Efficient Model Points Selection in Insurance by Parallel Global Optimization Using Multi CPU and Multi GPU

Abstract

In the insurance sector, Asset Liability Management refers to the joint management of the assets and liabilities of a company. The liabilities mainly consist of the policies portfolios of the insurance company, which usually contain a large amount of policies. In the article, the authors mainly develop a highly efficient automatic generation of model points portfolios to represent much larger real policies portfolios. The obtained model points portfolio must retain the market risk properties of the initial portfolio. For this purpose, the authors propose a risk measure that incorporates the uncertain evolution of interest rates to the portfolios of life insurance policies, following Ferri (Optimal model points portfolio in life, 2019, arXiv:1808.00866). This problem can be formulated as a minimization problem that has to be solved using global numerical optimization algorithms. The cost functional measures an appropriate distance between the original and the model point portfolios. In order to solve this problem in a reasonable computing time, sequential implementations become prohibitive, so that the authors speed up the computations by developing a high performance computing framework that uses hybrid architectures, which consist of multi CPUs together with accelerators (multi GPUs). Thus, in graphic processor units (GPUs) the evaluation of the cost function is parallelized, which requires a Monte Carlo method. For the optimization problem, the authors compare a metaheuristic stochastic differential evolution algorithm with a multi path variant of hybrid global optimization Basin Hopping algorithms, which combines Simulated Annealing with gradient local searchers (Ferreiro et al. in Appl Math Comput 356:282–298, 2019a). Both global optimizers are parallelized in a multi CPU together with a multi GPU setting.

Appendices

Appendix 1: LIBOR Market Model

In this section, following Brigo and Mercurio (2006) we describe the risk-free dynamics of the discounted bond price, when considering the LIBOR Market Model governing the time evolution of the forward rates.

Let N be a positive integer and hence define the finite set $\mathcal {T}=\lbrace T_0,T_1,\ldots ,T_N\rbrace$ to be a fixed tenor structure, with $T_0=1$ and $T_0< T_1< \cdots < T_N$, so that $T_n$ corresponds to a specific maturity time. For $n=1,\ldots ,N$, let $\tau _n = T_n-T_{n-1}$ be the corresponding accruals. Moreover, we set $\mathcal {I}=(0,1)$ to be the unit interval on the real line, which corresponds to the period of one year.

We use $B_n(t)$ to denote the risk-neutral discounted price at time $t\in \mathcal {I}$ of a (zero-coupon) bond expiring at the tenor date $T_n$, for any $n=0,\ldots ,N$. Moreover, we denote by $F_n(t)$ the value at time $t\in \mathcal {I}$ of the LIBOR forward rate associated to the accrual period $(T_{n-1},T_n]$, for $n=1,\ldots ,N$. Therefore, $F_n(t)$ satisfies the following condition:

$$\begin{aligned} B_{n}(t)(1+F_n(t)\tau _n)=B_{n-1}(t). \end{aligned}$$

Hence, for any $n=1,\ldots ,N$ we can write

$$\begin{aligned} B_n(t)= B_0(t) \prod _{k=1}^n \frac{1}{1+F_k(t)\tau _k}, \qquad \qquad {\text {for any }} t\in \mathcal {I}. \end{aligned}$$

It is worth to be highlighted that since $t< T_n$, for any $t\in \mathcal {I}$ and $n=0,\ldots ,N$, the price $B_n(t)$ is always well defined.

We will consider the price $B_0(t)$ of the bond expiring at the tenor date $T_0=1$, for $t\in \mathcal {I}$, as the reference numeraire process. Hence, we denote by $\mathbb {Q}$ the forward measure related to $T_0$, i.e., the martingale measure associated to the numeraire process $B_0(t)$, for $t\in \mathcal {I}$.

Next, we fix a N-dimensional Wiener process $W(t)=(W_1(t),\ldots ,W_N(t))$, for $t\in \mathcal {I}$, defined on the suitable complete probability space $(\varOmega ,\mathscr {F},\mathbb {Q})$, and we write $\varrho =(\varrho _{nk})_{nk}$ to denote the corresponding (positive defined) correlation matrix, i.e.,

$$\begin{aligned} dW_n(t)dW_k(t) = \varrho _{nk} dt. \end{aligned}$$

In particular, we shall assume constant correlation coefficients given by the usual parameterization:

$$\begin{aligned} \varrho _{nk} = \exp ( - \beta \mid T_n-T_k \mid ), \end{aligned}$$

with $\beta =0.01$ in the numerical examples. Note that these coefficients will correspond to the correlation between LIBOR forward rates.

For any given $h=(h_1,\ldots ,h_N) \in {\mathbb R}$, we define the following norm:

$$\begin{aligned} \Vert h \Vert _W = \bigg \lbrace \sum _{n,k=1}^N \varrho _{nk}h_nh_k \bigg \rbrace ^{1/2}. \end{aligned}$$

For each $n=1,\ldots ,N$, let $\sigma _n(t)$, for $t\in \mathcal {I}$, be a given deterministic function, representing the volatility of $F_n$. According to the LIBOR Market Model, given any fixed $n=1,\ldots ,N$, the corresponding forward rate $F_n$ is a martingale with respect the risk-neutral measure induced by the numeraire $B_n$. When we consider the same measure $\mathbb {Q}$ (associated to the numeraire $B_0$) to write the dynamics of all $F_n$, then Girsanov theorem implies that the risk-neutral dynamics of the process $F_n(t)$, for $t\in \mathcal {I}$ is given by

$$\begin{aligned} dF_n(t)= \mu _n(t)dt + \sigma _n(t)F_n(t)dW_n(t), \end{aligned}$$

