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I can't visualize the profit/loss of a position described in terms of its Up-Gamma and Down-Gamma. The question arise from pag. 193 of Dynamic Hedging by Taleb.

How would you describe a position that is long up-gamma and short down-gamma in terms of profit and loss? How the signs of the up-gamma and down-gamma affect the position?

My idea: P/L increases "more than linearly" if the underlying price increase thanks to the positive effect of positive and greater value of up-gamma (long up-gamma). The P/L are affected positively by smaller value of down-Gamma, possibly negative values, that implies a P/L where the losses are "capped".

Edit

My second interpretation is that "short" Down-Gamma only means that the Down-Gamma of the position is negative. If I am right, this is coherent with a risk reversal but not with a diagonal ratio spread, the two examples presented by the author. Or it could be coherent provided that the P/L is concave when the asset price is declaning. Basically, to visualize the P/L more information is required is my opinion.

If I am not clear, please, let me know. Thanks for the help.

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  • $\begingroup$ I hope someday someone publishes an annotated edition of this book, with the original text on the left page and explanations of what the author really means, definition of terms, derivations etc. on the right hand page. $\endgroup$
    – nbbo2
    Commented Jul 1 at 11:50
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    $\begingroup$ That would be great, even if by trying to decrypt what he means I'm learning a lot. And what about the dialogues in the "Option Wizard" segments? Those are hard! $\endgroup$
    – Enrico
    Commented Jul 1 at 12:14

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Your understanding is spot on!

Just elaborating more, basis my understanding.

Up-Gamma denotes the rate of change of an option's Delta in response to increases in the underlying asset's price. A portfolio that is long Up-Gamma benefits from rising prices as its Delta becomes more sensitive, amplifying potential profits. This characteristic allows for gains that escalate "more than linearly" with underlying price increases, reflecting a positive convexity in the profit profile.

Conversely, Down-Gamma measures the rate of change of Delta when the underlying price decreases. A position that is short Down-Gamma mitigates losses in declining markets by reducing Delta sensitivity as prices fall. This dampens the impact of downward movements on the portfolio's value, effectively capping potential losses.

Combining a long Up-Gamma position with a short Down-Gamma position creates a strategy that balances these dynamics. Such a strategy benefits from both increased profit potential during market rallies and reduced downside risk during downturns.

This strategic approach is favored in volatile markets where price swings are more pronounced. It provides you with a structured framework to optimize risk-adjusted returns, leveraging the dual benefits of positive convexity during bullish phases and downside protection during bearish phases.

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  • $\begingroup$ Thanks. Is this coherent with a position where we are buying OTM call and selling OTM puts, as the author says? I do not think so since this position has a short "down-gamma", where "short" means simply negative, but the losses could accelerate. Basically I do not understand what is the meaning of "short" for the author. From this example it seems just "negative" $\endgroup$
    – Enrico
    Commented Jul 1 at 8:20

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