Developing Bid-Ask Probabilities for High-Frequency Trading
DOI:
https://doi.org/10.34021/ve.2020.03.02(1)Keywords:
path integral, financial markets, high-frequency tradingAbstract
Methods of path integrals are used to develop multi-factor probabilities of bid-ask variables to be used in high-frequency trading (HFT). Adaptive Simulated Annealing (ASA) is used to fit the nonlinear forms, so developed to a day of BitMEX tick data. Maxima algebraic code is used to develop the path integral codes into C codes, and a sampling code is used for the fitting process. After these fits, the resultant C code is very fast and useful for forecasting upcoming ‘ask’, bid, midprice, etc., when narrow and wide windows of incoming data are used. A bonus is the availability of canonical momenta indicators (CMI) useful to forecast direction and strengths of these variables.
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References
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Baradely, N., & Bouchardz, B., Evangelista, D., & Mounjid, O. (2018). Optimal inventory management and order book modelling. Technical Report arXiv:1802.08135v2 [q-fin.TR]. Paris: Paris-Dauphine, Paris Universite.
Cartea, A., Jaimungal, S., & Ricci, J. (2014). Buy low sell high: a high frequency trading perspective. SIAM Journal Financial Mathematics, 5(1), 415–444.
Cont, R., Stoikov, S., & Talreja, R. (2009). A stochastic model for order book dynamics. Operations Research, 58(3), 1–21.
Cont, R. (2011). Statistical modelling of high-frequency financial data. IEEE Signal Processing, 28(5), 16–25.
Dekker, H. (1980). On the most probable transition path of a general diffusion process. Physics Letters A, 80, 99–101.
Fodra, P., & Labadie, M. (2012). High-frequency market-making with inventory constraints and directional bets. Technical Report arXiv:1206.4810v1 [q-fin.TR]. Paris: EXQIM.
Gueant, O. (2017). Optimal market making.Technical Report arXiv:1605.01862v5 [q-fin.TR]. Paris: Universite Paris.
Huang, W., Lehalle, C.A., & Rosenbaum, M. (2014). Simulating and analyzing order book data: The queue-reactive model. Technical Report arXiv:1312.0563v2 [q-fin.TR]. Paris: University Pierre et Marie Curie.
Ingber, L., Chen, C., Mondescu, R., Muzzall, D., & Renedo, M. (2001). Probability tree algorithm for general diffusion processes. Physical Review E, 64(5), 056702–056707. Retrieved from https://www.ingber.com/path01_pathtree.pdf
Ingber, L., Fujio, H., & Wehner, M. (1991a). Mathematical comparison of combat computer models to exercise data. Mathematical Computer Modelling, 15(1), 65–90. Retrieved from https://www.ingber.com/combat91_data.pdf
Ingber, L., & Mondescu, R. (2001). Optimization of trading physics models of markets. IEEE Transactions Neural Networks, 12(4), 776–790. Retrieved from https://www.ingber.com/markets01_optim_trading.pdf
Ingber, L., & Mondescu, R. (2003). Automated internet trading based on optimized physics models of markets. In R. Howlett, N. Ichalkaranje, L. Jain, & G. Tonfoni (Eds.), Intelligent Internet-Based Information Processing Systems (pp. 305–356). Singapore: World Scientific. Retrieved from https://www.ingber.com/markets03_automated.pdf
Ingber, L., Nunez, P. (1995). Statistical mechanics of neocortical interactions: High resolution path-integral calculation of short-term memory. Physical Review E, 51(5), 5074–5083. Retrieved from https://www.ingber.com/smni95_stm.pdf
