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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"?

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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.

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  • $\begingroup$ Agree with your answer, except for the reference to "replication method". Replicating portfolios are used for valuing some insurance contracts (and more often for determining capital), but for valuation of complex contracts monte carlo is at least as common (and analytical formulas for the simple options described by OP). $\endgroup$ – Bram Jun 27 at 22:06

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