Yes We Can (Price Derivatives on Survivor Indices)

• Author: M. Martin Boyer,Lars Stentoft
• Published date: 01 March 2017
• DOI: http://doi.org/10.1111/rmir.12073
• Date: 01 March 2017

Risk Management and Insurance Review

C

Risk Management and Insurance Review, 2017, Vol.20, No. 1, 37-62

DOI: 10.1111/rmir.12073

FEATURE ARTICLE

YES WECAN (PRICE DERIVATIVES ON SURVIVOR INDICES)

M. Martin Boyer

Lars Stentoft

ABSTRACT

We propose a simulation approach to value derivatives when the underlying

dynamics are estimated using the survivor indices directly. Our results show

that survivor forward and swap premiums increase with maturity and with the

market price of risk. Our results also confirm that taking the optionality into

consideration is important from a pricing perspective, for both U.S. women and

men. Wecompare our results to what is obtained using an alternative modeling

approach in which a Lee–Carter model is used to indirectly model the survivor

index. Compared to this method, our estimated premiums and prices arehigher

for all longevity products.Moreover, comparing American-style with European-

style options we find that, although the early exercise option has value when

using survivor indices directly,the relative value of the early exercise option is

significantly less than when the Lee–Carter model is used to indirectly model the

survivor index. It follows that the assumed mortality dynamics have important

implications for the term structure of forward and swap premiums and for the

effect that changes in the market price of risk has on them.

INTRODUCTION

For the life insurance industry,longevity risk is systematic in nature since global increases

in life expectancy (and general mortality improvements) cannot be mitigated by a large

portfolio of individual risks. For the overall capital market, however, longevity risk

is not systematic since it is industry specific. The obvious industry-specific feature of

longevity risk has led researchers and investment bankers to look for a way to transfer

to the capital markets a risk, which is thought of as having little upside. The recent 2015

survey in Insurance: Mathematics and Economics relates the short, albeit volatile, history of

M. Martin Boyer is at the Department of Finance, HEC Montr´

eal (Universit´

edeMontr

´

eal), 3000,

chemin de la Cˆ

ote-Ste-Catherine, Montr´

eal QC, H3T 2A7, Canada and CIRANO. Lars Stentoft

is at the Department of Economics and Department of Statistical and Actuarial Sciences, Social

Science Centre, University of WesternOntario, London, Ontario, N6A 5C2, Canada. The authors

thank Enrico Biffis, AndrewCairns, Am ´

elie Favaro, Joanna Mejza, and participants at the Seventh

International Longevity Risk and Capital Markets Solutions Conference for valuable comments.

This research is financially supported by the Social Science and Humanities Research Council of

Canada, by the Structured Product Institute (IFSID) at HEC Montr´

eal, as well as by CIRANO.

Lars Stentoft gratefully appreciates financial support from CREATES (Center for Research in

Econometric Analysis of Time Series, funded by the Danish National Research Foundation).

37

38 RISK MANAGEMENT AND INSURANCE REVIEW

the longevity risk hedging market since 2005 (see also Tan et al., 2015). Although not all

longevity risk transfers are reported in that survey, one does have the impression that the

market is not picking up steam as much as what was first believed in the early years of

the 21st century. Biffis et al. (2014) report that from 2007 to 2014, there was a grand total

of 25 longevity swap transactions that were publicly announced in the United Kingdom,

for an average of 3 transactions per year (see also Tan et al., 2015).

Enhanced liquidity on the market for hedging longevity risk is a laudable goal. It would

be welfare enhancing if all market participants had access to a method that would allow

them to value survivor derivatives like forwards, swaps, and options of both European

and American styles. It would be even better if the method worked even though market

participants have heterogeneous beliefs concerning mortality risk and the price of risk

in the economy.Boyer and Stentoft (2013) propose such a framework that solely relies on

the ability to simulate from the risk-neutral process.1The authors provide an application

of the methodology to the situation where the mortality distribution follows a Lee–Carter

(1992) model. Although they do not provide any evidence, they argue that the flexibility

of the simulation method is such that it can be applied to the case where mortality does

not follow a Lee–Carter model.

We offer in this article evidence that their simulation approach can be applied to

other types of time-series processes that explain the mortality dynamic. Specifically,

we propose to use a risk-neutral simulation process to value derivatives on a sur-

vivor index akin to the Life & Longevity Markets Association (LLMA) index. To do

so, we consider the situation where the underlying dynamics are estimated using sim-

ple time-series models for the survivor indices “directly.” Conceptually, the pricing of

derivatives based on longevity indices resembles the problems encountered in valuing

inflation-linked derivatives as in Glasserman and Wang (2000), Jarrow and Yildirim

(2003), and Mercurio (2005). We discuss how the method should be implemented, and

provide pricing results for survivor derivatives like forwards, swaps, and options of

both European and American styles. Finally, we compare our results with the special

case of using the Lee–Carter model estimated on the mortality experience of the entire

population.2

In neither the Tanet al. (2015) survey paper nor the Haberman et al. (2014) position paper

is there a mention of the risk-neutral simulation process technique that we propose to

use in this article. In fact, according to our understanding, the longevity risk typology

proposed in Tan et al. does not include at all the technique that we use in this article.

The closest would be to fit our present contribution in the “management and hedging

1Simulation-based valuation of financial derivatives dates back at least to Boyle (1977), and it

has been applied quite extensively within the financial literature. Recent methods for pricing

derivatives with early exercise include Barraquand and Martineau (1995), Broadie and Glasser-

man (1997, 2004), Carriere (1996), Longstaff and Schwartz (2001), Tilley (1993), and Tsitsilis and

Van Roy (2001). For an excellent review ofsimulation techniques, see Glasserman (2003).

2It should be noted that while the payoffs examined in this article are standard, the pricing ma-

chinery we develop can be adapted straightforwardly to any payoff structurethat counterparties

in longevity risk transactions are actually looking for.