Construct a replicating portfolio for the options in 7(b) using delta-hedging.

A variable annuity with a Guaranteed Minimum Maturity Benefit (GMMB) is a put option on the policyholder's fund. The insurer sold it; the insurer must hedge it. Delta-hedging tells you exactly how many shares to hold and how much cash to lend, period by period.

A GMMB pays $\max(G - F_T, 0)$ at maturity $T$ if the policyholder is alive. A GMDB pays $\max(G - F_{\tau}, 0)$ at the time of death $\tau$. A GMAB resets the guarantee at fixed dates. Each is a put (or strip of puts) on the separate-account fund $F$, which under the risk-neutral measure follows geometric Brownian motion:

$$dF_t = (r - m) F_t \, dt + \sigma F_t \, dW_t$$

Setting $m = 0$ reduces to standard Black-Scholes. The exam usually states a margin offset already absorbed in $F$.

Let $\tau = T - t$. The risk-neutral put price on a non-dividend underlying is:

$$P_t = G e^{-r\tau} N(-d_2) - F_t N(-d_1)$$

Common mistakes

• Sign on the stock leg. Writing $+N(-d_1)$ shares long instead of short. Put delta is negative; the replicator is SHORT $N(-d_1)$ units.
• Swapping $d_1$ and $d_2$ in the cash leg. Cash multiplier is $N(-d_2)$, not $N(-d_1)$. $d_1$ goes with the stock leg only.
• Survival from issue age. Using ${}_{T}p_x$ at every reset. Reanchor to current age and remaining term: ${}_{T-t}p_{x+t}$.

Bottom line

• GMMB payoff: $\max(G - F_T, 0)$ per surviving policyholder. It is a European put on the separate-account fund with strike $G$.
• Put delta: $\Delta_t = -N(-d_1) = N(d_1) - 1$, always between $-1$ and $0$.
• Replicating portfolio for a long put: short $N(-d_1)$ shares of the fund plus lend $G e^{-r(T-t)} N(-d_2)$ in cash. Total cost equals the Black-Scholes put price.
• Mortality scaling: multiply both the share count and the cash leg by ${}_{T-t}p_{x+t}$ per policy in force at time $t$.

Exam shortcut

Memorize the put replication as two legs paired by their $d$: stock leg pairs with $d_1$, cash leg pairs with $d_2$. Whenever you write $N(-d_1)$ you are writing a share count; whenever you write $N(-d_2)$ you are writing a money multiplier. DECISION: Single guarantee at fixed $T$ means one put scaled by ${}_{T}p_{x}$. Death benefit at random time means a strip of puts, scaling each by the deferred-mortality probability.

The full lesson (about 2,074 words, 14 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 7c

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