# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2022062
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## Asset prices with investor protection and past information

 1 Department of Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan 610074, China 2 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610074, China 3 Institute of Financial Research, People's Bank of China, Beijing, China

*Corresponding author: Nan-Jing Huang

Received  December 2021 Revised  March 2022 Early access April 2022

Fund Project: This work was supported by the National Natural Science Foundation of China (11671282, 11801462, 12171339)

In this paper, we consider a dynamic asset pricing model in an approximate fractional economy to address empirical regularities related to both investor protection and past information. Our newly developed model features not only a controlling shareholder who diverts a fraction of the output, but also good (or bad) memory in his budget dynamics which can be well-calibrated by a pathwise way from the historical data. We find that poorer investor protection leads to higher stock holdings of the controlling shareholder, lower gross stock returns, lower interest rates, and lower modified stock volatilities if the ownership concentration is sufficiently high. More importantly, by establishing an approximation scheme for good (bad) memory of investors on the historical market information, we conclude that good (bad) memory would increase (decrease) aforementioned dynamics and reveal that good (bad) memory strengthens (weakens) investor protection for the minority shareholder when the ownership concentration is sufficiently high, while good (bad) memory inversely weakens (strengthens) investor protection for the minority shareholder when the ownership concentration is sufficiently low. Our model's implications are consistent with a number of interesting facts documented in the recent literature.

