Mathematical problems in algorithmic trading and financial regulation

We study algorithmic trading strategies in order driven markets. We make three contributions to the literature. One, we show how a market maker employs information about the momentum in the price of the asset to design liquidity provision strategies. The momentum in the midprice of the asset depends on the arrival of liquidity taking orders and the arrival of news. Buy market orders (MOs) exert a short-lived upward pressure on the midprice and sell MOs exert a downward pressure of the price. We employ high-frequency data to estimate model parameters and show the performance of the market making strategy. Two, we model the trading strategy of an investor who spoofs the limit order book (LOB) to increase the revenue she obtains from selling a position in an asset. The strategy employs, in addition to sell limit orders (LOs) and sell market orders (MOs), a large number of spoof buy LOs to manipulate the volume imbalance of the LOB. Our results show that spoofing considerably increases the revenues from liquidating a position. The spoof strategy employs, on average, fewer sell MOs (than a strategy without spoof LOs) and from executing roundtrip trades that are initiated by buy spoof LOs that are inadvertently filled and subsequently unwound with sell LOs. Spoofing is illegal and difficult to detect. We show that as the financial penalty for spoofing increases, the spoof strategy relies less on spoof LOs. There is a critical point where the gains from spoofing are outweighed by the financial penalty, so it is optimal no not to spoof the LOB. Three, we show how the supply of liquidity in order driven markets is affected if LOs are forced to rest in the LOB for a minimum resting time (MRT) before they can be cancelled. The bid-ask spread increases as the MRT increases because market makers (MMs) increase the depth of their LOs to protect them from being picked off by other traders. The expected profits of the MMs increase when the MRT increases. The intuition is as follows. As the MRT increases, there are two opposing forces at work. (i) The longer the MRT, the more likely the LOs are to be filled and, on average, shares are sold at a loss. (ii) because the depth of the posted LOs increases, the probability that the LO is picked off by other traders before the end of the MRT decreases. The net effect is that a longer MRT leads to a higher expected profit. We also show that the depth of LOs increases when the volatility of the price of the asset increases. Also, the depth of LOs increases when the arrival rate of market orders increases because it is less likely that LOs will be picked off by the end of the MRT. Finally, our model also makes predictions about the overall liquidity of the market. We show that MMs choose to supply the minimum amount of shares per LO allowed by the exchange because expected profits are maximised when liquidity provided is lowest