# 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

### 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.

• Basically I need to fit 1/c to make deltaP proportional to c*ln(Q), based on historical data? Commented Dec 14, 2022 at 18:52
• @OscarMorales You would need to fit the end result of the equation ie. $\lambda(\delta)=A \exp(-k \delta)$, with $k$ and $A$ be defined as above. You then try to estimate the params in this equation, $\alpha$, $K$, $\Lambda$ (AFAIK). I answer the question of where $K$ originates from, which might be from the definition of proportionality. Remember that 1/c is still a constant (for $c$ being a constant) and can be redefined as $K$ to fit the premise of the above equation. I've only read the paper and never tried to fit the model. Therefore, I don't have much insight into this premise.
– Pleb
Commented Dec 14, 2022 at 19:28
• Got it this makes sense, thank you. Commented Dec 14, 2022 at 20:50