doi: 10.3934/dcdsb.2020010

Asset price volatility and price extrema

1. 

Economic Science Institute, Chapman Unviersity, Orange, CA 92866

2. 

Mathematics Department, University of Pittsburgh, Pittsburgh, PA 15260

Received  February 2019 Revised  June 2019 Published  December 2019

The relationship between price volatility and expected price market extremum is examined using a fundamental economics model of supply and demand. By examining randomness through a microeconomic setting, we obtain the implications of randomness in the supply and demand, rather than assuming that price has randomness on an empirical basis. Within a general setting of changing fundamentals, the volatility is maximum when expected prices are changing most rapidly, with the maximum of volatility reached prior to the maximum of expected price. A key issue is that randomness arises from the supply and demand, and the variance in the stochastic differential equation governing the logarithm of price must reflect this. Analogous results are obtained by further assuming that the supply and demand are dependent on the deviation from fundamental value of the asset.

Citation: Carey Caginalp, Gunduz Caginalp. Asset price volatility and price extrema. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020010
References:
[1]

P. Billingsley, Probability and Measure, Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken, NJ, 2012.  Google Scholar

[2]

F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062.  Google Scholar

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Z. Bodie, A. Kane and A. Marcus, Investments, Ed. McGraw Hill, New York, 2010. Google Scholar

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G. Caginalp and D. Balenovich, Asset flow and momentum: Deterministic and stochastic equations, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), 2119-2133.  doi: 10.1098/rsta.1999.0421.  Google Scholar

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C. Caginalp and G. Caginalp, The quotient of normal random variables and application to asset price fat tails, Physica A, 499 (2018), 457-471.  doi: 10.1016/j.physa.2018.02.077.  Google Scholar

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C. Caginalp and G. Caginalp, Stochastic asset price dynamics and volatility using a symmetric supply and demand price equation, Physica A, 523 (2019), 807-824.  doi: 10.1016/j.physa.2019.02.049.  Google Scholar

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G. Caginalp and M. Desantis, Multi-group asset flow equations and stability, Disc. and Cont. Dynam. Systems B, 16 (2011), 109-150.  doi: 10.3934/dcdsb.2011.16.109.  Google Scholar

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G. CaginalpD. Porter and V. Smith, Initial cash/asset ratio and asset prices: An experimental study, Proc. Nat. Acad. Sciences USA, 95 (1998), 756-761.  doi: 10.1073/pnas.95.2.756.  Google Scholar

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E. Díaz-Francés and F. J. Rubio, On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables, Stat. Papers, 54 (2013), 309-323.  doi: 10.1007/s00362-012-0429-2.  Google Scholar

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D. V. Hinkley, On the ratio of two correlated normal random variables, Biometrika, 56 (1969), 635-639.  doi: 10.1093/biomet/56.3.635.  Google Scholar

[11]

J. Hirshleifer and A. Glazer, Price Theory and its Applications, Prentice Hall, Engelwood Cliffs, NJ, 1997. Google Scholar

[12]

J. Hull, Risk Management and Financial Institutions, John Wiley & Sons, 2012. Google Scholar

[13]

H. H. Merdan and M. Alisen, A mathematical model for asset pricing, Applied Mathematics and Computation, 218 (2011), 1449-1456.  doi: 10.1016/j.amc.2011.06.028.  Google Scholar

[14]

H. Merdan and H. Cakmak, Liquidity effect on the asset price forecasting, Journal of Nonlinear Systems and Applications, (2012), 82–87. Google Scholar

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M. Nerouppos, D. Saunders, C. Xiouros and S. A. Zenios, Risk Management in Emerging Markets: Practical Methodologies and Empirical Tests, 2015. Google Scholar

[16]

C. R. Plott, Markets as information gathering tools, Southern Economic Journal, 10 (2006), 179-221.   Google Scholar

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C. Plott and T. Salmon, The simultaneous, ascending auction: Dynamics of price adjustment in experiments and in the UK3G spectrum auction, Journal of Economic Behavior and Organization, 53 (2004), 353-383.   Google Scholar

[18]

W. Rudin, Real and Complex Analysis, Third edition, McGraw-Hill Book Co., New York, 1987.  Google Scholar

[19]

Z. Schuss, Theory and Applications of Stochastic Processes, An analytical approach. Applied Mathematical Sciences, 170. Springer, New York, 2010. doi: 10.1007/978-1-4419-1605-1.  Google Scholar

[20]

H. Shefrin, A Behavioral Approach to Asset Pricing, Elsevier, 2008. Google Scholar

[21]

V. L. Smith, G. L. Suchanek and A. W. Williams, Bubbles, crashes, and endogenous expectations in experimental spot asset markets, Econometrica, (1988), 1119–1151. Google Scholar

[22]

D. Sornette, P. Cauwels and G. Smilyanov, 2017, Can We Use Volatility to Diagnose Financial Bubbles?, Lessons from 40 Historical Bubbles, Swiss Finance Institute Research Paper No. 17-27, (2017), Available at SSRN: https://ssrn.com/abstract=3006642 or http://dx.doi.org/10.2139/ssrn.3006642. Google Scholar

[23]

S. Stojanovic, Computational Financial Mathematics Using Mathematica: Optimal Trading in Stocks and Options, SBirkhäuser Boston, Inc., Boston, MA, TELOS. The Electronic Library of Science, Santa Clara, CA, 2003. doi: 10.1007/978-1-4612-0043-7.  Google Scholar

[24]

S. D. Stojanovic, Risk premium and fair option prices under stochastic volatility: The HARA solution, C. R. Math. Acad. Sci. Paris, 340 (2005), 551-556.  doi: 10.1016/j.crma.2004.11.002.  Google Scholar

