Estimating Market-Maker Spreads from Competing Quotes
Summary
The document develops a statistical way to relate an individual market maker’s quoted spread to the best executable bid–ask spread observed across multiple makers. It assumes each maker uses the same unknown spread and that uncertainty in each maker’s estimate of the market midpoint follows a normal distribution. Competition then selects the lowest ask and highest bid, narrowing the tradeable spread relative to an individual quote.
The answer gives expressions for one, two, and three makers, then uses an extreme-value approximation to generalize the expected spread reduction as the number of makers increases. It illustrates the implication with an interest-rate swap example, suggesting individual quotes could be wider than the resulting market spread. These are model-based expectations, not a procedure for recovering an exact spread from an arbitrary order book. The shared-spread assumption, normal distribution, and treatment of makers’ midpoint uncertainty are strong simplifications; real quotes may differ and depend on inventory, liquidity, and market conditions.
Key ideas
- The model assumes all market makers quote the same unknown individual spread.
- It represents uncertainty in makers’ midpoints with a normal distribution.
- The best bid and ask among competing makers can produce a narrower tradeable spread.
- An extreme-value approximation estimates how spread compression changes with maker count.
- The estimates depend on simplified assumptions and do not identify exact quotes in every order book.
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Full text
# Estimating the spread of a market maker
# Estimating the spread of a market maker
If we have an order book and we assume that we know there is only one market maker, how can we determine exactly the spread of the market maker? What if there are more than one market makers?
## Answer by Attack68 (score 4)
https://quant.stackexchange.com/a/55703
### Model Building
I will answer this question from a statistical perspective since that has a definitive answer (under stated assumptions)
> assumption 1) Suppose that each market-maker applies the same, unknown, spread, $X$ to quote a specific product.
> assumption 2) Suppose that the uncertainty in the mid-market price of each marker is determined by a normal distribution about a mean with a variance, $\mathcal{N}(\mu, \sigma^2)$
Then the quoted bid-ask spread is determined by:
The expectation of this can be assessed and depends upon the number, $n$, of market makers. This answer: https://math.stackexchange.com/questions/473229/expected-value-for-maximum-of-n-normal-random-variable is helpful. We can then derive the result:
- $n = 1$: the tradeable bid-ask is $X$
- $n = 2$: the expected tradeable bid-ask is: $$\underbrace{\left ( \mu - \sigma \pi^{-\frac{1}{2}} + \frac{X}{2} \right )}_{ask} - \underbrace{\left ( \mu + \sigma \pi^{-\frac{1}{2}} - \frac{X}{2} \right )}_{bid} = X - 2\sigma\pi^{-\frac{1}{2}}$$
- $n = 3$: the expected tradeable bid-ask is: $X - 3\sigma\pi^{-\frac{1}{2}}$
The approximation given in the above link suggests that, in general, the expected tradeable bid-ask is given by:
$$ X - \sqrt{8 log(n)} \sigma$$
### Reflection
I actually think this is quite interesting. Take for example a market I know well - interest rate swaps - where there might be say 10 market makers and the tradeable bid-ask is typically 0.1 bps. If the uncertainty standard deviation is considered the same value then using the above approximation that each market maker would have an individual bid-ask of 0.38bps, which might be a bit wide but not unreasonably.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.