Module black76 

Black-76 (1976): European options on a future/forward price F.

Two settlement styles:

• Discounted (standard Black-76): the premium is paid up front and the payoff is discounted, call = e^{-rT}[F N(d1) - K N(d2)].
• Margined (futures-style / “future-style”): the option premium is itself margined daily like the future, so there is no discounting, call = F N(d1) - K N(d2). Common for options on futures on many non-US derivatives exchanges (e.g. Eurex, ICE, ASX).

F is the futures price directly — Black-76 has no spot, dividend or carry, since a future already embeds the cost of carry. All Greeks are sensitivities with respect to F (delta/gamma), sigma, r and time.

Enums

FuturesSettlement

How an option on a future is settled.

Functions

charm

Charm, the change in futures delta per year of calendar time.

delta

Delta with respect to the futures price F.

gamma

Gamma with respect to the futures price F (same for calls and puts).

gamma_p

Delta elasticity (also called percentage gamma), F * gamma / delta. It is undefined when delta is zero and returns NaN in that case.

price

Black-76 price of a European option on a future.

rho

Rho (sensitivity to the risk-free rate). Zero for margined options, which have no discounting; -T * price for discounted options (the futures price is exogenous, so r enters only through the discount).

theta

Theta (calendar time decay, dV/dt = -dV/dT).

vanna

Vanna, the change in futures delta per unit change in volatility.

vega

Vega (per unit of vol; same for calls and puts).

volga

Volga (also called vomma), the change in vega per unit change in volatility, d(vega)/d(sigma) = vega * d1 * d2 / sigma. Same for calls and puts, since put-call parity is volatility-independent. It is negative near the money (vega is concave in vol there) and positive in the wings.

zomma

Zomma, the change in futures gamma per unit change in volatility.