A Sonnet of Intraday High-Frequency Stock Volatility: Measurement and Decomposition

Chinese

Sonnet of Intraday Stock Volatility: Metrics and Decomposition.

This article discusses several measures of high-frequency stock volatility within a day, and summarizes its various decomposition methods from a unified perspective, providing a reference for volatility-related investment practices.

Measurement

Moment

Assume that the logarithmic price of a stock satisfies the following Jump-Diffusion Model:

$$ dp_t = \mu p_t dt + \sigma p_t dW_t + J_t dp_t $$

\(dp_t\) is the change of stock price, \(\mu\) is the annualized average return of the diffusion part, \(\sigma\) is the volatility of the diffusion part of the stock price, \(dW_t\) is the stochastic differential term of Brownian motion, \(J_t\) is the stochastic differential term of jump, representing the impact of jump events.

The well-known realized volatility (Realized Variance) measures the intraday stock price volatility, which converges to the sum of the integrated volatility and the intraday logarithmic price jump according to probability.

$$ \sum_{i=1}^{N} r_{i}^{2} \stackrel{\mathcal{P}}{\rightarrow} \int_0 ^T \sigma_s^2 d s+\sum_{0 \le \tau \leq T}\left(\Delta p_{\tau}\right)^2 $$

If we consider higher-order moments, we can go one step further and measure the shape and asymmetry of the intraday return distribution. The realized x-order moment of intraday return rate (Realized Moment x) is defined as:

$$ RM_{x}=N^{x/2-1} \frac{\sum_{i=1}^{N} r_{t, i}^{x}}{R V_{t}^{x / 2}} $$

The realized skewness factor RM3 is the asymmetry of returns described by high-frequency data. When the factor value is negative, it means that the high-frequency returns are left skewed. At this time, the higher expected returns are compensation for the left skew. In addition, it is generally believed that stocks with extreme positive returns during the day will trigger the gambling preferences of irrational investors, so when the factor is positive, the expected excess returns are negative. The overall direction of the factor is negative.

The realized kurtosis factor RM4 is the kurtosis and tail shape of returns described by high-frequency data. When the factor value is low, the corresponding tail heavy tail phenomenon is not obvious, that is, the tail risk is smaller and tends to have better expected performance. Logically, the factor direction is also negative.

Superskewness RM5 is similar to skewness and describes asymmetry; while supertaility RM6 is similar to kurtosis and describes the kurtosis and the relative information of the tail and center. However, the two have greater integral weight on the tail and describe the tail risk more deeply.

Tail risk

Risk measures that describe tail risk include value at risk (VaR) and conditional value at risk (cVaR/Expected shortfall). VaR is the stock return rate corresponding to a specific quantile (such as 5%, 95%), and cVaR is the expected return outside the quantile point. Compared with the central moment factor mentioned above, VaR/cVaR focuses on the return distribution under extreme circumstances, and using CVaR for risk measurement can more robustly estimate possible risks.

Logically, we believe that stocks with smaller tail risk fluctuations, or stocks whose risk levels remain relatively stable, are better than stocks whose tail risks fluctuate significantly, or stocks whose risk levels are difficult to estimate or predict, because the size of the risk and the degree of change in risk reflect the excessive speculation of investors.

VOV

The volatility of volatility (Volatility of volatility) constructed by Northeast Securities describes the uncertainty of risk,

$$ \operatorname{VOV}_{t}=\frac{\sqrt{\frac{1}{\tau-1} \sum_{s=0}^{\tau-1}\left(R V_{t-s}-\overline{R V_{t}}\right)^{2}}}{\overline{R V_{t}}} $$

The numerator part is the standard deviation of the realized volatility in the past period. The denominator part not only makes the factor cross-sections comparable, but also enhances the factors from the perspective of double sorting. Within the high-risk group, the negative correlation between realized variance volatility and expected returns is more significant, which indicates that for assets with higher levels of risk, investors will pay more attention to the uncertainty of the risk itself, and the VOV effect is enhanced.

break down

Up and down decomposition

Based on the positive and negative returns, it can be said that realized volatility is decomposed into upward and downward realized volatility, and the asymmetry of upward and downward volatility is calculated. \(RV=RV^++RV^-\)

$$ RV^+=\sum_{i=1}^{N}{r_i^+}^2 $$

$$ RV^-=\sum_{i=1}^{N}{r_i^-}^2 $$

The volatility asymmetry factor, normalized by RV, is the RSJ factor.

