Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2021 / 2 / d

If $p>0$ and $K>0$, the payoff is positive exactly when $S_T>k:=K^{1/p}$. Splitting it into two truncated moments givesFor $p=0$, the payoff is the constant $(1-K{)}^{+}$, so its price is $e^{-r\tau }(1-K{)}^{+}$. For $K=0$ and $p>0$, its price is $e^{-r\tau }{s}^p{M}_p(\tau )$. These boundary cases avoid taking $K^{1/p}$ when $p=0$.