Closed-form Approximations in Multi-asset Market Making

ABSTRACT

A large proportion of market making models derive from the seminal model of Avellaneda and Stoikov. The numerical approximation of the value function and the optimal quotes in these models remains a challenge when the number of assets is large. In this article, we propose closed-form approximations for the value functions of many multi-asset extensions of the Avellaneda–Stoikov model. These approximations or proxies can be used (i) as heuristic evaluation functions, (ii) as initial value functions in reinforcement learning algorithms, and/or (iii) directly to design quoting strategies through a greedy approach. Regarding the latter, our results lead to new and easily interpretable closed-form approximations for the optimal quotes, both in the finite-horizon case and in the asymptotic (ergodic) regime.

KEYWORDS:

• Algorithmic trading
• market making
• stochastic optimal control
• closed-form approximations
• Monte-Carlo methods

Acknowledgments

The authors would like to thank Bastien Baldacci (Ecole Polytechnique) and Iuliia Manziuk (Ecole Polytechnique) for their careful reading of an initial version of the paper. Olivier Guéant would like to thank the Research Initiative ‘Nouveaux traitements pour les données lacunaires issues des activités de crédit’ financed by BNP Paribas under the aegis of the Europlace Institute of Finance for its support. A special thank goes to Laurent Carlier (BNP Paribas) for his vivid interest in academic questions around market making throughout the years.

Disclosure statement

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

Notes

1. There is an economic literature on market making, for instance the seminal paper (Grossman and Miller 1988) by Grossman and Miller. The results in this literature are, however, more interesting for understanding the price formation process than for building market making algorithms.

2. There was also from the very beginning a focus on options markets – see for instance (Stoikov and Sağlam 2009) (cf. (Baldacci, Bergault, and Guéant 2021) and (Aoud and Abergel 2015) for more recent papers).

3. For market making, the influence of the parameters has already been studied in (Guéant, Lehalle, and Fernandez-Tapia 2013) (one-asset case) and (Guéant 2017) (multi-asset case).

4. $Q^i$ is assumed to be a multiple of $z^i$. It corresponds to the risk limit of the market maker for asset $i$.

5. It is often assumed in the literature that the point processes are independent of the Brownian motions. In that case, the quote processes $({\delta }_t^{i,b}{)}_{t\in {\mathbb{R}}_{+}}$ and $({\delta }_t^{i,a}{)}_{t\in {\mathbb{R}}_{+}}$ have to be independent of prices. In fact, the optimal control problem can be written in a weak form to show that this assumption in not necessary – see Appendix A for more details on the construction of the processes in that case.

6. In the literature there is sometimes a penalty function applied to the inventory at terminal time $T$ to ‘force’ liquidation. Here, as we shall focus on the asymptotic regime of the optimal quotes, there is no point considering such a penalty. However, it is noteworthy that most of our non-asymptotic results could be generalized to the case of a quadratic terminal penalty.

7. Given a positive number $z\in {\mathbb{R}}_{+}^{\ast }$, $z\mathbb{Z}$ denotes the set of multiples of $z$, i.e., $z\mathbb{Z}=\{\dots ,-2z,-z,0,z,2z,\dots \}.$.

8. It is noteworthy that our definition of $H_{\xi }^{i,b}$ and $H_{\xi }^{i,a}$ differs from that of (Guéant 2017) (by a factor $z^i$). The alternative definition we use in this paper is also that of (Bergault and Guéant 2020) for $\xi =0$.

9. The true optimal quotes correspond to the greedy strategy with respect to the value function $u$ (in Model A) or $v$ (in Model B) deduced from the true $\theta$.

10. We omit the subscript $\xi$ in the definition of $\stackrel{⌣}{H}{ }^{i,b}$ and $\stackrel{⌣}{H}{ }^{i,a}$. In particular, although the subscript $\xi$ is not written, the coefficients ${\alpha }_0^{i,b}$, ${\alpha }_1^{i,b}$, ${\alpha }_2^{i,b}$, ${\alpha }_0^{i,a}$, ${\alpha }_1^{i,a}$, and ${\alpha }_2^{i,a}$ do depend on $\xi$.

11. $S_d^{+}$ (resp. $S_d{ }^{++}$) is throughout this paper the set of positive semi-definite (resp. definite) symmetric $d$-by-$d$ matrices.

12. It is noteworthy that in the case of Model B the bid-ask spread is a nondecreasing function of the risk aversion parameter $\gamma$.

13. There can also be tiers to proxy the existence of trading platforms with different clients and/or different costs.

14. $\omega$ is omitted in what follows.

15. This additional constraint of a fixed lower bound is just a technical one to be able to state theorems in the general case where request sizes are distributed (Bergault and Guéant 2020).

16. Proportional transaction costs can be considered in the initial model through shifts in the intensity functions.

17. The results would be similar for Model A.