## The Annals of Applied Probability

### Financial markets with a large trader

#### Abstract

We construct a large trader model using tools from nonlinear stochastic integration theory and an impact function. It encompasses many well-known models from the literature. In particular, the model allows price changes to depend on the size as well as on the speed and timing of the large trader’s transactions. Moreover, a volume impact limit order book can be studied in this framework. Relaxing a condition about existence of a universal martingale measure governing all resulting small trader models, we can show absence of arbitrage for the small trader under mild conditions. Furthermore, a case study on utility maximization from terminal wealth highlights new phenomena that can arise in our framework. Finally, an outlook on further research provides insights on (no) arbitrage opportunities for the large trader and how different levels of information may affect our analysis.

#### Article information

Source
Ann. Appl. Probab., Volume 27, Number 6 (2017), 3735-3786.

Dates
Revised: November 2016
First available in Project Euclid: 15 December 2017

https://projecteuclid.org/euclid.aoap/1513328713

Digital Object Identifier
doi:10.1214/17-AAP1295

Mathematical Reviews number (MathSciNet)
MR3737937

Zentralblatt MATH identifier
06848278

Subjects
Primary: 60H05: Stochastic integrals 91G10: Portfolio theory

#### Citation

Blümmel, Tilmann; Rheinländer, Thorsten. Financial markets with a large trader. Ann. Appl. Probab. 27 (2017), no. 6, 3735--3786. doi:10.1214/17-AAP1295. https://projecteuclid.org/euclid.aoap/1513328713

#### References

• [1] Alfonsi, A. and Schied, A. (2010). Optimal trade execution and absence of price manipulations in limit order book models. SIAM J. Financial Math. 1 490–522.
• [2] Almgren, R. and Chriss, N. (1999). Value under liquidation. Risk 12 61–63.
• [3] Almgren, R. and Chriss, N. (2000). Optimal execution of portfolio transactions. J. Risk 3 5–39.
• [4] Bank, P. and Baum, D. (2004). Hedging and portfolio optimization in financial markets with a large trader. Math. Finance 14 1–18.
• [5] Biagini, S. and Frittelli, M. (2005). Utility maximization in incomplete markets for unbounded processes. Finance Stoch. 9 493–517.
• [6] Biagini, S. and Frittelli, M. (2008). A unified framework for utility maximization problems: An Orlicz space approach. Ann. Appl. Probab. 18 929–966.
• [7] Biagini, S. and Sîrbu, M. (2012). A note on admissibility when the credit line is infinite. Stochastics 84 157–169.
• [8] Bogachev, V. I. (2007). Measure Theory, Vols. I, II. Springer, Berlin.
• [9] Carmona, R. A. and Nualart, D. (1990). Nonlinear Stochastic Integrators, Equations and Flows. Stochastics Monographs 6. Gordon & Breach Science Publishers, New York.
• [10] Çetin, U., Jarrow, R. A. and Protter, P. (2004). Liquidity risk and arbitrage pricing theory. Finance Stoch. 8 311–341.
• [11] Choulli, T. and Stricker, C. (1996). Deux applications de la décomposition de Galtchouk–Kunita–Watanabe. In Séminaire de Probabilités, XXX. (J. Azéma, M. Émery and M. Yor, eds.). Lecture Notes in Math. 1626 12–23. Springer, Berlin.
• [12] Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300 463–520.
• [13] Delbaen, F. and Schachermayer, W. (1995). The existence of absolutely continuous local martingale measures. Ann. Appl. Probab. 5 926–945.
• [14] Emery, M. (1979). Une topologie sur l’espace des semimartingales. In Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78). Lecture Notes in Math. 721 260–280. Springer, Berlin.
• [15] Frey, R. (1998). Perfect option hedging for a large trader. Finance Stoch. 2 115–141.
• [16] Gatheral, J. (2010). No-dynamic-arbitrage and market impact. Quant. Finance 10 749–759.
• [17] Gatheral, J. and Schied, A. (2013). Dynamical models of market impact and algorithms for order execution. In Handbook on Systemic Risk (J. P. Fouque and J. A. Langsam, eds.) 579–602. Cambridge Univ. Press, Cambridge.
• [18] Hulley, H. and Schweizer, M. (2010). $\mathrm{M}^{6}$—On minimal market models and minimal martingale measures. In Contemporary Quantitative Finance (C. Chiarella and A. Novikov, eds.) 35–51. Springer, Berlin.
• [19] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
• [20] Jarrow, R. (1992). Market manipulation, bubbles, corners, and short squeezes. J. Financ. Quant. Anal. 27 311–336.
• [21] Jeulin, T. and Yor, M. (1979). Inégalité de Hardy, semimartingales, et faux-amis. In Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78). Lecture Notes in Math. 721 332–359. Springer, Berlin.
• [22] Jeulin, T. and Yor, M. (1990). Filtration des ponts browniens et équations différentielles stochastiques linéaires. In Séminaire de Probabilités, XXIV, 1988/89. Lecture Notes in Math. 1426 227–265. Springer, Berlin.
• [23] Kardaras, C. (2013). On the closure in the Emery topology of semimartingale wealth-process sets. Ann. Appl. Probab. 23 1355–1376.
• [24] Kühn, C. (2006). Optimal investment in financial markets with different liquidity effects. Preprint, MathFinance Institute, Goethe Univ., Frankfurt, Germany.
• [25] Kunita, H. (1997). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge. Reprint of the 1990 original.
• [26] Kyle, A. (1985). Continuous auctions and insider trading. Econometrica 53 1315–1335.
• [27] Mémin, J. (1980). Espaces de semi martingales et changement de probabilité. Z. Wahrsch. Verw. Gebiete 52 9–39.
• [28] Protter, P. E. (2005). Stochastic Integration and Differential Equations, 2nd ed. Stochastic Modelling and Applied Probability 21. Springer, Berlin. Version 2.1, corrected third printing.
• [29] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
• [30] Schachermayer, W. (1993). A counterexample to several problems in the theory of asset pricing. Math. Finance 3 217–229.
• [31] Schied, A. (2013). Robust strategies for optimal order execution in the Almgren–Chriss framework. Appl. Math. Finance 20 264–286.
• [32] Schied, A. and Schöneborn, T. (2009). Risk aversion and the dynamics of optimal liquidation strategies in illiquid markets. Finance Stoch. 13 181–204.
• [33] Schied, A., Schöneborn, T. and Tehranchi, M. (2010). Optimal basket liquidation for CARA investors is deterministic. Appl. Math. Finance 17 471–489.
• [34] van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge Univ. Press, Cambridge.