Black-Scholes and Black-76

The idea

Black, Scholes and Merton showed in 1973 that an option can be replicated by continuously trading the underlying, so its price can't depend on anyone's view of direction, only on how much the underlying moves. Assume the price follows a lognormal random walk with a constant volatility, and the option's fair value comes out in closed form.

Black's 1976 version writes the same formula on the forward F for the option's expiry instead of spot. Crypto options are priced that way, because each expiry has its own forward and the hedge is a future for that date:

call = F × N(d1) − K × N(d2),  put = K × N(−d2) − F × N(−d1)

with d1 = (ln(F / K) + σ²T / 2) / (σ√T), d2 = d1 − σ√T, N the standard normal distribution, σ the volatility and T the years to expiry. (With a non-zero rate, both prices are also multiplied by a discount factor.)

The inputs

• Forward and strike: known.
• Time to expiry: known; crypto trades every day, so it is counted in calendar days over 365.
• Interest rate: only discounts the payoff; over weeks it changes the price very little (rho).
• Volatility: the only unknown. The market solves the formula backwards for it, and that is the implied volatility.

A worked example

A 30-day at-the-money BTC call: F = K = $100,000, σ = 50%, T = 30 / 365.

• σ√T = 0.50 × √(30/365) ≈ 0.1433;
• d1 = (0 + 0.1433² / 2) / 0.1433 ≈ 0.0717 and d2 ≈ −0.0717;
• N(d1) ≈ 0.5286 and N(d2) ≈ 0.4714;
• call = $100,000 × (0.5286 − 0.4714) ≈ $5,714.

The put at the same strike is also $5,714: with F = K, put-call parity says call minus put is zero. A quick check: an at-the-money option costs about 0.4 × F × σ√T, here $5,734. Raise the volatility to 55% and the call is worth $6,284, $570 more over five points, about $114 per volatility point, which is its vega.

Assumptions crypto breaks

The model assumes a lognormal price with one constant volatility, continuous trading without jumps, and frictionless hedging. BTC violates all of it: volatility changes all the time, prices jump on news and liquidation cascades, returns have fat tails, and hedging costs spreads and fees. The market doesn't fix the model; it uses a different volatility for every strike and expiry so that the model reproduces real prices. That pattern is the smile and the term structure.

So Black-76 is best read as a translator between prices and implied volatilities, not as a forecast.

Where BOSS uses it

Live prices and live Greeks in BOSS are the venues' own. The model comes in where the market can't answer: the Greek curves across a range of prices, and the model prices behind the portfolio margin estimate's scenarios, are Black-76 with a zero rate, at each option's own implied volatility (shocked up and down in the margin scenarios). A zero rate is a deliberate simplification: crypto has no single riskless rate, and over the horizons these charts cover the curves' shape barely depends on it.