Value the following options embedded in insurance and annuity contracts, using the Black-Scholes model: guaranteed minimum death benefit, and guaranteed minimum accumulation/maturity benefit.

A variable annuity policyholder hands you $100,000, you invest it in a unitised equity fund, and you promise that on death or maturity she gets back at least the original $100,000. You have just written a put option. Black-Scholes prices it.

The acronym GMMB stands for guaranteed minimum maturity benefit, the insurer's promise that a policyholder reaching the maturity date will receive no less than a stated floor regardless of fund performance. The acronym GMDB stands for guaranteed minimum death benefit, the same kind of floor but triggered by death rather than survival. Both riders exist because policyholders demand downside protection on equity-linked products, and insurers monetize that protection through option-style charges.

A GMMB pays $\max (F_T,G)$ at maturity $T$ if the policyholder is alive. A GMAB (accumulation benefit) is the same idea with periodic resets; for exam pricing you treat it as a GMMB to the next reset date. A GMDB pays $\max (F_{\tau },G_{\tau })$ at the time of death $\tau$, if death occurs within the guarantee term.

Common mistakes

• Discounting at $r-m$ instead of $r$. The charge $m$ belongs in the fund drift only. Discount payoffs at $r$.
• Using real-world drift $\mu$. Embedded option pricing is risk-neutral. Replace $\mu$ with $r$ (less $m$). Candidates who substitute equity expected return inflate $d_1$, $d_2$ and underprice the put.
• Pricing the call instead of the put. The insurer's liability is $\max (G-F,0)$, a put. $\max (F,G)=F+$ put, not call.

Bottom line

• GMMB and GMDB are put options on the policyholder's fund $F_t$ struck at the guarantee $G$, weighted by the relevant survival or death probability.
• Risk-neutral pricing: fund drifts at $r-m$ under $Q$ (where $m$ is the continuous M&E and rider charge deducted from the fund), but payoffs discount at $r$.
• GMMB value: ${}_T{p}_x\cdot [G{e}^{-rT}N(-d_2)-F_0{e}^{-mT}N(-d_1)]$.
• GMDB value: sum of put values across each possible death year, weighted by deferred mortality probabilities ${}_{k|}{q}_x={}_k{p}_x\cdot q_{x+k}$.

Exam shortcut

If $F_0=G$ and $r=m$, then $d_1=\frac{1}{2}\sigma \sqrt{T}$ and $d_2=-\frac{1}{2}\sigma \sqrt{T}$. The put collapses to $F_0{e}^{-rT}[N(\frac{1}{2}\sigma \sqrt{T})-N(-\frac{1}{2}\sigma \sqrt{T})]$, a fast sanity check. DECISION: Maturity benefit → one put, weight by survival. Death benefit → strip of puts, weight by deferred mortality. Accumulation benefit with one reset → treat as GMMB to reset date.

The full lesson (about 5,159 words, 34 min read) adds 5 worked examples, all 10 common mistakes, a self-check, free in the app.

Learning objectives

• 7b

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