# American Institute of Mathematical Sciences

June  2021, 6(2): 117-138. doi: 10.3934/puqr.2021006

## An infinite-dimensional model of liquidity in financial markets

 1 Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA 2 Institute of Mathematical Sciences, Claremont Graduate University, Claremont, CA 91711, USA 3 Drucker School of Management, Claremont Graduate University, Claremont, CA 91711, USA

lototsky@usc.edu

Received  September 2019 Accepted  May 17, 2021 Published  June 2021

Fund Project: The authors gratefully acknowledge very helpful suggestions from the editors, referees, and the editorial staff.

We consider a dynamic market model of liquidity where unmatched buy and sell limit orders are stored in order books. The resulting net demand surface constitutes the sole input to the model. We model demand using a two-parameter Brownian motion because (i) different points on the demand curve correspond to orders motivated by different information, and (ii) in general, the market price of risk equation of no-arbitrage theory has no solutions when the demand curve is driven by a finite number of factors, thus allowing for arbitrage. We prove that if the driving noise is infinite-dimensional, then there is no arbitrage in the model. Under the equivalent martingale measure, the clearing price is a martingale, and options can be priced under the no-arbitrage hypothesis. We consider several parameterizations of the model and show advantages of specifying the demand curve as a quantity that is a function of price, as opposed to price as a function of quantity. An online appendix presents a basic empirical analysis of the model: calibration using information from actual order books, computation of option prices using Monte Carlo simulations, and comparison with observed data.