Ingber, L., & Nunez, P. (2010). Neocortical dynamics at multiple scales: EEG standing waves, statistical mechanics, and physical analogs. Mathematical Biosciences, 229, 160–173. Retrieved from https://www.ingber.com/smni10_multiple_scales.pdf
Ingber, L., Pappalepore, M., & Stesiak, R. (2014). Electroencephalographic field influence on calcium momentum waves. Journal of Theoretical Biology, 343, 138–153. Retrieved from https://www.ingber.com/smni14_eeg_ca.pdf
Ingber, L., Srinivasan, R., & Nunez, P. (1996). Path-integral evolution of chaos embedded in noise: Duffing neocortical analog. Mathematical Computer Modelling, 23(3), 43–53. Retrieved from https://www.ingber.com/path96_duffing.pdf
Ingber, L., Wehner, M., Jabbour, G., & Barnhill, T. (1991b). Application of statistical mechanics methodology to term-structure bond-pricing models. Mathematical Computer Modelling, 15(11), 77–98. Retrieved from https://www.ingber.com/markets91_interest.pdf
Ingber, L., & Wilson, J. (1999). Volatility of financial markets. Mathematical Computer Modelling, 29(5), 39–57. Retrieved from https://www.ingber.com/markets99_vol.pdf
Ingber, L., & Wilson, J. (2000). Statistical mechanics of financial markets: Exponential modifications to black-scholes. Mathematical Computer Modelling, 31(8/9), 167–192. Retrieved from https://www.ingber.com/markets00_exp.pdf
Ingber, L. (1984). Statistical mechanics of nonlinear nonequilibrium financial markets. Mathematical Modelling, 5(6), 343–361. Retrieved from https://www.ingber.com/markets84_statmech.pdf
Ingber, L. (1989). Very fast simulated re-annealing. Mathematical Computer Modelling, 12(8), 967–973. Retrieved from https://www.ingber.com/asa89_vfsr.pdf
Ingber, L. (1990). Statistical mechanical aids to calculating term structure models. Physical Review A, 42(12), 7057–7064. Retrieved from https://www.ingber.com/markets90_interest.pdf
Ingber, L. (1991). Statistical mechanics of neocortical interactions: A scaling paradigm applied to electroencephalography. Physical Review A, 44(6), 4017–4060. Retrieved from https://www.ingber.com/smni91_eeg.pdf
Ingber, L. (1992). Generic mesoscopic neural networks based on statistical mechanics of neocortical interactions. Physical Review A, 45(4), R2183–R2186. Retrieved from https://www.ingber.com/smni92_mnn.pdf
Ingber, L. (1993a). Statistical mechanics of combat and extensions. In C. Jones (Ed.), Toward a Science of Command, Control, and Communications (pp. 117–149). Washington, D.C.: American Institute of Aeronautics and Astronautics. Retrieved from https://www.ingber.com/combat93_c3sci.pdf
Ingber, L. (1993b). Adaptive simulated annealing (ASA). Technical Report
Global optimization C-code. Pasadena, CA: Caltech Alumni Association. Retrieved from https://www.ingber.com/#ASA-CODE
Ingber, L. (1994). Statistical mechanics of neocortical interactions: Path-integral evolution of short-term memory. Physical Review E, 49(5B), 4652–4664. Retrieved from https://www.ingber.com/smni94_stm.pdf
Ingber, L. (1995). Path-integral evolution of multivariate systems with moderate noise. Physical Review E, 51(2), 1616–1619. Retrieved from https://www.ingber.com/path95_nonl.pdf
Ingber, L. (1996a). Canonical momenta indicators of financial markets and neocortical EEG. In S.I. Amari, L. Xu, I.
