May  2020, 25(5): 1935-1958. 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  May 2020 Early access  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 and Continuous Dynamical Systems - B, 2020, 25 (5) : 1935-1958. 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.

[2]

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

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[12]

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

[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.

[14]

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

[15]

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

[16]

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

[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. 

[18]

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

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[26]

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

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

show all references

References:
[1]

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

[2]

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

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[12]

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

[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.

[14]

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

[15]

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

[16]

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

[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. 

[18]

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

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[26]

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

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