Construct a replicating portfolio for the options in 7(b) using delta-hedging. | Free Exam ALTAM Lesson

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 $max (G - F_{τ}, 0)$ at the time of death $τ$ . 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:

$d F_{t} = (r - m) F_{t} d t + σ F_{t} d W_{t}$

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

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

$P_{t} = G e^{- r τ} N (- d_{2}) - F_{t} N (- d_{1})$

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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: $Δ_{t} = - N (- d_{1}) = N (d_{1}) - 1$ , 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.

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