King, & K.S. Leung (Eds.), Progress in Neural Information Processing (pp. 777–784). Springer: New York. Retrieved from https://www.ingber.com/markets96_momenta.pdf
Ingber, L. (1996b). Statistical mechanics of nonlinear nonequilibrium financial markets: Applications to optimized trading. Mathematical Computer Modelling, 23(7), 101–121. Retrieved from https://www.ingber.com/markets96_trading.pdf
Ingber, L. (1996c). Statistical mechanics of neocortical interactions: Multiple scales of EEG. In R. Dasheiff, & D. Vincent (Eds.), Frontier Science in EEG: Continuous Waveform Analysis (Electroencephal. Clin. Neurophysiol. Suppl. 45) (pp. 79–112). Elsevier: Amsterdam. Retrieved from https://www.ingber.com/smni96_eeg.pdf
Ingber, L. (1996d). Adaptive simulated annealing (ASA): lessons learned. Control and Cybernetics, 25(1), 33–54. Retrieved from https://www.ingber.com/asa96_lessons.pdf
Ingber, L. (1997). Statistical mechanics of neocortical interactions: Applications of canonical momenta indicators to electroencephalography. Physical Review E, 55(4), 4578–4593. Retrieved from https://www.ingber.com/smni97_cmi.pdf
Ingber, L. (1998a). Statistical mechanics of neocortical interactions: Training and testing canonical momenta indicators of EEG. Mathematical Computer Modelling, 27(3), 33–64. Retrieved from https://www.ingber.com/smni98_cmi_test.pdf
Ingber, L. (1998b). Data mining and knowledge discovery via statistical mechanics in nonlinear stochastic systems. Mathematical Computer Modelling, 27(3), 9–31. Retrieved from https://www.ingber.com/path98_datamining.pdf
Ingber, L. (2000). High-resolution path-integral development of financial options. Physica A, 283(3-4), 529–558. Retrieved from https://www.ingber.com/markets00_highres.pdf
Ingber, L. (2006). Statistical mechanics of neocortical interactions: Portfolio of physiological indicators. Technical Report Report 2006:PPI. Ashland, OR: Physical Studies Institute, Retrieved from https://www.ingber.com/smni06_ppi.pdf
Ingber, L. (2009a). Statistical mechanics of neocortical interactions: Portfolio of physiological indicators. The Open Cybernetics Systemics Journal, 3(14), 13–26. https://dx.doi.org/10.2174/1874110x00903010013
Ingber, L. (2009b). Statistical mechanics of neocortical interactions: Nonlinear columnar electroencephalography. NeuroQuantology Journal, 7(4), 500–529. Retrieved from https://www.ingber.com/smni09_nonlin_column_eeg.pdf
Ingber, L. (2010). Trading in risk dimensions. In G. Gregoriou (Ed.), The Handbook of Trading: Strategies for Navigating and Profiting from Currency, Bond, and Stock Markets (pp. 287–300). New York: McGraw-Hill.
Ingber, L. (2012a). Columnar EEG magnetic influences on molecular development of short-term memory. In G. Kalivas, & S. Petralia (Eds.), Short-Term Memory: New Research (pp. 37-72). NY: Nova, Hauppauge. Retrieved from https://www.ingber.com/smni11_stm_scales.pdf
Ingber, L. (2012b). Influence of macrocolumnar EEG on ca waves. Current Progress Journal, 1(1), 4–8. Retrieved from https://www.ingber.com/smni12_vectpot.pdf
Ingber, L. (2012c). Adaptive simulated annealing. In J. Oliveira, A. Petraglia, L. Ingber, M. Machado, & M. Petraglia (Eds.), Stochastic global optimization and its applications with fuzzy adaptive simulated annealing (pp. 33–61). New York: Springer. Retrieved from https://www.ingber.com/asa11_options.pdf
Ingber, L. (2015). Calculating consciousness correlates at multiple scales of neocortical interactions. In A. Costa, & E. Villalba (Eds.), Horizons in Neuroscience Research (pp. 153-186). Hauppauge, NY: Nova. Retrieved from https://www.ingber.com/smni15_calc_conscious.pdf
