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Martingale part of the discontinuous put payoff

I am looking for need the martingale part of the put payoff (therefore not $C^2$). C^2$..). Where $S_t=exp(\sigma W_t -\frac{\sigma^2t}{2})$

$d[(S_t -K)^+ ]$ ??

I guess I need to use local times but how?

Thank you for your help!

show/hide this revision's text 1

Martingale part of put payoff

I am looking for the martingale part of the put payoff (therefore not $C^2$). Where $S_t=exp(\sigma W_t -\frac{\sigma^2t}{2})$

$d[(S_t -K)^+ ]$ ?? I guess I need to use local times but how?

Thank you for your help!