Why Binomial Trees Model Futures Prices Directly for Options
Summary
The document explains why a binomial tree for an option on a futures contract can model futures prices directly rather than modeling spot and converting it at every node. Under the risk-neutral measure, futures prices are martingales, so they can be represented as the tree’s underlying process. This can make the model simpler when the relevant volatility is known.
The answers add that converting spot to futures through a deterministic interest-rate formula is valid only when rates are deterministic. Direct futures modeling is also useful when spot and futures dynamics differ, as can happen with energy contracts. For American-style futures options, a binomial tree can be used to infer a volatility that reproduces the observed option price. These points explain modeling choices rather than prescribe a single tree specification; the appropriate process still depends on the contract and assumptions.
Key ideas
- Risk-neutral futures prices are martingales and can be modeled directly in a binomial tree.
- The spot-to-futures relationship given in the question assumes deterministic interest rates.
- Spot and futures processes can differ, particularly for some commodity contracts.
- A binomial tree can help infer implied volatility for American options on futures.
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# How to price an option on future using a binomial tree? # How to price an option on future using a binomial tree? While trying to price an option on a future using a binomial tree, I found out in a manual (Actuarial Finance by Boudreault and Renaud) that we need to calculate $F_0$, and then apply the up and down factor on this value. I am wondering why we don't model the stock value instead and use the formula $F_t^T = S_t\exp(r(T-t))$? For me this seems more intuitive; it would give us the same value for an option on a stock and an option on a future with the same maturity which makes sense (same payout = same initial value). ## Answer by João (score 1) https://quant.stackexchange.com/a/85453 Because under the risk neutral, future prices are already a martingale so you can model them directly. Why work on a spot and mapping them each node if that doesn't bring more precision/efficiency (e.g., Black76) ## Answer by Frido (score 1) https://quant.stackexchange.com/a/85522 In addition to the existing answer and comments, two more points why directly working with futures may be preferable: - Interest rates are stochastic, the expression you wrote in your question for the futures price holds only when interest rates are deterministic (an idealization). However, regardless of whether interest rates are stochastic or not, futures prices are martingales under the risk-neutral measure, hence quite 'easy' to model, if you know the vol that is. - Options on futures are sometimes (often?) American type options. The binomial model is then a nice way to 'de-Americanize' these options, i.e. for each strike find the (implied) volatility such that when you value the American option using a binomial tree with this volatility you reproduce the market price of the options on the futures price. ## Answer by Rylan (score 1) https://quant.stackexchange.com/a/85532 Building a bit on @João and @AKdemy's arguments: for some underlyings, the spot process is quite different from the futures process. For stocks, because we can hold a stock at ~0 cost, we can define a simple arbitrage-free relationship between the stock and the future. This is not the case with, for example, futures on electricity or gas. Essentially, a lot of the time, the futures process is actually much simpler to work with.
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