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.  Google Scholar

[2]

Cartea, Á., Jaimungal, S. and Penalva, J., Algorithmic and High-frequency Trading, Cambridge University Press, Cambridge, 2015. Google Scholar

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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.  Google Scholar

[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. Google Scholar

[6]

Kyle, A.S., Continuous auctions and insider trading, Econometrica, 1985, 53(3): 1315-1335. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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Jarrow, R., Derivative securities markets, market manipulation and option pricing theory, Journal of Financial and Quantitative Analysis, 1994, 29(3): 241-261. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[20]

Gökay, S. and Soner, M. H., Hedging in an illiquid binomial market, Nonlinear Analysis: Real World Applications, 2014, 1(3): 1-16. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

Blümmel, T., Rheinländer, T., Financial markets with a large trader, Annals of Applied Probability, 2017, 27(6): 3735-3786. Google Scholar

[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.  Google Scholar

[25]

Cont, R. and Mueller, M. S., A stochastic pde model for limit order book dynamics, Working Paper, 2019. Google Scholar

[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.  Google Scholar

[27]

O’Hara, M., Liquidity and financial market stability, National Bank of Belgium Working Paper No. 55, 2004, https://ssrn.com/abstract=1691574. Google Scholar

[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.  Google Scholar

[29]

Rosu, I., A dynamic model of the limit order book, Review of Financial Studies, 2009, 22(3): 4601-4641. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[34]

Lototsky, S. V. and Rozovsky, B. L., Stochastic Partial Differential Equations, Springer, Cham, 2017. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

Kunita, H., Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge, 1997. Google Scholar

[38]

Protter, P., Stochastic Integration and Differential Equations, Springer, New York, 2004. Google Scholar

[39]

Shreve, S., Stochastic Calculus for Finance II, Springer, Newe York, 2004. Google Scholar

[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.  Google Scholar

[41]

Chow, P.-L., Stochastic Partial Differential Equations, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, Chapman & Hall, Boca Raton, FL, 2007. Google Scholar

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.  Google Scholar

[2]

Cartea, Á., Jaimungal, S. and Penalva, J., Algorithmic and High-frequency Trading, Cambridge University Press, Cambridge, 2015. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[6]

Kyle, A.S., Continuous auctions and insider trading, Econometrica, 1985, 53(3): 1315-1335. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

Jarrow, R., Derivative securities markets, market manipulation and option pricing theory, Journal of Financial and Quantitative Analysis, 1994, 29(3): 241-261. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

Gökay, S. and Soner, M. H., Hedging in an illiquid binomial market, Nonlinear Analysis: Real World Applications, 2014, 1(3): 1-16. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

Blümmel, T., Rheinländer, T., Financial markets with a large trader, Annals of Applied Probability, 2017, 27(6): 3735-3786. Google Scholar

[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.  Google Scholar

[25]

Cont, R. and Mueller, M. S., A stochastic pde model for limit order book dynamics, Working Paper, 2019. Google Scholar

[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.  Google Scholar

[27]

O’Hara, M., Liquidity and financial market stability, National Bank of Belgium Working Paper No. 55, 2004, https://ssrn.com/abstract=1691574. Google Scholar

[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.  Google Scholar

[29]

Rosu, I., A dynamic model of the limit order book, Review of Financial Studies, 2009, 22(3): 4601-4641. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[34]

Lototsky, S. V. and Rozovsky, B. L., Stochastic Partial Differential Equations, Springer, Cham, 2017. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

Kunita, H., Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge, 1997. Google Scholar

[38]

Protter, P., Stochastic Integration and Differential Equations, Springer, New York, 2004. Google Scholar

[39]

Shreve, S., Stochastic Calculus for Finance II, Springer, Newe York, 2004. Google Scholar

[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.  Google Scholar

[41]

Chow, P.-L., Stochastic Partial Differential Equations, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, Chapman & Hall, Boca Raton, FL, 2007. Google Scholar

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