本文研究四個市場(KOSPI200、NIKKEI225、FTSE100、ESTX50)的日資料中，估計買賣價差出現負價差的比例。我們使用Cowin and Schultz (2012) 提出的CS模型以及Li, Lambe, and Adegbite (2018) 提出的Basic High and Low (BHL) 模型來估計買賣價差，我們發現這兩種模型在我們的四個市場中出現負價差的比例主要介於50%-60%之間。因為許多研究表示全距是一個很好拿來衡量波動度的變數，他可以更好的反映市場波動，並且更加貼近常態分佈。因此我們採用Alizadeh, Brandt, and Diebold (2002) 提出的Range-based Stochastic Volatility (SV) 模型，並且我們使用Kalman filter估計出每日波動度的條件期待值以及每兩天為單位的波動度條件期待值，再將估計出來的條件期待值分別帶入上述兩個買賣價差的模型當中，去觀察這兩個買賣價差模型在四個市場上的負價差比例。

This study examines the proportion of negative bid-ask spreads in the daily data of four markets (KOSPI200, NIKKEI225, FTSE100, ESTX50). We use the CS model proposed by Cowin and Schultz (2012) and the Basic High and Low (BHL) model proposed by Li, Lambe, and Adegbite (2018) to estimate the bid-ask spreads. We find that the proportion of negative bid-ask spreads in these four markets ranges between 50% and 60% for both models. As many studies indicate that range is a good variable for measuring volatility, it can better reflect market fluctuations and is closer to a normal distribution. Hence, we adopt the Range-based Stochastic Volatility (SV) model proposed by Alizadeh, Brandt, and Diebold (2002). Using the Kalman filter, we estimate the conditional expectation of daily volatility and the conditional expectation of volatility over two-day intervals. These conditional expectations are then inputted into the two bid-ask spread models mentioned above to observe the proportion of negative bid-ask spreads in the four markets.

論文審定書 i
摘要 ii
Abstract iii
Table of Contents iv
List of Figures vi
List of Tables vii
1 Introduction 1
2 Literature review 5
2.1 Bid-Ask Spread Model 5
2.2 Range as Volatility Proxy 10
2.3 Stochastic Volatility Model 15
3 Data 20
4 Model and Method 33
4.1 CS Model 33
4.2 BHL Model 34
4.3 Range-based Standard Volatility Model (Range-based SV model, One-factor SV model) 36
5 Simulation 40
5.1 Monte Carlo I 40
5.2 Monte Carlo II 41
6 Empirical Results 47
6.1 Summary of parameters estimation 47
6.2 The initial results of the CS and BHL models 48
6.3 Adding the SV-daily model 48
6.4 Adding the SV-bidaily model 49
6.5 Discussion and Restriction 49
7 Conclusion 54
References 55

Abdi, F., & Ranaldo, A. (2017). A simple estimation of bid-ask spreads from daily close, high, and low prices. The Review of Financial Studies, 30(12), 4437-4480.
Alizadeh, S., Brandt, M. W., & Diebold, F. X. (2002). Range‐based estimation of stochastic volatility models. The Journal of Finance, 57(3), 1047-1091.
Beckers, S. (1983). Variances of security price returns based on high, low, and closing prices. Journal of Business, 97-112.
Chan, J. C., & Grant, A. L. (2016). Modeling energy price dynamics: GARCH versus stochastic volatility. Energy Economics, 54, 182-189.
Cobandag Guloglu, Z., & Ekinci, C. (2022). Liquidity measurement: A comparative review of the literature with a focus on high frequency. Journal of Economic Surveys, 36(1), 41-74.
Corwin, S. A., & Schultz, P. (2012). A simple way to estimate bid‐ask spreads from daily high and low prices. The Journal of Finance, 67(2), 719-760.
Enoksen, F. A., Landsnes, C. J., Lučivjanská, K., & Molnár, P. (2020). Understanding risk of bubbles in cryptocurrencies. Journal of Economic Behavior & Organization, 176, 129-144.
Garman, M. B., & Klass, M. J. (1980). On the estimation of security price volatilities from historical data. Journal of Business, 67-78.
Glosten, L. R., Jagannathan, R., & Runkle, D. E. (1993). On the relation between the expected value and the volatility of the nominal excess return on stocks. The Journal of Finance, 48(5), 1779-1801.
Goyenko, R. Y., Holden, C. W., & Trzcinka, C. A. (2009). Do liquidity measures measure liquidity?. Journal of Financial Economics, 92(2), 153-181.
Hansson, B., & Hördahl, P. (2005). Forecasting variance using stochastic volatility and GARCH. The European Journal of Finance, 11(1), 33-57.
Harris, L. (2003). Trading and exchanges: Market microstructure for practitioners. Oxford University Press, USA.
Harris, L. (1990). Statistical properties of the Roll serial covariance bid/ask spread estimator. The Journal of Finance, 45(2), 579-590.
Harvey, A., Ruiz, E., & Shephard, N. (1994). Multivariate stochastic variance models. The Review of Economic Studies, 61(2), 247-264.
Hasbrouck, J. (2009). Trading costs and returns for US equities: Estimating effective costs from daily data. The Journal of Finance, 64(3), 1445-1477.
Lesmond, D. A., Ogden, J. P., & Trzcinka, C. A. (1999). A new estimate of transaction costs. The Review of Financial Studies, 12(5), 1113-1141.
Le, H., & Gregoriou, A. (2020). How do you capture liquidity? A review of the literature on low‐frequency stock liquidity. Journal of Economic Surveys, 34(5), 1170-1186.
Li, Z., Lambe, B., & Adegbite, E. (2018). New bid-ask spread estimators from daily high and low prices. International Review of Financial Analysis, 60, 69-86.
Lin, C. C. (2014). Estimation accuracy of high–low spread estimator. Finance Research Letters, 11(1), 54-62.
Liu, W. (2006). A liquidity-augmented capital asset pricing model. Journal of Financial Economics, 82(3), 631-671.
Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica: Journal of the econometric society, 347-370.
Parkinson, M. (1980). The extreme value method for estimating the variance of the rate of return. Journal of Business, 61-65.
Rogers, L. C. G., & Satchell, S. E. (1991). Estimating variance from high, low and closing prices. The Annals of Applied Probability, 504-512.
Roll, R. (1984). A simple implicit measure of the effective bid‐ask spread in an efficient market. The Journal of Finance, 39(4), 1127-1139.
Sandmann, G., & Koopman, S. J. (1998). Estimation of stochastic volatility models via Monte Carlo maximum likelihood. Journal of Econometrics, 87(2), 271-301.
Tan, S. K., Ng, K. H., Chan, J. S. K., & Mohamed, I. (2019). Quantile range-based volatility measure for modelling and forecasting volatility using high frequency data. The North American Journal of Economics and Finance, 47, 537-551.
Taylor S.J. (2005) Asset price dynamics, volatility, and prediction: Princeton University Press
Taylor, S. J. (1994). Modeling stochastic volatility: A review and comparative study. Mathematical finance, 4(2), 183-204.
Tiwari, A. K., Kumar, S., & Pathak, R. (2019). Modelling the dynamics of Bitcoin and Litecoin: GARCH versus stochastic volatility models. Applied Economics, 51(37), 4073-4082.