Analytic option pricing and risk measures under a regime-switching generalized hyperbolic model with an application to equity-linked insurance

Abstract

Option pricing and managing equity linked insurance (ELI) require the proper modeling of stock return dynamics. Due to the long duration nature of equity-linked insurance products, a stock return model must be able to deal simultaneously with the preceding stylized facts and the impact of market structure changes. In response, this article proposes stock return dynamics that combine Lévy processes in a regime-switching framework. We focus on a non-Gaussian, generalized hyperbolic distribution. We use the most popular linked equity of ELIs, the S&P 500 index, as an example. The empirical study verifies that the proposed regime-switching generalized hyperbolic (RSGH) model gives the best fit to data. In investigating the effects of stock return modeling on pricing and risk management for financial contracts, we derive the characteristic function, embedded option price, and risk measure of equity-linked insurance analytically. More importantly, we demonstrate that the regime-switching generalized hyperbolic (RSGH) model is realistic and can meet the stylistic facts of stock returns, which in turn can be employed in option pricing and risk management decisions.

Keywords:

• Regime switching model
• Generalized hyperbolic process
• VaR
• CTE
• Equity-linked insurance

JEL Classification:

• C52
• C59
• G12
• G23

Notes

¹ These are also known as segregated fund contracts in Canada, unit-linked insurance in Britain, and variable annuities in the United States.

² We could consider additional states, though at a cost of a vast expansion of the parametric dimension and rising complexity of the subsequent study on the time series of the estimated parameters. In addition, using the likelihood ratio test (Klugman et al. 1998), Hardy (2001) compares regime-switching models with different regimes according to S&P500 data and finds that the model with two regimes provides the best goodness of fit. As a result, we consider only two states in this study.

³ About the ω restriction and shift into the complex domain appear in the work of Chorro et al. (2012) proposition 3.4.

⁴ Or, equivalently, adheres to the GH distribution.

⁵ For more details about regimes and an evaluation of the likelihood function, see Hamilton (1994).

⁶ Hamilton (1994) notes that the Markov chain is ergodic, and as such, the unconditional probabilities π can be used as initial value.

⁷ The parameters of stock return models are estimated on daily basis. In the numerical calculation using the derived formulas, we use the time step is also on daily basis. Assume 21 trading days per month and 252 trading days per year. Thus, for the duration of a three-month and a one-year put option, we use T = 63 and 252 in the formulas.

⁸ Similar to mutual funds, segregated funds charge a management fee for their management services. For example, for Seymour Investment Management the management fee structure for segregated funds is 1% on assets of less than $1 million, 0.75% on assets between $1 million and $2 million, and 0.5% on assets greater than $2 million. Without loss of generality, we assume 1% management fee as an example.

⁹ Because the regulator has required the insurer to measure the risk of such product on an individual policy basis (see CIA 2001, American Academy of Actuaries 2005), for example, Hardy (2001) and Ho et al. (2010) both also demonstrate the calculation of the VaR and CTE of the equity-linked insurance on a single policy.