Deriving Continuous Dividend Yield from Discrete Dividends

Summary

The document explains how dividend yield enters equity option pricing through the stock forward price. It distinguishes a forward yield reported by a data source from trailing yield, which uses past dividends, and indicated yield, which annualizes an expected dividend rate. These measures need not agree, so simply dividing last year’s payments by today’s stock price may not give the yield appropriate for an option’s maturity.

For known or estimated cash dividends before expiry, it discounts each payment to today and converts the resulting present value into an equivalent continuous yield by matching the discrete-dividend forward price to the continuous-yield forward formula. This gives a maturity-specific way to represent discrete payments. The method depends on dividend estimates, payment dates, the interest rate, and the option horizon; the example gives no empirical validation or guidance for uncertain dividend forecasts.

Key ideas

• Dividend yield measures differ depending on whether dividends are backward-looking or forward-looking.
• A stock forward price with discrete dividends subtracts the future payments’ appropriately discounted value.
• An equivalent continuous yield can be derived by matching discrete and continuous dividend forward prices.
• The implied yield depends on the dividend estimates and the maturity being priced.

Tags

# How to calculate dividend yield - option pricing


# How to calculate dividend yield - option pricing












Hey how do you calculate the dividend rate if you want to price your stock options eg apple? Just take the dividends paid last year and divide by today's share price? This page reports 0.85% (https://finance.yahoo.com/quote/AAPL?p=AAPL)

## Answer by ir7 (score 2, accepted)

https://quant.stackexchange.com/a/57037

Note that yahoo is posting the forward dividend yield. Other yields, trailing (that you seem to describe) and indicated, are described (Investopedia) here and (Wikipedia) here. See also this, this, and related links on Stack Exchange.

In general, for (estimated) discrete dividends $ (D_i)_{1\leq i\leq n} $ at future times $(0<)t_1<\ldots<t_n (\leq T)$, the $0$-time forward stock price for $T$-maturity is

$$ F_{0,T} = \mathrm{e}^{rT}S_0 - \sum_{i=1}^n \mathrm{e}^{r(T-t_i)} D_i$$

With continuous dividend yield over the same period $[0,T]$, the forward price is:

$$ F_{0,T} = \mathrm{e}^{(r-q)T}S_0 $$

By equating them, we get:

$$ q = -T^{-1} \ln \left(1 - S_0^{-1} \left( \sum_{i=1}^n\mathrm{e}^{-rt_i} D_i\right) \right) $$

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.