Pricing European Options on Deferred Annuities

39 Pages Posted: 15 Feb 2012 Last revised: 4 Feb 2013

See all articles by Jonathan Ziveyi

Jonathan Ziveyi

University of New South Wales; ARC Centre of Excellence in Population Ageing Research and School of Risk & Actuarial Studies

Craig Blackburn

University of New South Wales (UNSW) - School of Actuarial Studies; ARC Centre of Excellence in Population Ageing Research (CEPAR)

Michael Sherris

UNSW Business School

Date Written: February 15, 2012

Abstract

This paper considers the pricing of European call options written on pure endowment and deferred life annuity contracts, also known as guaranteed annuity options. These contracts provide a guaranteed value at the maturity of the option. The contract valuation is dependent on the stochastic interest rate and mortality processes. We assume single-factor stochastic square-root processes for both the interest rate and mortality intensity, with mortality being a time-inhomogeneous process. We then derive the pricing partial differential equation (PDE) and the corresponding transition density PDE for options written on the pure endowment and deferred annuity contracts. The general solution of the pricing PDE is derived as a function of the transition density function. We solve the transition density PDE by first transforming it to a system of characteristic PDEs using Laplace transform techniques and then applying the method of characteristics. Once an explicit expression for the density function is found, we then use sparse grid quadrature techniques to generate European call option prices on the pure endowment and deferred annuity contracts. This approach can easily be generalised to other contracts which are driven by similar stochastic processes presented in this paper. We test the sensitivity of the option prices by varying independent parameters in our model. As option maturity increases, the corresponding option prices significantly increase. The effect of mispricing the guaranteed annuity value is analysed, as is the benefit of replacing the whole-life annuity with a term annuity to remove volatility of the old age population.

Keywords: Mortality risk, Deferred insurance products, European options, Laplace Transforms

JEL Classification: C63, G22, G13, G12

Suggested Citation

Ziveyi, Jonathan and Blackburn, Craig and Sherris, Michael, Pricing European Options on Deferred Annuities (February 15, 2012). UNSW Australian School of Business Research Paper No. AIPAR 2012/1, Available at SSRN: https://ssrn.com/abstract=2005461 or http://dx.doi.org/10.2139/ssrn.2005461

Jonathan Ziveyi

University of New South Wales; ARC Centre of Excellence in Population Ageing Research and School of Risk & Actuarial Studies ( email )

School of Risk and Actuarial Studies
UNSW Business School
Sydney, NSW 2000
Australia
+61 2 9065 8254 (Phone)
+61 2 9385 1883 (Fax)

Craig Blackburn (Contact Author)

University of New South Wales (UNSW) - School of Actuarial Studies ( email )

Sydney, NSW 2052
Australia

ARC Centre of Excellence in Population Ageing Research (CEPAR) ( email )

Level 6, Central Lobby (enter via East Lobby)
Australian School of Business Building
Sydney, New South Wales NSW 2052
Australia

Michael Sherris

UNSW Business School ( email )

Sydney, NSW 2052
Australia

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