Valuing benefit guarantees using simulation: formula sheet

Key formulas

GMDB payoff

Death benefit = max(F, G) = F + max(G − F, 0)

F is fund value at death and G is the guaranteed amount. The extra cost to the insurer is a put-type payoff max(G − F, 0).

GMAB payoff at maturity

Maturity benefit = max(F_T, G) = F_T + max(G − F_T, 0)

Paid only if the policyholder survives to T. Looks like a European put with strike G on the fund, weighted by the survival probability.

Cost of the guarantee as a put option

Guarantee cost at 0 = E_Q[ v(T) × max(G − F_T, 0) ] × survival probability

E_Q is expectation under the risk-neutral measure, with v(T) the discount factor. Survival is assumed independent of the fund here. For GMDB, weight by the probability of death at each time.

Rolled-up guarantee

G_t = P × (1 + i)^t

P is the premium and i the guaranteed roll-up rate. Check whether the roll-up is simple or compound, and whether it stops at a given age.

Ratchet (highest anniversary value) guarantee

G_t = max(G_(t−1), F_t)

The guarantee steps up to the fund value on each anniversary and never falls.

GMWB basis

Annual withdrawal = w × P, for 1/w years in total

Total withdrawals guaranteed equal P. Fund exhaustion before then leaves the insurer paying the remainder.

Benefit with a maturity guarantee

Benefit = max(F(T), G) = F(T) + max(G − F(T), 0)

The second term is the payoff of a put option with strike G on the fund.

Guarantee cost as a put

Guarantee cost = P(0) = value of a European put on F with strike G, expiry T

Value it under the risk-neutral measure, not the real-world measure.

Black-Scholes put (no charge or dividend)

P = G e^(−rT) N(−d2) − S0 N(−d1)

S0 is the initial fund value, r the continuously compounded risk-free rate, T the term in years, N the standard normal distribution function.

d1 and d2

d1 = [ln(S0 ÷ G) + (r + σ²÷2) T] ÷ (σ√T); d2 = d1 − σ√T

σ is the annual volatility of the fund. Use the same d1 and d2 in both terms of the put.

Black-Scholes put with a continuous charge or yield q

P = G e^(−rT) N(−d2) − S0 e^(−qT) N(−d1), with d1 = [ln(S0 ÷ G) + (r − q + σ²÷2) T] ÷ (σ√T)

If a charge of q per year is taken continuously from the fund, it acts like a dividend yield. d2 = d1 − σ√T as before.

Put-call parity (European, no yield)

C − P = S0 − G e^(−rT)

Use it to get the put from the call, or to check your answer.

Normal symmetry

N(−x) = 1 − N(x)

Needed to read N(−d1) and N(−d2) from tables that show positive values.

Continuous rate from annual effective rate

r = ln(1 + i)

Convert before using the formula if the question gives an effective annual rate.

Lognormal model (GBM) solution

S(t) = S(0) × exp[(μ − σ²/2)t + σ√t × Z], Z ~ N(0,1)

Real-world uses μ as the expected return. Under risk-neutral valuation replace μ by r (net of any continuous dividend or charge).

Step-by-step simulation

S(t+h) = S(t) × exp[(μ − σ²/2)h + σ√h × Z]

Use a fresh independent Z for each step. h is the time step in years.

Mean and variance of log return

ln[S(t+h)/S(t)] ~ N[(μ − σ²/2)h, σ²h]

Expected price is E[S(t)] = S(0) × exp(μt).

Inverse transform for normals

Z = Φ⁻¹(U), U ~ U(0,1)

Each uniform gives one standard normal.

Box-Muller

Z₁ = √(−2 ln U₁) × cos(2πU₂); Z₂ = √(−2 ln U₁) × sin(2πU₂)

Two independent uniforms give two independent standard normals.

Regime-switching transition

P(next state = j | current state = i) = pᵢⱼ, with Σⱼ pᵢⱼ = 1

Each row of the transition matrix sums to 1. Draw a uniform to pick the next state.

Monte Carlo estimate and standard error

Estimate = (1/N) Σ v^T × payoffᵢ; SE = s ÷ √N

s is the sample standard deviation of discounted payoffs. To halve the error you need four times the paths.

Guarantee payoff (maturity guarantee)

Payoff = max(G − F(T), 0)

G is the guaranteed amount, F(T) the fund value at maturity. Add deaths or surrenders if the benefit is paid on those events.

