Option Implied Stock BuySide and SellSide Market Depths
Abstract
:1. Introduction
2. Model
2.1. The Economic Foundations
2.2. Parsimonious Stochastic VolatilityLiquidity Models
2.2.1. A stochastic Liquidity Model (SL)
2.2.2. A Stochastic VolatilityLiquidity Model (SVL)
2.2.3. A Stochastic VolatilityLiquidity Model with the Leverage Effect (SVLL)
2.3. Short Summary
2.4. Extension
2.5. Generalization
2.6. Valuation
3. Data and Methodology
3.1. Data Description
3.2. Methodology
3.2.1. Estimation for Illiquidity in Intraday Markets
3.2.2. Estimation for the Behavior of Statistical Price Process
3.2.3. Calibration of Option Models
4. Results
4.1. Estimates in Intraday Markets
4.2. Estimates in Daily Markets
4.3. Calibration in Index Option Markets
4.4. Event Days
4.5. MarkettoModel
4.6. Comparison between Two Events
4.7. Short Summary
4.8. Implication for Asymmetric Liquidity
5. Conclusions
Funding
Acknowledgments
Conflicts of Interest
Appendix A
References
 Abudy, Menachem, and Yehuda Izhakian. 2013. Pricing stock options with stochastic interest rate. International Journal of Portfolio Analysis and Management 1: 250–77. [Google Scholar] [CrossRef]
 Acharya, Viral, and Lasse Heje Pedersen. 2005. Asset pricing with liquidity risk. Journal of Financial Economics 77: 375–410. [Google Scholar] [CrossRef] [Green Version]
 Amihud, Yakov, Haim Mendelson, and Lasse Heje Pedersen. 2013. Market Liquidity: Asset Pricing, Risk, and Crises. Cambridge: Cambridge University Press. [Google Scholar]
 Ane, Thierry, and Hélyette Geman. 2000. Order flow, transaction clock and normality of asset returns. Journal of Finance 55: 2259–84. [Google Scholar] [CrossRef]
 Bakshi, Gurdip, Charles Cao, and Zhiwu Chen. 1997. Empirical performance of alternative option pricing models. Journal of Finance 52: 2003–49. [Google Scholar] [CrossRef]
 Black, Fischer. 1976. Studies of stock price volatility changes. In Proceedings of the 1976 Meetings of the Business and Economic Statistics Section. Washington: American Statistical Association, pp. 177–81. [Google Scholar]
 Brennan, Michael, and Avanidhar Subrahmanyam. 1996. Market microstructure and asset pricing: on the compensation for illiquidity in stock returns. Journal of Financial Economics 41: 441–64. [Google Scholar] [CrossRef]
 Brennan, Michael, Tarun Chordia, Avanidhar Subrahmanyam, and Qing Dong. 2012. Sellorder Liquidity and the CrossSection of Expected Stock Returns. Journal of Financial Economics 105: 523–41. [Google Scholar] [CrossRef]
 Carr, Peter, and Dilip B. Madan. 1999. Option valuation using the fast Fourier transform. Journal of Computational Finance 2: 61–73. [Google Scholar] [CrossRef] [Green Version]
 Carr, Peter, Hélyette Geman, Dilip B. Madan, and Marc Yor. 2003. Stochastic volatility for Levy processes. Mathematical Finance 13: 345–82. [Google Scholar] [CrossRef]
 Cetin, Umut, Robert A. Jarrow, and Philip Protter. 2004. Liquidity risk and arbitrage pricing theory. Finance and Stochastics 8: 311–41. [Google Scholar] [CrossRef]
 Cetin, Umut, Robert A. Jarrow, Philip Protter, and M. Warachka. 2006. Pricing options in an extended Black Scholes economy with illiquidity: theory and empirical evidence. The Review of Financial Studies 19: 493–529. [Google Scholar] [CrossRef]
