Deriving Rho and Theta for the Bachelier Option Model

Summary

The document concerns Greeks for the Bachelier model, which assumes normally distributed price changes. The author says delta, vega, and gamma are straightforward to derive, then focuses on deriving rho and theta when the interest rate is nonzero. They consult a paper that treats the zero-rate case and question a step in its theta derivation, while also seeking a reference for the nonzero-rate formulas.

The text provides context for a derivation problem, but it does not include the proposed expressions for rho or theta, a worked derivation, or a confirmed correction to the cited paper. It therefore identifies which sensitivities need attention and where a derivation may be unclear, without resolving the mathematics. Readers should treat it as a question motivating further analysis rather than as a source of validated Greek formulas.

Key ideas

• The Bachelier model uses a normal price process for option valuation.
• The author seeks rho and theta formulas when the interest rate is nonzero.
• A cited derivation covers theta in the zero-rate case and raises a question about one step.
• The document offers no final formulas or verification of the proposed derivation.

Tags

# Deriving Bachelier Greeks


# Deriving Bachelier Greeks












I am working on the Bachelier Model with r not equal to 0 as described in the first and most upvoted answer in following link:

Bachelier model call option pricing formula

This is fairly easy to code and to derive delta, vega and gamma. I want to derive however other first order greeks, namely: rho and theta.

I found the following paper that derived these greeks for a Bachelier model with r = 0 and I am using it as support:

https://www.nottingham.ac.uk/business/who-we-are/centres-and-institutes/gcbfi/documents/cris-reports/cris-paper-2007-7.pdf

In their derivation of theta I do not understand equation A32 on page 16. They write:

Shouldn't it be ? If not, why ?:

On a side note, if anyone has a reference where these bachelier with r != 0 greeks are derived I'll take it

EDIT:

Here are theta and rho that I derived. Hope someone can tell me if it's correct or not.

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