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Derivatives Pricing and Computational Finance / Index, Currency, Futures and Multi-Asset Options

Futures Options and Black's Model

Commodity desks, equity-index futures books, and rates traders often hedge with options on futures — the right to enter a futures position at strike KK — rather than options on spot. Under no-arbitrage, the futures price FtF_t for a fixed maturity TT is a Q\mathbb{Q}-martingale (zero cost of carry in the risk-neutral drift). Fischer Black's 1976 model prices European options on FtF_t with the same Gaussian machinery as Black–Scholes, but with F0F_0 replacing discounted spot. This note states Black's formula, connects it to the spot formula via cost of carry, and notes American exercise on futures options.
Learning objectives. By the end of this note you should be able to:
  • distinguish options on spot from options on futures and state when their payoffs coincide at delivery;
  • derive the risk-neutral SDE dFt=σFt dWtQdF_t = \sigma F_t\, dW_t^{\mathbb{Q}} and the resulting Black PDE in FF;
  • price European calls and puts on futures with Black's formula;
  • convert between spot and futures option prices using F0=S0e(r−q)TF_0 = S_0 e^{(r-q)T} and explain margining conventions.

In this lesson

  1. 2.1 Futures, forwards, and the cost of carry
  2. 2.2 Black's model
  3. 2.3 American options on futures

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