Pricing and Hedging Variable Annuities in a Lévy Market: A Risk Management Perspective

Abstract

Pricing and hedging life insurance contracts with minimum guarantees are major areas of concern for insurers and researchers. In this article, we propose a unified framework for pricing, hedging, and assessing the risk embedded in the guarantees offered by Variable Annuities in a Lévy market. We address these questions from a risk management perspective. This method proves to be fast, accurate, and efficient. For hedging, we use a local risk minimization to provide a concise formula for the optimal hedging ratio. We also consider hedging strategies that use a portfolio of standard options. For assessing risk, we introduce an accumulated discounted loss function that takes mortality, transaction costs, and fees into account. We apply our resulting unified framework to the Minimum Guarantees for Maturity Benefit, Death Benefit, and Accumulation Benefit contracts. We illustrate the whole method with CGMY and Kou processes, which prove to offer a realistic modeling for financial prices. From this application, we draw important practical implications. In particular, we show that the assumption of geometric Brownian motion leads to undervalue the actual economic capital necessary to hedge and gives an illusion of safety.Abdou Kélani and François Quittard-Pinon are at the CEFRA (Center For Financial Risk Analysis), EMLYON Business School, France. Kélani and Quittard-Pinon can be contacted via

Valuation of Guaranteed Unitized Participating Life Insurance under MEGB2 Distribution

“…In this section, we study the optimal hedging decision for this kind of guaranteed participating life insurance product when it implements the dynamic hedging strategy. The problem of hedging in insurance markets has been extensively studied, such as Carr and Picron [42], Møller [43], and Vandaele and Vanmaele [44] et al In this article, following the example of Kélani and Quittard-Pinon [27], we use Delta dynamic optimal hedging strategy. We select riskless asset and the underlying assets of our proposed insurance contract to construct our hedging portfolio.…”

“…Therefore, asymmetries and leptokurtic have to be taken into account. The research has developed models with regime-switching schemes [17] to cope with these stylized facts, or with method of Lévy process, such as Cont [18], Ballotta [19], Coleman et al [20], Siu et al [21], Fard and Siu [22], Yu [23], Huang and Yu [24], Lin et al [25], Orozco-Garcia and Schmeiser [26], and Kélani and Quittard-Pinon [27]. However, Melick and Thomas [28] mentioned that it was more natural to begin with an assumption about the future distribution of the underlying asset, rather than the particular stochastic process by which it evolved and to use option prices to directly recover the parameters of that distribution.…”

Risk-Based Capital for Variable Annuity Under Stochastic Interest Rate

“…Feng and Volmer (2014) build up a framework to evaluate the liability of the GMWB and determine the premium charge from the insurer's side. Kolkiewicz and Liu (2012), Kelani and Quittard-Pinon (2017), Mackay (2014) study the hedging strategy for the VA. Due to a large number of existing VA contracts, the liability is crucial for the insurer to manage the risk and calculate the regulatory capitals.…”

Optimal surrender of guaranteed minimum maturity benefits under stochastic volatility and interest rates

“…The authors note that the valuations vary substantially depending on the modelling framework used. Kélani and Quittard-Pinon (2017) develop a unified valuation framework for pricing and hedging various GMLBs under the Lévy market and note that traditional modelling assumption of using the GBM framework undervalues economic capital required by providers to hedge such guarantees.…”