Citation: Sergey V Lototsky, Henry Schellhorn, Ran Zhao. An infinite-dimensional model of liquidity in financial markets. Probability, Uncertainty and Quantitative Risk, 2021, 6 (2) : 117-138. doi: 10.3934/puqr.2021006
##### References:
 [1] Santa-Clara, P. and Sornette, D., The dynamics of the forward interest rate curve with stochastic string shocks, Review of Financial Studies, 2001, 14(1): 149-185. doi: 10.1093/rfs/14.1.149. [2] Cartea, Á., Jaimungal, S. and Penalva, J., Algorithmic and High-frequency Trading, Cambridge University Press, Cambridge, 2015. [3] Brogaard, J., Hendershott, T. and Riordan, R., High-frequency trading and price discovery, Review of Financial Studies, 2014, 27(8): 2267-2306. doi: 10.1093/rfs/hhu032. [4] Kirilenko, A., Kyle, A. S., Samadi, M. and Tuzun, T., The flash crash: high-frequency trading in an electronic market, Journal of Finance, 2017, 72(3): 967-998. doi: 10.1111/jofi.12498. [5] Bongaerts, D. and Van Achter, M., Competition among liquidity providers with access to high-frequency trading technology, Journal of Financial Economics, 2021, 140: 220-249. [6] Kyle, A.S., Continuous auctions and insider trading, Econometrica, 1985, 53(3): 1315-1335. [7] Bank, P. and Baum, D., Hedging and portfolio optimization in financial markets with a large trader, Mathematical Finance, 2004, 14(1): 1-18. doi: 10.1111/j.0960-1627.2004.00179.x. [8] Bank, P. and Kramkov, D., A large model for a large investor trading at market indifference prices II: continuous-time case, Annals of Applied Probability, 2015, 25(5): 2708-2742. [9] Bank, P. and Kramkov, D., A model for a large investor trading at market indifference prices. I: single-period case, Finance and Stochastics, 2015, 19(2): 449-472. doi: 10.1007/s00780-015-0258-y. [10] Frey, R. and Stremme, A., Market volatility and feedback effects from dynamic hedging, Mathematical Finance, 1997, 7(4): 351-375. doi: 10.1111/1467-9965.00036. [11] Jarrow, R., Market manipulation, bubbles, corners and short squeezes, Journal of Financial and Quantitative Analysis, 1992, 27(3): 311-336. doi: 10.2307/2331322. [12] Jarrow, R., Derivative securities markets, market manipulation and option pricing theory, Journal of Financial and Quantitative Analysis, 1994, 29(3): 241-261. [13] Papanicolaou, G. and Sircar, R., General Black-Scholes models accounting for increased market volatility from hedging strategies, Applied Mathematical Finance, 1998, 5(3): 45-82. [14] Platen, E. and Schweizer, M., On feedback effects from hedging derivatives, Mathematical Finance, 1998, 8(1): 67-84. doi: 10.1111/1467-9965.00045. [15] Rogers, C. and Singh, S., The cost of illiquidity and its effects on hedging, Mathematical Finance, 2010, 20(4): 597-615. doi: 10.1111/j.1467-9965.2010.00413.x. [16] Schönbucher, J. P. and Wilmott, P., The feedback effect of hedging in illiquid markets, SIAM Journal on Applied Mathematics, 2000, 61(2): 232-272. [17] Çetin, U. and Rogers, C., Modeling liquidity effects in discrete time, Mathematical Finance, 2007, 17: 15-29. doi: 10.1111/j.1467-9965.2007.00292.x. [18] Çetin, U., Jarrow, R. and Protter, P., Liquidity risk and arbitrage pricing theory, Finance and Stochastics, 2004, 8(3): 311-341. doi: 10.1007/s00780-004-0123-x. [19] Çetin, U., Soner, M. H. and Touzi, N., Option hedging for small investors under liquidity costs, Finance and Stochastics, 2010, 14(3): 317-341. doi: 10.1007/s00780-009-0116-x. [20] Gökay, S. and Soner, M. H., Hedging in an illiquid binomial market, Nonlinear Analysis: Real World Applications, 2014, 1(3): 1-16. [21] Roch, A., Liquidity risk, price impacts and the replication problem, Finance and Stochastics, 2011, 15: 399-419. doi: 10.1007/s00780-011-0156-x. [22] Ku, H. and Zhang, H., Option pricing for a large trader with price impact and liquidity costs, Journal of Mathematical Analysis and Applications, 2018, 459(1): 32-52. doi: 10.1016/j.jmaa.2017.10.072. [23] Blümmel, T., Rheinländer, T., Financial markets with a large trader, Annals of Applied Probability, 2017, 27(6): 3735-3786. [24] Buckdahn, R., Li, J. and Peng, S., Nonlinear stochastic differential games involving a major player and a large number of collectively acting minor agents, SIAM Journal on Control and Optimization, 2014, 52(1): 451-492. doi: 10.1137/130933174. [25] Cont, R. and Mueller, M. S., A stochastic pde model for limit order book dynamics, Working Paper, 2019. [26] Pohl, M., Ristig, A., Schachermayer, W. and Tangpi, L., The amazing power of dimensional analysis: quantifying market impact., Market Microstructure and Liquidity, 2017, 3(03n04): 1850004. doi: 10.1142/S2382626618500041. [27] O’Hara, M., Liquidity and financial market stability, National Bank of Belgium Working Paper No. 55, 2004, https://ssrn.com/abstract=1691574. [28] Obizhaeva, A. A. and Wang, J., Optimal trading strategy and supply/demand dynamics, Journal of Financial Markets, 2013, 16(1): 1-32. doi: 10.1016/j.finmar.2012.09.001. [29] Rosu, I., A dynamic model of the limit order book, Review of Financial Studies, 2009, 22(3): 4601-4641. [30] Mueller, C., Some tools and results for parabolic stochastic partial differential equations, In: A Minicourse on Stochastic Partial Differential Equations—Lectures Notes in Mathematics 1962, 2009, 111-144. [31] Walsh, J. B., An introduction to stochastic partial differential equations, In: École d’ Été de Probabilités de Saint Flour XIV-1984, Springer, Berlin, 1986: 265-439. [32] Da Prato, G. and Zabczyk, J., Stochastic Equations in Infinite Dimensions (2 eidtion), Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2014: 152. [33] Krylov, N. V., An analytic approach to SPDEs, In: Stochastic Partial Differential Equations: Six Perspectives. Math. Surveys Monogr., Amer. Math. Soc., Providence, RI, 1999, 64: 185-242. [34] Lototsky, S. V. and Rozovsky, B. L., Stochastic Partial Differential Equations, Springer, Cham, 2017. [35] Allouba, H., Different types of SPDEs in the eyes of Girsanov’s theorem, Stochastic Analysis and Applications, 1998, 16(5): 787-810. doi: 10.1080/07362999808809562. [36] Krylov, N. V., On the Itô-Wentzell formula for distribution-valued processes and related topics, Probability Theory and Related Fields, 2011, 150(1-2): 295-319. doi: 10.1007/s00440-010-0275-x. [37] Kunita, H., Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge, 1997. [38] Protter, P., Stochastic Integration and Differential Equations, Springer, New York, 2004. [39] Shreve, S., Stochastic Calculus for Finance II, Springer, Newe York, 2004. [40] Heath, D., Jarrow, R. and Morton, A., Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation, Econometrica, 1992, 60(1): 77-105. doi: 10.2307/2951677. [41] Chow, P.-L., Stochastic Partial Differential Equations, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, Chapman & Hall, Boca Raton, FL, 2007.

show all references

1There is no loss of generality in that statement. A buy market order can be specified in our model as a buy limit order with a limit price equal to infinity. Since we model assets with only positive prices, a sell market order can be specified in our model as a sell limit order with a limit price equal to zero.

