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## Friday, June 1, 2012

### 2nd differential is the highest differential we usually need

When I first encountered the concept of the 2nd derivative, I thought maybe people will be equally interested in the 3rd derivative or 4th. Now I feel outside physics (+ math itself), folks mostly use first derivative and 2nd derivative. In classical physics, 2nd derivative is useful -- acceleration. Higher derivatives are less used.

Note on notation. f'' is (inherently) a function in the black-box sense that for each input value, there's an output value. This function derives from the original function f. We write f''(x) in that context. However, f'' can be usefully treated as a independent variable just like x,y and t, so we write f'' without the (x). In this context, we aren't concerned about how f'' depends on x. That dependency might be instrumental in our domain, but at least for the time being we endeavor to ignore that and concentrate on how f'' as an independent variable affects "downstream"

Graphically, 2nd derivative describes concavity or convexity --
^ When f'' is positive, curve is concave upwards, regardless whether f(x) is positive or negative, whether f(x) is rising or falling, whether f'(x) is positive or negative. However, f' is rising for sure.
^ When f'' is Negative, curve is concave Downwards, regardless.

This observation is relevant to portfolio gamma. Take a short put for example. This position's delta is always Positive but Falling with S, towards 0. The PnL graph is concave downward, so this gamma is always Negative. (See https://www.thinkorswim.com/tos/displayPage.tos?webpage=lessonGreeks) It's important to clarify a few points and assumptions implicit in the above context --

* This PnL graph is purely theoretical. Underlier (S) has just one price of \$88 right now, and it won't become \$1 even though the graph includes that price on the S axis.
* PnL graph is about the Current valuation (and PnL) but with Imaginary S prices. It shows "what-if" underlier price S becomes \$1 in the next moment -- that's the meaning of the \$1 on the x-axis.
** However, the most useful part of the PnL curve is the region around the current S -- \$88. This region reveals our sensitivity to underlier moves. It shows how much our short put valuation (and PnL) would gain or suffer When (not "if") underlier moves a tiny bit in the next moment.

* The delta curve is purely theoretical. At the current S = \$88, our delta is, say 0.51 or 0.47 or whatever. It won't suddenly become 0.01 even though you may see this delta value at a high S. That 0.01 delta means "if S becomes so high tomorrow, our delta would be 0.01"

* there's no "evolution over time" depicted in any of these graphs. Time is not an axis. These curves are pure mathematical functions describing "what if S is at this level". In this sense the delta curve is very similar to the price/yield curve. Even the standard yield curve and forward curve are similarly Unrelated to so-called time-series graphs.

If you are confused about "on the far right put is OTM, but on a smile curve OTM puts are on the far Left", read my other blog posts about OTM put.