A semi-supervised learning approach for variance reduction in life insurance

Abstract

Monte-Carlo based valuation in life insurance involves the simulation of various components of the balance sheet: portfolios, guarantees, assets mix, market conditions and many other specific risk factors; which can be time-consuming. In view of the time needed to achieve this task, insurers are then facing a trade-off of balancing the number of simulations against the uncertainty surrounding the estimated quantity. In the current paper, we propose a variance-reduction methodology using a machine learning technique. It roots from the unsupervised learning literature in conjunction with the quantization of random processes. The goal is to reduce the number of simulated Brownian paths, using auxiliary scenarios that can be seen as path clusters, which efficiently and accurately approximate the initial ones with regards to an adequate measure of distance. Moreover, we introduce penalty to accommodate for various inputs of the initial conditions of risk factors. By doing so, we implicitly assign labels to the scenarios and thus advocate using a semi-supervised learning to enhance the performance of the scenarios reduction by imposing an additional impurity condition on the clusters based on these labels. This is made possible thanks to a decomposition property of the insurers cash-flows, which allows to disentangle the initial conditions of risk factors from the Brownian motions driving their dynamics. The training of the proposed learning algorithm is based on an adaptation of the well-known k-means algorithm. An intensive numerical study is carried out over a range of simulation setups to compare the performances of the proposed methodology. We show, numerically, that the proposed methodology outperforms some classical variance reduction approach. Also, using a real-life dataset, we show that our methodology outperforms some conventional variance reduction used by life insurance practitioners.

A Appendices

1.1 A.1 Analytical variance of the call option

We will derive the theoretic variance of the cash flow $f(Z,x)=e^{-rT}(x\exp ((r-\sigma ^2/2)T+\sigma \sqrt{T}Z)-C)^+$, where Z is a random Gaussian variable. First, we consider

$$\begin{aligned} \mathbb {E}\left[ f(Z,x)\right]= & {} \mathbb {E}\left[ e^{-rT}(xe^{(r-\sigma ^2/2)T+\sigma \sqrt{T}Z}-C)^+\right] \\= & {} \mathbb {E}\left[ (x\exp (-\sigma ^2 T/2+\sigma ^2\sqrt{T}Z)-C e^{-rT})1_{Z+ d_2\ge 0}\right] , \end{aligned}$$

where $d_2=\left( \log (C/x)+(r-\sigma ^2/2)T\right) /\sigma \sqrt{T}$. Hence, we have the classic Black & Scholes analytic formula

$$\begin{aligned} \mathbb {E}\left[ f(Z,x)\right]= & {} \int _{-\infty }^{d_2}\left( x e^{-\sigma \sqrt{T}y - \sigma ^2 T/2}-e^{-rT}C\right) \dfrac{1}{\sqrt{2\pi }}e^{-y^2/2}dy\\= & {} \int _{-\infty }^{d_2}x e^{-\sigma \sqrt{T}y - \sigma ^2 T/2}\dfrac{1}{\sqrt{2\pi }}e^{-y^2/2}dy -\dfrac{e^{-rT}C}{\sqrt{2\pi }}\int _{-\infty }^{d_2}e^{-y^2/2}dy \end{aligned}$$

Hence, letting $u=y+\sigma \sqrt{T}$, we come up the well-celebrated Black & Scholes formula

$$\begin{aligned} \mathbb {E}\left[ f(Z,x)\right]= & {} \dfrac{x}{\sqrt{2\pi }}\int _{-\infty }^{d_2} e^{-\sigma \sqrt{T}(u-\sigma \sqrt{T}) -\dfrac{\sigma ^2 T}{2}-\dfrac{(u-\sigma \sqrt{T})^2}{2}}du-Ce^{-rT}\phi (d_2)\\= & {} \dfrac{x}{\sqrt{2\pi }}\int _{-\infty }^{d_1}e^{-u^2/2}du-Ce^{-rT}\phi (d_2)\\= & {} x \phi (d_1)-Ce^{-rT}\phi (d_2), \end{aligned}$$

where $d_1=d_2+\sigma \sqrt{T}$, and $\phi$ is the standard normal cumulative distribution function. In order to come with the variance of cash flows, we need to compute the following quantity

$$\begin{aligned} \mathbb {E}\left[ f(Z,x)^2\right]= & {} \mathbb {E}\left[ \left( e^{-rT}(xe^{(r-\sigma ^2/2)T+\sigma \sqrt{T}Z}-C)^+\right) ^2\right] \\= & {} \mathbb {E}\left[ \left( x\exp (-\sigma ^2 T/2+\sigma \sqrt{T}Z)-Ce^{-rT}\right) ^21_{Z+ d_2\ge 0}\right] ,\\= & {} \int _{-\infty }^{d_2}\Big [\left( x^2\exp (-2\sigma ^2 T+2\sigma \sqrt{T}z)+C^2 e^{-2rT}\right. \\&\quad -\left. 2xC\exp (-\sigma ^2 T/2+\sigma \sqrt{T}z-rT)\right) \dfrac{e^{-z^2/2}}{\sqrt{2\pi }} \Big ]dy. \end{aligned}$$

