Differentiating a Payoff
Okay this is probably going to be an extremely easy/straightforward question but I thought I should post it here just to double check. Suppose I have a payoff $ \Phi = (S_{T}-K)^{+} $ . Now let’s say I now have an equation: $ u = s\partial_{s}\Phi - \Phi $ , this means that given a payoff $ \Phi $ as given above then, substituting this payoff into the equation and assuming $ S_T = S_{0}\exp((r-1/2)T+\sigma\sqrt{T}Z_i)) $ then I should get:
$ u = max(S_{T},0) - max(S_{T}-K,0) $ , right?
And from this equation, the possible solutions should be:
If $ S_{T} > K $ , $ u = K $ , if $ S_{T} < K $ and $ S_{T} > 0 $ , $ u = S_{T} $ , and if $ S_{T} < K $ and $ S_{T} < 0 $ then $ u = 0 $ .
Is all of this correct? I know this is really trivial but I just thought I should check…
$$ s\partial_{s}\Phi = S_T I_{S_T>K}. $$ so no.
(I am not absolutely sure whether you want to differentiate wrt S_T or S_0 however.)