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

• DOI: http://doi.org/10.1111/jori.12087
• Published date: 01 March 2017
• Date: 01 March 2017

©2015 The Journal of Risk and Insurance. Vol.84, No. 1, 209–238 (2017).

DOI: 10.1111/jori.12087

Pricing and Hedging Variable Annuities in a L´evy

Market: A Risk Management Perspective

Abdou K´

elani

Franc¸ois Quittard-Pinon

Abstract

Pricing and hedging life insurance contracts with minimum guarantees are

major areas of concern for insurersand researchers. In this article, we propose

a unified framework for pricing, hedging, and assessing the risk embedded

in the guarantees offered by VariableAnnuities in a L´

evy market. Weaddress

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. Wealso consider

hedging strategies that use a portfolio of standard options. For assessing

risk, we introduce an accumulated discounted loss function that takes mor-

tality,transaction costs, and fees into account. We apply our resulting unified

framework to the Minimum Guarantees for Maturity Benefit, Death Bene-

fit, and Accumulation Benefit contracts. Weillustrate the whole method with

CGMY and Kou processes, which proveto offer a realistic modeling for finan-

cial 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.

Introduction

Variable Annuities (VAs) are life insurance contracts linked to financial markets.

Companies design VAs so that policyholders can benefit from favorable movements

in the markets, yet remain protected when prices plummet. They are often grouped

under the acronym GMxB. The G refers to guarantee, M to minimum, B to benefit,

and x to a particular contract type: for example, M refers to maturity, D to death, A to

accumulation, I to income, and W to withdrawal (see Hardy, 2003; Kalberer and Ravin-

dran, 2009). These contracts have had great success in the United States, the United

Abdou K´

elani and Franc¸ois Quittard-Pinon are at the CEFRA (Center For Financial Risk Analy-

sis), EMLYONBusiness School, France. K ´

elani and Quittard-Pinon can be contacted via e-mail:

abdou.kelani@gmail.com, quittardpinon@em-lyon.com. The authors would like to thank the

participants of the AFFI 2013, in particular Yue-Kuen Kwok, and the participants of the MAF

2014, in particular Pietro Millossovich. Weare indebted to two anonymous referees whose de-

tailed comments have improved the paper significantly. In addition, we would like to thank

Areski Cousin, Jean-Paul Décamps, Monique Jeanblanc, and Olivier Le Courtois for many

fruitful discussions and the editor Keith J. Crocker for his constructive recommendations.

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210 The Journal of Risk and Insurance

Kingdom, Japan, and, to a lesser extent, in continental Europe. This success stems from

the fact that VAsoffer the opportunity to manage long-term savings and to potentially

provide postretirement income. Specific tax advantages also provide incentives for

investing in these products. VAs, together with similar contracts such as Equity Index

Annuities (EIAs), represent a huge market. Taking into account the world’s aging

population, and the contribution that VAs can bring to the financing of postretire-

ment income, it is likely that these markets will continue to expand over the coming

years.

The pricing, hedging, and risk management of these products are the main areas of

concern for insurers and represent a challenge for researchers. VAs involve different

risks whose methods of interaction are unknown (Bacinello et al., 2011). Furthermore,

these contracts contain nonstandard embedded options, for example, the Guarantee

Minimum for Death Benefit (GMDB), in which the optional rider is an option whose

expiration date is random (the policyholder’s death). Milevsky and Posner (2001)

name these options Titanic, give the fair fees for this type of contract, and compare

their calculations with the fees actually charged by insurers.

A key assumption in the theoretical analysis of these contracts concerns the model-

ing of financial prices, and particularly the dynamics of the value of the referenced

financial portfolio or market index. Many studies assume a Gaussian hypothesis for

financial returns. However, the research now widely accepts that the distribution of

these returns is not normal. Asymmetries and leptokurticities have to be taken into

account. To cope with these stylized facts, the research has developed models with

regime-switching schemes (Hardy, 2003), or with stochastic volatility (Heston, 1993).

Furthermore, recent studies, such as Cont (2001) and A¨

ıt-Sahalia and Jacod (2009),

show that jumps are present in financial prices. The embedded options in VAs are

very sensitive to the tails of the underlying distribution, so jumps and/or stochas-

tic volatility have to be taken into account. Pricing and hedging thus become more

complex and must be performed in incomplete markets. This is not an easy task. The

aim of this article is to suggest a general methodology for both pricing and hedging a

subclass of VAs in a general L´

evy context so that insurers can evaluate fair mortality

and expense fees, and can hedge the risk contained in the embedded option. This

analysis is developed from an operational risk management point of view.

Pricing these contracts is a familiar theme in the literature. After the seminal article of

Brennan and Schwartz (1976) using the principle of arbitrage in continuous time, an

impressive flow of articles has been devoted to the pricing of life insurance contracts,

in particular VAsand EIAs. Many risks have been considered—mortality risk, market

risk, stochastic interest rate risk, default risk, and surrender risk—in the usual Black

and Scholes economy, and more recently in a L´

evy environment (Kassberger, Kiesel,

and Liebmann, 2008; Ballotta, 2005). The issue of hedging, extremely important for

risk management purposes, has been far less frequently analyzed in the insurance lit-

erature than the issue of valuation. The theoretical analysis of hedging in incomplete

markets is a well-established area of study in the world of mathematical finance.

Many solutions have been proposed, for example, the well-known methods of

total risk minimization or local risk minimization (Schweizer, 1991, 2001); however,