[25]

Y. L. Tong, The Multivariate Normal Distribution, Springer Series in Statistics, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4613-9655-0.  Google Scholar

[26]

D. Watson and M. Getz, Price Theory and its Uses, 5th Ed., University Press of America, Lanham, MD, 1981. Google Scholar

[27] P. WilmottS. Howison and J. Dewynne, The Mathematics of Financial Derivatives. A Student Introduction, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511812545.  Google Scholar

show all references

References:
[1]

P. Billingsley, Probability and Measure, Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken, NJ, 2012.  Google Scholar

[2]

F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062.  Google Scholar

[3]

Z. Bodie, A. Kane and A. Marcus, Investments, Ed. McGraw Hill, New York, 2010. Google Scholar

[4]

G. Caginalp and D. Balenovich, Asset flow and momentum: Deterministic and stochastic equations, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), 2119-2133.  doi: 10.1098/rsta.1999.0421.  Google Scholar

[5]

C. Caginalp and G. Caginalp, The quotient of normal random variables and application to asset price fat tails, Physica A, 499 (2018), 457-471.  doi: 10.1016/j.physa.2018.02.077.  Google Scholar

[6]

C. Caginalp and G. Caginalp, Stochastic asset price dynamics and volatility using a symmetric supply and demand price equation, Physica A, 523 (2019), 807-824.  doi: 10.1016/j.physa.2019.02.049.  Google Scholar

[7]

G. Caginalp and M. Desantis, Multi-group asset flow equations and stability, Disc. and Cont. Dynam. Systems B, 16 (2011), 109-150.  doi: 10.3934/dcdsb.2011.16.109.  Google Scholar

[8]

G. CaginalpD. Porter and V. Smith, Initial cash/asset ratio and asset prices: An experimental study, Proc. Nat. Acad. Sciences USA, 95 (1998), 756-761.  doi: 10.1073/pnas.95.2.756.  Google Scholar

[9]

E. Díaz-Francés and F. J. Rubio, On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables, Stat. Papers, 54 (2013), 309-323.  doi: 10.1007/s00362-012-0429-2.  Google Scholar

[10]

D. V. Hinkley, On the ratio of two correlated normal random variables, Biometrika, 56 (1969), 635-639.  doi: 10.1093/biomet/56.3.635.  Google Scholar

[11]

J. Hirshleifer and A. Glazer, Price Theory and its Applications, Prentice Hall, Engelwood Cliffs, NJ, 1997. Google Scholar

[12]

J. Hull, Risk Management and Financial Institutions, John Wiley & Sons, 2012. Google Scholar

[13]

H. H. Merdan and M. Alisen, A mathematical model for asset pricing, Applied Mathematics and Computation, 218 (2011), 1449-1456.  doi: 10.1016/j.amc.2011.06.028.  Google Scholar

[14]

H. Merdan and H. Cakmak, Liquidity effect on the asset price forecasting, Journal of Nonlinear Systems and Applications, (2012), 82–87. Google Scholar

[15]

M. Nerouppos, D. Saunders, C. Xiouros and S. A. Zenios, Risk Management in Emerging Markets: Practical Methodologies and Empirical Tests, 2015. Google Scholar

[16]

C. R. Plott, Markets as information gathering tools, Southern Economic Journal, 10 (2006), 179-221.   Google Scholar

[17]

C. Plott and T. Salmon, The simultaneous, ascending auction: Dynamics of price adjustment in experiments and in the UK3G spectrum auction, Journal of Economic Behavior and Organization, 53 (2004), 353-383.   Google Scholar

[18]

W. Rudin, Real and Complex Analysis, Third edition, McGraw-Hill Book Co., New York, 1987.  Google Scholar

[19]

Z. Schuss, Theory and Applications of Stochastic Processes, An analytical approach. Applied Mathematical Sciences, 170. Springer, New York, 2010. doi: 10.1007/978-1-4419-1605-1.  Google Scholar

[20]

H. Shefrin, A Behavioral Approach to Asset Pricing, Elsevier, 2008. Google Scholar

[21]

V. L. Smith, G. L. Suchanek and A. W. Williams, Bubbles, crashes, and endogenous expectations in experimental spot asset markets, Econometrica, (1988), 1119–1151. Google Scholar

[22]

D. Sornette, P. Cauwels and G. Smilyanov, 2017, Can We Use Volatility to Diagnose Financial Bubbles?, Lessons from 40 Historical Bubbles, Swiss Finance Institute Research Paper No. 17-27, (2017), Available at SSRN: https://ssrn.com/abstract=3006642 or http://dx.doi.org/10.2139/ssrn.3006642. Google Scholar

[23]

S. Stojanovic, Computational Financial Mathematics Using Mathematica: Optimal Trading in Stocks and Options, SBirkhäuser Boston, Inc., Boston, MA, TELOS. The Electronic Library of Science, Santa Clara, CA, 2003. doi: 10.1007/978-1-4612-0043-7.  Google Scholar

[24]

S. D. Stojanovic, Risk premium and fair option prices under stochastic volatility: The HARA solution, C. R. Math. Acad. Sci. Paris, 340 (2005), 551-556.  doi: 10.1016/j.crma.2004.11.002.  Google Scholar

[25]

Y. L. Tong, The Multivariate Normal Distribution, Springer Series in Statistics, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4613-9655-0.  Google Scholar

[26]

D. Watson and M. Getz, Price Theory and its Uses, 5th Ed., University Press of America, Lanham, MD, 1981. Google Scholar

[27] P. WilmottS. Howison and J. Dewynne, The Mathematics of Financial Derivatives. A Student Introduction, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511812545.  Google Scholar
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