Market makers and liquidity

A stock chart shows one price per moment, but nobody trades at "the price". A buyer pays one number, a seller receives a lower one, and between the two sits a firm that stands ready to take either side. This lesson starts the track on market microstructure by explaining what that firm earns, what it risks, and how to measure the gap between the two prices. It uses returns from "Returns and volatility" and the idea that price changes behave like a random walk from "Random walks and Brownian motion".

Bid, ask, mid and spread

A market quotes two prices at any moment. The bid $b$ is the highest price at which someone is willing to buy; the ask $a$ is the lowest price at which someone is willing to sell. The bid is always below the ask, otherwise the two sides would already have traded. A trader who wants to buy right now pays the ask, and a trader who wants to sell right now receives the bid.

The mid price is the midpoint, $m = (a + b)/2$, and is the usual stand-in for "the price". The spread is the gap $a - b$. It is quoted in three ways: in currency, in ticks (the smallest allowed price step, so a spread of 4 cents on a 1-cent tick is 4 ticks), and in basis points of the mid, which makes stocks with different prices comparable:

With a bid of 99.98 and an ask of 100.02 the mid is 100.00 and the spread is 0.04, or $10^4 \times 0.04 / 100 = 4$ basis points. A tight spread means trading is cheap; a wide spread means a trader loses more just by crossing from one side to the other.

Liquidity and immediacy

Liquidity is the ability to trade a meaningful size quickly without moving the price much. It comes from a simple mismatch in time. Some traders want to trade now and accept a worse price for it; others are willing to wait and post the price they want. The first group sends market orders, which execute immediately against the best quote on the other side. The second group posts limit orders, which rest in the book until someone trades against them.

The trader who sends the market order is the taker: she buys at the ask or sells at the bid and so pays half the spread relative to the mid. The trader whose limit order was waiting is the maker: he earns that half-spread. Many exchanges reinforce this with fees: the taker pays a small fee per share and the maker receives a smaller rebate, both a fraction of a cent. Either way, the spread is the price of immediacy, and the people who sell immediacy are the market makers.

The round trip

A market maker is a firm that continuously posts both a bid and an ask in the same asset and trades against whoever arrives. It does not want a position; it wants to buy at the bid, sell at the ask, and repeat. Take one round trip: a customer buys $q$ shares at the ask, so the maker is now short $q$, and later another customer sells $q$ shares at the bid, which closes the position. Suppose the maker keeps its quotes symmetric around the mid, $a = m + h$ and $b = m - h$, where $h$ is the half-spread. If the mid is $m_1$ at the first trade and $m_2$ at the second, the maker sold at $m_1 + h$ and bought back at $m_2 - h$, so per share

If the mid has not moved, the maker keeps the whole spread $a - b$. If the mid has moved, the maker also gains or loses $\Delta m$ per share on the position it carried, with a minus sign because it was short: a rising mid hurts. The first term is spread income; the second is inventory P&L. The maker controls the first by choosing the spread and has little control over the second.

The culprit in the last step is the position the maker carried through the move, not the spread. A 4 bp spread earns 40 USD on this size, while a 10-cent move of the mid, which is only 10 bp, is enough to erase it two and a half times over.

Inventory risk and adverse selection

Two separate risks sit behind the inventory term. The first is inventory risk. While the maker is short, its profit depends on the mid price, and the mid moves unpredictably. If the mid's change per step has standard deviation $\sigma$ and steps are independent, as in a random walk, the change over $k$ steps has standard deviation $\sigma\sqrt{k}$. The longer the maker holds a position, the more that risk grows, and it grows with the size of the position too. Later lessons in this track develop what a maker does about it.

The second is adverse selection: the customer may know something. If a buyer knows the price is about to rise, the maker sells exactly before the mid moves up, and the loss is not luck but a consequence of who chose to trade. A maker cannot see who is informed, so it must charge every customer a spread wide enough to cover the losses to the informed ones. Later lessons develop it; here it is enough to see that the spread pays for both risks, not just for waiting.

The figure below plays out the round trip. A mid price follows a random walk for 30 steps; the maker always quotes half-spread $h$ around the current mid; a customer buys at the ask at step 5 and another sells at the bid at step 20, for 100 shares. Drag the ask line, or use the half-spread slider, and watch the spread income grow in proportion. Then raise the volatility, press New path a few times and compare the sign of the inventory P&L over different paths. At the defaults (half-spread 2¢, volatility 1¢ per step, first path) the spread income is 4.00 USD, the mid happens to rise 4.47¢ between the two trades, the inventory P&L is −4.47 USD and the round trip ends at −0.47 USD. Over 15 steps of 1¢ the standard deviation of the mid's change is $\sqrt{15} \approx 3.9$¢, about as large as the whole spread, so a typical single round trip is close to a coin flip around the income.

Types of market maker

Some market makers carry an obligation: a designated market maker on an exchange agrees to quote continuously, within a maximum spread and with a minimum size, and in return gets fee advantages or a privileged role. Most liquidity today comes from electronic liquidity providers, trading firms with no obligation whose software posts and cancels quotes thousands of times a second and leaves when conditions are bad. In over-the-counter markets, such as bonds and currencies, dealers quote to customers on request and keep the position on their own books. The three differ in obligations and in technology, but share the economics above: spread income in return for inventory risk and adverse selection.

Summary

• The bid is the best price to sell at, the ask the best price to buy at, and the spread $a - b$ is the cost of crossing; it is measured in currency, ticks or basis points $10^4 (a - b)/m$.
• Takers pay the spread for immediacy; makers who post limit orders earn it.
• A round trip at symmetric quotes earns $(a - b) - \Delta m$ per share: the spread, minus the rise of the mid while the maker was short.
• The mid's move is inventory risk, which grows like $\sigma\sqrt{k}$ with holding time; informed customers add adverse selection, and the spread must pay for both.

The next lesson, "The limit order book", shows where the quotes of all makers and traders are stored and how a market order consumes them.