Quantitative Derivatives Lab

Black-76 Futures Option & Swaption Lab

Price options on futures contracts, commodities, interest rate caplets/floorlets, and swaptions using Fischer Black's (1976) forward valuation model.

Market Archetypes:

Contract & Market Inputs

Use 1.0 for commodity/futures options; use PVBP for swaptions or accrual τ for caplets.

Valuation Summary & Futures Greeks

Black-76 Price
$0.00
Call Premium
Intrinsic Value
$0.00
Discounted Payoff
Time Value
$0.00
0.0% of Premium
Discount Factor
0.0000
e^(-rT)
Call Price (C) Put Price (P) C - P Spread Parity: e^(-rT) × (F - K) Parity Discrepancy
$0.00 $0.00 $0.00 $0.00 0.0000

Futures & Swaption Greeks

Futures Delta (Δ) 0.0000
Gamma (Γ) 0.0000
Vega (ν) 0.0000
Theta (Θ / day) 0.0000
Rho (ρ) 0.0000
Forward Delta Δ = e^(-rT) · N(d1). In futures markets, delta measures position change with respect to the forward/futures contract, requiring zero upfront initial outlay.

Visual Analytics & Sensitivity Profile

Blue: Black-76 Option Value | Green Dash: Discounted Intrinsic Payoff | Gold Dot: Current Forward Price (F)

Matrix 1: Forward Price (F) vs. Volatility (σ) on Option Price

Evaluates option valuation across forward price movements (±20%) and volatility shocks (±10%).

Matrix 2: Strike (K) vs. Expiry (T) on Option Vega (ν)

Analyzes volatility sensitivity (Vega per 1% vol shift) across strike moneyness and maturity horizons.

Mathematical Foundation: Fischer Black (1976)

1. Forward Dynamics & Closed-Form Pricing

In Fischer Black's (1976) formulation, the underlying is the forward or futures price F_t, which follows a driftless geometric Brownian motion under the forward risk-neutral measure: dF_t = σ F_t dW_t. Because forward contracts require no initial capital investment, the continuous carrying cost is zero:

d_1 = [ln(F / K) + (1/2)σ² T] / [σ√T]
d_2 = d_1 - σ√T
Call: C = e^{-rT} · [F · N(d_1) - K · N(d_2)] · A
Put: P = e^{-rT} · [K · N(-d_2) - F · N(-d_1)] · A

The multiplier A represents the contract scaling: A = 1 for single commodity/equity futures, A = τ (accrual day-count fraction) for interest rate caplets, and A = ∑ τ_i P(0, T_i) (PVBP annuity) for European swaptions.

2. Futures Greeks & Put-Call Parity

Because both legs of the payoff are discounted from maturity, the Greeks display unique properties compared to spot options:

Futures Delta: Δ_fwd = ∂C / ∂F = e^{-rT} · N(d_1)
Gamma: Γ = e^{-rT} · N'(d_1) / [F · σ√T]
Vega: ν = e^{-rT} · F · √T · N'(d_1)
Put-Call Parity: C - P = e^{-rT} · (F - K) · A

Under put-call parity on futures, holding a long call and short put synthesizes a discounted forward contract e^{-rT} · (F - K). This identity holds regardless of volatility or distributional skew.

Black-76 Futures Option Mastery Quiz

Frequently Asked Questions

The Black-76 model, developed by Fischer Black in 1976, is a variant of the Black-Scholes formula designed specifically to price options on futures contracts, commodity forwards, interest rate caplets/floorlets, and swaptions where the underlying asset is a forward or futures price F rather than a spot price S.

In Black-Scholes, the underlying is a spot asset requiring continuous funding costs and dividend adjustments (cost of carry b = r - q). In Black-76, the underlying is already a forward price F which incorporates the cost of carry up to expiration. Therefore, the drift of F is zero under the forward risk-neutral measure, and both terms in the option payoff are discounted back from maturity using e^(-rT).

Put-call parity for futures options states: C - P = e^(-rT) * (F - K). Unlike spot options where the spot price is undiscounted (C - P = S - K * e^(-rT)), both the forward price and the strike price are discounted by e^(-rT) because entering a futures contract requires zero upfront cash investment.

For interest rate caplets, Black-76 treats the forward benchmark rate (e.g., SOFR or Euribor) as F, multiplying the option payoff by the accrual fraction and notional principal. For European swaptions, the underlying is the forward swap rate S_0, and the option payoff is multiplied by the forward swap annuity (PVBP), which represents the present value of a basis point across all payment dates.

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