Chapter 18: Futures Options and Black's Model

Nature of Futures Options

A futures option gives the holder the right to enter into a futures contract at a specified futures price (the strike). Upon exercise:

• Call: Holder receives a long futures position + cash equal to $F_T-K$
• Put: Holder receives a short futures position + cash equal to $K-F_T$

where $F_T$ is the most recent settlement futures price.

Futures options are typically American-style.

Reasons for Popularity

• More liquid and easier to trade than options on the physical underlying
• Futures prices are immediately observable
• Easier to deliver (cash or futures, not physical commodity)
• No issues with physical delivery logistics
• Often lower transaction costs

Futures options are available on: commodities, bonds, currencies, stock indices, and interest rates.

European Spot and Futures Options

For European options, when the options and futures mature at the same time, the futures option value equals the spot option value. When they mature at different times, they can differ.

Put–Call Parity for Futures Options

$c+K{e}^{-rT}=p+F_0{e}^{-rT}$

where $F_0$ is the current futures price.

Bounds for Futures Options

• Call: $c\ge \max (F_0-K,0)e^{-rT}$
• Put: $p\ge \max (K-F_0,0)e^{-rT}$

Black's Model (1976)

In the risk-neutral world, the expected growth rate of a futures price is zero (futures contracts cost nothing to enter). Therefore, the drift is zero.

Fisher Black showed that for European futures options:

$c=e^{-rT}[F_{0}N(d_1)-KN(d_2)]$

$p=e^{-rT}[KN(-d_2)-F_{0}N(-d_1)]$

where:

$d_1=\frac{\ln (F_0/K)+{\sigma }^{2}T/2}{\sigma \sqrt{T}},d_2=d_1-\sigma \sqrt{T}$

Black's model is widely used — not just for futures options, but also for:

• Interest rate caps and floors
• Swaptions
• Bond options
• Any situation where the underlying is a forward/futures price

Valuation Using Binomial Trees

When using binomial trees for futures options:

$p=\frac{1-d}{u-d}$

(the risk-neutral probability of an up move — note: no $r$ in the formula since futures have zero drift in a risk-neutral world)

American Futures vs. American Spot Options

When futures mature after the option, American futures options can be worth more than their European counterparts, especially:

• Call futures options when the market is in contango ($F>S$)
• Put futures options when the market is in backwardation ($F<S$)

When futures mature before or at the same time as the option, it's never optimal to exercise an American call futures option early. American put futures options may still be exercised early.

Futures-Style Options

In futures-style options (as opposed to regular "equity-style" options):

• No upfront premium is paid
• The option is marked to market daily like a futures contract
• At exercise, the cash payoff equals the option's intrinsic value

Futures-style options are common in some markets (e.g., S&P 500 options on certain exchanges).