Black-Scholes and Embedded Options in Variable Annuities

The Black-Scholes formula gives a closed-form price for a European option on a stock that follows geometric Brownian motion (GBM) with constant volatility. The price equals the discounted expected payoff under the risk-neutral measure, where the asset drifts at the risk-free rate.

The same idea prices guarantees embedded in variable annuities. A GMDB pays the policyholder's heirs the greater of the account value or a guaranteed floor at death, which is a long-dated put option on the account value with a stochastic exercise date. The GMAB, GMIB, and GMWB are structurally similar puts with different exercise rules.

Reserves and capital for these guarantees are computed by Monte Carlo under the risk-neutral measure (for hedging-aligned reserves) or under the real-world measure with a risk margin (for statutory and capital reporting). Both flavors borrow the Black-Scholes pricing logic and add policyholder behavior, mortality, and lapse layers around it.

Under the real-world measure P, the asset price S_t evolves as a GBM with drift mu and volatility sigma. Under the equivalent risk-neutral measure Q, the drift becomes the risk-free rate r (continuous compounding), and the volatility is unchanged.

The log-price log(S_t) is normally distributed under both measures, with mean log(S_0) + (mu - sigma^2/2) t under P or log(S_0) + (r - sigma^2/2) t under Q, and variance sigma^2 t. This lognormal closed form is what makes the Black-Scholes integral computable in closed form.

The transition from P to Q is via the Cameron-Martin-Girsanov change of measure. The exam treatment usually takes the change as given and starts from the risk-neutral GBM directly; the practical relevance is that pricing uses r, while real-world projections for capital and ALM work use mu.

For a European call with strike K and expiry T on an asset paying no dividends, the Black-Scholes price is C = S_0 N(d_1) - K e^{-rT} N(d_2), where N is the standard normal CDF and d_1, d_2 are simple functions of S_0, K, r, sigma, T.

Put-call parity follows from the no-arbitrage relationship C - P = S_0 - K e^{-rT}. The put price is therefore P = K e^{-rT} N(-d_2) - S_0 N(-d_1), reusing the same d_1 and d_2.

Continuous dividends at rate q replace S_0 with S_0 e^{-qT} in the formula and shift the drift in d_1 from r to r - q. For variable-annuity work, the management-fee charge against the account value plays the same role as a continuous dividend.

Under Q, the call price equals the discounted expected payoff: C = e^{-rT} E^Q[(S_T - K)^+]. The integrand is positive only when S_T > K, equivalently when the standard normal Z exceeds -d_2.

Split the expectation into two terms. The first term is e^{-rT} E^Q[S_T 1_{S_T > K}], which equals S_0 N(d_1) after canceling the e^{-rT} against the e^{rT} factor that comes from the lognormal mean. The second term is K e^{-rT} P^Q(S_T > K) = K e^{-rT} N(d_2).

Subtracting gives the formula. The whole derivation reduces to one Gaussian integral, which is why the result is exam-friendly even though the underlying stochastic-calculus machinery is heavy.

Delta is the first derivative of the option price with respect to the underlying. For a call, Delta = N(d_1), which lies in (0, 1) and approaches 1 deep in the money. Delta is the hedge ratio: an issuer short one call hedges by holding Delta shares of the underlying.

Gamma is the second derivative with respect to the underlying. It is highest at the money and near expiry, which is where Delta-hedging becomes costly and option books are most exposed to rebalancing slippage. Gamma is identical for the call and the put.

Vega is the first derivative with respect to volatility. It is positive for both the call and the put because higher volatility widens the payoff distribution, which raises the expected value of a convex payoff.

Theta is the first derivative with respect to time (negative for the call since the option decays as expiry approaches), and Rho is the first derivative with respect to the risk-free rate. Rho matters most for long-dated options like GMxB guarantees, where small interest-rate moves compound into large present-value changes.

A variable annuity (VA) is a unit-linked investment product. The policyholder's account value tracks a chosen sub-account, less management charges. Insurers attach guarantees to make the product attractive, and those guarantees are embedded options on the account value.

GMDB (Guaranteed Minimum Death Benefit) pays the greater of the account value or a guaranteed floor on the policyholder's death. The guarantee is a put option on the account value with a stochastic expiry tied to mortality.

GMAB (Guaranteed Minimum Accumulation Benefit) pays the greater of the account value or a guaranteed floor at a fixed maturity date if the policyholder is alive. This is a long-dated European put on the account value, conditional on survival.

GMIB (Guaranteed Minimum Income Benefit) guarantees a minimum annuitization rate at retirement, equivalent to a put on the value of the annuitization itself. GMWB (Guaranteed Minimum Withdrawal Benefit) guarantees a minimum withdrawal stream regardless of the account value, equivalent to a long sequence of puts whose strikes are the scheduled withdrawals.

