Correcting the Volatility Term in Bachelier Spread Options

Summary

The note examines the volatility term for a spread option under a Bachelier model, where the underlying prices are modeled with arithmetic Brownian motion. The questioner expects the spread’s volatility to combine the two futures volatilities and their correlation, and asks whether a square root is missing from a published formula. The answer confirms that the square root is missing, correcting the expression for the spread volatility.

For an exchange option with a zero strike, the answer identifies the pricing result as Margrabe’s formula using Black–Scholes as numeraire, and also points to Kirk’s formula as potentially useful. The exchange is brief and offers no derivation, numerical example, or conditions for applying those related formulas. It resolves the specific algebraic concern but leaves implementation details and model suitability to the reader. In particular, the response does not explain how to estimate the input volatilities or correlation from market data.

Key ideas

• Spread volatility under the stated Bachelier setup requires a square root in its combination formula.
• The answer confirms that the cited expression omits that square root.
• A zero-strike exchange option is associated with Margrabe’s formula under a Black–Scholes numeraire.
• Kirk’s formula is mentioned as another potentially useful approximation.

Tags

# Bachelier pricing of exchange futures option volatility mistake


# Bachelier pricing of exchange futures option volatility mistake












I'm pricing exchange future options with Bachelier formula following "Spread Options, Exchange Options and Arithmetic Brownian Motion" by Michael Poitras 1998, and I think there is a mistake or something I'm clearly missing.

The article shows the formula as:

But from my understanding, Bachelier assumption is that prices are normal processes, and therefore sigma of the difference is $\sqrt{\sigma_{F_1}^2+\sigma_{F_2}^2-2\rho\sigma_{F_1}\sigma_{F_2}}$, but the article mentions no square root. Is there an error in my logic? Is there an error in the article?

Additionally, I use it for an exchange option so my strike is always $0$, I am allowed to use the formula as it is? (I think yes, but since I'm asking for the vol I thought I could double check)

## Answer by João (score 2)

https://quant.stackexchange.com/a/82085

Yes the square root is missing.

With a strike = 0 the pricing formula is known as the Margrabe´s formula with the BS as numeraire

Also , Kirk´s Formula should come in handy for you

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