Interpreting Put Gamma as Curvature in Option Sensitivity

Summary

The document explains that the second derivative of an option’s value with respect to the underlying price is gamma. It measures how quickly delta, the option’s price sensitivity to the underlying, changes as the underlying moves. This curvature matters because an option’s exposure is nonlinear, so its delta does not remain constant across price changes.

For a long put, the document identifies negative delta and positive gamma; a short put has the opposite exposures. These signs describe how directional sensitivity and its rate of change behave, and are useful when considering an option position’s response to movements in the underlying. The explanation is conceptual and brief: it names gamma and outlines its interpretation, but does not provide a formula, a numerical example, or a broader discussion of how gamma changes with time, volatility, or moneyness.

Key ideas

• Gamma is the second derivative of option value with respect to the underlying price.
• Gamma measures how delta changes as the underlying price moves.
• A long put has negative delta and positive gamma.
• A short put has positive delta and negative gamma.

Tags

# What is the second derivative with respect to price of a put option?


# What is the second derivative with respect to price of a put option?












What is the reasoning/meaning behind the second derivative of a put option

## Answer by AlRacoon (score 3)

https://quant.stackexchange.com/a/44260

It is the rate at which the price of the option changes with respect to the change of the delta (the rate of change with respect to the underlying). As by design, options are non-linear in order to provide protection (limit loss) as well as provide some exposure to the underlying, their value will change its sensitivity to changes in the underlying. Due to curvature, so will this sensitivity to changes in the underlying. The second derivative is a measure of this change in sensitivity. It is a measure of realized volatility and is commonly referred to as gamma, among the option “greeks.”

As for a put option, if you are long the put option you are short delta and long gamma. If you are short the put, you are long delta and short gamma.

## Answer by Bob Jansen (score 1)

https://quant.stackexchange.com/a/44259

It's called Gamma one of the option Greeks.

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