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 cal- culation 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.

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.

(a) 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?

(b) 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. 3 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.

(c) Because market makers are strategic that is, they both take into account the behavior of the other when making their own decisions in the end, both of them will want to submit lots of limit orders at exactly the same bid and ask prices. These are the prices you solved in part (b) for market maker 2.

Now, since both market makers subunit at the same prices, price priority becomes irrelevant for ranking the priority of their limit orders. "Time priority" becomes important. When limit order submission becomes possible (the market opens) whoever can submit her limit orders before the other one will pet priority.

Suppose market maker I can invest into colocation: by spending $100, she can move her trading firm to be at the same location with the exchange. As a consequence, every day when trading market opens, her orders will be received 1 second earlier than the other market maker's orders. Will market maker 1 make this investment?

Hint: please use profit-and-loss calculations to support your conclusion. What would the daily trading profits be for market maker 1 if she does, or does not, make the investment? Does the difference justify paying a one-time colocation investment cost of $100

(d) Suppose market maker I has already invested into collocation. Now, suppose market maker 2 can also spend $100 to colocate, after which her orders will be received earlier or later than market maker 1's orders with equal chance, rather than always be 1 second later. (Recall the price-time priority rule determines who gets to trade with the market orders). Will market maker 2 decide to make this investment?

Hint: same as the hint for part (c). Given market maker I has already made the investment, calculate the profits for market maker 2 with or without making the investment, and examine whether the difference is worth the colocation instruct