# Market Making and Mean Reversion

The paper by Tanmoy Chakraborty and Michael Kearns, “Market Making and Mean Reversion”, June 2011 (the “CK Paper” and the authors “CK“) purportedly proved that market making is generally profitable on mean reverting time series without requiring additional assumptions about the stochastic price process. Some journalistic websites (such as quantnews.com) referenced this paper as proof that market making is mathematically certain to make money when prices are mean reverting, even if only to a slight degree.

Here I show, using the same basic market-making algorithm in the CK Paper but with a somewhat more realistic model of how markets work, that market making in the manner envisaged by CK is almost always unprofitable. This is so regardless of the mean reversion present in the time series and even when generous assumptions are made regarding the fill rates of the limit orders.

The CK Paper assumes a market exists in which a security can be bought and sold at prices determined exogenously by a single time series , where . A buy (respectively sell) limit order placed in this market at time and price will be filled at the earliest time such that (respectively for sell orders). This means the limit order will be filled once the limit price is touched or penetrated by .

Since prices in reality are quoted in integral multiples of a minimum tick (e.g. one penny for most U.S. stocks or 0.25 for E-mini S&P 500 future contracts), can be assumed to be a positive integer by rescaling the unit of money and expressing price in number of ticks. The CK Paper considers the following market-making algorithm: ladders of buy limit orders at and sell limit orders at are placed at time , where the ladder depth is a positive integer chosen such that the price fluctuation shall not exceed for all (this is possible in practice if the time step is small enough). Unfilled orders are canceled and new orders are placed at every time step, which is necessary only if .

CK shows that the following result holds:

Let and . If the above market making algorithm is applied with order size of one share for each buy and sell order, then at time the inventory is , and the profit is .

Intuitively, if spends a lot of time around its initial price and only goes off a bit at the end, then should be much larger than and therefore the market making profit should be positive. CK formalizes this intuition by proving that the expected profit of the market making algorithm is positive for any biased random walk in which the probability of a positive price change is greater (respectively smaller) if the current price is below (respectively above) a given equilibrium level. CK also shows that for the Ornstein-Unlenbeck process, the expectation grows linearly with while is bounded by a constant, implying that the algorithm will eventually be profitable if it is traded long enough.

Real markets are of course a far cry from the one assumed by CK. Still one would hope the conclusions regarding the profitability of market making would remain largely valid even after accounting for bid/ask spreads, transaction costs, etc. Unfortunately this is not the case as I show using a simple argument below. In fact, it appears that bid/ask spreads exist precisely to defeat the CK type of market making algorithms, rendering them unprofitable for any price time series, mean reverting or not.

We assume, like CK but more realistically, that prices at which transactions occur are determined exogenously, except now there are two exogenous time series, viz. bid and ask prices. Let and be the bid and ask series respectively. Assume , i.e. the bid/ask spread is constant and equal to 1. At time , the market making algorithm observes and and places a ladder of buy limit orders at and another of sell limit orders at , where is a positive integer chosen such that and for all . Thus the market making algorithm joins the bid and ask at the inside market and brackets it sufficiently on both sides to ensure all price fluctuations are captured. We assume that a buy (respectively sell) limit order placed at time and price will be filled at the earliest time such that (respectively for sell orders). This means the buy order (respectively sell order) will be filled once its limit price is touched or penetrated by the ask price (respectively bid price).