 Chacko, George C., Jakub W. Jurek, and Erik Stafford. 2008. The price of immediacy. The Journal of Finance 63: 1253–90. [Google Scholar] [CrossRef]
 Chang, Carolyn W., Jack S. K. Chang, and Kian Guan Lim. 1998. Informationtime option pricing: Theory and empirical evidence. Journal of Financial Economics 48: 211–42. [Google Scholar] [CrossRef]
 Christoffersen, Peter, Ruslan Goyenko, Kris Jacobs, and Mehdi Karoui. 2018. Illiquidity premia in the equity options market. The Review of Financial Studies 31: 811–51. [Google Scholar] [CrossRef]
 Clark, Peter K. 1973. A subordinated stochastic process model with finite variance for speculative prices. Econometrica 41: 135–55. [Google Scholar] [CrossRef]
 Easley, David, Soeren Hvidkjaer, and Maureen O’Hara. 2002. Is information risk a determinant of asset returns? Journal of Finance 57: 2185–221. [Google Scholar] [CrossRef]
 Engle, Robert. 2004. Risk and volatility: Econometric models and financial practice. The American Economic Review 94: 405–20. [Google Scholar] [CrossRef]
 Gradshetyn, Izrail Solomonovich, and Iosif Moiseevich Ryzhik. 1980. Table of Integrals, Series, and Products. New York: Academic Press. [Google Scholar]
 Harris, Larry. 2003. Trading and Exchanges: Market Microstructure for Practitioners. New York: Oxford University Press. [Google Scholar]
 Hasbrouck, Joel. 2007. Empirical Market Microstructure: The Institutions, Economics, and Econometrics of Securities Trading. New York: Oxford University Press. [Google Scholar]
 Jagannathan, Raj. 2008. A class of asset pricing models governed by subordinate processes that signal economic shocks. Journal of Economic Dynamics and Control 32: 3820–46. [Google Scholar] [CrossRef]
 Kyle, Albert S. 1985. Continuous auctions and insider trading. Econometrica 53: 1315–36. [Google Scholar] [CrossRef]
 Lee, Charles M., and Mark J. Ready. 1991. Inferring trade direction from intraday data. Journal of Finance 46: 733–46. [Google Scholar] [CrossRef]
 Luciano, Elisa, and Wim Schoutens. 2006. A multivariate jumpdriven financial asset model. Quantitative Finance 6: 385–402. [Google Scholar] [CrossRef] [Green Version]
 Madan, Dilip B., Peter Carr, and Eric C. Chang. 1998. The variance gamma process and option pricing. European Finance Review 2: 79–105. [Google Scholar] [CrossRef]
 Pastor, Ľuboš, and Robert F. Stambaugh. 2003. Liquidity risk and expected stock returns. Journal of Political Economy 111: 642–85. [Google Scholar] [CrossRef]
 Schoutens, Wim. 2003. Levy Processes in Finance: Pricing Financial Derivatives. West Sussex: John Wiley & Sons Ltd. [Google Scholar]
 Watanabe, Masahiro. 2007. A Model of Stochastic Liquidity. Yale ICF Working Paper No. 0318; EFA 2003 Glasgow Annual Conference Paper. Available online: https://ssrn.com/abstract=413983 (accessed on 26 October 2019).
1  I also run the following regression with fixed cost and the conclusion does not change:
$$\mathrm{log}({S}_{t})\mathrm{log}({S}_{t1})=a+{\lambda}_{b}{x}_{t}{\lambda}_{s}{y}_{t}+c[{d}_{t}{d}_{t1}]+{\epsilon}_{t}$$

