Dividend Treatment in Index Option Pricing
Summary
The document explains how dividends enter vanilla index-option valuation under the Black-Scholes framework. For options written on a spot index, one response describes using a continuous dividend yield inferred from put-call parity. Another notes that actual index dividends may be discrete and concentrated at particular dates, so a smooth yield is an approximation. It outlines an alternative for known deterministic cash dividends: subtract their present value from spot and use a zero-yield formulation, counting only payments before expiry.
For options on an index futures contract, the responses say to use the matching futures price as the underlying and set dividend yield to zero. They also caution that a dividend forecast based on projected growth can produce inconsistencies with option prices, while a constant-yield model is easier to implement consistently. The discussion offers several modeling conventions rather than a single universal rule; the appropriate treatment depends on whether the option is priced from spot or futures and how dividends are represented.
Key ideas
- Spot index options can incorporate dividends through a continuous yield inferred from put-call parity.
- A continuous yield approximates index dividends that may occur as discrete payments.
- Known deterministic dividends can be represented by subtracting their present value from spot and using zero yield.
- For options on index futures, use the corresponding futures price and set dividend yield to zero.
- The dividend convention should match the underlying instrument and modeling assumptions.
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Full text
# When valuing a vanilla option on an index, should we take dividend into account?
# When valuing a vanilla option on an index, should we take dividend into account?
When valuing a vanilla option on an index (eg FTSE 100), should we take index dividend yield into account?
$$ c=Se^{-q\tau}N\left(d_1\right)-Ke^{-r\tau}N\left(d_2\right) $$ $$ d_1=\frac{\ln\left(\frac{S}{K}\right)+\left(r-q+\frac{1}{2}\sigma^2\right)\tau}{\sigma\sqrt{\tau}} $$ $$ d_2=d_1-\sigma\sqrt{\tau} $$
Using the above formulae, should $q$ be 0 or "equivalent index dividend yield" when $S$ is an index (eg FTSE 100)?
## Answer by Randor (score 1)
https://quant.stackexchange.com/a/19504
Ftse100 would not have a smooth dividend yield, as your formula has, it would be discrete, being much higher on certain days of year than others. In pricing options on ftse, u need to take into account implied dividends (dividends that are implied by put call parity)
## Answer by Eli (score 1)
https://quant.stackexchange.com/a/19508
When valuing a plain index option, there are two options in terms of index dividend:
(1) The underlying price is a spot price like in the FTSE 100 case (option is valued off the index): you can use continuous dividend yield. You can imply a dividend yield from a linearized call-put parity:
The present value of the dividend payment is
$PV(div)=P-C+(S-K)+K(e^{rt} - 1)$,
then the implied dividend yield is $d = \frac {PV(div)}{T*S}$
(2) The underlying instrument is a future contract on the index (for example, IBEX35 or TAIEX), you'd set index dividend yield to zero and use future price of corresponding maturity as an underlying price.
For the purposes of calculating option prices or implied volatilities, the use of a dividend forecasting model based on projected actual dividend growth rates can lead to an option model which is internally inconsistent. In contrast, the use of a model based on constant dividend yields is not only consistent, but also easier to implement.
## Answer by MattBecker82 (score 0)
https://quant.stackexchange.com/a/19509
To expand on Randor's answer, the standard Black-Scholes formula as you've given it assumes a constant continuous dividend yield of $q$. To adapt this to cope with discrete deterministic (absolute) dividends $d_i$ at known times $\tau_i$, you could recast the formula in terms of the "dividend-free" stock price:
$$S^* := S - \sum_i d_i e^{-r\tau_i}$$
and set $q$ to zero.
N.B. You only include dividends between spot and the option expiry.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.