Ingber, L. (2016). Path-integral quantum PATHTREE and PATHINT algorithms. International Journal of Innovative Research in Information Security, 3(5), 1–15. Retrieved from https://www.ingber.com/path16_quantum_path.pdf
Ingber, L. (2017a). Options on quantum money: Quantum path-integral with serial shocks. International Journal of Innovative Research in Information Security, 4(2), 7–13. Retrieved from https://www.ingber.com/path17_quantum_options_shocks.pdf
Ingber, L. (2017b). Quantum path-integral qPATHINT algorithm. The Open Cybernetics Systemics Journal, 11, 119–133. Retrieved from https://www.ingber.com/path17_qpathint.pdf
Ingber, L. (2017c). Evolution of regenerative ca-ion wave-packet in neuronal-firing fields: Quantum path-integral with serial shocks. International Journal of Innovative Research in Information Security, 4(2), 14–22. Retrieved from https://www.ingber.com/path17_quantum_pathint_shocks.pdf
Ingber, L. (2018a). Quantum variables in finance and neuroscience. Technical Report
Lecture Plates 2018:QVFN. Ashland, OR: Physical Studies Institute. Retrieved from https://www.ingber.com/path18_qpathint_lecture.pdf
Ingber, L. (2018b). Quantum variables in finance and neuroscience II. Technical Report Report 2018:FNQV. Ashland, OR: Physical Studies Institute. Retrieved from https://www.ingber.com/path18_qpathint.pdf
Ingber, L. (2018c). Quantum calcium-ion interactions with EEG. Sci, 1(7), 1–20. Retrieved from https://www.ingber.com/smni18_quantumCaEEG.pdf
Langouche, F., Roekaerts, D., & Tirapegui, E. (1979). Discretization problems of functional integrals in phase space. Physical Review D, 20, 419–432.
Langouche, F., Roekaerts, D., & Tirapegui, E. (1982). Functional Integration and Semiclassical Expansions. The Netherlands: Reidel, Dordrecht.
Nunez, P., Srinivasan, R., Ingber, L. (2013). Theoretical and experimental electrophysiology in human neocortex: Multiscale correlates of conscious experience. In M. Pesenson (Ed.), Multiscale Analysis and Nonlinear Dynamics: From genes to the brain (pp. 149–178).
New York: Wiley. https://dx.doi.org/10.1002/9783527671632.ch06
Schulman, L. (1981). Techniques and Applications of Path Integration. New York: J. Wiley and Sons.
Avellaneda, M., & Stoikov, S. (2008). High-frequency trading in a limit order book. Quantitative Finance, 8(3), 217–224.
Baradely, N., & Bouchardz, B., Evangelista, D., & Mounjid, O. (2018). Optimal inventory management and order book modelling. Technical Report arXiv:1802.08135v2 [q-fin.TR]. Paris: Paris-Dauphine, Paris Universite.
Cartea, A., Jaimungal, S., & Ricci, J. (2014). Buy low sell high: a high frequency trading perspective. SIAM Journal Financial Mathematics, 5(1), 415–444.
Cont, R., Stoikov, S., & Talreja, R. (2009). A stochastic model for order book dynamics. Operations Research, 58(3), 1–21.
Cont, R. (2011). Statistical modelling of high-frequency financial data. IEEE Signal Processing, 28(5), 16–25.
Dekker, H. (1980). On the most probable transition path of a general diffusion process. Physics Letters A, 80, 99–101.
Fodra, P., & Labadie, M. (2012). High-frequency market-making with inventory constraints and directional bets. Technical Report arXiv:1206.4810v1 [q-fin.TR]. Paris: EXQIM.
Gueant, O. (2017). Optimal market making.Technical Report arXiv:1605.01862v5 [q-fin.TR]. Paris: Universite Paris.
Huang, W., Lehalle, C.A., & Rosenbaum, M. (2014). Simulating and analyzing order book data: The queue-reactive model. Technical Report arXiv:1312.0563v2 [q-fin.TR]. Paris: University Pierre et Marie Curie.
