By: Satyen Kale
In a paper appearing in NIPS 2013 in collaboration with Jacob Abernethy of the University of Michigan, Ann Arbor, we study market makers in financial markets and investigate how the choice of the bid-ask spread influences the payoff of market makers. We also provide an algorithm to adaptively choose the bid-ask spread to obtain almost as high payoff as that of the best spread-based strategy without knowledge of how prices evolve in the market.
When a trader enters a market, say a stock or commodity market, with the desire to buy or sell a certain quantity of an asset, how is this trader guaranteed to find a counterparty to agree to transact at a reasonable price? This is not a problem in a liquid market, with a deep pool of traders ready to buy or sell at any time, but in a thin market the lack of counterparties can be troublesome. A rushed trader may even be willing to transact at a worse price in exchange for immediate execution.
This is where a market maker (MM) can be quite useful. A MM is any agent that participates in a market by offering to both buy and sell the underlying asset at any time. To put it simply, a MM consistently guarantees liquidity to the marketplace by promising to be counterparty to any trader. The act of market making has both potential benefits and risks. The MM bears the risk of transacting with better-informed traders that may know much more about the movement of the asset’s price, and in such scenarios the MM can take on a large inventory of shares that it may have to offload at a worse price. On the positive side, the MM can profit as a result of the bid-ask spread, the difference between the MM’s buy price and sell price. In other words, if the MM buys 100 shares of a stock from one trader at a price of p, and then immediately sells 100 shares of stock to another trader at a price of p + x, the MM records a profit of 100x.
This describes the central goal of a profitable market making strategy: minimize the inventory risk of large movements in the price while simultaneously aiming to benefit from the bid-ask spread. The MM strategy has a state, which is the current inventory or holdings, receives as input order and price data, and must decide what quantities and what prices to offer in the market.
We assume that at discrete time steps t = 1, 2,…,T, the market maker is provided the current price pt of the financial commodity. The price sequence can be adversarially generated and we make only a modest bounded volatility assumption on it; i.e. the change in price from one time step to the next is bounded by some value.
A spread-based strategy operates in the following intuitive way. It is parameterized by a spread size b. At each time step, it maintains a window [at, at + b], and places buy orders for 1 share each for all price levels below at, and a sell order for 1 share each for each price level above at + b. If the current price pt falls in the window, then no orders are executed and the window doesn’t move:
If the price is lower than at, then at – pt buy orders are executed, and the window shifts so that the price lies in the window again:
Finally, if the price is higher than at + b, then pt – (at + b) sell orders are executed, and the window shifts so that the price lies in the window again:
The upshot of moving the window is that any share that is bought at some price is immediately offered for sale at a price b units higher; and so the market maker stands to make a profit of b on each such transaction. A similar argument holds when a share is bought; the market maker again stands to make a profit of b.
The downside is that if the price sequence fluctuates very little and spends most of its time in the window, then the market maker makes no profit at all.
Characterization of payoff
Our first theorem is a characterization of the payoff of a spread based strategy.
Theorem: the payoff of a strategy using a spread size of b is at least
This clearly shows that increasing b means that the market maker stands to make more profit; but on the other hand the window might not shift much and so the |at+1 – at| term might be small.
Learning the best spread
Now that we have seen that the tradeoff between spread-size and payoff, one might want to adaptively learn the best spread for a price sequence as it unfolds. In particular, fix a set B of different spread sizes. Our goal is to design an algorithm that adaptively chooses the spread so as to obtain payoff that is almost as high as the that of the best spread in the set B. Performance is measured in terms of regret, which is the difference between the payoff of the adaptive algorithm and the payoff of the best spread-based strategy in hindsight.
Treating the strategies as experts, one might be tempted to use one of the well-known algorithms for learning from expert advice (such as Multiplicative Weights (MW), Follow the Perturbed Leader (FPL), etc). However, there is a significant challenge to their direct application. All these algorithms assume that the experts have no state, whereas in our setting the spread-based strategies do have significant state: viz., the amount of stock they’re holding (this could be positive or negative).
In our paper we prove certain combinatorial properties of spread-based strategies, which allows us to show that we can actually change state from one strategy to another with bounded cost (this is where the bounded price volatility assumption is used). This means that we can apply standard expert learning algorithms such as MW and FPL, as long as we can prove that these algorithms change experts infrequently. This is easy to prove for MW and FPL, and this allows us to prove the following:
Theorem: under the bounded price volatility assumption, there is an algorithm to adaptively choose spread sizes with regret bounded by O(√T) after T rounds.