Any time you are long gamma, the shape of your P/L curve could be described as a smile shape, or a U shape, or a saucer shape, etc. Any time you are short gamma, your P/L curve could be described as a frown shape, or an upside-down U shape, or upside-down saucer shape. The U or smile or frown or saucer could be tilted to the left or right depending on the position's delta. But any option trader can tell at a glance by looking at a P/L curve if it represents a long-gamma or short-gamma position. My question is this - is there a simple mathematical way of describing a long-gamma (smile) curve vs a short-gamma (frown) curve? Thanks.

Hi Dmo, Your daily pnl is about : 0.5*gamma*ds*ds-theta (gamma could BS's one or a modified one and ds=daily move). So what did you mean when you wrote:"a simple mathematical way of describing a long-gamma (smile) curve vs a short-gamma (frown) curve?"

If I understand what you're asking correctly, they're just opposite positions. So Profit(long gamma) = -1 * Profit(short gamma).

Maw, I'm sure you can answer this question if only I can express what it is I'm asking. So far I haven't done a good job, based on the answers I've gotten. I'm attaching a word file. On page 1 are long gamma P/L curves - all with that characteristic "smile" shape. On page 2 are short-gamma P/L curves, all with the characteristic "frown" shape. So in pointing out what the long-gamma curves have in common, I could say that they all sport a "smile shape," and that the short-gamma curves all have a "frown shape." But if I wanted to avoid using the terms "smile shape" and "frown shape" and substitute a mathematical term, what would it be? In mathematical terms, I guess I could say that the long-gamma curves are such that as you move to the right, the slope of the tangent becomes more and more positive. The opposite is true of the short-gamma curves. But that's long-winded - I'm looking for something brief to describe such a curve.

I guess you may want to express it in terms of convexity. What you claim about the slope of the tangent can be expressed with delta. Would it be something like "positive convexity (long gamma) means an increase in profit wrt absolut delta" ?

Yes, I think "positive convexity" and "negative convexity" are exactly what I'm looking for. Thanks. Your point about delta is exactly what I'm trying to show.

One more question - from a mathematical point of view, is negative convexity (the "frown-shaped" curve) the same thing as concavity? And if so, would a pure mathematician who has no connection with finance be more likely to refer to a "frown-shaped" curve as negative convexity or concavity? Thanks.