Why Black–Scholes Futures and Stock Option Values May Differ

Summary

This question examines whether a call option on a futures contract has the same theoretical value as a call option on the underlying stock. It compares the discounted futures-option pricing expression with the stock-option expression and uses the cost-of-carry relation between the futures price and stock price to motivate the comparison.

The displayed substitution alone does not settle equivalence: the option pricing inputs, including the definitions of d₁ and d₂ and the underlying exposure, must be compared consistently. A futures option is a derivative on a futures contract, while a stock option is written on the stock; their values can align under appropriate assumptions and pricing conventions, but the question provides no answer or conditions establishing that result. It therefore serves as a starting point for checking model conventions, carrying costs, and contract structure rather than evidence that the two instruments always have identical values.

Key ideas

• A futures option pricing expression discounts the value based on the futures price and strike.
• The futures price is related to the spot price through carrying costs in the stated setup.
• Substitution into a pricing formula requires consistent definitions of its inputs and terms.
• The question does not specify assumptions sufficient to establish that stock and futures options always have the same value.

Tags

# Option on Futures vs. Stocks


# Option on Futures vs. Stocks












The Black-Scholes call on a Futures is valued as: $$ C_t=e^{-r(T-t)}[F_tN(d_1)-KN(d_2)] $$ It holds: $F_t=S_te^{r(T-t)}$.

If I plug this back in, I get the Black-Scholes call on a stock:

$$ C_t=S_tN(d_1)-e^{-r(T-t)}KN(d_2) $$ Does this mean that the option on a futures has the same value as the option on a stock in theory?

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