Pricing long-dated insurance contracts with stochastic interest rates and stochastic volatility

We consider the pricing of long-dated insurance contracts under stochastic interest rates and stochastic volatility. In particular, we focus on the valuation of insurance options with long-term equity or foreign exchange exposures. Our modeling framework extends the stochastic volatility model of Schِbel and Zhu (1999) by including stochastic interest rates. Moreover, we allow all driving model factors to be instantaneously correlated with each other, i.e. we allow for a general correlation structure between the instantaneous interest rates, the volatilities and the underlying stock returns. As insurance products often incorporate long-term exposures, they are typically more sensitive to changes in the interest rates, volatility and currencies. Therefore, having the flexibility to correlate the underlying asset price with both the stochastic volatility and the stochastic interest rates, yields a realistic model which is of practical importance for the pricing and hedging of such long-term contracts. We show that European options, typically used for the calibration of the model to market prices, and forward starting options can be priced efficiently and in closed-form by means of Fourier inversion techniques. We extensively discuss the numerical implementation of these pricing formulas, allowing for a fast and accurate valuation of European and forward starting options. The model will be especially useful for the pricing and risk management of insurance contracts and other exotic derivatives involving long-term maturities.