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Pricing European Options in a Discrete Time Model for the Limit Order Book

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Abstract

In this paper we build a discrete time model for the structure of the limit order book, so that the price per share depends on the size of the transaction. We deduce the value of a portfolio when the investor trades using market orders and a bank account with different interest rates for lending and borrowing. We also deduce conditions to rule out arbitrage and solve the problem of pricing and hedging an European call option with physical delivery. It is shown that contrary to the perfectly liquid setting, the price of a European call is not given by an expectation, but can be expressed as an optimization problem on a set of equivalent probability measures.

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References

  • Agliardi R, Gençay R (2014) Hedging through a limit order book with varying liquidity. J Deriv 22(23):32–49

    Article  Google Scholar 

  • Bank P, Baum D (2004) Hedging and portfolio optimization in illiquid financial markets with a large trader. Math Financ 14:1–18

    Article  MATH  Google Scholar 

  • Bayraktar E, Ludkovski M (2011) Optimal trade execution in illiquid markets. Math Financ 21(4):681–701

    MathSciNet  MATH  Google Scholar 

  • Boyd S, Vandenberghe L (2009) Convex optimization, 7th edn. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Cetin U, Jarrow R, Protter P (2004) Liquidity risk and arbitrage pricing theory. Finance Stochast 8:311–341

    Article  MathSciNet  MATH  Google Scholar 

  • Çetin U, Rogers L (2007) Modeling liquidity effects in discrete time. Math Finance 17(1):15–29

    Article  MathSciNet  MATH  Google Scholar 

  • Cetin U, Soner H, Touzi N (2010) Option hedging for small investors under liquidity costs. Finance Stochast 14:317–341

    Article  MathSciNet  MATH  Google Scholar 

  • Cox J, Ross S, Rubinstein M (1979) Option pricing: A simplified approach. J Financ Econ 7:229–263

    Article  MathSciNet  MATH  Google Scholar 

  • Cvitanić J, Ma J (1996) Hedging options for a large investor and forward-backward SDE’s. Ann Appl Probab 6:370–398

    Article  MathSciNet  MATH  Google Scholar 

  • Delbaen F, Schachermayer W (1994) A general version of the fundamental theorem of asset pricing. Math Ann 300(3):463–520

    Article  MathSciNet  MATH  Google Scholar 

  • Dolinsky Y, Soner H (2013) Duality and convergence for binomial markets with friction. Finance Stochast 17:447–475

    Article  MathSciNet  MATH  Google Scholar 

  • Foucault T, Kadan O, Kendel E (2005) Limit order book as a market for liquidity. Rev Financ Stud 18(4):1171–1217

    Article  Google Scholar 

  • Gökay S, Soner HM (2012) Liquidity in a binomial market. Math Financ 22 (2):250–276

    Article  MathSciNet  MATH  Google Scholar 

  • Gökay S, Soner HM (2013) Hedging in an illiquid binomial market. Nonlinear Anal: Real World Appl 16:1–16

    MathSciNet  MATH  Google Scholar 

  • Harrison J, Pliska S (1981) Martingales and stochastic integrals in the theory of continuous trading. Stochast Process Appl 11:215–260

    Article  MathSciNet  MATH  Google Scholar 

  • Harrison JM, Kreps DM (1979) Martingales and arbitrage in multiperiod securities markets. J Econ Theory 20:381–408

    Article  MathSciNet  MATH  Google Scholar 

  • Jarrow R (1994) Derivative security markets, market manipulation and option pricing. J Financ Quant Anal 29:241–261

    Article  Google Scholar 

  • Obizhaeva A, Wang J (2013) Optimal trading strategy and supply/demand dynamics. J Financ Mark 16:1–32

    Article  Google Scholar 

  • Predoiu S, Shaikhet G, Shreve S (2011) Optimal execution in a general one-sided limit-order book. SIAM J Financ Math 2:183–212

    Article  MathSciNet  MATH  Google Scholar 

  • Roch A (2011) Liquidity risk, price impacts and the replication problem. Finance Stochast 15(3):399–419

    Article  MathSciNet  MATH  Google Scholar 

  • Shreve SE (2004) Stochastic calculus for finance. II. Springer Finance. Springer, New York. Continuous-time models

    Book  MATH  Google Scholar 

  • Simard C (2014) General model for limit order books and market orders. Social Science Research Network. Available at SSRN: http://ssrn.com/abstract=2435198 or https://doi.org/10.2139/2435198.ssrn

  • Simard C, Rémillard B (2014) Option pricing in a discrete time model for the limit order book. Social Science Research Network. Available at SSRN: http://ssrn.com/abstract=2472410

  • Simard C, Rémillard B. (2016) Supplementary to option pricing in a discrete time model for the limit order book. Methodology and Computing in Applied Probability, Link to the supplementary material

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Authors and Affiliations

  1. Département de Mathématiques, Université du Québec à Montréal, 201 av. Président-Kennedy, Montréal, H2X 3Y7, Canada

    Clarence Simard

  2. Department of Decision Sciences, HEC Montreal, 3000, Cote-Ste-Catherine, Montreal, H3T 2A7, Canada

    Bruno Rémillard

Authors
  1. Clarence Simard
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  2. Bruno Rémillard
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Correspondence to Clarence Simard.

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Simard, C., Rémillard, B. Pricing European Options in a Discrete Time Model for the Limit Order Book. Methodol Comput Appl Probab 21, 985–1005 (2019). https://doi.org/10.1007/s11009-017-9610-3

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  • DOI: https://doi.org/10.1007/s11009-017-9610-3

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