Interpreting Trade-Size Uncertainty in a Kalman Filter Market-Making Model

Summary

The question examines a Kalman-filter price update proposed for market making, where the gain controls how much a new measurement changes the current estimate. It relates estimation and measurement uncertainty to trade size, then asks why the stated uncertainty appears to become negative and how that should affect a quoted spread in a mean-reverting market-making strategy.

The equations as presented contain a likely notation or algebra issue: substituting the stated measurement-uncertainty expression into the gain formula produces the displayed positive expression for the Kalman gain, not for the measurement uncertainty itself. The document does not include an answer, so it does not establish which formula or interpretation was intended, nor recommend a corrected market-making model. It also gives no empirical evidence or guidance on calibrating uncertainty from trade sizes.

Key ideas

• The Kalman gain determines how strongly a new price measurement updates the current estimate.
• The question links uncertainty to trade size and interprets estimation uncertainty as a market-making spread.
• The displayed positive expression follows from substituting the stated relation into the gain formula, not directly as a measurement-uncertainty formula.
• The document leaves the intended equations and any suitable mean-reverting market-making approach unresolved.

Tags

# How is it possible that the measurement uncertainty in Kalman Filter is less than 0?


# How is it possible that the measurement uncertainty in Kalman Filter is less than 0?












In Euan Sinclair's Option Trading, Pricing and Volatility Strategies and Techniques, it mentions that the true value of the price can be estimated via Kalman Filter:

$$S_\mathrm{new} = S + k (S_b − S),$$

where $S_\mathrm{new}$ denotes the new price, $S$ the estimation of the price, $S_b$ the new measurement and $k$ the Kalman gain.

The Kalman gain is defined as $k=\frac{e^2}{e^2 + m^2}$, where $m$ denotes the measurement uncertainty and $e$ the estimation uncertainty.

The estimation uncertainty is a function of trade size, $T$, i.e. $m = e \left( \frac{T_\mathrm{max}}{T} - 1\right)$.

When e is placed into Kalman Gain equation, we get:

$m = ( \frac{1}{1+(\frac{T_\mathrm{max}}{T}-1)^2})$.

However, if we plot this equation and also the estimate uncertainty as a function of T:

It doesnt make sense that the estimation uncertainty goes below 0 since estimation uncertainty is the spread we want to quote in the market as a market maker. And Kalman gain tells us how much to care about the new measurement. As you can see Kalman gain is around 0.5 when uncertainty goes below zero which also does not make sense to me.

Is this a problem in the equation or am I just misunderstanding? Also if it is a problem, what is a better way of using Kalman filters in mean reversing market making?

Edit:

Also the graph should be measurement uncertainty vs T. I am sorry

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