FRM Exam Part I · The Black-Scholes-Merton Model

Options on Dividend-Paying Stocks and Other Underlyings

Updated 11 October 2026 · Fact-checked

To price European options on dividend-paying stocks, indices, currencies or futures, keep the Black-Scholes-Merton formula and replace the spot price with S0e^(-qT). For known discrete dividends, subtract their present value from S0. For currencies set q to the foreign rate. For futures options set q equal to r and use F0 as the price.

Understand Options on Dividend-Paying Stocks and Other Underlyings

The basic BSM formula assumes the stock pays no income. A dividend lowers the stock price when it is paid, but the holder of a call does not receive it. So a dividend-paying stock is worth less to an option holder than a non-paying one. The fix is to remove the part of the price that goes to dividends before you apply the formula.

There are two cases. With known discrete dividends, you subtract the present value of all dividends paid before expiry from S0 and use that reduced price as the stock price in the standard formula. With a continuous dividend yield q, you use S0e^(-qT) in place of S0. This second case is Merton's model. It suits stock indices, where many small dividends arrive through the year.

Currencies and futures follow the same logic. A foreign currency earns the foreign risk-free rate rf, which acts like a dividend yield, so you set q = rf. This is the Garman-Kohlhagen model. A futures contract costs nothing to enter, and under the risk-neutral measure the futures price has zero expected growth, so it behaves like an asset with yield q = r. This is Black's model, and you use F0 in place of S0.

The unifying idea is the forward price. For any of these underlyings, F0 = S0e^((r - q)T). If you write the formula in terms of F0, d1 becomes [ln(F0 / K) + σ²T / 2] ÷ (σ√T). This is why one formula covers all four cases. All of these formulas price European options only.

Key formulas to remember

Merton call (continuous yield q): c = S0e^(-qT)N(d1) - Ke^(-rT)N(d2); Use for stock indices, and for stocks with a continuous yield. For a currency, q = rf. For a futures option, S0e^(-qT) becomes F0e^(-rT).
Merton put (continuous yield q): p = Ke^(-rT)N(-d2) - S0e^(-qT)N(-d1); Mirror image of the call. Note the signs on d1 and d2.
d1 and d2 with yield q: d1 = [ln(S0 / K) + (r - q + σ²/2)T] ÷ (σ√T); d2 = d1 - σ√T; The only change from plain BSM is r - q in place of r. Rates and σ are annual and continuously compounded.
Known discrete dividends: Use S* = S0 - PV(dividends) in place of S0, with q = 0; Discount each dividend at the risk-free rate from its payment date to today. Only dividends paid before expiry count. Volatility is that of S*, the stock price net of the dividends.
Currency options (Garman-Kohlhagen): c = S0e^(-rf T)N(d1) - Ke^(-r T)N(d2), with d1 = [ln(S0 / K) + (r - rf + σ²/2)T] ÷ (σ√T); S0 is the spot rate in domestic currency per unit of foreign currency. r is the domestic rate and rf the foreign rate.
Black's model for futures options: c = e^(-rT)[F0N(d1) - KN(d2)]; p = e^(-rT)[KN(-d2) - F0N(-d1)]; d1 = [ln(F0 / K) + σ²T/2] ÷ (σ√T); F0 is the futures price today. The option expires at T, and the formula assumes the futures matures at or after T. σ is the volatility of the futures price.
Put-call parity with yield q: c + Ke^(-rT) = p + S0e^(-qT); For futures options, c + Ke^(-rT) = p + F0e^(-rT). For discrete dividends, replace S0e^(-qT) with S0 - PV(dividends).
Forward price link: F0 = S0e^((r - q)T); Substituting this shows that q = r gives Black's model with F0 as the underlying price.

How to solve Options on Dividend-Paying Stocks and Other Underlyings questions

Use the same routine for every question. The only real decision is how to convert the underlying into an adjusted price and a yield.

