Optimal Tender Price for a Block Trade

A seller will accept your tender price $P$ for a block of shares (size $Q$) with probability $F(P)$, where $F$ is a strictly increasing CDF on $\mathbb{R}$ with density $f$. If the tender is accepted, your mark-to-market gain is $V - P$, where $V \sim N(\mu, \sigma^2)$ is the true value of the block, independent of $F$. You want to choose $P$ to maximize your expected profit. 1. Write down the expected gain as a function of $P$ and derive the first-order condition (FOC). 2. Rewrite the FOC in terms of the hazard rate $h(P) = \frac{f(P)}{1 - F(P)}$. Interpret the optimality condition economically. 3. Solve explicitly for $P^{*}$ when $F$ is the logistic CDF with location $\mu_0$, scale

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