Citation: Jia Yue, Ming-Hui Wang, Nan-Jing Huang, Ben-zhang Yang. Asset prices with investor protection and past information. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022062
##### References:
 [1] R. Albuquerue and N. Wang, Agency conflicts, investment, and asset pricing, J. Finance, 63 (2008), 1-40. [2] E. Alòs, O. Mazet and D. Nualart, Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than $1/2$, Stochastic Process. Appl., 86 (2000), 121-139.  doi: 10.1016/S0304-4149(99)00089-7. [3] R. Anderson and Y. Liu, How low can you go? Negative interest rates and investors' flight to safety, The Regional Economist, 1 (2013), 12-13. [4] S. Basak, G. Chabakauri and M. D. Yavuz, Investor protection and asset prices, The Review of Financial Studies, 32 (2019), 4905-4946. [5] L. A. Bebchuk, R. H. Kraakman and G. G. Triantis, Stock pyramids, cross-ownership, and dual class equity: The creation and agency costs of separating control from cash flow rights, In Concentrated Corporate Ownership, (ed. R. K. Morck), The University of Chicago Press, (2000), 295–315. doi: 10.2139/ssrn.147590. [6] F. Biagini, Y. Hu, B. Øksendal and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer-Verlag, London, 2008. doi: 10.1007/978-1-84628-797-8. [7] M. Bru, Wishart process, J. Theoret. Probab., 4 (1991), 725-751.  doi: 10.1007/BF01259552. [8] G. Chabakauri, Dynamic equilibrium with two stocks, heterogeneous investors, and porfolio constraints, The Review of Financial Studies, 26 (2013), 3104-3141. [9] J. Chen, D. Ma, X. Song and M. Tippett, Negative real interest rates, The European J. Finance, 23 (2017), 1447-1467. [10] J. C. Cox, J. E. Ingersoll and S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242. [11] C. Doidge, A. Karolyi and R. Stulz, Why are foreign firms listed in the U. S. worth more?, J. Financial Economics, 71 (2004), 205-238. [12] J. Dow, G. B. Gorton and A. Krishnamurthy, Equilibriun investment and asset prices under imperfect corporate control, American Economic Review, 95 (2005), 659-681. [13] N. T. Dung, Semimartingale approximation of fractional Brownian motion and its applications, Comput. Math. Appl., 61 (2011), 1844-1854.  doi: 10.1016/j.camwa.2011.02.013. [14] J. D. Fonseca, M. Grasselli and C. Tebaldi, Option pricing when correlations are stochastic: An analytical framework, Review of Derivatives Research, 10 (2007), 151-180. [15] J.-P. Fouque and R. Hu, Optimal portfolio under fast mean-reverting fractional stochastic environment, SIAM J. Financial Math., 9 (2018), 564-601.  doi: 10.1137/17M1134068. [16] J.-P. Fouque and R. Hu, Optimal portfolio under fractional stochastic environment, Math. Finance, 29 (2019), 697-734.  doi: 10.1111/mafi.12195. [17] J. Franke, W. K. Härdle and C. M. Hafner, Statistics of Financial Markets: An Introduction, 5$^th$ edition, Universitext. Springer, Cham, 2019. doi: 10.1007/978-3-030-13751-9. [18] M. Giannetti and Y. Koskinen, Investor protection, equity returns, and financial globalization, J. Financial and Quantitative Analysis, 45 (2010), 135-168. [19] P. Gompers, J. Ishii and A. Metrick, Corporate governance and equity prices, Quarterly J. Economics, 118 (2003), 107-156. [20] G. B. Gorton, P. He and L. Huang, Agency-based asset pricing, J. Economic Theory, 149 (2014), 311-349.  doi: 10.1016/j.jet.2012.09.017. [21] M. Guidolin and A. Timmermann, Economic implications of the bull and bear regimes in UK stock and bond returns, The Economic Journal, 115 (2005), 111-143. [22] A. Jentzen, P. E. Kloeden and A. Neuenkirch, Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coefficients, Numer. Math., 112 (2009), 41-64.  doi: 10.1007/s00211-008-0200-8. [23] A. Jentzen and P. E. Kloeden, Taylor Approximations for Stochastic Partial Differential Equations, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611972016. [24] C. P. Jones, M. D. Wlker and J. W. Wilson, Analyzing stock market volatility using extreme-day measures, J. Financial Research, 27 (2004), 585-601. [25] P. E. Kloeden, A. Neuenkirch and R. Pavani, Multilevel Monte Carlo for stochastic differential equations with additive fractional noise, Ann. Oper. Res., 189 (2011), 255-276.  doi: 10.1007/s10479-009-0663-8. [26] E. Kole and D. Van Dijk, How to identify and forecast bull and bear markets?, J. Appl. Econometrics, 32 (2017), 120-139.  doi: 10.1002/jae.2511. [27] H. Kraft and F. Weiss, Consumption-portfolio choice with preferences for cash, J. Econom. Dynam. Control, 98 (2019), 40-59.  doi: 10.1016/j.jedc.2018.09.006. [28] Y. K. Kwok, Mathematical Models of Financial Derivatives, 2$^nd$ edition, Springer Finance. Springer, Berlin, 2008. [29] R. H. Kwon, A stochastic semidefinite programming approach for bounds on option pricing under regime switching, Annals of Operations Research, 237 (2016), 41-75. [30] R. La Porta, F. Lopez-de-Silanes and A. Shleifer, Corporate ownership around the world, The Journal of Finance, 54 (1999), 471-517. [31] R. La Porta, F. Lopez-de-Silanes, A. Shleifer and R. W. Vishny, Investor protection and corporate valuation, The Journal of Finance, 57 (2002), 1147-1170. [32] M. Y. Li, R. Gençay and Y. Xue, Is it Brownian or fractional Brownian motion?, Econom. Lett., 145 (2016), 52-55.  doi: 10.1016/j.econlet.2016.05.012. [33] A. W. Lo and A. C. Mackinley, Stock market prices do not follow random walks: Evidence from a simple specification test, The Review of Financial Studies, 1 (1988), 41-66. [34] J. M. Maheu and T. H. McCurdy, Identifying bull and bear markets in stock returns, J. Bus. Econom. Statist., 18 (2000), 100-112.  doi: 10.2307/1392140. [35] B. B. Mandelbrot and J. W. van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437.  doi: 10.1137/1010093. [36] Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-75873-0. [37] M. Ohnishi and Y. Osaki, The comparative statics on asset prices based on bull and bear market measure, European J. Oper. Res., 168 (2006), 291-300.  doi: 10.1016/j.ejor.2004.07.005. [38] G. Perez-Quiros and A. Timmermann, Firm size and cyclical variations in stock returns, J. Finance, 55 (2000), 1229-1262. [39] J. Picard, Representation formulae for the fractional Brownian motion, In Séminaire de Probabilités XLIII, (eds. C. Donati-Martin et al.), Lecture Notes in Mathematic 2006 (2011), 3–70. doi: 10.1007/978-3-642-15217-7_1. [40] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer-Verlag, Berlin, 1994. [41] S. Rostek, Option Pricing in Fractional Brownian Markets, Springer-Verlag, Berlin, Heidelberg, 2009. doi: 10.1007/978-3-642-00331-8. [42] G. W. Schwert, Stock volatility during the recent financial crisis, European Financial Management, 17 (2011), 789-805. [43] A. Shleifer and D. Wolfenzon, Investor protection and equity markets, J. Financial Economics, 66 (2002), 3-27. [44] F. Shokrollahi and A. Kılıçman, The valuation of currency options by fractional Brownian motion, SpringerPlus, 5 (2016). [45] T. H. Thao, A note on fractional Brownian motion, Vietnam J. Math., 31 (2003), 255-260. [46] T. H. Thao and T. T. Nguyen, Fractal Langevin equation, Vietnam J. Math., 30 (2002), 89-96. [47] M. I. M. Wahab and C.-G. Lee, Pricing swing options with regime switching, Ann. Oper. Res., 185 (2011), 139-160.  doi: 10.1007/s10479-009-0599-z. [48] L. Xu and Z. Li, Doubly perturbed neutral stochastic functional equations driven by fractional Brownian motion, J. Partial Differ. Equ., 28 (2015), 305-314.  doi: 10.4208/jpde.v28.n4.2. [49] J. Yue and N. J. Huang, Fractional Wishart processes and $\varepsilon$-fractional Wishart processes with applications, Comput. Math. Appl., 75 (2018), 2955-2977.  doi: 10.1016/j.camwa.2018.01.024.

