Economic Modelling · Valuing benefit guarantees using simulation

Guarantees as Options and Black-Scholes Valuation

Q: Why is a maturity guarantee a put option and not a call?

The insurer pays only when the fund ends below the guaranteed sum G, and pays the shortfall G − F(T). That payoff is max(G − F(T), 0), which is a put on the fund with strike G. The fund value itself is paid in any case.

Q: Do I use the real-world expected return in the Black-Scholes guarantee value?

No. Black-Scholes is a risk-neutral valuation. The fund grows at the risk-free rate (less any charge) and cash flows are discounted at the risk-free rate. The real-world return matters for projecting claims, not for pricing the hedge.

Q: When should I use simulation instead of the closed-form formula?

Use simulation when the guarantee depends on the path of the fund, when volatility or interest rates are stochastic, when there are jumps, or when lapses and charges depend on fund performance. Closed form needs lognormal prices, constant parameters and a payoff only at maturity.

Q: How do annual charges affect the value of the guarantee?

A charge taken from the fund lowers its growth, so you treat it as a continuous yield q. The put then uses r − q in d1 and the factor e^(−qT) on the initial fund value. A higher charge makes the fund more likely to fall short, so the guarantee is worth more.

Updated 11 October 2026 · Fact-checked

A maturity guarantee pays the shortfall if the fund ends below the guaranteed sum, so it is a European put option on the fund with strike equal to the guarantee. If the fund is lognormal with constant interest rate and volatility, value it with the Black-Scholes put formula. Otherwise use simulation.

Understand Guarantees as Options and Black-Scholes Valuation

Take a unit-linked policy. You pay a premium, it is invested in a fund, and at maturity you get the fund value. The insurer adds a maturity guarantee: you get at least a stated sum G. Your benefit at time T is therefore the larger of F(T) and G.

Write that as F(T) + max(G − F(T), 0). The first part is just the fund, which the policyholder already owns. The second part is the guarantee cost. It is exactly the payoff of a European put option on the fund with strike G and expiry T. The insurer has sold this put to the policyholder.

If the fund follows geometric Brownian motion (lognormal prices) and the risk-free rate and volatility are constant, the put has a closed-form value from the Black-Scholes formula. The value is the price of the guarantee under risk-neutral valuation. The fund grows at the risk-free rate (less any charge taken from it) and you discount at the risk-free rate. The real-world expected return does not enter.

The closed form is fast and exact, but only if the assumptions hold. It breaks down when guarantees depend on the path of the fund (ratchets, lookbacks), when volatility or rates vary, when charges or policyholder behaviour depend on the fund, or when there are jumps. Then you use Monte Carlo simulation, which can handle any payoff but gives an estimate with sampling error. The Black-Scholes value is also a useful check on a simulation.

For a policy where the benefit is paid only on survival to T, the cost per policy issued is the put value times the probability of surviving to T. This assumes mortality is independent of the fund. Per policy in force at time T, use the put value directly.

Key rules to remember

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.

How to solve Guarantees as Options and Black-Scholes Valuation questions

Use this method for any question on valuing a guarantee as an option. State assumptions as you go, since examiners give marks for them.

1Write the benefit as max(F(T), G) and split it into the fund plus max(G − F(T), 0). Say that the extra part is a European put with strike G and expiry T.
2List the inputs: S0 (fund invested at outset, after any initial charges), G, T, r, σ and any continuous charge q. Convert an effective annual rate to r = ln(1 + i).
3State the assumptions: fund lognormal, constant r and σ, guarantee only at maturity, risk-neutral valuation, no dependence on mortality or lapses unless told.
4Calculate d1 = [ln(S0 ÷ G) + (r − q + σ²÷2) T] ÷ (σ√T), then d2 = d1 − σ√T.
5Find N(−d1) and N(−d2) using N(−x) = 1 − N(x) and the normal tables.
6Compute P = G e^(−rT) N(−d2) − S0 e^(−qT) N(−d1). Scale to the amount invested, for example per ₹1,00,000.
7If the question involves survival, multiply by the survival probability to T. Then comment: is the closed form suitable, or would simulation be needed because of path dependence, stochastic volatility or dynamic behaviour?

Quickest way: Fast route for a maturity guarantee put

When to use it: Use when the exam gives S0, G, T, r and σ for a single maturity guarantee and asks for the cost. It also suits MCQs that need only a rough size or direction.

Write S0, G, r, q, σ, T on one line. If S0 = G, then ln(S0 ÷ G) = 0 and d1 = (r − q + σ²÷2) √T ÷ σ.
Get d1, then d2 = d1 − σ√T. Check that d2 < d1.
Read N(−d1) and N(−d2) from tables. N(−d2) is always larger than N(−d1).
Put = G e^(−rT) N(−d2) − S0 e^(−qT) N(−d1). Check it is positive and below G e^(−rT).
For direction questions, use the facts: the put rises with σ, rises with G, rises with a higher charge q, and falls with a higher r.

