I promise this question isn't as long as it looks, it's just wordy. Please HELP and if more info is needed I have a fast response rate. Thank you, it's so appreciated.
Question 3 (high-frequency trading (HFT) arms-race). You want to make sure you understand
how trading on limit order book works before attempting this question.
Suppose there is one security being traded on an exchange where trading operates via a limit order book. The exchange opens at 9 am every day and closes at 4 pm. There is no uncertainty about the value of the security: each share is worth $10.5. Suppose there are two HFTs, 1 and 2, who compete by submitting limit orders. Other traders, collectively, submit one market order for one share every minute. Each market order buys or sells with equal probability.
The goal of HFTs is to maximize expected profits. HFT profit calculation goes as follows: if my limit buy order of Q shares at a price $P is traded (met by a market sell order), my profit (possibly negative) is Q ×$(10.5 −P ).4 Symmetrically, If my limit sell order of Q shares at a price $P traded, then my profit (possibly negative)
is Q ×$(P −10.5). The exchange rule says that the minimum tick size is $1, so limit orders can only be submitted at integer prices (...$8, $9, $10, $11, ...). When market orders arrive, the execution of limit orders follows the “price-time priority” principle.
(a) (1 point) Out of all possible prices, what is the lowest price at which HFTs are willing to sell (so that they will not suffer losses when trading)? What is the highest price at which HFTs are willing to buy?
(b) (1 point) Let’s start thinking about how HFTs compete with each other. Suppose that the current limit order book only has limits orders already submitted by HFT 1: limit buy orders of 1,000,000 shares at the price of $9 and limit sell orders of 1,000,000 shares at a price of $12. Now it is HFT 2’s opportunity to submit limit orders. To make profits, at what prices should HFT 2 submit limit buy and sell orders?
Hint: what would happen if HFT 2 submits limit orders at the same prices ($9 and $12)? Well, HFT 1’s orders will take priority due to earlier submission. Further, because HFT 1’s orders are larger and won’t be depleted any time soon (1,000,000 shares!), for all practical purposes, submitting at the same price means that HFT 2’s orders won’t get a chance to be executed.
(c) (1 point) Because HFTs 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 end up submitting lots of limit orders at exactly the same bid and ask prices — the prices you solved in part (b).
Since both HFTs submit at the same prices, “price priority” becomes irrelevant for who wins the business. “Time priority” becomes important. When limit order submission becomes possible (the market opens at 9 am each day), whoever can submit her limit orders before the other one — even if just by a split of a second — will get priority. However, because both HFTs have access to the same technology, none of them have an advantage over the other. Therefore, on half of the days, HFT 1 manages to be faster (and gets all the business on that day), while HFT 2 is faster the other half of the time.
Question: suppose HFT 1 can invest in colocation: by spending $100, she can move her trading firm to be at the same location as the exchange. As a consequence, when the exchange opens, her orders will be received 1 second earlier than the other HFT’s orders. Is this a profitable investment?
Hint: Please use profit-and-loss calculations to support your conclusion. What
would the expected profits be for HFT 1 if she does, or does not, make the in-
vestment? Does the difference justify paying a one-time colocation investment
cost of $100?(d) (1 point) Suppose HFT 1 has already invested in collocation. Now, suppose HFT 2 can also spend $100 to colocate, after which her orders will again be received earlier than HFT 1’s orders with half chance, rather than always being 1 second later. Is this a profitable investment for HFT 2?
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