Algorithmic market making for options

Abstract

In this article, we tackle the problem of a market maker in charge of a book of options on a single liquid underlying asset. By using an approximation of the portfolio in terms of its vega, we show that the seemingly high-dimensional stochastic optimal control problem of an option market maker is in fact tractable. More precisely, when volatility is modeled using a classical stochastic volatility model—e.g. the Heston model—the problem faced by an option market maker is characterized by a low-dimensional functional equation that can be solved numerically using a Euler scheme along with interpolation techniques, even for large portfolios. In order to illustrate our findings, numerical examples are provided.

Keywords:

• Market making
• Algorithmic trading
• Options
• Stochastic optimal control

Acknowledgements

Bastien Baldacci gratefully acknowledges the support of the ERC Grant 679836 Staqamof. Olivier Guéant thanks the Research Initiative “Modélisation des marchés actions, obligations et dérivés” financed by HSBC France under the aegis of the Europlace Institute of Finance for their support regarding an early version of the paper (entitled “Algorithmic market making: the case of equity derivatives”). The authors would like to thank Lorenzo Bergomi (Société Générale), Bruno Bouchard (Université Paris-Dauphine), Rama Cont (Oxford University), Renaud Delloye (HSBC), Thomas de Garidel (HSBC), Nicolas Grandchamp des Raux (HSBC), Iuliia Manziuk (Université Paris 1 Panthéon-Sorbonne), Ben Nasatyr (Citi), Jiang Pu (Institut Europlace de Finance), and Mathieu Rosenbaum (Ecole Polytechnique) for the discussions they had on the topic. The readers should nevertheless be aware that the views, thoughts, and opinions expressed in the text belong solely to the authors.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 The models are also suited for assets traded in limit order books if the bid-ask spread to tick size ratio is large.

2 For references, see for instance Gatheral (2011).

3 In applications, we always choose ${\delta }_{\mathrm{\infty }}$ negative enough so that this lower bound is never binding.

4 See Appendix A.3 for more details on the construction of those processes.

5 The frontier of this set defines the risk limits of the market maker.

6 In Appendix A.2 we propose a method to relax the constant-vega assumption. This method is based on a Taylor expansion around the constant-vega case. The curse of dimensionality is tamed by the reduction of the problem to a Monte-Carlo simulation.

7 This plot has been computed using ${10}^5$ Monte-Carlo simulations for each option.

8 This is only an approximation as trade sizes are in number of options and option prices move.

9 As noted when presenting this paper to a wide audience, our method could easily be extended to the case of multiple underlying assets using the same method as in Bergault and Guéant (2019) if the instantaneous variance processes of the different assets are driven by a few factors.

Additional information

Funding

Bastien Baldacci gratefully acknowledges the support of the ERC Grant 679836 Staqamof. Olivier Guéant thanks the Research Initiative “Modélisation des marchés actions, obligations et dérivés” financed by HSBC France under the aegis of the Europlace Institute of Finance for their support regarding an early version of the paper (entitled “Algorithmic market making: the case of equity derivatives”).