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. Suppose in the current limit order book, only market maker 1 has submitted limited orders. Specifically, market maker 1 has limit buy orders of 1,000,000 shares at the price of $9 and limit sell orders of 1,000,000 shares at price of $12.^[2]Now it is market maker 2s opportunity to submit limit orders. To maximize profits, at what prices (out of all possible integer prices ...,$8, $9, $10, $11, ...) should market maker 2 submit limit buy and sell orders?

Hint: remember that market maker 2s limit orders will not displace market maker 1s limit orders. If both market makers submit limit orders at the same prices, because market maker 1s orders have been submitted earlier, due to the price-time priority, market maker 2s orders will not be executed.

[2] In other words, market maker 1s limit orders will be depleted any time soon.