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Why Adding the Underlying Does Not Change Portfolio Gamma

Article Quant Q&A · Author: luca dibo
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Summary

This brief explanation answers whether adding a holding in the underlying asset changes the gamma of a portfolio containing options. It defines portfolio gamma as the second derivative of the portfolio’s value with respect to the underlying price. A position consisting of a fixed quantity of the underlying contributes a linear term to portfolio value, so its second derivative with respect to price is zero. Adding that term therefore leaves the portfolio’s gamma unchanged.

The result is a direct calculus argument and applies to the stated setup of adding a constant quantity of the underlying. It clarifies the distinction between changing exposure to price movements and changing curvature: the underlying position affects the portfolio’s value and delta, while gamma measures curvature. The answer is concise and does not discuss complications such as multiple risk factors, nonlinear instruments, or how gamma may change as market inputs or prices move.

Key ideas

  • Portfolio gamma is the second derivative of portfolio value with respect to the underlying price.
  • A fixed holding in the underlying adds a linear price exposure to portfolio value.
  • The second derivative of that linear exposure is zero, so it does not change portfolio gamma.
  • Adding underlying can change delta even though it leaves gamma unchanged in this setup.

Tags

OptionsDerivatives pricingRisk management
Full text
# Proof that adding some quantity of stocks in a portfolio of option does not change the portfolio Gamma


# Proof that adding some quantity of stocks in a portfolio of option does not change the portfolio Gamma












I would like to proof mathematically and intuitively that adding some quantity of underlying to a portfolio of option does not change the overall gamma. Can you help me?

## Answer by alexprice (score 3, accepted)

https://quant.stackexchange.com/a/50448

overall gamma is second derivative of whole portfolio over underlying. adding any function (such as underlying*constant) which second derivative is 0 does not alter overall second derivative.

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