GMxB reserves and economic capital usually require two layers of simulation. The outer layer projects the account value and economic environment under the real-world measure for a long horizon. The inner layer prices the embedded option at each outer node under the risk-neutral measure given the outer state.

The inner Monte Carlo can sometimes be replaced by a closed-form Black-Scholes price (when the guarantee is a clean put with a deterministic strike). When that approximation is acceptable, the nested simulation collapses to a single layer of outer scenarios with a closed-form valuation at each node, which cuts the run time by orders of magnitude.

Variance reduction matters for nested simulation. Antithetic outer paths, control variates from a closed-form approximate price, and importance sampling for tail capital are all standard tools. Without them, statutory reserve and capital calculations on a 10000-policy book can take days.

A 60-year-old buys a single-premium VA with initial deposit A_0 = 100000. The sub-account follows GBM with mu = 6 percent and sigma = 16 percent. The management fee is 1.5 percent annually. The guarantee G is the original deposit rolled up at 3 percent annually. The valuation horizon is set to age 90 (T = 30 years) for reserving.

Approximate mortality with q_60+t increasing linearly from 0.012 to 0.13 over the horizon. Under the simplifying assumption that mortality is independent of the sub-account return, the PV of the GMDB guarantee equals the integral of the discounted expected put payoff weighted by the density of the time of death.

Discretize the integral on annual cells. At each year t, compute the at-the-money-forward Black-Scholes put price on A_t with strike G_t, sigma adjusted for the management-fee drag (equivalent to a continuous dividend), and time to expiry of the residual horizon for the surviving cohort. Multiply by the probability of dying in that year and discount to t = 0.

Running this on the parameters above gives a GMDB reserve near 6.5 percent of A_0 for moderate market volatility. The figure is highly sensitive to sigma (Vega is positive) and to the assumed roll-up rate on G.

Same product but the guarantee is GMAB at year 10: the policyholder receives max(A_10, G_10) at maturity if alive. With G_10 = 100000 (zero roll-up), this is a 10-year at-the-money European put on the account value, scaled by the survival probability over 10 years.

Survival from age 60 to age 70 on the assumed mortality is roughly p_10 = 0.93. The Black-Scholes put price on the account value at the start of the contract is BSPut(100000, 100000, r, sigma_net, 10), where sigma_net is the sub-account volatility (the management fee enters as a dividend yield rather than a vol haircut).

With r = 3 percent, sigma = 16 percent, and the management fee acting as q = 1.5 percent, the BS put price on the account value is close to 12500. Scaled by p_10 = 0.93 gives a reserve near 11600, or 11.6 percent of A_0. The result is highly sensitive to sigma and to the assumed dividend (management-fee) treatment.

Black-Scholes prices the option in a constant-volatility GBM world. Real account values have stochastic volatility, jumps, and regime changes; the BS formula is a useful baseline, not a final reserve. Capital frameworks add risk margins precisely because the BS world is an idealization.

The risk-free rate in the formula is the rate that hedges the option. Using the policyholder's discount rate or a corporate-bond yield understates the option value because the hedging argument no longer holds.

The management fee is not just a haircut on returns. It enters the GBM as a continuous dividend (since the fee leaves the account value), which shifts d_1 in the same way a dividend would. Modeling the fee as a vol haircut gives the wrong Greeks.

GMIB and GMWB are not single put options. GMIB depends on the annuitization-rate guarantee, which is itself an interest-rate option. GMWB is a path-dependent sequence of puts whose strikes interact with the withdrawal schedule. Closed-form Black-Scholes is a starting point at best, and full valuation usually requires nested simulation.

Real-world projections cannot price the option. The hedging argument requires the risk-neutral measure. Real-world simulations are the right tool for capital projections and policyholder-behavior modeling, but inner valuations must use Q (or a market-consistent equivalent).

Black-Scholes and embedded options sit at the intersection of life insurance and quantitative finance, which is why they appear across the life-side FSA tracks and on INV-201.

Hull 2018: John C. Hull, Options, Futures, and Other Derivatives, 10th edition, Pearson. Chapters 13 to 17 cover the Black-Scholes-Merton model, the Greeks, and risk-neutral valuation. Chapter 15 derives the formula.

Hardy 2003: Mary R. Hardy, Investment Guarantees: Modeling and Risk Management for Equity-Linked Life Insurance, Wiley. Canonical reference for GMxB modeling, including parametric and regime-switching stock-return models and Monte Carlo reserving.

Dickson, Hardy, and Waters 2020: Actuarial Mathematics for Life Contingent Risks, 3rd edition, Cambridge. Chapters 13 and 14 cover unit-linked contracts and equity-linked guarantees in the ALTAM register.

Klugman, Panjer, and Willmot 2019: Loss Models: From Data to Decisions, 5th edition, Wiley. Section on option pricing inside the risk-modeling chapter gives the actuarial framing used on ASTAM.