Why Black’s Futures Option Formula Needs No Dividend Adjustment

Summary

The document explains how dividends enter the Black 1976 model for options on futures. For a typical index future, the futures holder does not receive the cash index’s dividends. Instead, the continuously compounded dividend yield and interest rate affect the relationship between the cash index and the futures price over the contract horizon.

Because Black’s futures option formula uses the observable futures price as its underlying, the dividend effect is already reflected in that price. The option formula therefore needs no separate dividend adjustment. The point is specific to futures whose holders do not receive dividends; it distinguishes futures options from options priced directly on a dividend-paying spot index. The discussion provides the pricing relationship as its explanation but does not explore contract designs with different dividend entitlements or other market conventions.

Key ideas

• Index futures holders generally do not receive the cash index’s dividends.
• Interest and dividend rates determine the relationship between spot index and futures prices.
• Black’s futures option formula takes the futures price as its underlying.
• A separate dividend adjustment is unnecessary in the formula when dividends are already reflected in the futures price.

Tags

# Dividend yield under Black 1976 formula for futures options?


# Dividend yield under Black 1976 formula for futures options?












I have a question regarding the BS 1976 formula for futures options.

https://www.glynholton.com/notes/black_1976/

How do I deal with dividends under this model, assuming that the dividend yield is continously compunded? Background is that I want to model the implied volatility of options on index futures given a deterministic continously compounded dividend yield!

Thank you very much in advance,

Best regards,

John

## Answer by Alex C (score 3, accepted)

https://quant.stackexchange.com/a/33354

For most index futures I am familiar with (for example S&P 500 futures) the holder of the future does NOT receive the dividends. The interest rate $r$ and the dividend rate $d$ on the cash index determine the relationship between the future $F$ and the spot $S$ (i.e. $F= S e^{(r-d)T}$).

The Black 1976 formula for options on futures is based on F, the observable future price, and since the future does not receive the dividend there is no need for a dividend adjustment in that formula.

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.