Binomial Treesadvanced

Dividend-adjusted Binomial Tree

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.

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.

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 $0.5147 $), the dividend yield **lowers$ 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

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

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