Pricing long-dated insurance contracts with stochastic interest rates and stochastic volatility |
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| Affiliation: | 1. Netspar/University of Amsterdam, Department of Quantitative Economics, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands;2. Delta Lloyd Insurance, Risk Management, Spaklerweg 4, PO Box 1000, 1000 BA Amsterdam, The Netherlands;3. Cardano, Financial Engineering, 27-28 Clements Lane, London EC4N 7AE, UK;4. Maastricht University, Department of Finance, Department of Quantitative Economics, PO Box 616, 6200 MD Maastricht, The Netherlands;5. ING Life Japan, Variable Annuity Market Risk Management, Japan;1. NYU Stern School of Business, 44W4th St., 9th floor, NY 10012, New York, United States;2. McGill University, Desautels School of Management, Canada;3. Harvard Business School, United States;1. School of Business, Jiangnan University, Wuxi City, Jiangsu 214100, China;2. School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia;3. School of Commerce, University of South Australia, SA, Australia;4. Shanghai Branch, Small Enterprise Finance Department, China Citic Bank, Shanghai 200120, China;1. Department of Statistical and Actuarial Sciences, The University of Western Ontario, London, Ontario, Canada;2. Economics and Management School, Wuhan University, Hubei, China;3. Division of Physical Sciences and Mathematics, University of the Philippines Visayas, Miagao, Iloilo, Philippines;4. Ningbo National Institute of Insurance Development (NIID), Wuhan University, Ningbo, China |
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| Abstract: | 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 Schöbel 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. |
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