Question

If you plot the numbers below, you get a “volatility smile”: the
numbers follow one linear slope (the left slope), and then change to
following another linear slope (the right slope).

I have several sets of data like this and want to know where the slope
changes. Notes:

• The slope change usually occurs between points

• I don’t know how many points have the left slope and how many have
  the right slope.

• There’s no guarentee about the sign/magnitude of either slope or
  the relation between the slopes. Each slope may be negative or
  positive, and either might be greater than the other.

• If the slopes are identical, the program should report this as a
  special case.

0.1613 
0.1596 
0.1579 
0.1561 
0.1544 
0.1528 
0.1511 
0.1495 
0.1478 
0.1462 
0.1446 
0.1431 
0.1415 
0.1416 
0.1418 
0.1419 
0.1421 
0.1422 
0.1424 
0.1425 
0.1426 
0.1428 
0.1429 
0.1431 

Answer 1

Slope(X) = f(x) - f(x-1)
Slope2(x) = Slope(x) - Slope(x-1)

you need second one. It shows the speed of changing of the slope itself. (acceleration in terms of physics)
I plotted both graphs in excel and check what i have:

alt text http://img691.imageshack.us/img691/6716/slopes.png

you see peak of slope2? this is the indicator and it can easily be found.