# Optimal market making under partial information with general intensities

## TL;DR

This paper develops a comprehensive model for optimal market making under partial information, accounting for unobservable market regimes, and demonstrates how regime uncertainty impacts optimal quoting strategies and P&L sensitivity.

## Contribution

It introduces a unified, general framework for market making under partial information using stochastic filtering and control, extending existing models and analyzing the effects of regime uncertainty.

## Key findings

- Optimal spreads are biased under regime uncertainty.
- Market maker adjusts for regime uncertainty in P&L sensitivity.
- Longer waiting times increase the impact of regime uncertainty.

## Abstract

Starting from the Avellaneda-Stoikov framework, we consider a market maker who wants to optimally set bid/ask quotes over a finite time horizon, to maximize her expected utility. The intensities of the orders she receives depend not only on the spreads she quotes, but also on unobservable factors modelled by a hidden Markov chain. We tackle this stochastic control problem under partial information with a model that unifies and generalizes many existing ones under full information, combining several risk metrics and constraints, and using general decreasing intensity functionals. We use stochastic filtering, control and piecewise-deterministic Markov processes theory, to reduce the dimensionality of the problem and characterize the reduced value function as the unique continuous viscosity solution of its dynamic programming equation. We then solve the analogous full information problem and compare the results numerically through a concrete example. We show that the optimal full information spreads are biased when the exact market regime is unknown, and the market maker needs to adjust for additional regime uncertainty in terms of P&L sensitivity and observed order flow volatility. This effect becomes higher, the longer the waiting time in between orders.