# Extending risk neutral measure to insurance/mortality filtration

In insurance mathematics, one often models the underlying of an insurance policy with a Black Scholes model on a filtered probability space $(\Omega,\mathbb{Q},\mathcal{F},\mathbb{F}=(\mathcal{F}_{t}))$ with $\mathbb{Q}$ being the risk-neutral measure. For example, one now would like to value a pure endowment product, i.e. at a fixed time $T$ the prelevant stock price $S(T)$ is paid out if the policyholder is alive at time $T$ else there is no payout. Additionally, such products often contain some further financial guarantee, i.e. minimal payout. But this is not essential for my questions.

Therefore, one also has to model mortality. For this, one often considers $T_{x}$ the future lifespan of a $x$-year old and sets $\mathcal{G}_{t}:=\sigma(\mathbb{1}_{\{T_{x}\leq s \}}\vert s\leq t)$, which defines the "insurance filtration" $\mathbb{G}=(\mathcal{G}_{t})$. Then the one considers the enlarged filtration $\mathbb{H}=\mathbb{F}\vee\mathbb{G}$ and works on the filtered space $(\Omega,\mathbb{Q},\mathcal{F},\mathbb{H})$. The survival probability is then defined as $p_{x+t}(t,T):=\mathbb{Q}(T_{x}>T\vert \mathcal{H}_{t})$.

Unfortunately, I never found a general good and formal account of this. Many things seem implicitly assumed. My questions:

1. Are there any good references for this general modeling approach?
2. Why can the risk-neutral measure even be extended to the enlarged space and in particular be used to measure mortality?
3. Or, are they any special conditions needed?
4. If we assume that mortality is independent from financial markets, do we need any of this anyways?

Thanks alot for the help.

• You might want to take a look at @Quantuple's answer and comments to my question, Change of measure's impact on parameter value. It is a different setting $-$ I model mortgage prepayment instead of mortality $-$ but the takeaway is that, mortality being a stochastic process exogenous to your economy, changing measures does not impact your mortality process distribution. Maybe somebody else can give further insight. – Daneel Olivaw Apr 9 '17 at 13:18
• Alternatively, if you include an insurance-linked derivative in your model $-$ e.g. a mortality or longevity swap $-$ then you must ensure that your insurance product is a martingale under the relevant pricing measure you are using and you might have to modify the dynamics of $T_x$. – Daneel Olivaw Apr 9 '17 at 13:31

I'm Phd student in insurance mathematics so I think I have a good position to answer your question.

As you said, many insurance products have a financial component and an actuarial component, i.e. some financial guarantees upon the survival or death of the insured.

Most insurance papers price this type of payoff via risk-neutral valuation. If you assume that the equity and mortality risks are independent under Q, you can decompose the payoff into a product of risk-neutral expectations. Hence, you decompose the problem into a pure financial payoff and a pure actuarial payoff.

For the insurance payoff, most papers assume via diverse arguments that the dynamics under P and Q are the same and hence, you can use your historical data for the actuarial part.

For the financial part, if you assume that the financial market is complete (like B-S economy), you have a unique Q and pricing is straightforward. Otherwise, you can "pick" one decent Q via a popular criterion (like the Esscher risk-neutral measure or the minimal risk-neutral measure)

Notice that all these things work well because you assume that financial and actuarial risks are independent under Q. However, independence under P DOES NOT imply independence under Q.

If you are interested in this type of research, I recommend you to read papers from my supervisor on his website:

Especially, the paper: On the (in-)dependence between financial and actuarial risks. (2013)