Ingber, L., Chen, C., Mondescu, R., Muzzall, D., & Renedo, M. (2001). Probability tree algorithm for general diffusion processes. Physical Review E, 64(5), 056702–056707. Retrieved from https://www.ingber.com/path01_pathtree.pdf
Ingber, L., Fujio, H., & Wehner, M. (1991a). Mathematical comparison of combat computer models to exercise data. Mathematical Computer Modelling, 15(1), 65–90. Retrieved from https://www.ingber.com/combat91_data.pdf
Ingber, L., & Mondescu, R. (2001). Optimization of trading physics models of markets. IEEE Transactions Neural Networks, 12(4), 776–790. Retrieved from https://www.ingber.com/markets01_optim_trading.pdf
Ingber, L., & Mondescu, R. (2003). Automated internet trading based on optimized physics models of markets. In R. Howlett, N. Ichalkaranje, L. Jain, & G. Tonfoni (Eds.), Intelligent Internet-Based Information Processing Systems (pp. 305–356). Singapore: World Scientific. Retrieved from https://www.ingber.com/markets03_automated.pdf
Ingber, L., Nunez, P. (1995). Statistical mechanics of neocortical interactions: High resolution path-integral calculation of short-term memory. Physical Review E, 51(5), 5074–5083. Retrieved from https://www.ingber.com/smni95_stm.pdf
Ingber, L., & Nunez, P. (2010). Neocortical dynamics at multiple scales: EEG standing waves, statistical mechanics, and physical analogs. Mathematical Biosciences, 229, 160–173. Retrieved from https://www.ingber.com/smni10_multiple_scales.pdf
Ingber, L., Pappalepore, M., & Stesiak, R. (2014). Electroencephalographic field influence on calcium momentum waves. Journal of Theoretical Biology, 343, 138–153. Retrieved from https://www.ingber.com/smni14_eeg_ca.pdf
Ingber, L., Srinivasan, R., & Nunez, P. (1996). Path-integral evolution of chaos embedded in noise: Duffing neocortical analog. Mathematical Computer Modelling, 23(3), 43–53. Retrieved from https://www.ingber.com/path96_duffing.pdf
Ingber, L., Wehner, M., Jabbour, G., & Barnhill, T. (1991b). Application of statistical mechanics methodology to term-structure bond-pricing models. Mathematical Computer Modelling, 15(11), 77–98. Retrieved from https://www.ingber.com/markets91_interest.pdf
Ingber, L., & Wilson, J. (1999). Volatility of financial markets. Mathematical Computer Modelling, 29(5), 39–57. Retrieved from https://www.ingber.com/markets99_vol.pdf
Ingber, L., & Wilson, J. (2000). Statistical mechanics of financial markets: Exponential modifications to black-scholes. Mathematical Computer Modelling, 31(8/9), 167–192. Retrieved from https://www.ingber.com/markets00_exp.pdf
Ingber, L. (1984). Statistical mechanics of nonlinear nonequilibrium financial markets. Mathematical Modelling, 5(6), 343–361. Retrieved from https://www.ingber.com/markets84_statmech.pdf
Ingber, L. (1989). Very fast simulated re-annealing. Mathematical Computer Modelling, 12(8), 967–973. Retrieved from https://www.ingber.com/asa89_vfsr.pdf
Ingber, L. (1990). Statistical mechanical aids to calculating term structure models. Physical Review A, 42(12), 7057–7064. Retrieved from https://www.ingber.com/markets90_interest.pdf
Ingber, L. (1991). Statistical mechanics of neocortical interactions: A scaling paradigm applied to electroencephalography. Physical Review A, 44(6), 4017–4060. Retrieved from https://www.ingber.com/smni91_eeg.pdf
Ingber, L. (1992). Generic mesoscopic neural networks based on statistical mechanics of neocortical interactions. Physical Review A, 45(4), R2183–R2186. Retrieved from https://www.ingber.com/smni92_mnn.pdf
Ingber, L. (1993a). Statistical mechanics of combat and extensions. In C. Jones (Ed.), Toward a Science of Command, Control, and Communications (pp. 117–149). Washington, D.C.: American Institute of Aeronautics and Astronautics. Retrieved from https://www.ingber.com/combat93_c3sci.pdf
Ingber, L. (1993b). Adaptive simulated annealing (ASA). Technical Report
Global optimization C-code. Pasadena, CA: Caltech Alumni Association. Retrieved from https://www.ingber.com/#ASA-CODE
Ingber, L. (1994). Statistical mechanics of neocortical interactions: Path-integral evolution of short-term memory. Physical Review E, 49(5B), 4652–4664. Retrieved from https://www.ingber.com/smni94_stm.pdf
Ingber, L. (1995). Path-integral evolution of multivariate systems with moderate noise. Physical Review E, 51(2), 1616–1619. Retrieved from https://www.ingber.com/path95_nonl.pdf
Ingber, L. (1996a). Canonical momenta indicators of financial markets and neocortical EEG. In S.I. Amari, L. Xu, I.
