Question: Let H(x)=1/x be the payoff function for a European style derivative security. Find a closed form expression for the price: $$ u(t,x)=e^{-r(t-t)}E[H(S_T)|S_t=x] $$ for this claim using Black Scholes dynamics, with r and sigma constants.
My attempt: Under Black Scholes dynamics: $$ dS_t=S_t({\sigma}dW_t+rdt) $$ Using Ito's on this, with f(x)=1/x, I get that: $$ \frac 1{S_T} =\frac 1{S_t}exp((\sigma^2-r)(T-t)-\sigma\sqrt{T-t}Z) $$ Where Z is standard normal. Finally, plugging into the pricing formula: $$ V_t=B_tE[B^{-1}_TX|S_t] $$ Which gives me a final answer: $$ \frac 1{S_t}exp((T-t)(\frac32\sigma^2-2r)) $$ However, not sure if this is correct, anyone have any ideas? I don't have the solutions to this question unfortunately..
Cheers