Generic Pricing of FX, Inflation and Stock Options Under Stochastic Interest Rates and Stochastic Volatility

Abstract

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 structure between 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 effective control variate for the general Heston (1993) model.

Pricing of foreign exchange options under the Heston stochastic volatility model and CIR interest rates

“…It is 30 notable that their model is based on Gaussian processes and thus it enjoys analytical tractability, even in the most general case of a full correlation structure. By contrast, when the squared volatility is driven by the CIR process and the interest rate is driven either by the Vasicek (1977) or the Cox et al 35 (1985) process, a full correlation structure leads to the intractability of equity options even under a partial correlation of the driving factors, as has been documented by, among others, Van Haastrecht and Pelsser (2011) and Grzelak and Oosterlee (2011), who examined, in particular, the Heston/Vasicek and 40 Heston/CIR hybrid models (see also Grzelak et al (2012), who study the Schöbel-Zhu/Hull-White and Heston/Hull-White models for equity derivatives).…”

Inflation Derivatives Under Inflation Target Regimes

“…In fact, markets for inflation derivatives exhibit a strong volatility skew or smile, implying that log index returns deviate from normality and suggesting the use of skewed and fat‐tailed distributions. The assumption of log‐normality is relaxed by introducing stochastic volatility (Mercurio ; van Haastrecht and Pelsser, ) or jump diffusion processes (Hinnerich, ). Although such models provide an elegant and computationally effective way for pricing inflation derivatives, they lack a deeper economic underpinning.…”

Interest rate risk in long‐dated commodity options positions: To hedge or not to hedge?

“…It has been shown that interest rate risk is relevant to equity derivatives with longer maturities and models with stochastic interest rates tend to improve pricing and hedging of such contracts (Bakshi, Cao, & Chen, ). A representative literature on spot option pricing models with stochastic interest rates includes Rabinovitch (), Amin and Jarrow (), Scott (), Kim and Kunitomo (), and van Haastrecht and Pelsser (). Fabozzi, Paletta, Stanescu, and Tunaru () propose a quasianalytic method for pricing and hedging long‐dated equity options.…”