Incorporating Continuous Dividends in a Binomial Option Model
Summary
The document considers how to include proportional dividends in a multi-period binomial lattice for an American put. The question lays out a stock-price tree, terminal put payoffs, and backward induction that compares immediate exercise with discounted risk-neutral continuation value. It asks how those calculations should change when dividends are paid during each period.
The response recommends expressing the proportional dividend as a continuous yield and accounting for it alongside the risk-free rate when calculating the risk-neutral probability of an up move. The option tree can then be valued by the same exercise-versus-continuation comparison, using probabilities consistent with the dividend yield. The exchange offers a direction rather than a full derivation or revised formulas, and it does not discuss discrete cash dividends, dividend timing conventions, or numerical validation. Those details matter when adapting the approach to a particular lattice.
Key ideas
- A proportional dividend can be represented as a continuous dividend yield in a binomial model.
- The risk-neutral up probability should reflect both the risk-free rate and the dividend yield.
- American put valuation still compares immediate exercise value with discounted expected continuation value at each node.
- The response does not provide a full recalculation or cover discrete dividend schedules.
Tags
Full text
# How do I incorporate dividends into options pricing # How do I incorporate dividends into options pricing -Hey all, recently I encountered the necessity to incorporate dividends into options pricing. Lets say I have the following american put option: Initial price - 100, T-0.25, Volatility is 30%, Number of periods is 3, Interest rate is 2%. Lets further say the u = 1.07 and d = 0.93458. Given the prior information I calculate that the risk neutral probabilities are q = 63.08% and q-1=36.92% First I built the three period lattice given the parameters above: The stock lattice was computed using the excel formula =B2*(B7 ^(0))*(B8^(0)), where B2 is the initial price of 100 and the B7 and B8 are u and d respectively that are exponentiated by the number of up or down movements at any time. Now the option lattice was obtained from the stock lattice, first using the formula =MAX( H2 - E16, 0) at t=3, where the H2 is the strike price and E16 is the spot price, and then I calculated the t2, t1 and t0 using =MAX(MAX($H$2 - $D$16, 0), ($B$9 *$E$24 + $B$10 *$E$25)/$B$6), which should be read as =MAX(MAX(strike-spot,0), q *priceU + q-1*priceD)/(1+interest rate). How would I modify these formulas in order to incorporate dividends, given that these are paid in each period and are proportional to the price of the stock. Given that the dividend is 1%. ## Answer by AfterWorkGuinness (score 1, accepted) https://quant.stackexchange.com/a/21308 You want to express your dividend as a continuous rate and subtract it from the risk free rate when calculating the probability of an up jump. See this Wikipedia article: https://en.wikipedia.org/wiki/Binomial_options_pricing_model#STEP_3:_Find_Option_value_at_earlier_nodes
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