Using the same arguments as the above formula up to a change of variable, we can write

$$\begin{aligned} \mathbb {E}\left[ f(Z,x)^2\right]= & {} x^2e^{\sigma ^2T}\phi (d_3)-2xCe^{-rT}\phi (d_1)+C^2e^{-2rT}\phi (d_2), \end{aligned}$$

where $d_3=d_2+2\sigma \sqrt{T}$. Hence, the variance of the cash-flows can be given explicitly, which will be used to quantify the variance underlying the Monte-Carlo simulations as follows

$$\begin{aligned} \text {Var}\left( \dfrac{1}{M}\sum _{i=1}^{M}f(Z^i, x)\right) =\dfrac{1}{M}\text {Var}(f(Z,x)). \end{aligned}$$

1.2 A.2 Variance reduction with antithetic variables technique

The antithetic variable techinque estimates the expected value of f(Z, x) by

$$\begin{aligned} \dfrac{2}{M}\sum _{i=1}^{M/2}\dfrac{f(Z^i,x)-f(-Z^i,x)}{2}. \end{aligned}$$

The variance of this estimator is given by:

$$\begin{aligned}&\text {Var}\left( \dfrac{2}{M}\sum _{i=1}^{M/2}\dfrac{f(Z^i,x)-f(-Z^i,x)}{2}\right) \\&\quad =\dfrac{2}{M}\text {Var}\left( \dfrac{f(Z^i,x)-f(-Z^i,x)}{2}\right) \\&\quad =\dfrac{1}{M}\Big (\text {Var}(f(Z,x))+\text {Cov}(f(Z,x)-f(-Z,x))\Big ) \end{aligned}$$

where

$$\begin{aligned} \text {Cov}(f(Z,x),f(-Z,x))={\mathbb {E}}[f(Z,x)f(-Z,x)]-{\mathbb {E}}[f(Z,x)]^2, \end{aligned}$$

and

$$\begin{aligned}&{\mathbb {E}}[f(Z,x)f(-Z,x)]\\&\quad ={\mathbb {E}}\left[ e^{-rT} \left( xe^{(r-\sigma ^2/2)T +\sigma \sqrt{T} Z}-C\right) ^+ e^{-rT}\right. \\&\qquad \left. \left( xe^{((r-\sigma ^2/2)T-\sigma \sqrt{T} Z)}-C\right) ^+ \right] \\&\quad =\Big ((x^2 e^{-\sigma ^2 T}+C^2 e^{-2rT} )(\phi (d_2 )\\&\qquad -\phi (-d_2 ))-2xCe^{-rT} (\phi (d_1 )-\phi (d_4 )\Big )1_{d_2>0} \end{aligned}$$

with $d_4=-d_2+\sigma \sqrt{T}$.

1.3 A.3 Analytical variance of the fixed strike lookback call option

Consider a fixed strike Lookback Call Option that starts at $t=0$. Define $X = \max (C,x)$ then

$$\begin{aligned} {\mathbb {E}}[f(W_t,x)]= & {} {\mathbb {E}}\left[ e^{-rT}\left( \max _{0 \le t \le T}(xe^{(r-\sigma ^2/2)t+\sigma W_t})-C\right) ^+\right] \\= & {} e^{-rT}(X-C) + x\phi (d_1) - Xe^{-rT}\phi (d_2) \\&+ xe^{-rT}\dfrac{\sigma ^2}{2r}\left( -\left( \dfrac{x}{X}\right) ^{\dfrac{-2r}{\sigma ^2}}\phi \left( d_1-\dfrac{2r}{\sigma }\sqrt{T}\right) +e^{rT}\phi (d_1)\right) \\ {\mathbb {E}}[f(W_t,x)^2]= & {} e^{-2rT}(X-C)^2 - e^{-rT}xC\left( 2 + \dfrac{\sigma ^2}{r}\right) \phi (d_1) + e^{-2rT}X(2C-X)\phi (d_2) \\&+ e^{-2rT}x\left( X\dfrac{\sigma ^2}{2r+ \sigma ^2}-(X-C)\right) \dfrac{\sigma ^2}{r} \left( \dfrac{x}{X}\right) ^{\dfrac{-2r}{\sigma ^2}}\phi \left( d_1-\dfrac{2r\sqrt{T}}{\sigma }\right) \\&+ x^2e^{\sigma ^2T}\dfrac{2r+3\sigma ^2}{2r+\sigma ^2}\phi \left( d_1+\sigma \sqrt{T}\right) \end{aligned}$$