(7)

jointly with some given initial condition $F_n(0)$, where, at any time $t\in \mathcal {I}$, the drift component $\mu _n(t)$ is completely determined by following identity:

$$\begin{aligned} \mu _n(t)= \sigma _n(t)F_n(t)\sum _{k=1}^n \frac{\varrho _{nk}\tau _k \sigma _k(t)F_k(t)}{1+F_k(t)\tau _k}. \end{aligned}$$

Concerning the modeling of volatilities, in this article we choose the widely used parameterization:

$$\begin{aligned} \sigma _n(t)=[a+b(T_n-t)] \exp {[(T_n-t)]}+d \end{aligned}$$

Furthermore, in the numerical examples we have chosen the constant parameters: $a=0.07, \, b=0.2,\, c=0.6$ and $d=0.075$.

For any fixed $t\in \mathcal {I}$, set $\mu (t)=(\mu _1(t),\ldots ,\mu _N(t))$ and thus define $\varSigma (t)$ to be the matrix whose components are given by

$$\begin{aligned} \varSigma _{nk}(t)= \sigma _n(t)F_n(t)\delta _{nk},\quad {\text {for any }} n,k=1,\ldots ,N, \end{aligned}$$

(8)

where $\delta _{nk}$ denotes the Kronecker delta. Moreover, we shall write $\varSigma _n(t)$ to denote the nth row of the matrix $\varSigma (t)$, for any $n=1,\ldots ,N$. Then, when setting $F(t)=(F_1(t),\ldots ,F_N(t))$, we may regard (7) as a N-dimensional dynamics by means of the following compact form notation:

$$\begin{aligned} dF(t)=\mu (t)dt + \varSigma (t)dW(t), \end{aligned}$$

(9)

jointly with the initial condition $F(0)=(F_1(0),\ldots ,F_N(0))$.

Concerning the tenor structure of the LIBOR model, in all the article we consider 100 tenors, with maturities ranging from 1 to 100 and initial rates given by $F_1(0)=0.01, \, F_2(0)=0.02, \, F_3(0)=0.03, \,F_4(0)=0.04$ and $F_n(0)=0.05$, for $n\ge 5$.

Moreover, for any $n=1,\ldots ,N$ we shall write

$$\begin{aligned} \tilde{B}_n(t) = \frac{B_n(t)}{B_0(t)},\quad {\text {for any }} t\in \mathcal {I}, \end{aligned}$$

to denote the discounted price processes associated to the bond expiring at the tenor date $T_n$.

The following result provides the risk-free dynamics for the discounted price of any bond expiring at some tenor date in $\mathcal {T}$. For any $n=1,\ldots ,N$, the discounted bond price process $\tilde{B}_n(t)$ admits the dynamics

$$\begin{aligned} d\tilde{B}_n(t) = - \varepsilon _n(t)\tilde{B}_n(t)dW(t), \end{aligned}$$

(10)

where we set

$$\begin{aligned} \varepsilon _n(t) = \sum _{k=1}^n \frac{\tau _k}{1+ F_k(t)\tau _k} \varSigma _k(t). \end{aligned}$$

(11)

Appendix 2: Biometric Survival Model

$S(x_i,T_n)$ is the survival index which is understood as the proportion of those individuals labelled by $x_i\in \mathcal {X}$ that survive to the age $x_i+T_n$. The survival index can be computed using past survival tables or from a model.

In our case we use the model:

$$\begin{aligned} S(x_i,T_n)= \exp \bigg \lbrace - \int _1^{T_n} \mu (s,x_i+s)ds \bigg \rbrace , \nonumber \\ {\text {for any }} x_i\in \mathcal {X} {\text { and }} n=1,\ldots ,N, \end{aligned}$$

(12)

which yields the proportion of those individuals with age $x_i$ that survive to the age $x_i+T_n$, and where $\mu (s,x_i+s)$ denotes the force of mortality at time $s\ge 0$ related to the class of individuals labelled by $x_i\in \mathcal {X}$. In this respect, we assume that $\mu (s,x_i+s)$ is a deterministic observable function, for any $x_i\in \mathcal {X}$ and $s\ge 0$.

In particular, we consider a Gompertz-type law modeling the force of mortality (Gompertz 1825), by setting

$$\begin{aligned} \mu (s,x_i + s)= a(s)\exp {\lbrace (x_i+s)b(s)\rbrace },\nonumber \\ \, {\text {for any }} s\ge 1 {\text { and }} i=1,\ldots ,I, \end{aligned}$$

where a(s) and b(s) are deterministic functions for $s\ge 1$, which are considered to be observables. Throughout, we write $S_T$ to denote the derivative of S in its second variable, which is given by

$$\begin{aligned} S_T(x_i,T_n)= - S(x_i,T_n)\mu (T_n,x_i+T_n). \end{aligned}$$

Concerning the force of mortality, we consider the Gompertz type law modeling with constant parameters, i.e., $\mu (x)=a \exp (bx)$, where we will take $a=0.0003$ and $b=0.06$.

Appendix 3: Code Listings

In this section, we include the code snippets, illustrating the GPU parallelization.