Risk-neutral pricing the “un”guaranteed benefits of an insurance policy

Question

I'd love to know if the model of Black-Scholes-Merton could be used to anything that replicates the payoff of a call or option, for example:

An insurance contract with participation ( meaning that you can have a right to discretionary benefits, an extra something you can earn provided some conditions).

Imagine an insurance contract in which the policyholder invests 100\$ cash to receive one year later 102\$ cash. However, in a good economic scenario he can get an extra %-gain if the investments outperform the liabilities, i.e if the growth of the invested capital is higher than the growth of the liability.

If I define $S_t-K$ as the payoff of the discretionary benefits to the policy holder ( $S_t$ being the asset growth and $K$ the liability growth say some assumed fixed %) would I be able to use Black-Scholes-Merton formula for a call to get the "expected discretionary benefit"?

Answer 1

Insurers do use derivative pricing models such as Black-Scholes to price the sort of guarantees you describe. As far as I know, this used to be known as the "replication method" in the industry jargon, and it allows insurers to price guarantees in a market-consistent manner, hence enabling them to efficiently hedge them with traded instruments. In particular, I think a few years ago there was much frenzy within the Actuarial community regarding "variable annuities", namely annuities with some sort of optionality tied to rates or equities; models à la Black-Scholes were implemented to price these sort of contracts.

Risk-neutral methods are also significantly used to calculate the Market-Consistent Enterprise Value (MCEV) of an insurer, which nowadays is one of the standard ways to measure the value of an insurance company $-$ see for example this Wikipedia article for a few more details on market-consistent valuation. There is also plenty of material on the internet.