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Dealing with the inventory risk: a solution to the market making problem

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Abstract

Market makers continuously set bid and ask quotes for the stocks they have under consideration. Hence they face a complex optimization problem in which their return, based on the bid-ask spread they quote and the frequency at which they indeed provide liquidity, is challenged by the price risk they bear due to their inventory. In this paper, we consider a stochastic control problem similar to the one introduced by Ho and Stoll (J Fin Econ 9(1): 47–73, 1981) and formalized mathematically by Avellaneda and Stoikov (Quant Fin 8(3):217–224, 2008). The market is modeled using a reference price S t following a Brownian motion with standard deviation σ, arrival rates of buy or sell liquidity-consuming orders depend on the distance to the reference price S t and a market maker maximizes the expected utility of its P&L over a finite time horizon. We show that the Hamilton–Jacobi–Bellman equations associated to the stochastic optimal control problem can be transformed into a system of linear ordinary differential equations and we solve the market making problem under inventory constraints. We also shed light on the asymptotic behavior of the optimal quotes and propose closed-form approximations based on a spectral characterization of the optimal quotes.

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References

  1. Amihud Y., Mendelson H.: Dealership market. market-making with inventory. J. Fin. Econ. 8, 31–53 (1980)

    Article  Google Scholar 

  2. Amihud Y., Mendelson H.: Asset pricing and the bid-ask spread. J. Fin. Econ. 17(2), 223–249 (1986)

    Article  Google Scholar 

  3. Avellaneda M., Stoikov S.: High-frequency trading in a limit order book. Quant. Fin. 8(3), 217–224 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bayraktar, E., Ludkovski, M.: Liquidation in limit order books with controlled intensity. Arxiv preprint arXiv:1105.0247 (2011)

    Google Scholar 

  5. Benston Robert L., George J.: Determinants of bid-asked spreads in the over-the-counter market. J. Fin. Econ. 1(4), 353–364 (1974)

    Article  Google Scholar 

  6. Cartea, Á., Jaimungal S.: Modeling asset prices for algorithmic and high frequency trading. Working Paper (2010)

  7. Cartea, Á., Jaimungal S.: Risk measures and fine tuning of high frequency trading strategies. Working Paper (2012)

  8. Cartea, Á., Jaimungal, S., Ricci, J.: Buy low sell high: a high frequency trading perspective. Working Paper (2011)

  9. Cohen K.J., Maier S.F., Schwartz R.A., Whitcomb D.K.: Market makers and the market spread: a review of recent literature. J. Fin. Quant. Anal. 14(04), 813–835 (1979)

    Article  Google Scholar 

  10. Cohen K.J., Maier S.F., Schwartz R.A., Whitcomb D.K.: Transaction costs, order placement strategy, and existence of the bid-ask spread. J. Polit. Econ. 89(2), 287–305 (1981)

    Article  Google Scholar 

  11. Garman M.B.: Market microstructure. J. Fin. Econ. 3(3), 257–275 (1976)

    Article  MathSciNet  Google Scholar 

  12. Guéant, O., Lehalle, C.A.: Existence and Uniqueness for the Avellaneda–Stoikov PDE. Working Paper. (2012)

  13. Guéant, O., Lehalle, C.A., Fernandez-Tapia, J.: Optimal Execution with Limit Orders. Working Paper. SIAM J. Fin. Math. (2011) (to appear)

  14. Guilbaud, F., Pham, H.: Optimal high frequency trading with limit and market orders. Quant. Fin. (2011) (to appear)

  15. Guilbaud F., Pham H.: Optimal high frequency trading in a pro-rata microstructure with predictive information. Arxiv preprint arXiv:1205-3051 (2012)

    Google Scholar 

  16. Hendershott, T., Menkveld, A.: Price Pressures. Manuscript, VU University, Amsterdam (2009)

  17. Ho, T.S.Y., Macris, R.G.: Dealer bid-ask quotes and transaction prices: An empirical study of some AMEX options. J. Fin. 39, 23–45 (1984)

    Google Scholar 

  18. Ho T., Stoll H.R.: Optimal dealer pricing under transactions and return uncertainty. J. Fin. Econ. 9(1), 47–73 (1981)

    Article  Google Scholar 

  19. Ho T.S.Y., Stoll H.R.: The dynamics of dealer markets under competition. J. Fin. 38(4), 1053–1074 (1983)

    Article  Google Scholar 

  20. Lehalle, C.A., Guéant, O., Razafinimanana, J. : High Frequency Simulations of an Order Book: a Two-Scales Approach. In: Abergel, F., Chakrabarti, B.K., Chakraborti, A., Mitra, M. (eds.) Econophysics of Order-Driven Markets, New Economic Windows, Springer, New York (2010)

    Google Scholar 

  21. Madhavan A., Smidt S.: An analysis of changes in specialist inventories and quotations. J. Fin. 48(5), 1595–1628 (1993)

    Article  Google Scholar 

  22. Menkveld, A.J.: High Frequency Trading and The New-Market Makers. Social Science Research Network Working Paper Series (2010)

  23. Mildenstein, E., Schleef, H.: The optimal pricing policy of a monopolistic marketmaker in the equity market. J. Fin. 38, 218–231, (1983)

    Google Scholar 

  24. O’Hara M., Oldfield G.S.: The microeconomics of market making. J. Fin. Quant. Anal. 21(04), 361–376 (1986)

    Article  Google Scholar 

  25. Roll R.: A simple implicit measure of the effective bid-ask spread in an efficient market. J. Fin. 39(4), 1127–1139 (1984)

    Article  Google Scholar 

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Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, UFR de Mathématiques, Université Paris-Diderot, 175, rue du Chevaleret, 75013, Paris, France

    Olivier Guéant

  2. Head of Quantitative Research, Crédit Agricole Cheuvreux, 9, Quai du Président Paul Doumer, 92400, Courbevoie, France

    Charles-Albert Lehalle

  3. LPMA, Université Pierre et Marie Curie, 4 Place Jussieu, 75005, Paris, France

    Joaquin Fernandez-Tapia

Authors
  1. Olivier Guéant
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  2. Charles-Albert Lehalle
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  3. Joaquin Fernandez-Tapia
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Correspondence to Olivier Guéant.

Additional information

This research has been conducted within the Research Initiative “Microstructure des Marchés Financiers” under the aegis of the Europlace Institute of Finance.

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Guéant, O., Lehalle, CA. & Fernandez-Tapia, J. Dealing with the inventory risk: a solution to the market making problem. Math Finan Econ 7, 477–507 (2013). https://doi.org/10.1007/s11579-012-0087-0

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  • DOI: https://doi.org/10.1007/s11579-012-0087-0

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