Optimal Market Making under Partial Information with General Intensities

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.

KEYWORDS:

• Market making
• high-frequency trading
• algorithmic trading
• stochastic optimal control
• hidden Markov model
• stochastic filtering
• viscosity solutions
• piecewise-deterministic Markov processes

Acknowledgments

The authors would like to thank Álvaro Cartea and Katia Colaneri for helpful discussions on the subject.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1. Also referred to by some authors as micro-price (Cartea, Jaimungal, and Penalva 2015) or efficient price (Delattre, Robert, and Rosenbaum 2013) in the martingale case.

2. The latter paper also considers the multi-asset case.

3. Bayraktar and Ludkovski (2009, Sect.3.3) (arXiv version) and Casgrain and Jaimungal (2019) also study optimal trade execution under partial information, albeit with uncontrolled intensities.

4. This is done at the expense of renouncing to uniqueness in the optimal strategy. We also assume the decay to be ‘fast enough’ in certain cases.

5. That is, with controlled probability measures.

6. Any constant size would work in the same way, but this convention simplifies the notation. See, e.g., Guéant (2017) for some formulas with arbitrary constant size under full information.

7. For simplicity and without loss of generality, only the first alternative is formalized in our model (as done in Guéant (2017), Guéant and Lehalle (2015), Guéant, Lehalle, and Fernandez-Tapia 2013) and this is reflected in the admissible spreads being real-valued. We could also allow for different constraints depending on the sign of the inventory, but we refrain from this to simplify the notation.

8. By possibly enlarging the space, one can consider a counting process $Z$ with no common jumps with $N^{-},N^{+}$ and stochastic intensity ${\lambda }^{\ast }-{\lambda }^{-}-{\lambda }^{+}\ge 0$. Then, the process $Z+N^{-}+N^{+}$ is a Poisson with intensity ${\lambda }^{\ast }$ that dominates $N^{+}+N^{-}$. The claim follows immediately.

9. Such a simpler model can be constructed, for example, as a product of canonical spaces, with the existence of the counting processes with the right intensities proved in Jacod and Protter (1982, Thm.24 and Cor.31). The finite-dimensional result has the same proof as in one dimension, starting from independent Poisson measures.

10. For any càdlàg bounded process $M$(not necessarily adapted) on a filtered probability space $(\mathrm{\Sigma },\mathcal{H},\mathbb{H},\mathbb{P})$ satisfying the usual conditions, the optional projection of $M$ on $\mathbb{H}$ is the unique càdlàg process ${ }^o{M}^{(\mathbb{P},\mathbb{H})}$ such that ${ }^o{M}_t^{(\mathbb{P},\mathbb{H})}={\mathbb{E}}^{\mathbb{P}}[M_t|{\mathcal{H}}_t]$ a.s. for each t. Its existence is guaranteed by the Optional Projection Theorem (see, e.g., Jeanblanc, Yor, and Chesney (2009, p.264) or Nikeghbali (2006), p.357–358).

11. For example, because they are increasing càdlàg processes admitting continuous compensators for (any one of) the physical probabilities (Jacod and Shiryaev 2002, p.70 Prop.1.19 or p.77 Prop.2.9).

12. By ‘close to optimal’ we mean that for each $\epsilon >0$ there exists a strategy such that the supremum in (3.1.2) is attained up to $\epsilon$.

13. Lateral derivatives are considered on the domain border. The strict inequality c < 2 is needed when $\gamma =0$ and spreads are unconstrained.

14. $\mathrm{s}\mathrm{g}\mathrm{n}$ denotes the sign function, with $\mathrm{s}\mathrm{g}\mathrm{n}(0)=0$.

15. As previously mentioned, these are technically only candidates for optimal (or $ϵ$-optimal) strategies. We do not rigorously prove their optimality character here, but merely note that well known results of discrete time dynamic programming, together with the convergence of the discrete solutions of (5.0.1) to the analytical one, suggest that they can be safely used as such.

16. Our numerical findings were indeed consistent with this intuitive statement.

17. Although our martingale problem is non-homogeneous in time, $(Q_t)$ is deterministic, so this does not represent a problem.

18 . In Colaneri et al. (2016a), the support is assumed to be a subset of $(-1,\mathrm{\infty })$, but this is only for a ‘return (or yield) process’ as in their case.

19. This was ultimately a consequence of the decomposition in Assumptions 1.2.2 (i), that allowed for the vanishing factors of the intensities to be secluded as the reference probability intensities.

20. An additional detail now is that Colaneri et al. (2016a, Asm.2.1) is also used in Colaneri et al. (2016a, Lem.4.1), which states that the drift coefficient in equation ((11)) is Lipschitz in the state variable, uniform in time and control. This is routinely verified in our case, under our new assumption: $-\mathrm{\infty }<{\delta }_{\ast }<{\delta }^{\ast }<\mathrm{\infty }$.