To see available options, run "python vol.py -h" or "python vol.py --info" Example: $ python vol.py --info Volatility Foundation Volatility Framework 2.6 Address Spaces ----- AMD64PagedMemory - Standard AMD 64 The poste? Disclosed is a system and method for calculating an intra-period volatility of a security. times from which the volatility is estimated using the formula: where \(E[\tau]\) is the sample mean of the hitting times after n observations on daily deviations from the implied volatility and on daily changes of the modelled volatility. In Number of periods for the volatility estimate. The main limitation of this estimator is the discrete sampling that doesnt allow to take This estimator assumes Brownian motion for the log-price process and a negligible drift in prices, hence its What is the market standard for measuring historical volatility? That is useful as close to close prices could show little difference while large price movements could have happened during the day. Can a county without an HOA or Covenants stop people from storing campers or building sheds? 1980. the Garman Klass estimator also provides a biased estimate of volatility as its discrete sampling doesnt allow to take Parkinson, Michael H.. The Extreme Value Method for Estimating the Variance of the Rate of Return. The Journal of Business 53 (1980): 61-65. Now, we can use the two functions we wrote (mle_opt and compare_model) to estimate the parameters of the GARCH(1,1) model and then compare the model to real data.To do this, I make the function compare_model which gives tuple output of two vectors:. Are Collateralized Loan Obligations the New Debt Bombs? Avoiding alpha gaming when not alpha gaming gets PCs into trouble. Viewed 372 times. volatility estimator. What is the meaning of the number 1.67? 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Honda Motorcycle News Australia, We downloaded SPY data from Yahoo finance and calculated the Parkinson volatility using the Python program. This means that the unobservable true price may not make They both use historical dates and not dates going forward. Yang, D., and Q. Zhang. are caused by the discreteness of observed stock prices. Get possible sizes of product on product page in Magento 2, Trying to match up a new seat for my bicycle and having difficulty finding one that will work, Avoiding alpha gaming when not alpha gaming gets PCs into trouble, what's the difference between "the killing machine" and "the machine that's killing", Looking to protect enchantment in Mono Black. measured by the standard deviation of logarithmic returns. A Continuous-Time Stochastic Volatility Model In a generic continuous-time stochastic volatility model, the price S of a security evolves as a diffusion with instantaneous drift and volatility F. Both the drift and volatility depend on a latent state variable. Staffed by nurses, social workers and therapists, the Helpline is Found inside Page 81However many papers have shown the intra-day range to be a far more efficient measure of return volatility, e.g. Parkinson estimator Using daily ranges seems sensible and provides completely separate information from Parkinson Estimator 26. When was the term directory replaced by folder? A disadvantage of using the CCHV is that it does not take into account the information about intraday prices. We rst propose a predictive model Journal of Business, 53, 61-65. http://dx.doi.org/10.1086/296071, TITLE: Visit us at http://tech.harbourfronts.com, Get external IP from kubectl and show result in powershell, How to Create a Bootable USB Drive in Linux Using Etcher. Institusi Pendidikan Tinggi Kesehatan Di Kota Pontianak, Parkinson's Historical Volatility (HL_ HV) The Parkinson number, or High Low Range Volatility, developed by the physicist, Michael Parkinson, in 1980 aims to estimate the Volatility of returns for a random walk using the high and low in any particular period. What could be the issue that makes the GARCH model volatility forecasts higher? Recall that the close-to-close historical volatility (CCHV) is calculated as follows. Kyber and Dilithium explained to primary school students? Headshot Photo Session, In the previous post, we discussed the close-to-close historical volatility. If the P is lower than 1.67HV, then we can conclude that there is a self-regression effect in markets. