Replicating a European Option on a Futures Contract

Summary

The document poses a one-period binomial pricing question for a European call on a futures contract. It asks how to construct a replicating portfolio when the futures price can move up or down, and the underlying asset is itself a futures contract rather than a stock. The example specifies the option terms, volatility, interest rate, and possible next-step futures prices.

The author notes that risk-neutral probabilities and backward calculation would be straightforward, but seeks a replication approach using futures and risk-free investments. No solution or derivation is included, so the example illustrates the modeling question rather than demonstrating a completed hedge. It leaves open how to represent the futures position and financing in the one-period setup.

Key ideas

• A futures option can be modeled in a one-period binomial framework using up and down futures-price outcomes.
• Replication requires matching the option payoff at the next time step with a portfolio involving futures and risk-free investments.
• The document provides an example's contract assumptions and node prices but no replication solution.

Tags

# Price futures option via replication


# Price futures option via replication












I ran into some difficulties when trying to price a futures option via replication in a simple one-period binomial model. I am quite aware that this is easy with risk-neutral probabilities and backward calculation but I cannot see how to price it via replication.

The question with which I am confronted - $\textit{no}$ homework, just to be sure - is to price via replication in a one-period binomial model a European call option which matures in 3 months on a futures contract with strike price $K = 20$ and current futures price of 30. The volatility of the futures price is 30% p.a. and the risk-free rate 5%.

What I did was to approximate the up- and down-factors $u$ and $d$ for the future price development which gives me 34.855 in the upper node and 25.8212 in the down node. What I somehow have to do now is to set up a replication portfolio of some underlying asset (e.g. stocks, futures, risk-free investments) which will have the same value in $t = 1$ as the derivative. But how do I do that with futures? I don't have any other information about a stock or the maturity of the given future contract.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.