1Identify the underlying: stock with known dividends, stock or index with a yield, currency, or futures. Check the option is European.
2Set the adjustment. Known dividends: S* = S0 - PV(dividends), q = 0. Index or yield: q is given. Currency: q = rf. Futures: use F0 and set q = r.
3Write down r, q, σ, T and K in annual, continuously compounded terms. Convert simple rates or percentages if needed.
4Compute d1 = [ln(S / K) + (r - q + σ²/2)T] ÷ (σ√T), where S is the adjusted price (or use F0 with q = r). Then d2 = d1 - σ√T.
5Look up N(d1) and N(d2), or N(-d1) and N(-d2) for a put. Use the normal table or a calculator's normal function.
6Compute the price with the right formula. Discount the strike by e^(-rT) and the underlying by e^(-qT).
7Sanity check: a call must be at least S0e^(-qT) - Ke^(-rT), and a put at least Ke^(-rT) - S0e^(-qT). For futures options, the lower bounds are e^(-rT)(F0 - K) for a call and e^(-rT)(K - F0) for a put. If a put price is asked and you have the call, use put-call parity.

Quickest way: Everything is BSM with one swap

When to use it: Use this when the question gives you a currency, index or futures option and you need a fast answer, or when answer options differ by small amounts.

Map the underlying to a pair (S, q): stock with dividends (S - PV(D), 0), index (S, q), currency (S, rf), futures (F, r).
Plug that pair into the standard formula. Do not memorise four separate formulas.
If the option is at the money forward, meaning K = F0, then ln(F0 / K) = 0 and d1 = σ√T / 2. Then d2 = -d1, and N(d1) - N(d2) = 2N(d1) - 1.
For a futures call struck at F0, price = e^(-rT) × F0 × [2N(d1) - 1]. This saves a step.
Use parity to flip between call and put instead of computing both from scratch.
Eliminate options that break bounds, for example a call priced below S0e^(-qT) - Ke^(-rT).

Common mistakes in Options on Dividend-Paying Stocks and Other Underlyings

Using r - q in d1 but forgetting to multiply S0 by e^(-qT) in the price formula (or the reverse).
Students treat q as a tweak to d1 only.
Fix: q appears in two places: in d1 as r - q, and in the price as S0e^(-qT). Check both before you finish.
Applying F0 = S0e^((r - q)T) again to a futures price that is already given, or leaving out the e^(-rT) factor in Black's formula.
Students confuse Black's model with the forward-price link F0 = S0e^((r - q)T).
Fix: In Black's model, F0 is already a forward price. Use it directly in d1 and discount the whole bracket by e^(-rT).
Swapping the two currency rates, setting q equal to the domestic rate.
Quote conventions are confusing, especially which currency is the unit.
Fix: S0 is in domestic per foreign. r is domestic, q = rf is foreign. The foreign rate is the 'yield' on the underlying asset.
Subtracting dividends at face value instead of present value, or including dividends paid after expiry.
Students rush and subtract the cash amount.
Fix: Discount each dividend from its payment date to today at r. Ignore dividends after expiry.
Applying the formulas to American options without comment.
The same formula appears in the question, so students assume it works.
Fix: These formulas price European options. American calls on dividend-paying stocks may be exercised early just before an ex-dividend date, and American futures options may also be exercised early. Treat the formula as a European value.
Using ln(S0 / K) for a futures option instead of ln(F0 / K).
Habit from plain BSM.
Fix: For futures options the underlying price is F0. If the futures price is given, use it directly in ln(F0 / K). With F0 as the underlying, d1 contains only ln(F0 / K) + σ²T/2, so do not add r or q to it.

Worked examples

Example 1

A European call on a stock index has index level 1,000, strike 1,000, 1 year to expiry. The risk-free rate is 5% and the dividend yield is 2%, both continuously compounded. Volatility is 20%. Given N(0.25) = 0.5987 and N(0.05) = 0.5199, find the call price.

