the volatility, the inflation index, the nominal and the real rates. We derive explicit option pricing formulas for various securities, like vanilla call/put options, forward starting options, inflation-indexed swaps and inflation caps/floors. All these options can be valued in closed-form under Schöbel-Zhu (1999) stochastic volatility, whereas we device an (Monte Carlo) approximation in the form of a very eective control variate for the general Heston (1993) model.