show all references

##### References:
 [1] R. Albuquerue and N. Wang, Agency conflicts, investment, and asset pricing, J. Finance, 63 (2008), 1-40. [2] E. Alòs, O. Mazet and D. Nualart, Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than $1/2$, Stochastic Process. Appl., 86 (2000), 121-139.  doi: 10.1016/S0304-4149(99)00089-7. [3] R. Anderson and Y. Liu, How low can you go? Negative interest rates and investors' flight to safety, The Regional Economist, 1 (2013), 12-13. [4] S. Basak, G. Chabakauri and M. D. Yavuz, Investor protection and asset prices, The Review of Financial Studies, 32 (2019), 4905-4946. [5] L. A. Bebchuk, R. H. Kraakman and G. G. Triantis, Stock pyramids, cross-ownership, and dual class equity: The creation and agency costs of separating control from cash flow rights, In Concentrated Corporate Ownership, (ed. R. K. Morck), The University of Chicago Press, (2000), 295–315. doi: 10.2139/ssrn.147590. [6] F. Biagini, Y. Hu, B. Øksendal and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer-Verlag, London, 2008. doi: 10.1007/978-1-84628-797-8. [7] M. Bru, Wishart process, J. Theoret. Probab., 4 (1991), 725-751.  doi: 10.1007/BF01259552. [8] G. Chabakauri, Dynamic equilibrium with two stocks, heterogeneous investors, and porfolio constraints, The Review of Financial Studies, 26 (2013), 3104-3141. [9] J. Chen, D. Ma, X. Song and M. Tippett, Negative real interest rates, The European J. Finance, 23 (2017), 1447-1467. [10] J. C. Cox, J. E. Ingersoll and S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242. [11] C. Doidge, A. Karolyi and R. Stulz, Why are foreign firms listed in the U. S. worth more?, J. Financial Economics, 71 (2004), 205-238. [12] J. Dow, G. B. Gorton and A. Krishnamurthy, Equilibriun investment and asset prices under imperfect corporate control, American Economic Review, 95 (2005), 659-681. [13] N. T. Dung, Semimartingale approximation of fractional Brownian motion and its applications, Comput. Math. Appl., 61 (2011), 1844-1854.  doi: 10.1016/j.camwa.2011.02.013. [14] J. D. Fonseca, M. Grasselli and C. Tebaldi, Option pricing when correlations are stochastic: An analytical framework, Review of Derivatives Research, 10 (2007), 151-180. [15] J.-P. Fouque and R. Hu, Optimal portfolio under fast mean-reverting fractional stochastic environment, SIAM J. Financial Math., 9 (2018), 564-601.  doi: 10.1137/17M1134068. [16] J.-P. Fouque and R. Hu, Optimal portfolio under fractional stochastic environment, Math. Finance, 29 (2019), 697-734.  doi: 10.1111/mafi.12195. [17] J. Franke, W. K. Härdle and C. M. Hafner, Statistics of Financial Markets: An Introduction, 5$^th$ edition, Universitext. Springer, Cham, 2019. doi: 10.1007/978-3-030-13751-9. [18] M. Giannetti and Y. Koskinen, Investor protection, equity returns, and financial globalization, J. Financial and Quantitative Analysis, 45 (2010), 135-168. [19] P. Gompers, J. Ishii and A. Metrick, Corporate governance and equity prices, Quarterly J. Economics, 118 (2003), 107-156. [20] G. B. Gorton, P. He and L. Huang, Agency-based asset pricing, J. Economic Theory, 149 (2014), 311-349.  doi: 10.1016/j.jet.2012.09.017. [21] M. Guidolin and A. Timmermann, Economic implications of the bull and bear regimes in UK stock and bond returns, The Economic Journal, 115 (2005), 111-143. [22] A. Jentzen, P. E. Kloeden and A. Neuenkirch, Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coefficients, Numer. Math., 112 (2009), 41-64.  doi: 10.1007/s00211-008-0200-8. [23] A. Jentzen and P. E. Kloeden, Taylor Approximations for Stochastic Partial Differential Equations, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611972016. [24] C. P. Jones, M. D. Wlker and J. W. Wilson, Analyzing stock market volatility using extreme-day measures, J. Financial Research, 27 (2004), 585-601. [25] P. E. Kloeden, A. Neuenkirch and R. Pavani, Multilevel Monte Carlo for stochastic differential equations with additive fractional noise, Ann. Oper. Res., 189 (2011), 255-276.  doi: 10.1007/s10479-009-0663-8. [26] E. Kole and D. Van Dijk, How to identify and forecast bull and bear markets?, J. Appl. Econometrics, 32 (2017), 120-139.  