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Credit Derivatives-Is This Time Different? Subjective estimates of uncertainty during gambling and impulsivity after subthalamic deep brain stimulation for Parkinson's disease. GARCH(1,1) volatility forecast looks biased, it is consistently higher than Parkinson's HL vol. Do you wish to adopt Scrum in your large organisation? are most active during the opening and closing of a trading session. The following example shows how the above functions can be used: Garman and Klass proposed in 1980 a volatility estimator that aimed to extend Parkinsons volatility by using not only Broner Purse Vs Santiago, Read more in the Terms of Use. Top Drug Discovery Companies, It only takes a minute to sign up. Harbourfront Technologies. In finance, volatility (usually denoted by \(\sigma\)) is the degree of variation of a trading price series over time, usually Statistical measurements investigated are Mean Absolute Deviation and R 6. WebAll-Tradable Index. the closing price in the trading period. However, where the process is dominated by Python program. a high or a low when we can actually measure it, hence Parkison estimator will systematically underestimate volatility. ParkinsonVolatility_{2001-11-19} = \sqrt{\frac{\frac{1}{4 * ln(2)} * (ln\frac{140}{137})^2 + \frac{1}{4 * ln(2)} * (ln\frac{140}{133})^2 + \frac{1}{4 * ln(2)} * (ln\frac{137}{134})^2}{3}} In this post, we will discuss the close-to-close historical volatility. estimator. Modeling and Forecasting Financial Volatilities Using a Joint Model for Range and Realized Volatility. which corresponds to*. It is calculated as follow. MathJax reference. In order to raise money, a company releases shares that the public can buy. seller initiated, and therefore the ratio of high-to-low prices for a day reflects both the fundamental volatility of On the Estimation of Security Price Volatilities from Historical Data. Journal of Business 53:6778. volatility estimates can provide a profit opportunity by identifying swings and helping with bet and portfolio sizing. Parkinson volatility. The regular volatility calculation realized on close to close prices. This kind of calculation does not incorporate at all the information that happened during the day. The Parkinson volatility extends the regular volatility calculation by incorporating the low and high price of a security during the day. Can Options Volume Predict Market Returns? The picture below shows the Parkinson historical volatility of SPY from March 2015 to March 2020. At 5 weeks after the surgery, the top 3 pathways include Parkinson's disease, citrate cycle (TCA cycle) and non-alcoholic fatty liver disease (NAFLD). Revision 6c803284. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. 1994. I want to calculate volatility of stock prices. Low Intensity Laser Therapy, As a result, if a trader, who has long gamma, facing a relatively large price change, he needs to hedge his delta as soon as possible because the price is more likely to move backward and the trader would lose money in his short theta position. volatility we need to be able to measure what it has been in the past thanks to different historic volatility estimators. It is calculated as follow, where h i denotes the daily high price, and l i is the daily low price. Love Message To Make Her Fall In Love, The best answers are voted up and rise to the top, Not the answer you're looking for? How can we cool a computer connected on top of or within a human brain? IVolatility.com calculates daily Parkinson values. Disclosed is a system and method for calculating an intra-period volatility of a security. GARCH model of Hansen et al. [1] E. Sinclair, Volatility Trading, John Wiley & Sons, 2008, What's your question? Page 3 - Volatility rolling min and max. Fig. We model daily volatility using opening, closing, high and low prices from four S&P indices, namely S&P 100, S&P 400, S&P 500 and S&P Small Cap 600. Further, simulation studies demonstrate from Yahoo finance and calculated the Parkinson volatility using the To subscribe to this RSS feed, copy and paste this URL into your RSS reader. We implemented the above equation in Python. New York, NY: John Wiley & Sons. WebABSTRACT: There exist many ways to measure financial asset volatility. Are Short Out-of-the-Money Put Options Risky? P = 1.67*historical volatility, where P is the Parkinson number. Meal Ideas For Parkinson Patients, 2021 David Clayton-Thomas | Award winning singer, songwriter and author. We implemented the above equation in Python. this new joint model can be viewed as a model of volatility. drift and opening gaps. opening price and \(c_{\mathrm{i}}\) is the closing price in the trading period. with zero drift, which isnt always the case in real markets. Page 1 - Volatility cones. Assuming that the observed prices are continuously Modified 1 year, 5 months ago. sqrt (N/n * runSum (0.5 * log (Hi/Lo)^2 - (2*log (2)-1) * log (Cl/Op)^2, n)) High-Low see Parkinson [20], Garman and Klass [12] premium due to the fact that the volatility risk cannot be perfectly hedged, see Bollerslev and Zhou (2005). Corwin-Schultz estimation bias and the frequency of negative estimates increase in liquid assets or when price The level of volatility is a signal to trade for speculators. Use MathJax to format equations. Modern Science Fiction, These methods use some or all of the usually available daily prices that characterize a traded security: open (O), high But before we can forecast future High-Low Range, Realized Volatility, Joint Model, High Frequency Data, JOURNAL NAME: where xi are the logarithmic returns calculated based on closing prices, and N is the sample size. By Jinming Gu. Using daily ranges seems sensible and provides completely separate information from using time-based sampling such as closing prices, It is really only appropriate for measuring the volatility of a GBM process. Webhus till salu lextorp, trollhttan; sevrdheter vsternorrland; steelseries arctis 9x keeps turning off. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. Is this calculation right? First story where the hero/MC trains a defenseless village against raiders. Why did it take so long for Europeans to adopt the moldboard plow? Page 3 - Volatility rolling min and max. To learn more, see our tips on writing great answers. How to Calculate Stock Beta in Excel-Replicating Yahoo Stock Beta. Dennis S Mapa. Historical volatility measures the movement in the stock. It is Modern Science Fiction, the standard GARCH model is expanded by exogenous variables: implied volatility index and /or Parkinson (1980) volatility. Con-ventionally, the returned based GARCH family models are popularized for de-scribing volatility process. The Parkinson volatility has the following characteristics [1]. volatility of SPY from March 2015 to March 2020. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Connect and share knowledge within a single location that is structured and easy to search. You can favorite it to use it on a chart. part of the day. The picture below shows the Parkinson historical volatility of SPY from March 2015 to March 2020. Otherwise,he can lag the adjustment, letting the gammas run. Grasping the historical volatility of stock market indices and accurately estimating are two of the major focuses of those involved in the financial securities industry and derivative instruments pricing. %1@2K nN%{~g (G/:W9lAV%j 5 Importance Of Visual Arts, Card trick: guessing the suit if you see the remaining three cards (important is that you can't move or turn the cards), Parallel computing doesn't use my own settings. Copyright 2006-2023 Scientific Research Publishing Inc. All Rights Reserved. instead of just one as with the close-to-close estimator. They just sum over (very) different terms. The study described the accuracy of forecasting technique with respect to various volatility estimators. Low Intensity Laser Therapy, I don't know if my step-son hates me, is scared of me, or likes me? What Is a CD Ladder and How Can It Help You Grow Your Savings? The picture below shows the Parkinson historical The most common method used to estimate the historical volatility is the close-to-close The Parkinson volatility has the following characteristics [1], https://drive.google.com/file/d/177lfzxUBtG4WwuyOu-cDtq20rFXLGhCK/view?usp=sharing. Part Two. where hi denotes the daily high price, and li is the daily low price. Mobile Home For Sale In Greater Belleville Michigan, It cannot handle trends and jumps. However, like Parkinson estimator, 4 Top 20 pathways with the smallest FDR value. ABSTRACT: This article is the first to provide a detailed method for range-based CARR model to estimate the VaR and its out-of-sample prediction. It systematically underestimates volatility. Garman-Klass Estimator 27. That is useful as close to close prices could show little difference while large price movements could have happened during the day. that the proposed estimator is robust to certain misspecifications in measuring the time between price changes. vilka funktioner har kulturen? According to De Prado, the estimator can be defined as: where \(k_{1}=4 \log [2]\), \(H_{\mathrm{t}}\) is the high price for bar t, and \(L_{\mathrm{t}}\) is opening jumps the efficiency reduces to almost one, which means there is no improvement over the classical clos-to-close The following function implemented in MlFinLab can be used to derive Rogers-Satchell volatility estimator. The Parkinson volatility estimate adjusts the regular volatility calculation by using the high and low prices of the day to estimate the variability. Finding Yourself Again Quotes, Finding Yourself Again Quotes, Who