Mathematical and Statistical Methods for Actuarial Sciences and Finance

288 F. Quittard-Pinon and R. Randrianarivony

where(a,x)=

∫∞

x

e−tta−^1 dtis the upper incomplete gamma function whereamust

be positive. This condition entails an upper limit on the possible value of the insurance
risk charge:

<

1

b

. (20)

The Makeham mortality model adds an age-independent component to the Gom-
pertz force of mortality (18) as follows:

λ(x)=A+B.Cx, (21)

whereB> 0 ,C>1andA≥−B.
In this case, a numerical quadrature was used to compute the M&E fees.

3.4 Valuation of the embedded GMDB option

The valuation of this embedded GMDB option is done in two steps:
First, taking the conditional expectation given the policyholder’s remaining life-
time, the option is valued in the context of a jump diffusion process with stochastic
interest rates, with the assumption that the financial asset in the investor subaccount
is correlated to the interest rates.
More precisely, let us recall the embedded GMDB option fair price, as can be
seen in (4):

G()=EQ

[

EQ

[

δT(S 0 egT−ST)+|T=t

]]

.

Using the zero-coupon bond of maturityTas a new numeraire, the inner expectation ́
ITcan be rewritten as:

IT=EQ

[

δT(S 0 egT−ST)+

]

=P( 0 ,T)EQT

[

(K−ST)+

]

.

Then this expectation is computed using an adaptation^2 of the generalised Fourier
transform methodology proposed by Boyarchenko and Levendorskiˇı[3].

4 Empirical study

This section gives a numerical analysis of jumps, stochastic interest rates and mortality
effects. To study the impacts of jumps and interest rates, a numerical analysis is
performed in a first section while a second subsection examines all these risk factors
together.

(^2) A detailed account is available from the authors upon request.