King, & K.S. Leung (Eds.), Progress in Neural Information Processing (pp. 777–784). Springer: New York. Retrieved from https://www.ingber.com/markets96_momenta.pdf
Ingber, L. (1996b). Statistical mechanics of nonlinear nonequilibrium financial markets: Applications to optimized trading. Mathematical Computer Modelling, 23(7), 101–121. Retrieved from https://www.ingber.com/markets96_trading.pdf
Ingber, L. (1996c). Statistical mechanics of neocortical interactions: Multiple scales of EEG. In R. Dasheiff, & D. Vincent (Eds.), Frontier Science in EEG: Continuous Waveform Analysis (Electroencephal. Clin. Neurophysiol. Suppl. 45) (pp. 79–112). Elsevier: Amsterdam. Retrieved from https://www.ingber.com/smni96_eeg.pdf
Ingber, L. (1996d). Adaptive simulated annealing (ASA): lessons learned. Control and Cybernetics, 25(1), 33–54. Retrieved from https://www.ingber.com/asa96_lessons.pdf
Ingber, L. (1997). Statistical mechanics of neocortical interactions: Applications of canonical momenta indicators to electroencephalography. Physical Review E, 55(4), 4578–4593. Retrieved from https://www.ingber.com/smni97_cmi.pdf
Ingber, L. (1998a). Statistical mechanics of neocortical interactions: Training and testing canonical momenta indicators of EEG. Mathematical Computer Modelling, 27(3), 33–64. Retrieved from https://www.ingber.com/smni98_cmi_test.pdf
Ingber, L. (1998b). Data mining and knowledge discovery via statistical mechanics in nonlinear stochastic systems. Mathematical Computer Modelling, 27(3), 9–31. Retrieved from https://www.ingber.com/path98_datamining.pdf
Ingber, L. (2000). High-resolution path-integral development of financial options. Physica A, 283(3-4), 529–558. Retrieved from https://www.ingber.com/markets00_highres.pdf
Ingber, L. (2006). Statistical mechanics of neocortical interactions: Portfolio of physiological indicators. Technical Report Report 2006:PPI. Ashland, OR: Physical Studies Institute, Retrieved from https://www.ingber.com/smni06_ppi.pdf
Ingber, L. (2009a). Statistical mechanics of neocortical interactions: Portfolio of physiological indicators. The Open Cybernetics Systemics Journal, 3(14), 13–26. https://dx.doi.org/10.2174/1874110x00903010013
Ingber, L. (2009b). Statistical mechanics of neocortical interactions: Nonlinear columnar electroencephalography. NeuroQuantology Journal, 7(4), 500–529. Retrieved from https://www.ingber.com/smni09_nonlin_column_eeg.pdf
Ingber, L. (2010). Trading in risk dimensions. In G. Gregoriou (Ed.), The Handbook of Trading: Strategies for Navigating and Profiting from Currency, Bond, and Stock Markets (pp. 287–300). New York: McGraw-Hill.