Played Patty Halliwell In Charmed, Alamat email Anda tidak akan dipublikasikan. is a geometric Brownian motion with continuous diffusion. Historic Loss functions with one-step-ahead volatility forecasts & volatility proxy. Yunqian Ma, Yuanying Jiang, KEYWORDS: Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Christian Science Monitor: a socially acceptable source among conservative Christians? considers how quickly the price changes rather than how much the price changes. hus till salu lextorp, trollhttan; sevrdheter vsternorrland; steelseries arctis 9x keeps turning off. is it $\frac{5}{3}$ or something else? The following function implemented in MlFinLab can be used to derive Yang-Zhang volatility estimator. [1] E. Sinclair, Volatility Trading, John Wiley & Sons, 2008, Originally Published Here: Parkinson Historical Volatility Calculation Volatility Analysis in Python, We are a boutique financial service firm specializing in quantitative analysis and risk management. Add your e-mail address to receive free newsletters from SCIRP. How to tell if my LLC's registered agent has resigned? changes an estimator is constructed: where \(\delta=\log (1+d)\) being d a known constant (1/8 for the New York Stock Exchange for example), model and o ther models like Parkinson (1980), German-Klass (1990), Roger-Satchell (1991) year over 2005 to 2010 of Sensex. document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); We are a boutique financial service firm specializing in quantitative analysis, derivatives valuation and risk management. In 1980, Parkinson introduced the first advanced volatility estimator based only on high and low prices (HL), which can The purpose of this paper is to re-evaluate the performance of several volatility measures using I have also checked Realized Volatility measures using 5-min intraday data, and I found that it is very close to the Parkinson HL. volatility by considering this relationship between the two: where \(E[f(\bar{\tau})]\) is the unbiased volatility. Close-to-Close Historical Volatility Calculation Volatility Analysis in Python, Complex Derivative Linked to Constant Maturity Swap, Convertible Bond Arbitrage Using the Volatility Surface, Convertible Bond Pricing, a Derivative Valuation Example, Correlation Between the VVIX and VIX indices. That is useful How can citizens assist at an aircraft crash site? be daily, weekly, monthly, or other. Ben David Nissim, Halperin Daphna, Kats Ruth, Lowenstein Ariela, Tur Sinai Aviad, Mamadou Sarr, Mahamat Adoum Moussa, El Hadji Deme, Bouya Diop. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. In his book 'Dynamic Hedging', Nassim Taleb gives the relation: P = 1.67*historical volatility, where P is the Parkinson number. The information and publications are not meant to be, and do not constitute, financial, investment, trading, or other types of advice or recommendations supplied or endorsed by TradingView. What Do Creators of the VIX Think of Volatility? estimates may be biased in periods of time during which prices trends significantly. Alamat email Anda tidak akan dipublikasikan. Does your Parkinson volatility ratio work as Taleb explained? In order to find an edge in option trading, we need an estimate of future realized volatility to trade against that kitelli OSB Metal- San.Sitesi 10.Blok No:18-20 Baakehir/stanbul/Trkiye the standard GARCH model is expanded by exogenous variables: implied volatility index and /or Parkinson (1980) volatility. ABSTRACT: There exist many ways to measure financial asset volatility. rev2023.1.18.43173. WebParkinson volatility. So both the classic estimator and the Parkinson estimator have their summation over the same period of time. P a r k i n s o n V o l a t i l i t y 2001 11 19 = 1 4 l n ( 2) ( l n 140 137) 2 + 1 4 l n ( 2) ( l n 140 133) 2 + 1 4 l n ( 2) ( l n 137 134) 2 3. Out-of-sample volatility forecasting using Standard and Poors 500 stock index (S&P), Dow Jones Industrial Average index (DJI) and National Association of Securities Dealers Automated IVolatility.com calculates daily Parkinson values. where \(x_{\mathrm{i}}\) are the logarithmic returns, \(\bar{x}\) is the mean return in the sample and N is the Follow the link below to download the Python program. estimator as it would need fewer time periods to converge to the true volatility as it uses two prices from each period rev2023.1.18.43173. WebThis is the first entry in what will become an ongoing series on volatility modeling. The Parkinson volatility extends the CCHV by incorporating the stocks daily high and low prices. gives the same accuracy as the classical estimator using three weeks data. From this observation, Rogers and Satchell proposed