doi: 10.1002/jae.2511. [27] H. Kraft and F. Weiss, Consumption-portfolio choice with preferences for cash, J. Econom. Dynam. Control, 98 (2019), 40-59.  doi: 10.1016/j.jedc.2018.09.006. [28] Y. K. Kwok, Mathematical Models of Financial Derivatives, 2$^nd$ edition, Springer Finance. Springer, Berlin, 2008. [29] R. H. Kwon, A stochastic semidefinite programming approach for bounds on option pricing under regime switching, Annals of Operations Research, 237 (2016), 41-75. [30] R. La Porta, F. Lopez-de-Silanes and A. Shleifer, Corporate ownership around the world, The Journal of Finance, 54 (1999), 471-517. [31] R. La Porta, F. Lopez-de-Silanes, A. Shleifer and R. W. Vishny, Investor protection and corporate valuation, The Journal of Finance, 57 (2002), 1147-1170. [32] M. Y. Li, R. Gençay and Y. Xue, Is it Brownian or fractional Brownian motion?, Econom. Lett., 145 (2016), 52-55.  doi: 10.1016/j.econlet.2016.05.012. [33] A. W. Lo and A. C. Mackinley, Stock market prices do not follow random walks: Evidence from a simple specification test, The Review of Financial Studies, 1 (1988), 41-66. [34] J. M. Maheu and T. H. McCurdy, Identifying bull and bear markets in stock returns, J. Bus. Econom. Statist., 18 (2000), 100-112.  doi: 10.2307/1392140. [35] B. B. Mandelbrot and J. W. van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437.  doi: 10.1137/1010093. [36] Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-75873-0. [37] M. Ohnishi and Y. Osaki, The comparative statics on asset prices based on bull and bear market measure, European J. Oper. Res., 168 (2006), 291-300.  doi: 10.1016/j.ejor.2004.07.005. [38] G. Perez-Quiros and A. Timmermann, Firm size and cyclical variations in stock returns, J. Finance, 55 (2000), 1229-1262. [39] J. Picard, Representation formulae for the fractional Brownian motion, In Séminaire de Probabilités XLIII, (eds. C. Donati-Martin et al.), Lecture Notes in Mathematic 2006 (2011), 3–70. doi: 10.1007/978-3-642-15217-7_1. [40] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer-Verlag, Berlin, 1994. [41] S. Rostek, Option Pricing in Fractional Brownian Markets, Springer-Verlag, Berlin, Heidelberg, 2009. doi: 10.1007/978-3-642-00331-8. [42] G. W. Schwert, Stock volatility during the recent financial crisis, European Financial Management, 17 (2011), 789-805. [43] A. Shleifer and D. Wolfenzon, Investor protection and equity markets, J. Financial Economics, 66 (2002), 3-27. [44] F. Shokrollahi and A. Kılıçman, The valuation of currency options by fractional Brownian motion, SpringerPlus, 5 (2016). [45] T. H. Thao, A note on fractional Brownian motion, Vietnam J. Math., 31 (2003), 255-260. [46] T. H. Thao and T. T. Nguyen, Fractal Langevin equation, Vietnam J. Math., 30 (2002), 89-96. [47] M. I. M. Wahab and C.-G. Lee, Pricing swing options with regime switching, Ann. Oper. Res., 185 (2011), 139-160.  doi: 10.1007/s10479-009-0599-z. [48] L. Xu and Z. Li, Doubly perturbed neutral stochastic functional equations driven by fractional Brownian motion, J. Partial Differ. Equ., 28 (2015), 305-314.  doi: 10.4208/jpde.v28.n4.2. [49] J. Yue and N. J. Huang, Fractional Wishart processes and $\varepsilon$-fractional Wishart processes with applications, Comput. Math. Appl., 75 (2018), 2955-2977.  doi: 10.1016/j.camwa.2018.01.024.
Past information $\Lambda_t$ with $\varepsilon = 10^{-5}$
Past information $\Lambda_t$ with $\varepsilon = 0.1$
Stock holdings $n_{Ct}^*$ with different investor protection
Stock holdings $n_{Ct}^*$ with different past information
The fraction $x_t^*$ with different investor protection
The fraction $x_t^*$ with different past information
Modified stock volatilities $\sigma^H$ with different investor protection
Modified stock volatilities $\sigma^H$ with different past information
Modified stock returns $\mu^H$ with different investor protection
Modified stock returns $\mu^H$ with different past information
Gross stock returns $\mu^G$ with different investor protection
Gross stock returns $\mu^G$ with different past information
Interest rates $r$ with different investor protection
Interest rates $r$ with different past information
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