2  The results are verified using the NelderMead simplex approach as well as the simulated annealing method. 
$\mathbf{Model}:\text{}\mathbf{log}({\mathit{S}}_{\mathit{t}})\mathbf{log}({\mathit{S}}_{\mathit{t}1})=\mathit{a}+{\mathit{\lambda}}_{\mathit{b}}{\mathit{x}}_{\mathit{t}}{\mathit{\lambda}}_{\mathit{s}}{\mathit{y}}_{\mathit{t}}+{\mathit{\epsilon}}_{\mathit{t}}$  

Company Name  ${\mathit{\lambda}}_{\mathit{b}}$  ${\mathit{\lambda}}_{\mathit{s}}$ 
IBM  0.0138 (9.45)  0.1414 (50.61) 
KO  0.0320 (13.25)  0.2064 (20.64) 
FDX  0.0433 (11.81)  0.1859 (17.34) 
BKS  0.0994 (15.46)  0.1814 (11.25) 
BA  0.0270 (20.27)  0.2722 (38.90) 
DIS  0.0447 (26.16)  0.2583 (45.71) 
$\mathbf{Density}\text{}\mathbf{Function}:\text{}\mathit{h}(\mathit{z})=\frac{2\mathbf{exp}(\mathit{\mu}{}^{\prime}\mathit{x}/{\mathit{\sigma}}^{2})}{\sqrt{2\mathit{\pi}}\mathit{\sigma}{}^{\prime}\mathit{\Gamma}(\mathit{\beta}{}^{\prime}\mathit{t})}{\left(\frac{{\mathit{x}}^{2}}{2\mathit{\sigma}{{}^{\prime}}^{2}+\mathit{\mu}{{}^{\prime}}^{2}}\right)}^{\frac{\mathit{\beta}{}^{\prime}\mathit{t}}{2}\frac{1}{4}}{\mathit{K}}_{\mathit{\beta}{}^{\prime}\mathit{t}\frac{1}{2}}\left(\frac{1}{{\mathit{\sigma}}^{2}}\sqrt{{\mathit{x}}^{2}(2\mathit{\sigma}{{}^{\prime}}^{2}+\mathit{\mu}{{}^{\prime}}^{2})}\right)$  

Parameter Estimated  IBM  KO 
$\mu {}^{\prime}$  −0.0670 (0.0003)  −0.0871 (0.0003) 
$\sigma {}^{\prime}$  0.1000 (0.0003)  0.1086 (0.0003) 
$\beta {}^{\prime}$  2.6168 (0.0001)  1.9070 (0.0002) 
m  −0.0652 (0.0003)  −0.3190 (0.0003) 
Implied ${\lambda}_{b}$/${\lambda}_{s}$  0.0447/0.1118  0.0447/0.1318 
Log likelihood value  103.9537  158.2823 
Parameter estimated  FDX  BKS 
$\mu {}^{\prime}$  −0.2178 (0.0003)  0.1615 (0.0001) 
$\sigma {}^{\prime}$  0.1615 (0.0003)  0.1021 (0.0001) 
$\beta {}^{\prime}$  2.3168 (0.0002)  3.4623 (0.0000) 
m  −0.1850 (0.0003)  0.2849 (0.0001) 
Implied ${\lambda}_{b}$/${\lambda}_{s}$  0.0489/0.2667  0.1891/0.0276 
Log likelihood value  107.1581  87.2402 
Parameter estimated  BA  DIS 
$\mu {}^{\prime}$  −0.0084 (0.0002)  0.0794 (0.0003) 
$\sigma {}^{\prime}$  0.0512 (0.0001)  0.2296 (0.0003) 
$\beta {}^{\prime}$  2.1419 (0.0000)  1.6606 (0.0001) 
m  0.0252 (0.0002)  0.1951 (0.0003) 
Implied ${\lambda}_{b}$/${\lambda}_{s}$  0.0322/0.0406  0.2068/0.1274 
Log likelihood value  136.8132  98.7784 
Model  Index  $\mathit{\sigma}$  $\mathit{\kappa}$  $\mathit{\theta}$  $\mathit{\eta}$  $\mathit{\rho}$  ${\mathit{\lambda}}_{1}$  ${\mathit{\lambda}}_{2}$  RMSE 

BS  SPX  0.1287  9.8557  
SL  SPX  0.0920  0.0354  0.1194  9.7068  
