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In Wikipedia, the formula for the forward price of a tradable underlying that pays discrete dividends is given as: enter image description here

My confusion is this: once a dividend $D_i$is paid at time $t_i$, it becomes cash and arguably could be reinvested at the risk‐free rate $r$. So I would expect the accumulation for that dividend to be $D_ie^{r(T-t_i)}$. However the formula uses $D_ie^{(r-q)(T-t_i)}$ i.e., effectively using cost‐of‐carry rate $c=r-q$ for the dividends. I understand that for the stock, it should use the net carry rate $c=r-q$. But why does the formula also use $c$ for the accumulation of $D_i$?

Could someone clarify the modelling assumptions or derivation that justify that choice? Thanks in advance!

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A few assumptions here:

  1. Proportional dividends are reinvested into the stock and we model those as a continuous yield $q$
  2. Absolute dividends are reinvested into the cash account and we assume one will occur at time $T_d$ with value $D$
  3. The value of the forward just before the stock goes ex-div is the same as the forward after it goes ex-div.
  4. The risk neutral expectation on a known cash flow $K$ at time $T$ valued at time $t$ is $e^{-r(T-t)}K$
  5. The risk neutral expectation of a risky cash flow $S_T$ at time $T$ valued at time $t$ is $e^{-q(T-t)}S_t$

With these assumptions we can calculate the value of the forward contract at the following times

  • At time $T$: $V(T) = S_T-K$
  • At time $T_d^+$: $V(T_d^+) = e^{-q (T-T_d)}S_{T_d^+}-e^{-r(T-T_d)}K$
  • At time $T_d^-$: $V(T_d^-) = e^{-q (T-T_d)}(S_{T_d^-}-D)-e^{-r(T-T_d)}K$
  • At time $t$: $V(t) = e^{-q (T-t)}S_t - e^{-q (T-T_d)}e^{-r (T_d-t)}D-e^{-r(T-t)}K$

Setting this to par shows that the fair forward strike is $$K(t,T) = e^{(r-q) (T-t)}S_t - e^{(r-q) (T-T_d)}D$$

So we can see that we are re-investing the absolute dividend in the cash account but we do not receive a full dividend $D$ but rather a proportional dividend $e^{-q (T-T_d)}D$ as we are only holding $e^{-q (T-T_d)}$ stock at the ex-div date in the hedge portfolio.

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Discrete dividends are a very interesting topic.

It is better to model them as discrete than continuous for pricing (obviously depends). Forward price comes from the usual no arbitrage pricing. Dividends being reinvested are a choice that imply unnecessary problems to the normal forward price calculation.(e.g., what rate will you use for special dividends that are not on the schedule?)

Also dividends reduce the spot price, thus being accounted on the cost of carry Instead, each dividend reduces the forward value of S_0 and for the relationship to remain arbitrage-free, its value must evolve under the same c=r−q that links the spot and the forward.

Some articles and models:

Klassen created a somewhat "new" forward model based on the SKA

bos and vandermark 2002 created the spot strike adjustment (cannot find the article)

Escrowed Model

Forward Model

Piecwise Lognormal Model

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