Dividend-adjusted Binomial Tree

Q: Adding a continuous dividend yield q > 0 to a binomial tree changes the risk-neutral probability formula by:

Replacing r with r - q. Holding the stock now earns total return r in the risk-neutral world, but part of that return (q) is paid out as a dividend rather than re-invested in the price. The capital appreciation must drift at r - q, so p = (e^(r-q)Δ t - d)/(u - d) (Hull 2022, §17.3).

When the underlying pays dividends, the binomial tree must be re-parameterised. For a continuous dividend yield $q$ , replace the growth rate $r$ with $r - q$ in the risk-neutral probability, $p = (e^{(r-q)\Delta t} - d)/(u - d)$ . For discrete cash dividends, the standard practice is to split the stock into a deterministic dividend stream plus a stochastic component and build the tree on the latter (Hull 2022, §21.3). Dividends reduce expected stock growth, which lowers call prices and raises put prices.

Try it yourself

Binomial option pricing tree

Price each node forward (S₀·uᵏ·dⁿ⁻ᵏ), then fold the payoffs back at the risk-neutral probability p. The asset price sits in each circle; the option value is beneath it in the accent colour.

Steps

Type

Circles show the asset price; the figure beneath, in accent, is the option value. Edges are up-moves (u) and down-moves (d).

Call value today (root)

10.48

Risk-neutral p0.40

1 − p0.60

Discount e⁻ʳ0.98

Spot S₀100.00

Strike K100.00

Up factor u1.20

Down factor d0.90

Rate r / step2.0%

Assumption: r is a continuously compounded per-step rate with one step per period (Δt = 1), so the per-step growth is e^r and the discount factor is e^−r. p = (e^r − d) / (u − d); each node = e^−r·[p·V_up + (1−p)·V_down].

Why it matters

Dividends drain value from the share. A stock paying 3% per year drifts more slowly than a non-paying one because some of the total return is paid out rather than reinvested in the share price. The risk-neutral probability reflects this slower drift. Discrete dividends are messier because the stock drops by the dividend amount on the ex-dividend date, which kinks the tree. Splitting the stock into a deterministic component (the present value of future dividends) plus a stochastic component keeps the tree recombining and tractable.

Before you read on — recall

Adding a continuous dividend yield $q > 0$ to a binomial tree changes the risk-neutral probability formula by:

Formulas

Risk-neutral $p$ with continuous yield

p = \frac{e^{(r-q)\Delta t} - d}{u - d}

Index option (Hull 2022, §17.5)

p = \frac{e^{(r-q)\Delta t} - d}{u - d},\quad q = \text{index dividend yield}

FX option (treats foreign rate as yield)

p = \frac{e^{(r_d - r_f)\Delta t} - d}{u - d}

Worked examples

Scenario

ASX 200 index option with $S_0 = 7500$ index points, $u = 1.2$ , $d = 0.85$ , $r = 0.05$ , dividend yield $q = 0.02$ , $\Delta t = 0.5$ .

Solution

$p = (e^{(0.05 - 0.02) \times 0.5} - 0.85)/(1.2 - 0.85) = (e^{0.015} - 0.85)/0.35 = (1.01511 - 0.85)/0.35 = 0.4718$ . Compared with the no-dividend value of $p$ from the same $u$ , $d$ but $q = 0$ (which would be 0.5009), the dividend yield lowers $p$ , raising the value of puts and lowering the value of calls on the index.

Common mistakes

✗Dividends do not affect option prices. They do. Higher dividends reduce call values (less expected growth) and raise put values (stock falls on ex-date). The Black–Scholes formulas for dividend-paying stocks use $S_0 e^{-qT}$ in place of $S_0$ (Hull 2022, §17.3).
✗Discrete dividends only matter near ex-date. They matter throughout the option's life because the present value of all dividends paid before expiry is what the call holder gives up by not exercising early. This is why discrete dividends can make early exercise of American calls optimal just before ex-date.

Revision bullets

•Continuous yield $q$ : use $r - q$ in $p$ formula
•Discrete dividends: subtract PV of dividends from $S_0$ first
•Dividends lower call values, raise put values
•Can trigger early exercise of American calls before ex-date
•FX options treat foreign rate as a yield

Quick check

Adding a continuous dividend yield $q > 0$ to a binomial tree changes the risk-neutral probability formula by:

Connected topics

2-Step Tree BSM + Divs

More in Binomial Trees

1-Step Tree Risk-Neutral p Backward Ind.Early Exercise Delta Δ Delta Hedge

In learning paths

Pricing Models Path

Sources

Hull (2022), §17.3 and §21.3
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Covers both the continuous-yield adjustment to risk-neutral $p$ and the discrete-dividend tree construction that keeps the lattice recombining.
Cox, Ross & Rubinstein (1979)
Cox, John C., Stephen A. Ross and Mark Rubinstein. "Option Pricing: A Simplified Approach." Journal of Financial Economics 7(3), 1979, pp. 229–263.
Original binomial paper that extends the basic model to dividend-paying stocks via an adjusted drift.
ASX 200 Index Options
Australian Securities Exchange. Index options on the S&P/ASX 200. ASX product page, accessed 2026.
ASX-listed European index options are priced with a dividend-yield adjustment to the underlying index level.

How to cite this page

Dr. Phil's Quant Lab. (2026). Dividend-adjusted Binomial Tree. Derivatives Atlas. https://phucnguyenvan.com/concept/dividend-adjusted-tree