2A notable exception is [10].

3We do not consider markets for swaps, where the price can be negative.

##### References:
 [1] Santa-Clara, P. and Sornette, D., The dynamics of the forward interest rate curve with stochastic string shocks, Review of Financial Studies, 2001, 14(1): 149-185. doi: 10.1093/rfs/14.1.149. [2] Cartea, Á., Jaimungal, S. and Penalva, J., Algorithmic and High-frequency Trading, Cambridge University Press, Cambridge, 2015. [3] Brogaard, J., Hendershott, T. and Riordan, R., High-frequency trading and price discovery, Review of Financial Studies, 2014, 27(8): 2267-2306. doi: 10.1093/rfs/hhu032. [4] Kirilenko, A., Kyle, A. S., Samadi, M. and Tuzun, T., The flash crash: high-frequency trading in an electronic market, Journal of Finance, 2017, 72(3): 967-998. doi: 10.1111/jofi.12498. [5] Bongaerts, D. and Van Achter, M., Competition among liquidity providers with access to high-frequency trading technology, Journal of Financial Economics, 2021, 140: 220-249. [6] Kyle, A.S., Continuous auctions and insider trading, Econometrica, 1985, 53(3): 1315-1335. [7] Bank, P. and Baum, D., Hedging and portfolio optimization in financial markets with a large trader, Mathematical Finance, 2004, 14(1): 1-18. doi: 10.1111/j.0960-1627.2004.00179.x. [8] Bank, P. and Kramkov, D., A large model for a large investor trading at market indifference prices II: continuous-time case, Annals of Applied Probability, 2015, 25(5): 2708-2742. [9] Bank, P. and Kramkov, D., A model for a large investor trading at market indifference prices. I: single-period case, Finance and Stochastics, 2015, 19(2): 449-472. doi: 10.1007/s00780-015-0258-y. [10] Frey, R. and Stremme, A., Market volatility and feedback effects from dynamic hedging, Mathematical Finance, 1997, 7(4): 351-375. doi: 10.1111/1467-9965.00036. [11] Jarrow, R., Market manipulation, bubbles, corners and short squeezes, Journal of Financial and Quantitative Analysis, 1992, 27(3): 311-336. doi: 10.2307/2331322. [12] Jarrow, R., Derivative securities markets, market manipulation and option pricing theory, Journal of Financial and Quantitative Analysis, 1994, 29(3): 241-261. [13] Papanicolaou, G. and Sircar, R., General Black-Scholes models accounting for increased market volatility from hedging strategies, Applied Mathematical Finance, 1998, 5(3): 45-82. [14] Platen, E. and Schweizer, M., On feedback effects from hedging derivatives, Mathematical Finance, 1998, 8(1): 67-84. doi: 10.1111/1467-9965.00045. [15] Rogers, C. and Singh, S., The cost of illiquidity and its effects on hedging, Mathematical Finance, 2010, 20(4): 597-615. doi: 10.1111/j.1467-9965.2010.00413.x. [16] Schönbucher, J. P. and Wilmott, P., The feedback effect of hedging in illiquid markets, SIAM Journal on Applied Mathematics, 2000, 61(2): 232-272. [17] Çetin, U. and Rogers, C., Modeling liquidity effects in discrete time, Mathematical Finance, 2007, 17: 15-29. doi: 10.1111/j.1467-9965.2007.00292.x. [18] Çetin, U., Jarrow, R. and Protter, P., Liquidity risk and arbitrage pricing theory, Finance and Stochastics, 2004, 8(3): 311-341. doi: 10.1007/s00780-004-0123-x. [19] Çetin, U., Soner, M. H. and Touzi, N., Option hedging for small investors under liquidity costs, Finance and Stochastics, 2010, 14(3): 317-341. doi: 10.1007/s00780-009-0116-x. [20] Gökay, S. and Soner, M. H., Hedging in an illiquid binomial market, Nonlinear Analysis: Real World Applications, 2014, 1(3): 1-16. [21] Roch, A., Liquidity risk, price impacts and the replication problem, Finance and Stochastics, 2011, 15: 399-419. doi: 10.1007/s00780-011-0156-x. [22] Ku, H. and Zhang, H., Option pricing for a large trader with price impact and liquidity costs, Journal of Mathematical Analysis and Applications, 2018, 459(1): 32-52. doi: 10.1016/j.jmaa.2017.10.072. [23] Blümmel, T., Rheinländer, T., Financial markets with a large trader, Annals of Applied Probability, 2017, 27(6): 3735-3786. [24] Buckdahn, R., Li, J. and Peng, S., Nonlinear stochastic differential games involving a major player and a large number of collectively acting minor agents, SIAM Journal on Control and Optimization, 2014, 52(1): 451-492. doi: 10.1137/130933174. [25] Cont, R. and Mueller, M. S., A stochastic pde model for limit order book dynamics, Working Paper, 2019. [26] Pohl, M., Ristig, A., Schachermayer, W. and Tangpi, L., The amazing power of