I consider pricing and risk analysis of derivatives on dividends of the members of equity indices (such as Dow Jones EuroStoxx). There are options but I focus on futures.

  1. What are common stochastic models for dividends that allow pricing of such derivatives respectively risk analysis?
  2. What are practitioner approaches to pricing and risk analysis of dividend futures/options?

Who has references, experiences, comments?

  • 2
    $\begingroup$ This is very broad (you have at least 5 questions). You'd likely get (better) answers if you split this into 2-4 more specific questions. $\endgroup$ Apr 2, 2013 at 2:13
  • $\begingroup$ @JoshuaUlrich You are right - I will split the first 2 points and the second up. Thanks for this suggestion. $\endgroup$
    – Richi Wa
    Apr 2, 2013 at 14:50
  • 1
    $\begingroup$ This is a very interesting topic: Hans Buehler has several papers and presentation on stochastic dividends and dividend derivatives that might be useful (quantitative-research.de). If you have some experience with dividend derivatives perhaps you might help me here: quant.stackexchange.com/questions/7841/… $\endgroup$
    – sets
    Jun 20, 2013 at 13:34
  • $\begingroup$ What exactly do you need. If it is just futures, it is fairly straightforward imho because index dividends futures have the exact same payoff as index dividends swap. E.g. CME, SPXDIV tracks the realized dividends from all components of SPX Index. For swaps you use options Implied divs (usually very noisy) or forecasts (e.g. Bloomberg BDVD). However, market quoted prices of index dividends futures are usually used to imply the index dividends. There is no delta, no gamma, no vega... The only real problem you have with divs is that they are not determined by markets but management. $\endgroup$
    – AKdemy
    Feb 11, 2022 at 21:49
  • $\begingroup$ For large indices that changes little, as long as corona doesn't show up, as you can see by looking at SPXDIV. $\endgroup$
    – AKdemy
    Feb 11, 2022 at 21:50

2 Answers 2


Futures on dividends are directly related to the expected value of the dividends. A model for the dividends is not going to help in pricing of those futures - they are the hedging instruments.

For Options on dividends, early stochastic dividend models assume the dividend to be lognormal, see Geske (1978). This has the neat property that the Black-Scholes formula remains valid for Options on stocks.

More recently, Markov-functional like models, on top of an Orstein-Uhlenbeck dynamic have been applied to represent the smile of the options on dividends, see Guennoun and Henry-Labordère (2019).


I will try to give you a way of "pricing" european call option on a stock that pays divided at t =1. You can extend it to American , more nodes etc In the end these 2 papers Paper 1 and Paper2 are quite good if you want a rigourous treatment using SDE.

Lets define the underlined stock by(price ,up and down factors) $$ S = 100 , u = 1.1, d = 0.9, (1+r ) = 1.05 $$

Assume that the UL pays a $5 dividend at t = 1

You can have a binomial tree such that

enter image description here

If it is a european Call option with maturity =2 and strike K = 90 then Node 2 is

MAX(115.5 - 90 , 0 ) = 25.5

MAX(94.5 - 90 , 0 ) = 4.5

MAX(93.5 - 90 , 0 ) = 3.5

MAX(76.5 - 90 , 0 ) = 0

Find risk nuetral probability of going up $$ q = (1+r)-d/(u-d)$$ and find price at node 1 and thereafter at node 0.

Risk is a big problem in itself which I will leave for now for someone else.

  • 1
    $\begingroup$ thank you for your detailed answer about options on dividend-paying assets. But my question is about derivatives (e.g. futures) on dividends. This is related but a totally different story ... it is about instruments like this: globalderivatives.nyx.com/stock-indices/nyse-liffe/… $\endgroup$
    – Richi Wa
    Apr 4, 2013 at 7:16
  • $\begingroup$ @Richard as long as you can synthically replicate the other securities the meathod should work. $\endgroup$
    – ash
    Apr 4, 2013 at 14:28
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    $\begingroup$ sorry, but this is binomial pricing ... this does not directly help with futures on dividends what my question is about. I hope for a more specific answer to the question. Thanks for posting but this is too basic and too little related to my question. $\endgroup$
    – Richi Wa
    Apr 4, 2013 at 14:54
  • $\begingroup$ If I apply this scheme to a dividend process then it would mean that dividends pay dividends ... I can hardly imagine this. So this is off-topic - sorry. $\endgroup$
    – Richi Wa
    Apr 4, 2013 at 14:55
  • $\begingroup$ @Richard ATM I dont have time to do anymore on this. I will wait for someone to cook the solution $\endgroup$
    – ash
    Apr 4, 2013 at 17:15

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