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
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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 
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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