dimensional analysis: quantifying market impact., Market Microstructure and Liquidity, 2017, 3(03n04): 1850004. doi: 10.1142/S2382626618500041. [27] O’Hara, M., Liquidity and financial market stability, National Bank of Belgium Working Paper No. 55, 2004, https://ssrn.com/abstract=1691574. [28] Obizhaeva, A. A. and Wang, J., Optimal trading strategy and supply/demand dynamics, Journal of Financial Markets, 2013, 16(1): 1-32. doi: 10.1016/j.finmar.2012.09.001. [29] Rosu, I., A dynamic model of the limit order book, Review of Financial Studies, 2009, 22(3): 4601-4641. [30] Mueller, C., Some tools and results for parabolic stochastic partial differential equations, In: A Minicourse on Stochastic Partial Differential Equations—Lectures Notes in Mathematics 1962, 2009, 111-144. [31] Walsh, J. B., An introduction to stochastic partial differential equations, In: École d’ Été de Probabilités de Saint Flour XIV-1984, Springer, Berlin, 1986: 265-439. [32] Da Prato, G. and Zabczyk, J., Stochastic Equations in Infinite Dimensions (2 eidtion), Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2014: 152. [33] Krylov, N. V., An analytic approach to SPDEs, In: Stochastic Partial Differential Equations: Six Perspectives. Math. Surveys Monogr., Amer. Math. Soc., Providence, RI, 1999, 64: 185-242. [34] Lototsky, S. V. and Rozovsky, B. L., Stochastic Partial Differential Equations, Springer, Cham, 2017. [35] Allouba, H., Different types of SPDEs in the eyes of Girsanov’s theorem, Stochastic Analysis and Applications, 1998, 16(5): 787-810. doi: 10.1080/07362999808809562. [36] Krylov, N. V., On the Itô-Wentzell formula for distribution-valued processes and related topics, Probability Theory and Related Fields, 2011, 150(1-2): 295-319. doi: 10.1007/s00440-010-0275-x. [37] Kunita, H., Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge, 1997. [38] Protter, P., Stochastic Integration and Differential Equations, Springer, New York, 2004. [39] Shreve, S., Stochastic Calculus for Finance II, Springer, Newe York, 2004. [40] Heath, D., Jarrow, R. and Morton, A., Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation, Econometrica, 1992, 60(1): 77-105. doi: 10.2307/2951677. [41] Chow, P.-L., Stochastic Partial Differential Equations, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, Chapman & Hall, Boca Raton, FL, 2007.
 [1] María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473 [2] Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 [3] Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control and Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401 [4] Alexander Schied, Iryna Voloshchenko. Pathwise no-arbitrage in a class of Delta hedging strategies. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 3-. doi: 10.1186/s41546-016-0003-2 [5] Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199 [6] Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 [7] Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515 [8] Guowei Dai, Ruyun Ma, Haiyan Wang, Feng Wang, Kuai Xu. Partial differential equations with Robin boundary condition in online social networks. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1609-1624. doi: 10.3934/dcdsb.2015.20.1609 [9] Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157 [10] Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435 [11] Ahmed Boudaoui, Tomás Caraballo, Abdelghani Ouahab. Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2521-2541. doi: 10.3934/dcdsb.2017084 [12] Min Yang, Guanggan Chen. Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1565-1581. doi: 10.3934/dcdsb.2019240 [13] Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021020 [14] Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295 [15] Sergio Albeverio, Sonia Mazzucchi. Infinite dimensional integrals and partial differential equations for stochastic and quantum phenomena. Journal of Geometric Mechanics, 2019, 11 (2) : 123-137. doi: 10.3934/jgm.2019006 [16] Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209 [17] Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5203-5219. doi: 10.3934/dcds.2015.35.5203 [18] Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks and Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014 [19] Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221 [20] Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061

Impact Factor:

Article outline