What are market makers paid for?

Question

A market maker quotes a price to buy and a price to sell and earns the gap. What sets that gap? How should quotes move as the market maker builds up inventory? And when a market maker sells an option and hedges it, what is it actually being paid for?

Method

• Quote and survive. A dealer quotes around a price that moves at random. Uninformed orders fill less often the further the quote is from the mid. Informed traders know the price some steps ahead and only trade when that beats the quote. P&L splits exactly into spread capture, next-step pick-off and inventory drift. Each half-spread is scored over 200 simulated runs, and every spread sees the same runs, so differences between spreads are judged with paired standard errors.
• Avellaneda-Stoikov. Quotes centre on a reservation price that moves against inventory, compared with a fixed symmetric spread of the same average width over the same simulated paths.
• Glosten-Milgrom. A one-trade model of why the spread exists. It is a formula, so it needs no simulation here.
• Gamma scalping. A delta-hedged option, priced and hedged at implied vol while the stock moves at realised vol.

A best spread counts as clearly better than another only if z is at least 2.

Results

1. Informed flow and the best spread

Expected P&L against half-spread for the three presets. The presets differ in informed share, how far ahead informed traders see, volatility and news jumps (Calm 5% informed seeing 1 step, News-heavy 20% seeing 3 steps with jumps, Toxic flow 45% seeing 8 steps).

25 half-spreads from 0.5c to 50c, 200 runs each, 500 steps, seed 1.

What it shows. More informed flow lowers expected profit at almost every spread, and turns very tight quoting into a loss. But only far-sighted informed traders clearly move the best spread out: Toxic flow's best spread beats both neighbours. News-heavy's best spread cannot be told apart from the spread below it, which is Calm's best. Calm's own best is only borderline ahead of its lower neighbour. The curves are flat near the top, so the honest answer for News-heavy is “about the same as Calm”, not “wider”.

2. Avellaneda-Stoikov against a fixed spread

Simulating…

Paper settings (s = 100, σ = 2, T = 1, 200 steps, γ = 0.1, k = 1.5, A = 140), 500 paths, seed 20081. The fixed spread uses half the A-S average spread (0.746) on each side.

What it shows. Skewing quotes against inventory keeps inventory near zero, which roughly halves the spread of outcomes for a slightly lower average. The market maker is trading a little expected profit for a lot less risk.

3. Why the spread exists

With no informed traders the spread is zero. Every informed trader adds to it, because the market maker has to win from everyone else what it loses to them. The spread exists to pay for losses to informed traders.

4. What gamma pays, and how good the shortcut is

A long option, delta-hedged, earns when the stock moves more than its implied vol said and pays time decay when it moves less. The usual shortcut freezes gamma at its starting value. Measured against the simulation:

• Realised 15%: simulating…
• Realised 25%: simulating…
• Realised 30%: simulating…

Settings for these lines: 1 long at-the-money call (100 options), spot and strike 100, 30 days, implied vol 20%, r 4%, q 0, 4 rebalances a day, 1,000 paths each, seed 20151.

What it shows. The shortcut has the sign right and is too optimistic: it overstates gains when realised vol is above implied and understates losses when it is below, by more the further the two vols are apart.

Limitations

• Order flow is simulated, not taken from market data.
• No queue priority, fees, rebates or latency, and one unit per fill.
• The informed-trader model is a rule of thumb. Informed traders always trade when their look-ahead beats the quote, and the pick-off line counts only the next step, so part of their cost shows up as inventory drift.
• Avellaneda-Stoikov assumes fills arrive at a rate that falls exponentially with distance from the mid, a Brownian mid with no jumps, and no adverse selection.
• Gamma scalping uses constant volatilities, no smile and no hedging costs.
• The three presets change more than one thing at a time, so they show a direction, not a clean effect of one input.

Conclusion

[Conclusion to be written.]