Discounted payoff on path i

Xᵢ = vᵢ × Payoffᵢ

Under a risk-neutral basis, vᵢ = e^(−rT) with r the risk-free force of interest, if rates are constant.

Monte Carlo cost estimate

Cost ≈ X̄ = (1/N) Σ Xᵢ

Mean of the N discounted payoffs. Include any probability of the benefit being paid (survival, in-force) in Xᵢ.

Standard error of the estimate

SE = s ÷ √N

s is the sample standard deviation of the Xᵢ. Use it to give a confidence interval X̄ ± 1.96 × SE (approximate 95%).

Lognormal fund step (risk-neutral)

S(t+h) = S(t) × exp[(r − σ²/2)h + σ√h × Z], Z ~ N(0,1)

Real-world version replaces r with the real-world drift μ. Charges reduce the fund as per the contract.

Value at Risk

VaRα = the α-quantile of the loss distribution

With N sorted outcomes, it is roughly the outcome at position αN in ascending order of loss.

Conditional Tail Expectation

CTE(α) = average of the worst (1 − α) × N outcomes

For CTE(99) with N = 1,000, average the worst 10 outcomes. CTE(α) ≥ VaRα for continuous losses.

Risk-neutral price

V₀ = E_Q[ e^(−rT) × Payoff ]

Expectation under the risk-neutral measure Q, with constant risk-free rate r. With a stochastic rate, discount along each path.

Risk-neutral drift for a share

dS = r S dt + σ S dW_Q

Drift is r (less dividend yield q if there is one). Volatility σ is unchanged.

Real-world drift for a share

dS = μ S dt + σ S dW_P

μ = r + risk premium. Same σ as above.

Market price of risk

λ = (μ − r) ÷ σ

Links the two measures: dW_Q = dW_P + λ dt.

Martingale test

E[ e^(−rt) S_t ] = S₀

Check by simulation: average the discounted value across paths and compare with S₀ using the standard error.

Standard error of simulated mean

SE = s ÷ √N

s = sample standard deviation of discounted values, N = number of scenarios. A result within about 2 SE of S₀ passes.

Sample mean estimate

x̄ = (1 ÷ n) Σ xᵢ

Estimate of the expected discounted payoff, taken as the guarantee cost.

Standard error

SE = s ÷ √n, where s² = Σ(xᵢ − x̄)² ÷ (n − 1)

Use the sample standard deviation of the n independent observations.

Approximate 95% confidence interval

x̄ ± 1.96 × s ÷ √n

Valid by the central limit theorem when n is large.

Simulations needed for a target standard error

n = (z × s ÷ ε)², with z = 1.96 for 95% confidence and ε the allowed half-width

Needs an estimate of s from a pilot run.

Antithetic estimator

Pair average Wᵢ = (Xᵢ + X′ᵢ) ÷ 2; Var(W) = (σ² ÷ 2)(1 + ρ)

ρ is the correlation between the two payoffs in a pair. Gain needs ρ < 0.

Control variate estimator

X* = X − b(Y − E[Y]); Var(X*) = Var(X) − 2b Cov(X,Y) + b² Var(Y)

E[Y] must be known exactly.

Optimal control coefficient

b = Cov(X, Y) ÷ Var(Y); Var(X*) = Var(X)(1 − ρ²)

ρ is the correlation between X and Y. Higher |ρ| gives a larger reduction.

Nested simulation cost

Total inner runs = N × M

N outer scenarios, M inner simulations per scenario.

Delta

Δ = ∂V ÷ ∂S

V is the value of the guarantee or option, S is the underlying price. For a European put under Black-Scholes, Δ = N(d1) − 1, which is negative.

Gamma

Γ = ∂²V ÷ ∂S²

Measures how quickly delta changes. High gamma means delta hedges need frequent rebalancing.

Vega

ν = ∂V ÷ ∂σ

Sensitivity to volatility. Delta hedging does not remove it.

Delta-hedged position

Insurer's position delta = −Δp × quantity (positive, since Δp < 0). Hedge holding in underlying = Δp × quantity (negative, i.e. short)

Δp is the put delta per unit, which is negative. The insurer is short the put, so its position delta is positive. To offset it, the insurer holds a short position and sells |Δp| × quantity units of the underlying.

