Robust Volatility Estimation with and Without the Drift Parameter
- 9 Downloads
Abstract
We find the closed form solution for the joint probability of the running maximum and the drawdown of the Brownian motion with a non-zero drift parameter at a random time that is exponentially distributed and independent of the Brownian motion. This characterization leads us to come up with a robust method of estimating volatility using open, high, low and closing prices. We rigorously show the independence of robust volatility estimators based on extreme values of asset prices relative to the standard robust volatility estimator based on closing price alone. We further prove that the proposed robust volatility ratio is unbiased with no drift parameter. Moreover, we find that the robust volatility ratio with a non-zero drift parameter has only a second order effect. We have shown that our proposed extreme value robust volatility estimator is 2–3 times relatively more efficient when compared to the classical robust volatility estimator based on Monte Carlo simulation experiment. On the empirical side, we test the proposed robust volatility ratio based on high and low prices on different asset classes like stock indices, exchange rate and precious metals.
Keywords
Robust volatility modeling Extreme value estimators Radon Nikodym derivative Brownian motion Drawdown Absolute returnsJEL Classification
C13 C51 C58 G12References
- Adriana, S.C., and K. Chris. 2014. Discrete stochastic autoregressive volatility. Journal of Banking and Finance 43: 160–178.CrossRefGoogle Scholar
- Ait-Sahalia, Y., and R. Kimmel. 2007. Maximum Likelihood Estimation of Stochastic Volatility Models. Journal of Financial Economics 83: 413–452.CrossRefGoogle Scholar
- Ait-Sahalia, Y., E. Manresa, and D. Ammengual. 2015. Market-based Estimation of Stochastic Volatility Models. Journal of Econometrics 187 (2): 418–435.CrossRefGoogle Scholar
- Aleksandar, M., and M.R. Pistorius. 2012. On the Drawdown of a Completely Asymmetric Levy Processes. Stochastic Processes and their Applications 122: 3812–3836.CrossRefGoogle Scholar
- Alizadeh, S., M.W. Brandt, and F.X. Diebold. 2002. Range-Based Estimation of Stochastic Volatility Models. Journal of Finance 57 (3): 1047–1091.CrossRefGoogle Scholar
- Andersen, T.G., and B. Sorensen. 1997. GML and QML Asymptotic Standard Deviations in Stochastic Volatility Models. Journal of Econometrics 76: 397–403.CrossRefGoogle Scholar
- Andersen, T.G., and T. Bollerslev. 1997. Heterogeneous Information Arrivals and Return Volatility Dynamics: Uncovering the Long-run in High Frequency Returns. Journal of Finance 52: 975–1005.CrossRefGoogle Scholar
- Andersen, T.G., X.D. BollerslevT, and T. Bollerslev. 2001. The Distribution of Realized Stock Return Volatility. Journal of Financial Economics 61: 43–76.CrossRefGoogle Scholar
- Asmussen, S. 2000. Applied Probability and Queues. Berlin: Springer.Google Scholar
- Ball, C.A., and W.N. Torous. 1984. The Maximum Likelihood Estimation of Security Price Volatility: Theory, Evidence, and an Application to Option Pricing. Journal of Business 57 (1): 97–112.CrossRefGoogle Scholar
- Barndorff-Nielsen, O.E. 1997. Normal Inverse Gaussian Distributions and the Modelling of Stock Returns. Scandinavian Journal of Statistics 24: 1–13.CrossRefGoogle Scholar
- Barnett, V., and T. Lewis. 1978. Outliers in Statistical data. Chichester: Wiley.Google Scholar
- Bates, D.S. 1996. Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutschemark Options. Review of Financial Studies 9: 69–107.CrossRefGoogle Scholar
- Black, F., and M. Scholes. 1973. The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81: 637–654.CrossRefGoogle Scholar
- Blattberg, R., and N. Gonnedes. 1974. A Comparison of Stable and Student Distribution as Statistical Models for Stock Prices. Journal of Business 47: 244–280.CrossRefGoogle Scholar
- Chekhlov, A., S.P. Uryasev, and M. Zabarankin. 2005. Drawdown Measure in Portfolio Optimization. International Journal of Theoretical and Applied Finance 8: 1358.CrossRefGoogle Scholar
- Chou, R.Y. 2005. Forecasting Financial Volatilities with Extreme Values: The Conditional Autoregressive Range (CARR) Model. Journal of Money, Credit, and Banking 37: 561–582.CrossRefGoogle Scholar
- Christoffersen, P., K. Mimouni, and K. Jacobs. 2010. Volatility dynamics for & p 500: Evidence from realized volatility, daily returns and option prices. Review of Financial Studies 23: 3141–3189.CrossRefGoogle Scholar
- Clark, P.K. 1973. A Subordinate stochastic process model with finite variance for speculative prices. Econometrica 41 (1): 135–155.CrossRefGoogle Scholar
- Cont, R., and P. Tankov. 2003. Financial Modelling with Jump Processes. London: Chapman & Hall.CrossRefGoogle Scholar
- Davidian, M., and R.J. Carroll. 1987. Variance Function Estimation. Journal of the American Statistical Association 82: 1079–1091.CrossRefGoogle Scholar
