What it means
An option gives its holder the right, but not the obligation, to buy or sell something at a fixed price (the strike price) by a set date. Options on futures give the right to enter a futures contract rather than to buy the underlying asset directly.
Pricing them requires a model that links the futures price, the strike, the time left, interest rates and expected volatility. Black's model does this with a formula similar to Black-Scholes.
The key change is that a futures contract has no cost to enter and pays no dividend, so the model uses the futures price directly and discounts the result back to today. That makes it simpler for assets such as oil, wheat or government bond futures.
Traders use the model in two ways. They can feed in a volatility estimate to calculate a fair price, or they can start with the market price and work backwards to find the implied volatility, which shows how much movement the market expects.
The second use is more common, because volatility is the one input that cannot be observed directly. For business managers, the model matters wherever companies hedge with options on futures.
An airline might buy call options on fuel futures to cap its costs, while a farmer might buy put options to set a minimum selling price. Knowing how the model works helps finance teams judge whether the premium they are quoted is fair.
Like all models, it relies on assumptions. It assumes the futures price moves randomly with steady volatility, that interest rates are constant and that the option can only be exercised at expiry.
Real markets show changing volatility and sudden jumps, so professionals treat the output as a guide.
In practice
Real-world examples.
Example
A grain trader buys call options on wheat futures to protect against a price spike. The dealer quotes $7.57 for each $100 of futures value, and the trader compares that with Black's model before accepting. The quote is in line with the model, so she goes ahead.
Example
A bank's treasury desk sells options on interest rate futures to corporate clients. It uses Black's model to price each option and adds a margin for risk and profit. The desk then hedges its own exposure by trading the underlying futures.
Example
An energy company wants to cap its natural gas costs for the winter. Its finance team uses the model to compare options with different strike prices. It chooses a strike that balances the cost of the premium against the level of protection.
Formula
Calculation
Call price = e^(-rT) x [F x N(d1) - K x N(d2)]
d1 = [ln(F / K) + (volatility squared x T / 2)] / (volatility x square root of T); d2 = d1 - volatility x square root of T
Here F is the futures price, K is the strike, r is the risk-free rate, T is the years to expiry and N( ) is the standard normal cumulative probability.
Suppose F = $100, K = $100, volatility = 20%, T = 1 year and r = 5%. Then d1 = [0 + (0.04 x 1 / 2)] / 0.20 = 0.02 / 0.20 = 0.10, and d2 = 0.10 - 0.20 = -0.10. From standard tables, N(0.10) = 0.5398 and N(-0.10) = 0.4602. Call price = e^(-0.05) x [100 x 0.5398 - 100 x 0.4602] = 0.9512 x 7.96 = about $7.57.Case study
Seen in the real world.
Redwater Foods is a fictional cereal producer that worried about rising wheat prices. Its treasurer was offered call options on wheat futures with a $100 strike and a one-year term, priced at $11 per $100 of futures value by a broker.
In this illustrative case, she used Black's model with a volatility of 20% and found a fair price of about $7.57, then asked a second dealer for a quote. The second dealer quoted $8.00, and Redwater bought from that dealer, saving $3.00 per $100 of cover. Over a $5,000,000 hedge, the difference came to about $150,000 in premium.
The treasurer wrote a short note for the board explaining that the model gave her a benchmark for negotiation, not a guaranteed price. She now asks at least two dealers for quotes on every option hedge and records the implied volatility of each.
Watch out
Common mistakes.
- Using the share-price version of Black-Scholes for futures options. Black's model is designed for the futures price and gives different results.
- Treating the volatility input as certain. Small changes in volatility can move the option price noticeably.
- Believing the model price is the price you will be quoted. Dealers add margins and may use their own volatility assumptions.
Questions
People also ask.
What is the difference between Black's model and Black-Scholes?
Black-Scholes prices options on a share that may pay dividends, while Black's model prices options on futures, which have no cost to hold and no income.
Who was Fischer Black?
He was an American economist who co-developed the Black-Scholes option pricing model with Myron Scholes and worked on its extensions.
Is Black's model used for interest rate products?
Yes, variations of it are used for caps, floors and swaptions in the interest rate market.
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