Why Futures Options Use a Different Hedging Argument from Stock Options

Summary

The question examines a European option on a futures contract and asks why its pricing should reflect a risk-neutral futures growth rate of zero, even though a stock-option hedging argument appears to produce the same partial differential equation. The apparent inconsistency comes from treating the futures position as if it were an asset purchased with cash.

The answer explains that a futures contract is an unfunded position, ignoring margin, and has a value of zero when entered. Consequently, the hedging portfolio does not include the cash investment term associated with holding the underlying futures price. The apparent interest-bearing contribution proportional to the option’s sensitivity and the futures level should therefore not be included. This resolves the stated mismatch at the conceptual level, but the short answer does not derive the full pricing equation or discuss margin, contract conventions, or other assumptions. It is a concise explanation of why the stock-option replication argument cannot be transferred unchanged to options on futures.

Key ideas

• A futures contract is treated as an unfunded position and has zero value at initiation, ignoring margin.
• The stock-option hedging argument assumes a financed underlying position, which does not apply in the same way to futures.
• The riskless portfolio should not include an interest term for cash invested in the futures contract.
• The explanation is conceptual and does not derive a complete option pricing model.

Tags

# Option price of a future


# Option price of a future












This must be a dumb question. Consider a European option $V$ on a (stock) futures $F$. The hedging condition seems to be the same as that for a stock $$d\Big(V-\frac{\partial V}{\partial F}F\Big)=r\Big(V-\frac{\partial V}{\partial F}F\Big)dt$$ for the riskless short interest rate $r$, since the portfolio $\Pi:=V-\frac{\partial V}{\partial F}F$ is a riskless traded asset and thus should grow at the riskless interest rate. I will get the same PDE as that of a stock. However, one should obtain the risk-neutral growth rate of $F$ as $0$. What is the catch?

## Answer by Ivan (score 2)

https://quant.stackexchange.com/a/41804

The catch is the future is an unfunded position (I’m disregarding the margin here) and as a result the term $r\frac{\partial{V}}{\partial{F}}Fdt$ does not exist. F is actually a contract worth 0.

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