SVL  SPX  0.0718  2.0205  0.0930  2.0032  0.0283  0.0912  7.6090  
SVLL  SPX  0.0131  1.6976  0.4792  0.6444  −0.20  0.0017  0.0514  6.4715 
MSVLL  SPX  0.0086  1.6019  0.7849  0.9085  −0.18  0.0009  0.0430  7.7160 
DJX  0.0089  2.4375  0.4602  0.4552  −0.36  0.0006  0.0688  
NDX  0.0198  0.4503  0.4895  0.4392  −0.47  0.0041  0.0474 
DateIndex  $\mathit{\beta}$  ${\mathit{\lambda}}_{1}$  ${\mathit{\lambda}}_{2}$  $\mathit{\kappa}$  $\mathit{\theta}$  $\mathit{\eta}$  $\mathit{\rho}$ 

2/26 SPX  3.0661  1.1183  1.1744  1.5414  $2.19\times {10}^{6}$  0.8265  −0.10 
2/26 DJX  3.0661  0.0915  0.1441  1.5393  $3.65\times {10}^{5}$  0.7729  −0.13 
2/26 NDX  3.0661  0.5553  0.5836  0.7651  $2.55\times {10}^{5}$  1.1170  −0.14 
2/27 SPX  4.1101  0.0071  0.0033  0.0124  0.5539  0.9766  −0.13 
2/27 DJX  4.1101  0.0525  0.0823  0.6733  $7.53\times {10}^{6}$  0.6754  −0.19 
2/27 NDX  4.1101  0.0205  0.0230  0.0270  0.8568  0.6662  −0.25 
2/28 SPX  2.8288  0.8360  0.8780  1.1523  0.0132  1.0733  −0.12 
2/28 DJX  2.8288  0.0886  0.1347  1.2108  $2.29\times {10}^{5}$  0.9183  −0.14 
2/28 NDX  2.8288  0.4334  0.4543  0.5979  0.1839  1.1247  −0.16 
Date  Maturity  $\mathit{\beta}$  ${\mathit{\lambda}}_{1}$  ${\mathit{\lambda}}_{2}$  RMSE  ARPE (%)  APE (%)  AAE 

10 September  40  11.00  0.0317  0.0883  0.6963  9.05  0.99  0.5540 
68  7.97  0.0349  0.0982  0.3794  6.02  0.92  0.3902  
103  5.50  0.0372  0.1158  0.2767  2.48  0.46  0.2910  
187  3.53  0.0457  0.1346  0.1668  0.78  0.27  0.1623  
285  2.77  0.0448  0.1476  0.2794  0.66  0.41  0.3139  
17 September  33  22.31  0.0300  0.0740  0.2825  3.75  1.07  0.4280 
61  15.98  0.0327  0.0763  0.1537  2.12  0.40  0.1939  
96  8.59  0.0362  0.1033  0.2641  1.79  0.93  0.5449  
180  5.99  0.0294  0.1167  0.2517  2.00  0.62  0.3708  
278  5.04  0.0035  0.1270  0.4307  1.59  1.13  0.9207  
18 September  32  17.25  0.0288  0.0841  0.2171  4.03  0.64  0.2870 
60  10.23  0.0352  0.1000  0.2137  3.37  0.54  0.3021  
95  7.72  0.0351  0.1096  0.3114  2.32  0.79  0.4279  
179  3.10  0.0546  0.1664  0.2564  1.76  0.71  0.3964  
277  2.16  0.0747  0.1869  0.3767  1.15  0.80  0.6142 
© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Tsai, F.T. Option Implied Stock BuySide and SellSide Market Depths. Risks 2019, 7, 108. https://doi.org/10.3390/risks7040108
Tsai FT. Option Implied Stock BuySide and SellSide Market Depths. Risks. 2019; 7(4):108. https://doi.org/10.3390/risks7040108
Chicago/Turabian StyleTsai, FengTse. 2019. "Option Implied Stock BuySide and SellSide Market Depths" Risks 7, no. 4: 108. https://doi.org/10.3390/risks7040108