Hedging error (discrete)

Long option, delta-hedged: P&L over an interval ≈ ½ × Γ × (ΔS)² + θ × Δt, with θ negative. Insurer, short the guarantee: P&L ≈ −½ × Γ × (ΔS)² − θ × Δt

Γ and θ are those of the put. The insurer is short gamma, so large price moves cause losses, which are offset by time decay gains. Use it qualitatively unless the question gives values.

Black-Scholes d1

d1 = [ln(S ÷ K) + (r + σ²÷2)T] ÷ (σ√T)

Needed for N(d1) when computing delta. Assumes no dividends and constant r and σ.

Quick revision

• A guarantee is an option written by the insurer, so its cost is the value of that option.
• A maturity guarantee on a fund is like a put option on the fund with strike equal to the guaranteed amount.
• Black-Scholes assumes lognormal prices, constant volatility and a constant risk-free rate, with no jumps.
• Monte Carlo estimate = average of discounted payoffs over all simulated scenarios.
• Standard error of the estimate = sample standard deviation of payoffs ÷ √N, so quadrupling N halves it.
• Risk-neutral valuation uses the risk-free rate as drift and as the discount rate, and gives a market-consistent price.
• Real-world valuation uses realistic expected returns and is used for risk, capital and projections.
• Do not discount real-world payoffs at the risk-free rate and call the result a market price.
• Antithetic variates pair each draw Z with −Z to reduce variance when the payoff is monotonic in Z.
• Control variates use a related quantity with a known value, such as a Black-Scholes price, to reduce error.
• Nested simulation runs real-world outer scenarios with risk-neutral inner scenarios, so it is costly.
• Hedging with delta reduces exposure but leaves basis, rebalancing and model risks.

Common mistakes

• Treating the guarantee cost as the full benefit G rather than the shortfall G − F. Fix: The insurer already holds the fund F. Its extra cost is max(G − F, 0), which is zero when F ≥ G.
• Mixing up GMAB and GMWB. Fix: GMAB guarantees a value at a specified date. GMWB guarantees withdrawals over time, even after the fund is exhausted.
• Using the expected real-world return of the fund instead of the risk-free rate in the formula. Fix: Black-Scholes is risk-neutral valuation. The only drift that matters is r (less q). The real-world return affects the probability of a claim, not the price of the hedge.
• Treating the guarantee as a call option, or valuing max(F(T), G) as if it were only the put. Fix: The shortfall max(G − F(T), 0) is a put. Sketch the payoff first. The total benefit is the fund plus the put, so the put alone is the guarantee cost.
• Forgetting the −σ²/2 term in the exponent. Fix: Always write (μ − σ²/2) in the exponent. Check that E[S(t)] = S(0)e^(μt).
• Using real-world μ when valuing a guarantee for pricing. Fix: For market-consistent cost, set drift to the risk-free rate and discount at the same rate. Use real-world drift only for real-world projections.
• Averaging fund values instead of payoffs Fix: Because the payoff is non-linear, apply max(G − F, 0) on every path first, then average. Average of payoffs does not equal payoff of the average.
• Mixing risk-neutral drift with real-world discounting Fix: State the basis first. Risk-neutral: drift r, discount at r. Real-world: real drift, then a stated discount rate.
• Saying risk-neutral scenarios predict that assets earn the risk-free rate. Fix: Say it is a pricing tool. Expected discounted payoff under Q equals the market price. It does not forecast returns.
• Using real-world scenarios to price a guarantee that is hedged in the market. Fix: Price for hedging with market-consistent scenarios. Use real-world only for capital and risk questions, or for unhedged risk.

Exam tips

• Always define F and G before using them, and state the trigger event. Examiners reward clear notation.
• Link every guarantee to an option payoff. Say 'put option on the fund' explicitly in written answers.
• For comparison questions, give the trigger, the guaranteed quantity and the risk to the insurer for each guarantee in a short list.
• In simulation questions, state the model for the fund, the measure used (risk-neutral for pricing), the number of scenarios and how mortality is handled.
• Mention both market risks and policyholder behaviour risks when asked why guarantees are risky.
• Always start by writing the payoff as fund plus max(G − F(T), 0) and name it a European put. This earns easy marks in both MCQs and written answers.
• Show d1, d2, both N values and the final formula line. In a written paper, a correct method with a small table error still scores well.
• State the risk-neutral assumption in words. Examiners often ask why the real-world return is not used.