Constructing Out-of-the-Money Longevity Hedges Using Parametric Mortality Indexes

Abstract

Proposed by Chan et al. (2014), parametric mortality indexes (i.e., indexes created using the time-varying parameters in a suitable stochastic mortality model) can be used to develop tradable mortality-linked derivatives such as K-forwards. Compared to existing indexes such as the LLMA's LifeMetrics, parametric mortality indexes are richer in information content, allowing the market to better concentrate liquidity. In this paper, we further study this concept in several aspects. First, we consider options written on parametric mortality indexes. Such options enable hedgers to create out-of-the-money longevity hedges, which, compared to at-the-money-hedges created with q-/K-forwards, may better meet hedgers' need for protection against downside risk. Second, using the properties of the time-series processes for the parametric mortality indexes, we derive analytical risk-neutral pricing formulas for K-forwards and options. In addition to convenience, the analytical pricing formulas remove the need for computationally intensive nested simulations that are entailed in, for example, the calculation of the hedging instruments' values when a dynamic hedge is adjusted. Finally, we construct static and dynamic Greek hedging strategies using K-forwards and options, and demonstrate empirically the conditions under which an out-of-the-money hedge is more economically justifiable than an at-the-money one.

Pricing longevity derivatives via Fourier transforms

“…The valuation of longevity-linked derivatives with non-linear payoff structures (e.g., longevity options) has received little attention in the literature compared to that of longevity derivatives with a linear payoff but is attracting increasing interest. Some exceptions are Lin and Cox (2007), who study the pricing of a longevity call option linked to a population longevity index for older ages, Cui (2008), who discusses the valuation of longevity options (floors and caps) using the Equivalent Utility Pricing Principle, Dawson et al (2010), who derive closed-form Black-Scholes-Merton-type prices for European swaptions, Boyer and Stentoft (2013), who price European and American type survivor options using a risk neutral simulation approach, Wang and Yang (2013), who price survivor floors using an extension of the Lee-Carter model, Yueh et al (2016), who develop valuation models for mortality calls and puts-employing the jump-diffusion model developed by Cox et al (2006)-, Bravo andde Freitas (2018) and Bravo (2019Bravo ( , 2020, who discuss the valuation of longevity options embedded in longevity-linked life annuities, Fung et al (2019), who derive closed-form solutions for the price of longevity caps under a two-factor Gaussian mortality model resembling the Black-Scholes formula for option pricing when the underlying stock price follows a geometric Brownian motion, and Li et al (2019) as well as Cairns and Boukfaoui (2019), who discuss the valuation of K-options and call-spreads, respectively.…”

The impact of long memory in mortality differentials on index-based longevity hedges

“…An index-based longevity hedge is constructed using one or more instruments whose payoffs are linked to a mortality index which tracks the mortality experience of a certain reference population, typically a national population. Researchers have studied index-based longevity hedges from different angles, ranging from the development of effective hedging strategies to the quantification of residual risks that still remains when a properly calibrated index-based longevity hedge is in place [Dahl et al (2008); Cairns (2011); Coughlan et al (2011); Li and Hardy (2011); Cairns et al (2014); Zhou and Li (2017); Li et al (2021)].…”

Pricing Survivor Bonds with Affine-Jump Diffusion Stochastic Mortality Models

“…In recent years, several capital-market-based solutions for mortality and longevity risk management have been proposed and, some, successfully launched. They include insurance securitization, mortality-or longevity-linked securities such as CAT mortality bonds, survivor/longevity bonds [5], and derivatives with both linear and nonlinear payoff structures, e.g., Index-based Capital-market longevity swaps [6][7], q-forwards [8], S-forwards, K-forwards [9], mortality options, survivor options [7], survivor swaptions [10], K-options [11] and call-spreads [12].…”