Risk-Neutral Pricing of Participation Benefits in Insurance Contracts
Summary
The document considers an insurance contract with a fixed payment and a discretionary additional benefit tied to investment growth exceeding liability growth. It asks whether the additional payoff can be valued like a call option using Black–Scholes–Merton. The response explains that insurers have used derivative-pricing models, including Black–Scholes-type approaches, to value guarantees and optional benefits in a market-consistent way. Replication with traded instruments can support hedging as well as valuation.
The discussion connects this practice to variable annuities, whose benefits may depend on rates or equity performance, and to risk-neutral methods used in market-consistent measures of insurer value. It does not derive a pricing formula or establish that the basic Black–Scholes assumptions fit any particular policy. Applying the framework requires specifying the contract payoff and a suitable model for the underlying assets and liabilities; discretionary terms, non-traded exposures, and model assumptions may affect whether replication is feasible. The response is a high-level overview rather than a worked valuation.
Key ideas
- A benefit triggered when assets outperform liabilities can have option-like features.
- Derivative-pricing models can value insurance guarantees on a market-consistent basis.
- Replication with traded instruments can connect valuation to hedging.
- Variable annuities are an example of insurance products with embedded optionality.
- A Black–Scholes-style approach requires a suitable payoff definition and model assumptions.
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Full text
# Risk-neutral pricing the "un"guaranteed benefits of an insurance policy # Risk-neutral pricing the "un"guaranteed benefits of an insurance policy 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 by Daneel Olivaw (score 3) https://quant.stackexchange.com/a/46312 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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