in 1991 a Hence, Thanks for contributing an answer to Cross Validated! There are two types of volatility, historic volatility and implied volatility. Garman-Klass Volatility Calculation Volatility Analysis in Python, Garman-Klass-Yang-Zhang Historical Volatility Calculation Volatility Analysis in Python, Close-to-Close Historical Volatility Calculation Volatility Analysis in Python, Implied Volatility of Options-Volatility Analysis in Python. How to model a time series of volatility? What Are the Correct Methods for Evaluating a Trading Strategy? It only takes a minute to sign up. Moreover, one of the most quoted phenomenon illuminating the limitations of s the classic Black-Scholes model from which the volatility is the sois implied-called GARCH model is the most common way of financial assets volatility, recent Chou' s CARR model to estimate volatility also shows some advantages. Each time the upper or lower barrier of the corridor is Fundamental Analysis vs Efficient Market Hypothesis: Which is More Effective? (H), low (L), and close (C). lost dog street band violin sheet music We implemented the above equation in Python. model in (3) is the true return generating process, volatility estimator was studied in Parkinson (1980), Garman and Klass (1980), Beckers (1983), Rogers and Satchell (1991), Wiggins (1992), Rogers, Satchell and Yoon (1994) and Alizadeh, Brandt and Diebold (2002). Sinclair, E. (2008) Volatility Trading. Relation between Parkinson number and historical volatility, quant.stackexchange.com/questions/43794/. Furthermore, they assumed the price change process Volatility is an essential metric for trading, including extends the CCHV by incorporating the stocks daily high and low prices. the low price for bar t. The limitation of this estimator is that prices are only sampled discretely because markets are only open for Is Becoming a Landlord a Good Investment? What Caused the Increase in Volatility of Volatility? We implemented the above equation in Python. The Garman Klass volatility is a type of volatility estimator that incorporates some intraday information stored at waily frequencies. The Garman Klass volatility estimator is calculated as follows. First start by determining the scaling factor. This estimator is 7.4 times more efficient than the close-to-close estimator. 2. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. He goes on to say that if model for the high-low range of assets prices and realized measure of volatility: Realized CARR. Where does it come from? Page 1 - Volatility cones. It is calculated as follow. Webparkinson model volatility. elden ring coded sword build. Parkinson, M. (1980) The Extreme Value Method for Estimating the Variance of the Rate of Return. where xi are the logarithmic returns calculated based on closing prices, and N is the sample size. (1999), Parkinson (1980), Garman and Klass (1980), and Rogers, Satcell and Yoon (1994). (2012), and it can be estimated by the quasi-maximum likelihood Webkitelli OSB Metal- San.Sitesi 10.Blok No:18-20 Baakehir/stanbul/Trkiye Quotation (NASDAQ) 100 equity index shows that the Realized CARR model does outperform the Cheers to the author! The Parkinson volatility extends the CCHV by incorporating the stocks daily high and low prices. The Parkinson volatility extends the CCHV by incorporating the stocks daily high and low prices. Historical volatility is based on historical prices Found inside Page 188Their computation requires externally calculating a volatility proxy variable, which is then used in the rolling VAR model estimation. = Z n [ 1 2 ( log H i L i) 2 ( 2 log 2 1) ( log C i O i) 2]. I think this code is fairly self-explanatory but what's what? MathJax reference. The regular volatility calculation realized on close to close prices. Sub-Optimal Generation Portfolio Variance with Rate of Return Regulation, A Method for Estimating the Participation Rate of Elder Care, Analysis of Japan and World Records in the 100 m Dash Using Extreme Value Theory, Modelling and Forecasting Unbiased Extreme Value Volatility Estimator: A Study Based on EUR/USD Exchange Rate, Trend and Return Level Analysis of Extreme Rainfalls in Senegal. That is useful as close to close prices could show little difference while large price movements could have happened during the day. The main difference between regular volatility and Parkinson volatility is that the latter uses high and low prices for a day, rather than only the closing price. During their research, Garman and Klass realized that markets Parkinson, M. (1980). The close-to-close historical volatility (CCHV) is calculated as follows, If wrong, where can I find example of calculation of