Choosing the Forward Input for Black Options on Futures

# options on futures


# options on futures












For options on futures in the black model, I do remember that $F$ appearing in the formula must be the forward at maturity of the option (and not the future price).

So, say we have a future maturing quarterly.

And say we have weekly and monthly options on that future.

Then the price of the option will depend on the one week forward, and 1 month forward respectively, and will not depend on the price of the future (which matures in 3 months).

Can you please confirm that is right and suggest a rigorous derivation?

## Answer by Soumirai (score 2)

https://quant.stackexchange.com/a/60076

In Black formula, F is the forward of the underlying for the maturity of the option. In the case of an option on a future, the underlying is the future (which has a specific expiry, e.g. an option on the jun21 future).

We can assume that the forward of a future is flat, i.e. equal to spot for any maturity. It is reasonable, because the cost of carry of a future is very small (margin cost but you have netting benefits etc). So you can safely use the current future price as F (in our example, F would be the current value of the jun21 future).

## Answer by dm63 (score 2)

https://quant.stackexchange.com/a/60082

Using the e mini example , supposing in Dec 2020 you are looking at 5month option on the ESZ1 (ie futures expiring 12mo from now). Then it is clear that you should be using the ESZ1 futures contract as the underlying, not the 5month futures contract ESK1. However there is a subtler question of whether today’s price of the ESZ1 is the correct rate to plug into Black model. What you want is the 5 month forward price on the ESZ1, which is not necessarily exactly equal to the spot price of the ESZ1.

It is well known that certain futures contracts have ‘convexity adjustments’ expressing the difference between forward rates and futures prices , derived from the covariance between the underlying of the futures contract and interest rates between now and expiration of the futures. This convexity adjustment vanishes at futures expiration by definition. So, this suggests that at intermediate times the forward rate on a futures contract admits a convexity adjustment that is less than the spot convexity adjustment. Having said this, I would expect this effect to be negligible for options on the e-mini. It would be more significant for long dated options on Eurodollar futures where there is a high correlation between interest rates and the value of the futures contracts.

## Answer by Edward Watson (score 1)

https://quant.stackexchange.com/a/60222

If there is only a spot market or maybe one or two actively traded futures it is acceptable to derive the futures price for that security from it's spot price and a product specific funding cost. You can use a funding spread over a benchmark curve's term structure to price longer futures. In the case of a single stock futures price, you will match the stock futures deliverable date to the option expiration date. If you are pricing an option on an equity futures contract you will price the option to it's expiration and the future to it's normal delivery cycle date (Quarterly etc).

## Answer by jherek (score 0)

https://quant.stackexchange.com/a/80780

Marc Henrard has a paper on option on Futures "EURODOLLAR FUTURES AND OPTIONS: CONVEXITY ADJUSTMENT IN HJM ONE-FACTOR MODEL".

While it's in a more specific framework, since it involves futures on interest rates, it shows a convexity adjustment as part of the forward price used in the Black formula, and not only a change of volatility.

It thus would appear that by using the futures price Fut(0,T) in the Black formula, we make the assumption that the covariance with the rates is zero, like Fisher Black in "The pricing of commodity contracts" (1976).

## Answer by Dorian B. (score 0)

https://quant.stackexchange.com/a/81025

There are two sets of formulas for the option prices under the Black model. One is expressed with the spot and one is expressed with the forward. The derivation is the same (assume a delta-neutral portfolio and no arbitration possible).

$$ C = S \cdot N(d_1) - K \cdot e^{-r(T-t)} \cdot N(d_2) $$ $$ C = e^{-r(T-t)} \cdot [F \cdot N(d_1) - K \cdot N(d_2)]$$ $$ F(t, T) = S \cdot e^{r(T-t)} $$

In order to properly account for this, you will need to consider $r$ actually includes not only the financing part but also (add) storage costs $u$ and (substract) convenience yield $y$ and any dividend or yield income $q$. $$r=r_f+u-y-q$$

(1) Now, even if the two formulas coincide, the assumptions are the ones that differentiate.

- Does the futures contract $F_*(t,T)$ trade at a premium or discount to the forward $E[S_T](t) = F(t, T)$? This relates to normal backwardation effects in the Keynesian sense.

- Will the futures contract converge at maturity $F_*(T,T)=S(T)$? Will it do so smoothly? This relates to basis risk and backwardation / contango effects.

(2) Another difference is in the greeks. If the underlying is $F$ instead of $S$, then for example the delta for an option on future is:

$$ \delta_C = \frac{\partial C}{\partial F} = e^{-r(T-t)} \cdot N(d_1) $$ vs $$ \delta^{spot}_C = \frac{\partial C}{\partial S} = N(d_1) $$

This has implications for the put-call parity with options on futures and the corrolary that the call and put deltas for options on futures do not add up to 1 in absolute value, but to the discount factor.

$$ \left . C - P = (F - K) \cdot e^{-r(T-t)} \right|_{\frac{\partial}{\partial F}} $$

$$ \delta_C - \delta_P = 1 \cdot e^{-r(T-t)} $$

vs

$$ \delta^{spot}_C - \delta^{spot}_P = 1 $$

(3) Regarding the part about expirations, as mentioned above the value of the option is related to the forward at the option's expiration. But we considered the assumption $F_*(t,T)=F(t,T)$. So in the more general case of weeklies the forward (used for the option) $F(t, T_1)$ and the underlying futures prices $F_*(t, T_2)$ have different expiration times (big T) with $T_1 < T_2$. The only thing changing is that the price greeks (delta, gamma) are shifted additionally by $e^{-r(T_2-T_1)}$.

$F(t, T_2)=F(t, T_1) \cdot e^{r(T_2-T_1)}$ so under the convergence assumption that $F(t, T_2)=F_*(t, T_2)$ the delta would be:

$$ \delta^{weekly}_C = \frac{\partial C}{\partial F_*(t, T_2)} = e^{-r(T_2-T_1)} \frac{\partial C}{\partial F(t, T_1)} = e^{-r(T_2-T_1)} e^{-r(T_1-t)} \cdot N(d_1) $$ $$ \delta^{weekly}_C = e^{-r(T_2-t)} \cdot N(d_1) $$

Net, weekly options add up to the discount factor of the underlying future but behaves price-wise (vs spot) like an option that expires weekly.