Hi Mugono,
We have an approximative equality as follows:
c_new ~ c_old + Delta*(S_new - S_old) + 0.5*Gamma*(S_new - S_old)^2
S_old - old current share price
S_new - hypothetical new current share price to test the sensitivity of the derivative
c_old & c_new - call prices respectively
Delta = first partial derivative of c with respect to S_t
Gamma = second partial derivative of c with respect to (S_t)^2
So Gamma is allowed to be negative and still creating positive c_new price. For example (hypothetical - I do not know whether it is really possible to reach this....):
S_old = 100, S_new = 99
Delta = 0.01, Gamma = -0.01
c_old = 2
Then we would have c_new = 2 - 0.01 -0.005 = 1.985 still positive.
What do you think?
With the approximation above I see it as follows:
c_new ~ c_old + Delta*(S_new - S_old)
This part gives a linear relationship in the current option price vs. current share price plain. The sign of the Gamma then decides what curvature will c_new have against the current share price. And that, the curvature, is basically driven by the risk profile of the investors. Therefore I think, as I wrote above, that theoretically for the market with risk-seeking investors Gamma for some options might be negative for some rare cases (with Delta still positive).
...after exam days... it was very tough...
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