- Ding, Z., C.W.J. Granger, and R.F. Engle. 1993. A Long Memory Property of Stock Market Returns and a New Model. Journal of Empirical Finance 1: 83–106.CrossRefGoogle Scholar
- Douady, R., A.N. Shiryaev, and M. Yor. 2000. On the Probability Characteristics of Downfalls in a Standard Brownian Motion. Theory of Probability and its Applications 44 (1): 29–38.CrossRefGoogle Scholar
- Eberlein, E., and PrauseK KellerU. 1998. New Insights into Smile, Mispricing and Value at Risk: The Hyperbolic Model. Journal of Business 71: 371–405.CrossRefGoogle Scholar
- Eddington, A. 1914. Stellar Movements and the Structure of the Universe. London: Macmillan.Google Scholar
- Ederington, L.H., and W. Guan. 2004. Forecasting Volatility. Journal of Futures Markets 25: 465–490.CrossRefGoogle Scholar
- Fama, E.F. 1963. Mandelbrot and the Stable Paretian Hypothesis. The Journal of Business 36 (4): 420–429.CrossRefGoogle Scholar
- Fama, E.F. 1965. The Behavior of Stock Market Prices. Journal of Business 38: 34–105.CrossRefGoogle Scholar
- Forsberg, L., and E. Ghysels. 2007. Why do Absolute Returns Predict Volatility So Well? Journal of Financial Econometrics 5 (1): 31–67.CrossRefGoogle Scholar
- Garman, M., and M.J. Klass. 1980. On the Estimation of Security Price Volatilities From Historical Data. Journal of Business 53: 67–78.CrossRefGoogle Scholar
- Hamelink, F., and M. Hoesli. 2004. Maximum Drawdown and the Allocation to Real Estate. Journal of Property Research 21: 5–29.CrossRefGoogle Scholar
- Heston, S.L. 1993. A Closed-form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Review of Financial Studies 6: 327–343.CrossRefGoogle Scholar
- Horst, E.T., A. Rodriguez, H. Gzyl, and G. Molina. 2012. Stochastic Volatility Models including Open, Close, High and Low prices. Quantitative Finance 12: 199–212.CrossRefGoogle Scholar
- Huber, P. 1981. Robust Statistics. New York: Wiley.CrossRefGoogle Scholar
- Hull, J.C., and A. White. 1987. The Pricing of Options on Assets with Stochastic Volatilities. Journal of Finance 42: 381–400.CrossRefGoogle Scholar
- Jarrow, R.A. 1998. Volatility. London: Risk Publications.Google Scholar
- Kon, S.J. 1984. Models of stock returns—a comparison. Journal of Finance 39 (1): 147–165.Google Scholar
- Kunitomo, N. 1992. Improving the Parkinson Method of Estimating Security Price Volatilities. Journal of Business 65: 295–302.CrossRefGoogle Scholar
- Lehoczky, J.P. 1977. Formulas for Stopped Diffusion Process with Stopping Times based on the Maximum. The Annals of Probability 5 (4): 601–607.CrossRefGoogle Scholar
- Magdon Ismail, M., and A.F. Atiya. 2003. A Maximum Likelihood Approach to Volatility Estimation for a Brownian motion using High, Low and Close price data. Quantitative Finance 3 (5): 376–384.CrossRefGoogle Scholar
- Magdon-Ismail, M., A.F. Atiya, and Abu-Mostafa Y.S. PratapA. 2004. On the Maximum Drawdown of a Brownian Motion. Journal of Applied Probability 41: 147–161.CrossRefGoogle Scholar
- Maheswaran, S., and D. Kumar. 2014. A reflection principle for a random walk with implications for volatility estimation using extreme values of asset prices. Economic Modelling 38: 33–44.CrossRefGoogle Scholar
- Maheswaran, S., G. Balasubramanian, and C.A. Yoonus. 2011. Post-Colonial Finance. Journal of Emerging Market Finance 10 (2): 175–196.CrossRefGoogle Scholar
- Mandelbrot, B. 1963. The Variation of Certain Speculative Prices. Journal of Business XXXVI: 392–417.Google Scholar
- Merton, R.C. 1969. Lifetime Portfolio Selection Under Uncertainty: The Continuous-time Case. Review of Economics and Statistics 51: 247–257.CrossRefGoogle Scholar
- Parkinson, M. 1980. The Extreme Value Method for Estimating the Variance of the Rate of Return. Journal of Business 53: 61–65.CrossRefGoogle Scholar
- Pospisil, L., and J. Vecer. 2010. Portfolio Sensitivity to Changes in the Maximum and the Maximum Drawdown. Quantitative Finance 10 (6): 617–627.CrossRefGoogle Scholar
- Qamarul, I.M. 2014. Estimation in multivariate nonnormal distributions with stochastic variance function. Journal of Computational and Applied Mathematics 255: 698–714.CrossRefGoogle Scholar
- Rogers, L.C.G., and S.E. Satchell. 1991. Estimating Variance From High, Low and Closing Prices. The Annals of Applied Probability 1 (4): 504–512.CrossRefGoogle Scholar
- Rogers, L.C.G., and F. Zhou. 2008. Estimating Correlation from High, Low, Opening and Closing prices. The Annals of Applied Probability 18 (2): 813–823.CrossRefGoogle Scholar
- Taylor, H.M. 1975. A Stopped Brownian Motion Formula. The Annals of Probability 3 (2): 234–246.CrossRefGoogle Scholar
- Taylor, S. 1986. Modeling Financial Time Series. New York: Wiley.Google Scholar
- Yang, D., and Q. Zhang. 2000. Drift-Independent Volatility Estimation based on High, Low, Open, and Closing prices. Journal of Business 73: 477–491.CrossRefGoogle Scholar