Cross the Spread or Wait in Line

A stock is quoted $49.98$ bid, $50.02$ offered, with $5{,}000$ shares showing on each side. You want to buy $5{,}000$ shares and you have a one-minute forecast: with probability $60\%$ the fair value one minute from now is $50.10$ , and with probability $40\%$ it is $49.92$ .

(a) Is it profitable to take the offer at $50.02$ ?

(b) Instead of taking, you could post a passive bid at $49.98$ and wait. Suppose -- realistically -- that the only world in which a seller comes to you is the world in which the price is heading down; that is, your passive bid fills only in the $49.92$ state. What is the expected value of posting?

Answer