1. Question 2 (high-frequency trading arms-race).

Suppose there are two market makers, 1 and 2, who compete by submitting limit orders in the same security. There is zero uncertainty about the value of the security: each share is worth $10.5.

The goal of market makers is to maximize expected profits. Market maker profit calculation goes like this: if ones limit buy order of Q shares at price $P is traded (met by a market order), then the profit (possibly negative) is Q $(10.5 P). If ones limit sell order of Q shares at price $P traded, then the profit is Q $(P 10.5).

The exchange rule says that minimum tick size is $1, so limit orders can only be submitted at integer prices (...$8, $9, $10, $11, ...). When market orders arrive, the priority of limit order executions follow the price-time priority principle covered in lecture 14.^[1]

The total volume of trading: one market order arrives every minute and trades one share. Each market order buys or sells with equal probability. Recall that market makers can make profits only when their limit orders trade against market orders.

1. Out of all possible integer prices, what is the lowest price at which market makers are willing to sell at? What is the highest price at which market makers are willing to buy at?

[1] For instance, when a market sell order arrives, suppose there are two limit buy orders. If they have different prices, then the one with more favorable price is executed first. If the two limit orders have the same prices, then the one that is submitted earlier is executed first.