$$ SJ={RV}^+-{RV}^- $$

$$ RSJ=\frac{SJ} {RV} $$

Continuous/jump partial decomposition

Intraday stock price volatility QV = integral volatility IV + jump volatility QJ

We can estimate the integral fluctuation IV by multipower variation using

$$ \hat{IV} = \mu_{2/3}^{-3} \sum_{i=3}^{n} {|r_i|^{\frac23}|r_{i-1}|^{\frac23}|r_{i-2}|^{\frac23}} $$

Among them

$$ \mu_{2/3} = 2^{1/3} \frac{\Gamma(5/6)}{\Gamma(1/2)} $$

RJV is an estimate of QJ

$$ RJV=max(RV-\hat{IV},0) $$

Long/short range decomposition of jump fluctuations

RJV=RLJV+RSJV, the return rate higher than the threshold \(\gamma\) is counted as a long-range jump, and vice versa is a short-range jump.

$$RLJV=min(RJV,\sum_{i=1, |r_i| > \gamma}^{N}{r_i^2 i})$$

Among them

$$\gamma = \alpha N^{-0.49} \sqrt{\hat{IV}}$$

Multiple decomposition

Combining the above three decompositions, we can get the up and down jump fluctuation decomposition:

$$RJVP=max(RV^+-\hat{IV}/2,0)$$

$$RJVN=max(RV^--\hat{IV}/2,0)$$

$$SRJV=RJVP-RJVN$$

And the decomposition of long and short range up and down jump fluctuations:

$$RLJVP = min(RJVP, \sum_{i=1, r_i > \gamma}^{N}{r_i^2}))$$

$$RLJVN = min(RJVN, \sum_{i=1, r_i < -\gamma}^{N}{r_i^2}))$$

$$SRLJV = RLJVP-RLJVN$$

Summary and Outlook

This article briefly introduces dozens of factors that measure intraday stock price volatility. The main directions are the overall and local measurement of intraday high-frequency fluctuations, the measurement of risk uncertainty, and the measurement of each sub-component. We hope to provide readers with a reference for in-depth research. Although this article has many factors and has significant performance in historical backtesting, many of the structural logics are similar. How to analyze the intrinsic connection between them, examine the correlation, and prevent a sharp retracement at the same time will be an important topic.

In the future, an obvious improvement idea is to replace all logarithmic returns with idiosyncratic returns, which has been discussed in previous article, and is also very promising to improve factor performance.

References

• Aït-Sahalia, Yacine, and Jean Jacod. 2012. Analyzing the Spectrum of Asset Returns: Jump and Volatility Components in High Frequency Data. Journal of Economic Literature 50 (4): 1007–50. https://doi.org/10.1257/jel.50.4.1007.
• Andersen, Torben G, Tim Bollerslev, Francis
• Barndorff-Nielsen, O. E. 2004. Power and Bipower Variation with Stochastic Volatility and Jumps. Journal of Financial Econometrics 2 (1): 1–37. https://doi.org/10.1093/jjfinec/nbh001.
  Bollerslev, Tim, Sophia Zhengzi Li, and Bingzhi Zhao. 2020. “Good Volatility, Bad Volatility, and the Cross Section of Stock Returns”. Journal of Financial and Quantitative Analysis 55 (3): 751–81. https://doi.org/10.1017/S0022109019000097.
• Duong, Diep, and Norman R. Swanson. 2015. Empirical evidence on the importance of aggregation, asymmetry, and jumps for volatility prediction. Journal of Econometrics, Econometric Analysis of Financial Derivatives, 187 (2): 606–21. https://doi.org/10.1016/j.jeconom.2015.02.042.
• Yu, Bo, Bruce Mizrach, and Norman Rasmus Swanson. 2019. “New Evidence of the Marginal Predictive Content of Small and Large Jumps”. SSRN Electronic Journal. https://doi.org/10.2139/ssrn.3440320.
• 20230227-Northeast Securities-Factor Stock Selection Series Four: Realized high-order moment factors and improvements under high-frequency data
• 20230601-Northeast Securities-Factor stock selection series No. 5: Risk uncertainty factors based on high-frequency data
• 20200204-Founder Securities-Market Microstructure Analysis Series 1: Tail Characteristics of the Minute Line
• 20220831-GF Securities-High-frequency data factor research series nine: Factor research based on stock price jump model