Deriving the Intensity Parameter in an Avellaneda Market-Making Model

Summary

The document explains the origin of the constant K in a derivation of order-fill probabilities used in an Avellaneda market-making setting. It starts from a stated relationship in which price change is proportional to the logarithm of market-order size. Replacing the proportionality with an equality introduces a constant, and rearranging the inequality for a price move produces the K-scaled threshold inside the probability expression.

The explanation clarifies the algebraic role of K: it is the reciprocal of the proportionality constant linking price change and log order size. This helps interpret the parameter in the displayed derivation, but does not provide an empirical procedure for estimating it, validate the assumed price-impact relationship, or develop the broader market-making model. The document therefore offers a narrow derivation rather than evidence that the model or its fill probabilities are accurate in a particular market.

Key ideas

• The derivation assumes price change is proportional to the logarithm of market-order size.
• Introducing a proportionality constant turns that relationship into an equality.
• The parameter K is defined as the reciprocal of the proportionality constant.
• Rearranging the relationship places K times the price threshold inside the fill-probability expression.

Tags

# Finding the trading intensity - Avellaneda Market Making


# Finding the trading intensity - Avellaneda Market Making












Where does the K term come from in Avellaneda's description of finding the probability an order gets filled. Please see the image below

## Answer by Pleb (score 0, accepted)

https://quant.stackexchange.com/a/74080

#### It is a consequence of (direct) proportionality from equation (2.10):

Equation (2.10) states that the change in price ($\Delta p)$ is proportional to the logarithm of the market order size $Q$:

$$ \Delta p \propto \ln(Q). $$

From the definition of direct proportionality, $\Delta p$ is directly proportional to $\ln(Q)$ if there exists a non-zero constant $c$ such that:

$$ \Delta p = c \cdot\ln(Q) $$ where $c$ is known as the proportionality constant. Replacing $\Delta p$ with $c \cdot\ln(Q)$, dividing with $1/c$ on both sides and setting $1/c = K$, yields the result of the second equation in the derivation:

\begin{align*} \mathbb{P}\left(\Delta p > \delta\right) &= \mathbb{P}\left(\ln(Q) \cdot c > \delta\right)\\ &=\mathbb{P}\left(\ln(Q) > \frac{1}{c} \delta\right)\\ &=\mathbb{P}\left(\ln(Q) > K \delta\right). \end{align*}

A discussion regarding estimation of the model can be found here.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.