Black-76 model
Also written Black-76 · BS-76 · Black 76 formula
The variant of the Black-Scholes model used to price Options on Futures, which substitutes the futures price F for the spot price and discounts the whole payoff at e^(-rT).
In plain language
The Black-Scholes model prices an option from five inputs: the underlying price, the strike, volatility, time to expiry and the risk-free interest rate. It was built for options on something you can hold today — a share, or a quantity of goods.
An option on futures has a futures contract as its underlying. The Black-76 model adapts the formula to that: it uses the futures price F where Black-Scholes uses the spot price S, and discounts the entire expected payoff back to today.
In India, Black-76 prices Options on Futures; the original Black-Scholes prices Options on Goods. The workbook calls them the same principle depicted with different symbols.
How it works
The link between the two. From the futures chapter, F = S x e^(rT), so e^(-rT) x F = S. Put e^(-rT) x F wherever Black-Scholes has S and the goods formula becomes the futures formula.
The symbols (Chapter 4):
| Symbol | Meaning |
|---|---|
| F | Underlying futures price (Options on Futures) |
| S | Underlying spot price (Options on Goods) |
| K | Strike price |
| T | Time to expiry, in years |
| r | Continuously compounded risk-free rate (MIBOR) |
| v | Annualised volatility of the underlying futures |
| N | Cumulative standard normal distribution |
Why prices can still differ in practice. On expiry day, with identical terms, a call on futures and a call on goods should cost the same. But futures have Daily Price Limits that cap volatility; spot commodity markets do not, and spot prices come from a polling process with its own data limitations. Higher expected volatility is fed into options on goods, which may therefore be priced higher.
The ATM consequence. Put-call parity for options on futures reads C - P = F/(1+rt) - K/(1+rt). At the money F = K, so C = P.
The formula
Call (C) = e^(-rT) x [F x N(d1) - K x N(d2)]
Put (P) = e^(-rT) x [K x N(-d2) - F x N(-d1)]
or P = C - [e^(-rT) x (F - K)]
d1 = [ln(F/K) + (v^2 / 2) x T] / (v x sqrt(T))
d2 = d1 - v x sqrt(T)
A worked example
Price an ATM silver call and put on futures. The futures price and strike are both Rs 60,000 per kg — the level in the workbook's Delta example. The other inputs are illustrative: volatility 25 per cent, 2 months to expiry, MIBOR 6.5 per cent.
T = 2/12 = 0.1667
v x sqrt(T) = 0.25 x 0.4082 = 0.1021
d1 = [ln(1) + (0.0625/2) x 0.1667] / 0.1021 = 0.0510
d2 = 0.0510 - 0.1021 = -0.0510
N(d1) = 0.5203 N(d2) = 0.4797
e^(-rT) = e^(-0.065 x 0.1667) = 0.9892
Call = 0.9892 x [60,000 x 0.5203 - 60,000 x 0.4797] ≈ Rs 2,416
Put = 0.9892 x [60,000 x 0.5203 - 60,000 x 0.4797] ≈ Rs 2,416
The call and the put are equal, exactly as put-call parity for options on futures predicts at the money.
Move the strike to Rs 58,000 with everything else unchanged and the model gives a call of about Rs 3,494 and a put of about Rs 1,516. Check with the workbook's shortcut:
C - P = e^(-rT) x (F - K) = 0.9892 x 2,000 ≈ Rs 1,978
3,494 - 1,516 = Rs 1,978
Why NISM asks about it
Chapter 4, section 4.4.2 (Options Pricing Models), sets out the Binomial model, Black-Scholes and the Black-76 derivation; sections 4.4.5 (Put-Call Parity) and 4.5 (Options on Commodity Futures) apply it, and section 4.6.1 repeats that Options on Goods continue to use the original Black-Scholes. Expect a match between model and product, the meaning of each symbol (r is MIBOR), and why an option on goods may cost more than an equivalent option on futures.
Common exam traps
- Black-76 is for Options on Futures; Black-Scholes is for Options on Goods. Swapping them is the standard question.
- F replaces S, and the whole bracket is discounted by e^(-rT) — not just the strike term.
- r is the continuously compounded risk-free rate, which the workbook identifies as MIBOR.
- ATM calls and puts on futures are priced equally. On a spot commodity or security the call is generally dearer at the money.
- The workbook's two formulas do not reconcile exactly as printed. Its Options on Goods d1 is written as [ln(S/K) + (v^2/2)T] / (v x sqrt(T)), with no interest-rate term. With that d1, substituting S = e^(-rT) x F does not reproduce Black-76 — in the example above it gives about Rs 2,402 rather than Rs 2,416. The conceptual point the workbook makes (same principle, e^(-rT) x F = S) is what is examined.
- Equal in theory is not equal in practice. Higher volatility input and the writer's delivery obligation can make options on goods dearer in shallow markets.
- Binomial (Sharpe, 1978) iterates; Black-Scholes (1973) does not.
Where this is taught
Free preparation for NISM Series XVIRelated terms
- Implied volatilityThe volatility figure that, put into an option pricing model, reproduces the option's actual market price — the market's consensus forecast of how much the underlying will move.
- Put-call parityThe arbitrage-free relationship binding a European call and put of the same strike and expiry to the spot and the discounted strike: c + X·e^(−rt) = p + S. Deviations create risk-free profit.
- Cost of carryThe cost of holding an asset until a future date — financing interest plus storage, less any income it pays. It links spot to futures: fair futures price = spot + cost of carry − inflows.
- Option premiumThe price an option buyer pays the seller for the right the contract carries — non-refundable, and made up of intrinsic value plus time value.
- Options on FuturesA European commodity option whose underlying is a futures contract, so exercise devolves into a futures position opened at the strike price rather than into delivery of the goods.
- Options on GoodsA European option whose exercise devolves directly into delivery of and payment for the physical commodity, rather than into a futures position.
- Alternative Risk Management FrameworkA tougher SEBI margining regime for cash-settled contracts prone to near-zero or negative prices — pre-expiry margins rise 5% a day for the last five trading days once triggered, and spread margin benefit is withdrawn.