Common mistakes in Guarantees as Options and Black-Scholes Valuation

Using the expected real-world return of the fund instead of the risk-free rate in the formula.
Students think the guarantee depends on how the fund is expected to perform.
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.
The word option is linked to calls, and the payoff shape is not drawn.
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.
Swapping N(d) and N(−d), or using d1 in the strike term and d2 in the fund term.
The call formula is memorised and the put is adapted wrongly.
Fix: For the put: strike term uses N(−d2), fund term uses N(−d1). Check with put-call parity or the bound 0 < P < G e^(−rT).
Ignoring the annual management charge, or adding it to r.
Charges look like an expense item separate from the guarantee.
Fix: A charge taken from the fund reduces its growth, so treat it as a continuous yield q. Use r − q in d1 and the factor e^(−qT) on S0. This raises the put value.
Using an effective annual rate directly as r in e^(−rT).
The question says 5% and students plug in 0.05 without checking the basis.
Fix: If 5% is an effective annual rate, r = ln(1.05). If the question says continuously compounded, use 0.05. State which you used.
Applying the closed form where it is not valid, such as for ratchet guarantees, stochastic volatility, or lapses linked to fund performance.
The formula is easy, so students use it for every case.
Fix: Check each assumption. Where the payoff is path-dependent or behaviour is dynamic, say that simulation (with risk-neutral scenarios) is the appropriate approach. Black-Scholes can still serve as a benchmark.

Worked examples

Example 1

A single-premium unit-linked policy invests ₹1,00,000 in a fund. At the end of 5 years the policyholder receives the greater of the fund value and ₹1,00,000. Ignore charges, mortality and lapses. The risk-free rate is 5% per year continuously compounded and the fund volatility is 20% per year. Value the guarantee using Black-Scholes. Use normal table values to 2 decimal places of d and interpolate.

Show the solution

The benefit is max(F(5), 1,00,000) = F(5) + max(1,00,000 − F(5), 0). The guarantee is a European put on the fund with S0 = 1,00,000, G = 1,00,000, T = 5, r = 0.05, σ = 0.20.
Assumptions: fund is lognormal, r and σ are constant, valuation is risk-neutral, and the guarantee applies only at maturity.
d1 = [ln(1) + (0.05 + 0.02) × 5] ÷ (0.2 × √5) = 0.35 ÷ 0.44721 = 0.7826.
d2 = 0.7826 − 0.4472 = 0.3354.
N(0.7826) ≈ 0.7831, so N(−d1) ≈ 0.2169. N(0.3354) ≈ 0.6314, so N(−d2) ≈ 0.3686.
e^(−0.25) = 0.77880.
P = 1,00,000 × 0.77880 × 0.3686 − 1,00,000 × 0.2169 = 28,710 − 21,690 ≈ 7,020.
Check: 0 < 7,020 < G e^(−rT) = 77,880. The value is reasonable.

Answer: The guarantee is worth about ₹7,000 (about 7% of the amount invested), as the put value on ₹1,00,000, allowing for rounding in tables.

Example 2

Repeat the previous valuation, but now an annual management charge is taken continuously from the fund and is equivalent to q = 1% per year. All other inputs are unchanged. Explain why the answer changes and state one reason why the closed form might still be unsuitable in practice.

Show the solution

The charge reduces fund growth, so treat it as a continuous yield q = 0.01. Use the put with yield: P = G e^(−rT) N(−d2) − S0 e^(−qT) N(−d1).
d1 = [ln(1) + (0.05 − 0.01 + 0.02) × 5] ÷ (0.2 × √5) = 0.30 ÷ 0.44721 = 0.6708.
d2 = 0.6708 − 0.4472 = 0.2236.
N(0.6708) ≈ 0.7489, so N(−d1) ≈ 0.2511. N(0.2236) ≈ 0.5885, so N(−d2) ≈ 0.4115.
e^(−0.25) = 0.77880 and e^(−0.05) = 0.95123.
First term: 1,00,000 × 0.77880 × 0.4115 = 32,048.
Second term: 1,00,000 × 0.95123 × 0.2511 = 23,890.
P = 32,048 − 23,890 ≈ 8,160.
Why it rises: the fund grows more slowly after charges, so it is more likely to finish below ₹1,00,000, and the put is worth more.
Why closed form may fail: the charge may not be a fixed continuous percentage, volatility is not constant in practice, and policyholder lapses may depend on fund performance. Simulation can model these.

Answer: The guarantee is worth about ₹8,160 per ₹1,00,000 invested, up from about ₹7,000, because the charge lowers the fund growth. Simulation is needed where charges, volatility or behaviour are not constant.

Exam tips

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.
When asked to compare closed form and simulation, give a balanced answer: speed and exactness against flexibility for path-dependent payoffs, stochastic parameters and dynamic policyholder behaviour. Mention simulation error.
In a computer-based paper, check your simulated average discounted payoff against the Black-Scholes value for the plain maturity guarantee. A large gap means a coding or scenario error.

Practice questions from Valuing benefit guarantees using simulation