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What is the reason (better if it is intuitive, and not too math heavy), that when we talk of Greeks, we consider second derivative with respect to price (gamma), but only first derivative with respect to time (theta) and volatility (vega).

At least, most brokerage platforms only publish values for these Greeks. Why not consider the second derivative with respect to time and volatility? Are they not important?

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I would consider the financial applications of the Greeks: hedging. The "main" greeks, viz. Delta, Gamma, Theta, Vega and Rho, all have intuitive financial meanings.

Gamma is the rate of change of your Delta (how many shares of stock to own) with respect to the stock price, so a high Gamma implies you will be rebalancing in large quantities (often happens near expiration, at least in theory) - probably an undesireable situation.

Consider, instead, $\frac{\partial \theta}{\partial t}$, i.e. the second derivative of price w.r.t. time. What sort of financial application might this have? By itself, $\theta$ tells us the rate of change of the option price w.r.t. time, and so $\frac{\partial \theta}{\partial t}$ would be the "acceleration" of the price. While this has a nice physical meaning, it isn't of much use from a hedging perspective, in that it isn't explicitly used to calculate hedges.

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