Show the solution

Underlying is an index, so use Merton's model: q = 2%.
d1 = [ln(1000 / 1000) + (0.05 - 0.02 + 0.20²/2) × 1] ÷ (0.20 × 1) = (0 + 0.03 + 0.02) ÷ 0.20 = 0.25.
d2 = 0.25 - 0.20 = 0.05.
S0e^(-qT) = 1000 × e^(-0.02) = 1000 × 0.980199 = 980.199.
Ke^(-rT) = 1000 × e^(-0.05) = 1000 × 0.951229 = 951.229.
c = 980.199 × 0.5987 - 951.229 × 0.5199 = 586.85 - 494.54 = 92.31.

Answer: The call price is about 92.3 index points.

Example 2

A European call on a futures contract has futures price 50, strike 50, 6 months to expiry, risk-free rate 4% (continuously compounded) and volatility 30%. Given N(0.1061) = 0.5422 and N(-0.1061) = 0.4578, find the call price.

Show the solution

Underlying is a futures contract, so use Black's model with F0 = 50.
d1 = [ln(50 / 50) + 0.30² × 0.5 / 2] ÷ (0.30 × √0.5) = 0.0225 ÷ 0.21213 = 0.1061.
d2 = 0.1061 - 0.21213 = -0.1061.
e^(-rT) = e^(-0.04 × 0.5) = e^(-0.02) = 0.980199.
c = e^(-rT)[F0N(d1) - KN(d2)] = 0.980199 × [50 × 0.5422 - 50 × 0.4578].
Bracket = 50 × 0.0844 = 4.22. So c = 0.980199 × 4.22 = 4.136.

Answer: The call price is about 4.14.

Exam tips

Spot the underlying first. Most questions in this topic are testing whether you choose q correctly: yield for an index, rf for a currency, r for a futures contract.
Know that Black's model uses F0 and discounts the entire expression at r. This is a frequent multiple-choice trap.
When a question gives a discrete dividend, check its timing. A dividend after expiry is irrelevant, and you must discount any dividend before expiry.
Use put-call parity with the correct yield term to move between calls and puts. It is faster than computing a second price.
Keep four decimals in N(d1) and N(d2) and compare to the answer options. Options are usually far enough apart that small rounding will not change your choice.

Practice questions from The Black-Scholes-Merton Model

Options on Dividend-Paying Stocks and Other Underlyings in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Actuarial Core Principles (IAI)Extensions: Dividends, Currencies and Other Underlyings

Options on Dividend-Paying Stocks and Other Underlyings: frequently asked questions

What is the Black-Scholes formula with a continuous dividend yield?

It is Merton's model: c = S0e^(-qT)N(d1) - Ke^(-rT)N(d2), with d1 = [ln(S0 / K) + (r - q + σ²/2)T] ÷ (σ√T) and d2 = d1 - σ√T. The put is p = Ke^(-rT)N(-d2) - S0e^(-qT)N(-d1). It is used for European options on stocks with a yield and on stock indices.

How do I adjust Black-Scholes for known discrete dividends?

Subtract the present value of all dividends paid before expiry from the current stock price. Then use that reduced price as S0 in the standard BSM formula with q = 0. Discount each dividend at the risk-free rate from its payment date.

How is the Garman-Kohlhagen model different from Black-Scholes?

It treats the foreign risk-free rate as a continuous dividend yield. So q = rf, r is the domestic rate, and S0 is the spot exchange rate in domestic currency per unit of foreign currency. The structure of the formula is otherwise the same as Merton's model.

How does Black's model price futures options?

It uses the futures price F0 as the underlying and discounts the whole payoff expression at r: c = e^(-rT)[F0N(d1) - KN(d2)], with d1 = [ln(F0 / K) + σ²T/2] ÷ (σ√T). It is the same as Merton's model with q = r.