How Gamma and Theta Relate in Option Positions

Summary

The document asks whether positive gamma is generally costly, negative gamma earns premium, and theta tends to have the opposite sign from gamma. It frames theta as the carrying cost of holding long gamma and asks whether these observations can be proved without specifying how the underlying price evolves.

It presents these as questions rather than supplying a derivation or evidence. The central topic is the connection between option price curvature, time decay, and the assumptions used in pricing models. Any answer would need to distinguish general model-based relationships from claims that hold across all market dynamics; the source itself does not resolve that distinction or establish a universal rule.

Key ideas

• The document asks whether long gamma typically requires paying premium while short gamma earns it.
• It raises the claim that theta and gamma usually have opposite signs.
• It describes theta as a possible carrying cost of long gamma.
• It asks whether these relationships can be established without assumptions about the underlying price process.
• The source poses the questions but provides no proof or empirical analysis.

Tags

# Relationship between gamma and theta


# Relationship between gamma and theta












I have read somewhere the following statements, which I have observed to be true most of the time. I want to know how accurate they are mathematically, and how to prove them?

> $\Gamma > 0$ is a good position to be in, and therefore you have to pay a premium for it, on the other hand $\Gamma < 0$ is a bad position to be in so you get paid premium for it.

> $\theta$ and $\Gamma$ have an opposite relationship i.e. if, let's say $\theta$ is positive, then $\Gamma$ will be negative and vice-versa. $\theta$ is like the rent you pay to be long $\Gamma$

Are these statements model-free i.e. can they be proven, without assuming anything about the stochastic process that the underlying stock price follows. Is this it, or is there any deeper relationship between $\theta$ and $\Gamma$?

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