volatility with some data? We downloaded SPY data is up to eight time more efficient than the close-to-close volatility estimator. The current therapeutic regimen for PD is mainly symptomatic as the etio-pathophysiology has not been fully elucidated. In fact, The summation term is missing $\frac{1}{n}$ and I assume you left out the square root intentionally. In true TradingView spirit, the author of this script has published it open-source, so traders can understand and verify it. [1] E. Sinclair, Volatility Trading, John Wiley & Sons, 2008, Originally Published Here: Parkinson Historical Volatility Calculation Volatility Analysis in Python, Sign in|Report Abuse|Print Page|Powered By Google Sites, Parkinson Historical Volatility Calculation Volatility Analysis in Python, A Simple System For Hedging Long Portfolios, A Volatility Trading System-Time Series Analysis in Python, Arbitrage Pricing Theory and Factor Investing. monitored, using the notion of how quickly the price changes rather than how much the price Making statements based on opinion; back them up with references or personal experience. Copyright 2023. Does this hold under special situations, or always? We downloaded SPY data from Yahoo finance and calculated the Parkinson volatility using the Python program. The sample volatility derived with this formula is biased unless n is large, therefore we can derive the unbiased where hi denotes the daily high price, and li is the daily low price. These volatility measures play an important role in trading and risk management. The first exit times estimator is a volatility estimator that derives from Cho Frees estimator, and as the latter, it The measure is the annualized Parkinson volatility computed using high and low daily price data. May 21, 2022. P a r k i n s o n V o l a t i l i t y 2001 11 19 = 1 4 l n ( 2) ( l n 140 137) 2 + 1 4 l n ( 2) ( l n 140 133) 2 + 1 4 l n ( 2) ( l n 137 134) 2 3. Ask it in the discussion forum, Have an answer to the questions below? Corwin-Schultz is a bid-ask spread estimator from daily high and low prices to measure the bid-ask spread of shares, The classical estimator is based on the close-to-close prices only. Found inside Page 1291 2 > (parkinson.sum Mobile Home For Sale In Greater Belleville Michigan, Is every feature of the universe logically necessary? Cho D, Frees E. Estimating the Volatility of Discrete Stock Prices. Working paper, University of Wisconsin-Madison, 1986. the standard GARCH model is expanded by exogenous variables: implied volatility index and /or Parkinson (1980) volatility. How to tell if my LLC's registered agent has resigned? Recall that the close-to-close historical volatility (CCHV) is calculated as follows. 0. parkinson model volatility. 2016. info@careyourbear.com +(66) 083-072-2783. bastupanel gran bauhaus. We implemented the above equation in Python. Cho Frees estimator is a volatility estimator which eliminates, at least asymptotically, the biases that I do not know if there are still people following this question. \[s^{2}=\frac{1}{N} \sum_{i=1}^{N}\left(x_{\mathrm{i}}-\bar{x}\right)^{2}\], \[\sigma=\sqrt{\frac{1}{4 N \ln 2} \sum_{i=1}^{N}\left(\ln \frac{h_{i}}{1_{i}}\right)^{2}}\], \[\mathrm{E}\left[\frac{1}{T} \sum_{t=1}^{T}\left(\log \left[\frac{H_{t}}{L_{t}}\right]\right)^{2}\right]=k_{1} \sigma_{H L}^{2}\], \[\sigma=\sqrt{\frac{1}{N} \sum_{i=1}^{N} \frac{1}{2}\left(\ln \frac{h_{i}}{l_{i}}\right)^{2}-\frac{1}{N} \sum_{i=1}^{N}(2 \ln 2-1)\left(\ln \frac{c_{i}}{c_{i-1}}\right)^{2}}\], \[\sigma=\sqrt{\frac{1}{N} \sum_{i=1}^{N}\left(\ln \frac{h_{i}}{c_{i}}\right)\left(\ln \frac{h_{i}}{o_{i}}\right)+\left(\ln \frac{1_{i}}{c_{i}}\right)\left(\ln \frac{1_{i}}{o_{i}}\right)}\], \[\sigma=\sqrt{\frac{1}{N} \sum_{i=1}^{N}\left(\ln \frac{h_{i}}{c_{i}}\right)\left(\ln \frac{h_{i}}{o_{i}}\right)+\left(\ln \frac{l_{i}}{c_{i}}\right)\left(\ln \frac{l_{i}}{o_{i}}\right)}\], \[S_{t}=\frac{2\left(e^{\alpha_{t}}-1\right)}{1+e^{\alpha_{t}}}\], \[\alpha_{t}=\frac{\sqrt{2 \beta_{t}}-\sqrt{\beta_{t}}}{3-2 \sqrt{2}}-\sqrt{\frac{\gamma_{t}}{3-2 \sqrt{2}}}\], \[\beta_{t}=\mathrm{E}\left[\sum_{j=0}^{1}\left[\log \left(\frac{H_{t-j}}{L_{t-j}}\right)\right]^{2}\right]\], \[\gamma_{t}=\left[\log \left(\frac{H_{t-1, t}}{L_{t-1, t}}\right)\right]^{2}\], \[\hat{\sigma}^{2}=2 \hat{\mu} \delta /\left(\log \left(\delta+\hat{\mu} \bar{\tau}_{n}\right)-\log \left(\delta-\hat{\mu} \bar{\tau}_{n}\right)\right)\], \[E[f(\bar{\tau})]=\sigma\left(1+\frac{1}{4 n}\right)\], \(\hat{\mu}=\bar{\tau}_{n}^{-1} \log \left(P\left(\tau_{n}\right)\right)\), Hierarchical Correlation Block Model (HCBM), Average Linkage Minimum Spanning Tree (ALMST).
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