Dividend-adjusted BSM

Q: In the dividend-adjusted BSM formula with continuous yield q, S_0 is replaced by

S_0 e^-qT. Continuous dividend yield reduces the stock's risk-neutral growth, so the dividend-stripped price S_0 e^-qT replaces S_0 in the call formula. The discrete-dividend case uses S_0 - PV(dividends), but that is a separate adjustment for known cash payments rather than a continuous yield.

Q: All else equal, raising the continuous dividend yield q on a stock should

Decrease the call price and increase the put price. Dividends leak value out of the stock during the option's life. The call holder does not receive them, so the call falls. The put gains because the dividend-stripped stock is more likely to be below strike at expiry. The effect is visible in the S_0 e^-qT multiplier inside both formulas.

For a stock with continuous dividend yield $q$ , the BSM call becomes $C = S_0 e^{-qT} N(d_1) - K e^{-rT} N(d_2)$ , with $d_1 = [\ln(S_0/K) + (r - q + \sigma^2/2)T] / (\sigma\sqrt{T})$ and $d_2 = d_1 - \sigma\sqrt{T}$ . For discrete dividends, subtract the present value of expected cash dividends from $S_0$ before applying standard BSM (Hull, 2022, §17.2). Merton (1973) introduced the continuous-yield extension.

Try it yourself

Black-Scholes explorer

Move the inputs and watch the call and put reprice. The smooth curve is the Black-Scholes value; the dashed kink is intrinsic value max(S−K, 0). The gap between them is time value (which can go slightly negative for a European put when rates are high).

Curve Black-ScholesDashed IntrinsicTime value A$6.63

Call valueA$6.63

Put valueA$4.65

d₁0.2121

d₂0.0707

N(d₁)0.584

N(d₂)0.528

Put-call parity: C − P = A$1.98 = S − Ke^−rT = A$1.98

Spot S100

Strike K100

Volatility σ20%

Maturity T0.50 yr

Rate r4.0%

Why it matters

Dividends leak value out of the stock during the option's life, so the call is worth less and the put is worth more than in the no-dividend case. The fix is to use the dividend-stripped price $S_0 e^{-qT}$ , which is what a holder of the stock at expiry actually expects to own after dividends have been paid away. The drift in $d_1$ also shifts from $r$ to $r - q$ , since the risk-neutral growth of the stock is reduced by the yield.

Before you read on — recall

In the dividend-adjusted BSM formula with continuous yield $q$ , $S_0$ is replaced by

Formulas

Dividend-adjusted call

C = S_0 e^{-qT} N(d_1) - K e^{-rT} N(d_2)

Same form as the basic call with

S_0

replaced by the dividend-stripped price

S_0 e^{-qT}

Dividend-adjusted $d_1$

d_1 = \frac{\ln(S_0/K) + (r - q + \sigma^2/2)T}{\sigma\sqrt{T}}

q = 0

recovers the basic formula. For discrete cash dividends, replace

S_0

with

S_0 - \text{PV(dividends)}

and use the original

r

Dividend-adjusted put

P = K e^{-rT} N(-d_2) - S_0 e^{-qT} N(-d_1)

Worked examples

Scenario

ASX 200 index option, index level 7800, $K = 7800$ , $r = 4\%$ , $q = 3.5\%$ (ASX dividend yield), $\sigma = 18\%$ , $T = 0.5$ .

Solution

Dividend-stripped index $= 7800 e^{-0.0175} = 7665$ . $d_1 = [\ln(1) + (0.04 - 0.035 + 0.0162)(0.5)] / (0.18 \sqrt{0.5}) = 0.01062 / 0.1273 = 0.0835$ . $d_2 = 0.0835 - 0.1273 = -0.0438$ . $N(d_1) \approx 0.5333$ , $N(d_2) \approx 0.4825$ . $C = 7665 \times 0.5333 - 7800 e^{-0.02} \times 0.4825 \approx 4087 - 3688 \approx 399$ index points.

Scenario

Six-month European call on a CBA share trading at A$110 with one A$2.40 dividend expected in 2 months. $r = 4.5\%$ , $\sigma = 22\%$ , $K =$ A$110, $T = 0.5$ .

Solution

PV of dividend $= 2.40 e^{-0.045 \times 2/12} = 2.382$ . Adjusted spot $S_0' = 110 - 2.382 = 107.618$ . Then apply the basic BSM call with $S_0'$ , $r$ unchanged. The call is worth less than if the dividend were ignored, which matches the intuition that the holder of the option does not collect the cash dividend.

Common mistakes

✗Dividends barely affect option prices. Even a 2-3% yield meaningfully lowers calls and raises puts on longer-dated options. On the ASX, where average dividend yields sit near 4-5%, ignoring $q$ on index options can mis-price by several percent.
✗You add the dividend to $S_0$ in $d_1$ . You subtract dividends. Continuous yield, multiply by $e^{-qT}$ . Discrete cash, subtract PV. The economic logic is the same in both cases, the option holder does not receive the dividend.

Revision bullets

•Replace $S_0$ with $S_0 e^{-qT}$ for continuous yield
•Replace $r$ with $r - q$ in $d_1$
•Discrete dividends, $S_0' = S_0 - \text{PV(divs)}$
•Dividends lower calls and raise puts
•Merton (1973) introduced the continuous-yield extension

Quick check

In the dividend-adjusted BSM formula with continuous yield $q$ , $S_0$ is replaced by

All else equal, raising the continuous dividend yield $q$ on a stock should

Connected topics

Time Value Div. Trees BSM Intro

More in Black-Scholes-Merton

The Greeks Normal Dist.Z-table d1 & d2 BSM Call BSM Put Impl. Vol

In learning paths

Pricing Models Path

Sources

Merton (1973), Bell J. Econ.
Merton, Robert C. "Theory of Rational Option Pricing." Bell Journal of Economics and Management Science 4, no. 1 (Spring 1973): 141-183.
Extends Black-Scholes to options on stocks paying a known continuous dividend yield, the formulation used by most index option pricing today.
Hull (2022), §17.2
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Standard textbook treatment of both continuous-yield and discrete-cash dividend adjustments, with numerical examples.
ASX, Index Options
Australian Securities Exchange. "Index Options on the S&P/ASX 200." ASX Listed Derivatives Brochure, accessed 2026.
Confirms that ASX 200 index options are priced as European options on a dividend-yielding index, the canonical use case for Merton's formula in Australia.

How to cite this page

Dr. Phil's Quant Lab. (2026). Dividend-adjusted BSM. Derivatives Atlas. https://phucnguyenvan.com/concept/bsm-dividends