N1

Given, for a fixed expiryT, the set of put prices for all strikesK: Put(K) we defined the implied distribution,pdf(u), as the density function forS(T)that satisfies this relationship for all K: \[ \operatorname{Put}(K)=\exp (-r(T-t)) \mathcal{S}_{0}^{K}(K-u) \operatorname{pdf}(u) d u \] Show that theE[S(T)]under this density function equals the forward price of the stock to delivery dateT. Hint: First use this density to price out a forward contract with invoice price IP and delivery date T.