Ingber, L. (2012a). Columnar EEG magnetic influences on molecular development of short-term memory. In G. Kalivas, & S. Petralia (Eds.), Short-Term Memory: New Research (pp. 37-72). NY: Nova, Hauppauge. Retrieved from https://www.ingber.com/smni11_stm_scales.pdf
Ingber, L. (2012b). Influence of macrocolumnar EEG on ca waves. Current Progress Journal, 1(1), 4–8. Retrieved from https://www.ingber.com/smni12_vectpot.pdf
Ingber, L. (2012c). Adaptive simulated annealing. In J. Oliveira, A. Petraglia, L. Ingber, M. Machado, & M. Petraglia (Eds.), Stochastic global optimization and its applications with fuzzy adaptive simulated annealing (pp. 33–61). New York: Springer. Retrieved from https://www.ingber.com/asa11_options.pdf
Ingber, L. (2015). Calculating consciousness correlates at multiple scales of neocortical interactions. In A. Costa, & E. Villalba (Eds.), Horizons in Neuroscience Research (pp. 153-186). Hauppauge, NY: Nova. Retrieved from https://www.ingber.com/smni15_calc_conscious.pdf
Ingber, L. (2016). Path-integral quantum PATHTREE and PATHINT algorithms. International Journal of Innovative Research in Information Security, 3(5), 1–15. Retrieved from https://www.ingber.com/path16_quantum_path.pdf
Ingber, L. (2017a). Options on quantum money: Quantum path-integral with serial shocks. International Journal of Innovative Research in Information Security, 4(2), 7–13. Retrieved from https://www.ingber.com/path17_quantum_options_shocks.pdf
Ingber, L. (2017b). Quantum path-integral qPATHINT algorithm. The Open Cybernetics Systemics Journal, 11, 119–133. Retrieved from https://www.ingber.com/path17_qpathint.pdf
Ingber, L. (2017c). Evolution of regenerative ca-ion wave-packet in neuronal-firing fields: Quantum path-integral with serial shocks. International Journal of Innovative Research in Information Security, 4(2), 14–22. Retrieved from https://www.ingber.com/path17_quantum_pathint_shocks.pdf
Ingber, L. (2018a). Quantum variables in finance and neuroscience. Technical Report
Lecture Plates 2018:QVFN. Ashland, OR: Physical Studies Institute. Retrieved from https://www.ingber.com/path18_qpathint_lecture.pdf
Ingber, L. (2018b). Quantum variables in finance and neuroscience II. Technical Report Report 2018:FNQV. Ashland, OR: Physical Studies Institute. Retrieved from https://www.ingber.com/path18_qpathint.pdf
Ingber, L. (2018c). Quantum calcium-ion interactions with EEG. Sci, 1(7), 1–20. Retrieved from https://www.ingber.com/smni18_quantumCaEEG.pdf
Langouche, F., Roekaerts, D., & Tirapegui, E. (1979). Discretization problems of functional integrals in phase space. Physical Review D, 20, 419–432.
Langouche, F., Roekaerts, D., & Tirapegui, E. (1982). Functional Integration and Semiclassical Expansions. The Netherlands: Reidel, Dordrecht.
Nunez, P., Srinivasan, R., Ingber, L. (2013). Theoretical and experimental electrophysiology in human neocortex: Multiscale correlates of conscious experience. In M. Pesenson (Ed.), Multiscale Analysis and Nonlinear Dynamics: From genes to the brain (pp. 149–178).
New York: Wiley. https://dx.doi.org/10.1002/9783527671632.ch06
Schulman, L. (1981). Techniques and Applications of Path Integration. New York: J. Wiley and Sons.
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Published
2020-04-30
How to Cite
Ingber, L. (2020). Developing Bid-Ask Probabilities for High-Frequency Trading. Virtual Economics, 3(2), 7–24. https://doi.org/10.34021/ve.2020.03.02(1)
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