Extending Avellaneda–Stoikov Market Making Beyond a Short Horizon
Summary
The document concerns adapting the Avellaneda–Stoikov market-making model to a longer time horizon. Its question centers on the reservation-price expression, where inventory, risk aversion, volatility, and remaining time shift the indifference price away from the mid price. With a larger horizon, the author worries that this shift may become implausibly large or even produce a negative reservation price for positive inventory, and asks whether the model can be made independent of the horizon or mid price.
The response points to work that develops a rigorous framework and a steady-state approximation for inventory risk. It gives a spread expression based on inventory-dependent solutions to a steady-state value equation, with liquidity-flow intensity and risk aversion as key inputs. It also mentions later research on ergodic formulations and numerical methods for multidimensional problems. The document does not provide implementation guidance or a comparison of these approaches, so applying them requires consulting the cited research.
Key ideas
- In the Avellaneda–Stoikov reservation price, remaining time, inventory, risk aversion, and volatility affect the quote relative to the mid price.
- Increasing the horizon can make the inventory adjustment large in the model discussed.
- A steady-state approximation offers a framework for computing spreads from inventory-dependent value functions.
- The response identifies later work on ergodic formulations and numerical approaches to multidimensional market-making problems.
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# How to propertly change time horizon in Avellaneda-Stoikov model?
# How to propertly change time horizon in Avellaneda-Stoikov model?
I'm working in the Avellaneda-Stoikov implementation using Python. My implementation reproduces the authors' results, but I don't know how to properly adapt the algorithm in order to consider a larger time horizon. From the equation
```
r = s - q * gamma * sigma**2 * (T-t)
```
if we use larger T, the indifference price computed could become too big compared with the mid price s, or even a negative value, when q is positive.
Is there an a-dimensional implementation? How to independize from T? Is there a possibility to independize also from the election of s?
## Answer by lehalle (score 3)
https://quant.stackexchange.com/a/53819
In Dealing with the inventory risk: a solution to the market making problem (preprint available at arxiv) we extend the approximation proposed by Marco and Sasha to a rigorous mathematical framework and provide exact steady-state approximation. The bid-ask spread that should be quoted is
$$\psi(q;k,\gamma):=-\frac{1}{k}\ln\frac{f^0_{q-1}f^0_{q+1}}{(f^0_{q})^2}+\frac{2}{\gamma}\ln\frac{\gamma+k}{k},$$ where $k$ is a characteristic of the liquidity consuming flow, $\gamma$ is the MM risk aversion, and the $f^0_q$ are solution of the steady-state version of the ODE driving the value of accepting an inventory of $q$.
The paper explain all this and has been followed by some others. I especially recommend the ones driven by the two other authors:
- Joaquin Fernandez-Tapia went towards ergodic versions of the problem during his PhD thesis (identifying connections with reinforcement learning), and later to versions for online bets (especially for internet ad banners)
- Olivier Guéant went to more formal versions of the problem, and later to reinforcement learning to numerically solve multi-dimensional versionsShown 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.