Understanding Gamma P&L from Delta Changes

Summary

The document raises a question about the relationship between the familiar gamma contribution to the P&L of a delta-hedged position and a proposed calculation based on the change in delta during an underlying price move. It presents the gamma-and-theta approximation and asks whether multiplying the estimated delta change by the price move gives the same dollar P&L when the move is treated as instantaneous and theta is ignored.

The author says that an average-delta perspective also seems to lead to the proposed expression, but the document contains no answer or worked derivation. It therefore serves as a focused conceptual prompt rather than a complete explanation. Readers should treat the proposed calculation as an unresolved question here; the text does not state the assumptions needed to compare the expressions, clarify the role of the underlying price convention, or address approximation error for larger moves.

Key ideas

• The question compares a gamma-based delta-hedged P&L approximation with an estimate built from delta change and price movement.
• It considers an instantaneous underlying move and sets aside theta for the comparison.
• The author also suggests reasoning from average delta across the move.
• No resolution, derivation, or discussion of approximation limits is included.

Tags

# Gamma trading PnL


# Gamma trading PnL












I understand this delta hedged portfolio pnl formula and how it's derived. $$ \text{P&L}=\dfrac{1}{2}\Gamma(\Delta S)^{2}-\theta\delta t $$ However, if I think it from a different angle like this: say if $\Gamma$ is known and stock dollar move is $\Delta S$, and then how much my delta would move is $\frac{\Delta S}{S}*\Gamma$. Can I argue that my dollar $\text{P&L}$ is just $\frac{\Delta S}{S} * \Gamma * \Delta S$? (assuming it's an instant move so ignoring $\theta$) I also tried to think from a portfolio perspective if I move underlying up and down, and just take an average on how many deltas has moved, it still lead me to this formula.

I am having a hard time connecting the two, not sure what I am missing.

Thanks!

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