Pricing a payoff of one over the terminal stock price

In the Black-Scholes model (spot $S_0$, rate $r$, volatility $\sigma$, no dividends), a derivative pays $1/S_T$ at time $T$. Find its price today.

$\dfrac{1}{S_0}e^{(\sigma^2-2r)T}$. Under the risk-neutral measure $\ln S_T$ is normal with mean $\ln S_0+(r-\tfrac12\sigma^2)T$ and variance $\sigma^2T$, so $\mathbb{E}[1/S_T]=\tfrac{1}{S_0}e^{-(r-\sigma^2/2)T+\sigma^2T/2}=\tfrac{1}{S_0}e^{(\sigma^2-r)T}$. Discounting by $e^{-rT}$ gives the result.