Asset return volatility is typically calculated as (annualized) standard deviation of returns over a sequence of periods, usually daily from close to close. However, this is neither the only nor necessarily the best method. For exchange-traded contracts, such as equity indices, one can use open, close, high, and low prices and even trading volumes. These provide different types of information on the dispersion of prices and support the calculation of different volatility metrics. A recent paper illustrates the application of the volatility concepts of Parkinson, Garman-Klass, Rogers-Satchell, and Yang-Zhang, as well as intrinsic entropy, a method of econophysics. Intrinsic entropy seems to be more suitable for estimating short-term fluctuations in volatility.

Ausloos, Marcel, Titus Felix Furtună, and Claudiu Vințe (2022) “A Volatility Estimator of Stock Market Indices Based on the Intrinsic Entropy Model”

The below post is based on the above paper.

This post ties in with this site’s summary on macro trends.

### Basics on exchange-traded equity data

“Intraday trading data [i.e.] execution data [are] generated by the stock exchange matching engine once one buy and one sell orders are put in correspondence. For each exchange-listed security, a trading day consists of a succession of transactions generated by the exchange matching engine when buy and sell orders meet the conditions for being partially or entirely executed. Each individual transaction, namely __a trade, consists of the following information: the price at which the trade was made, the executed quantity, and the timestamp__ at which the order matching occurred and the trade was generated.”

“The __dispersion of the price values across…time…provides multiple types of information__: (i) the amplitude between the lowest and highest values…(ii) the deviation from a reference level [such as] the average price value for the interval for instance…(iii) the traded volume at a given price level…[and] (iv) the amplitude of the price changes.”

“Price dispersion over time is identified as the historical volatility [and used for] for assessing portfolio risk and pricing derivative products. Many __different methods have been developed to estimate the historical volatility__. These methods use some or all of the usually available daily prices that characterize a traded security: open (O), high (H), low (L), and close (C).”

### Six concepts

#### Standard close-to-close

“The most common method used to estimate the historical volatility is the close-to-close method. In this approach, the historical volatility is defined as either the annualized variance or standard deviation of log returns. The standard deviation is the root mean-squared deviation from the average log return.”

*Here μ is the drift, the average of log returns, Ci the closing price of trading day I, and Di the dividend paid on trading day i.*

“Based on close-to-close approach, the trading interval T is…the time frame between two consecutive closing prices: from the previous day’s closing price until the current day’s closing price. Since __within this interval T there is an ‘overnight’ period…during which the market is closed, __regardless of the meridian on which the stock exchange is located, this duration is commonly modelled as a fraction f of the trading interval T. Hence, there is an interval of length f x T between the previous day’s closing and the current day’s opening, and an interval of length (1-f) x T between the current opening and the current closing, during which the market is open for trading.”

“[Considering the division of the trading interval] the classical close-to-close volatility estimator [can be re-written with a useful notation that makes it comparable to alternatives, based on the opening price of the trading day (Oi) and the closing prices of trading days (Ci)].”

“The classical close-to-close estimator does handle drift ( may not be necessarily equal to zero) and quantifies potential opening jumps.”

#### Parkinson

“In 1980, Parkinson introduced the first advanced __volatility estimator based only on high and low prices__, [i.e. the current day’s high during the trading interval (Hi) and the current day’s low during the trading interval (Li)].”

“As [the measure] does not take into account the opening jumps, the Parkinson volatility estimator tends to underestimate the volatility. On the other hand, since it does not handle drift [and assumes that drift is zero], in a trendy market it may overestimate the volatility in the pertinent time interval.”

#### Garman-Klass

“Garman and Klass proposed [an] estimator that is based on all commonly available prices of the current day of trading (OHLC).”

“The Garman–Klass estimator includes opening and closing prices for the current trading day. From this perspective, the estimator extends and improves the performance offered by the Parkinson estimator. It does not include the overnight jumps though; therefore, it may underestimate the volatility.”

#### Rogers-Satchell

“Both the Parkinson and Garman–Klass advanced volatility estimators assume that there is no drift. In reality, securities may have a noticeable trend for periods of time. In order to overcome this deficiency of the previous estimators, Rogers and Satchell proposed __a volatility estimator that handles non-zero drifts and which takes into account all of the prices that synthetically characterize a day of trading __(OHLC)…The Rogers–Satchell estimator does not handle opening jumps; therefore, it underestimates the volatility. It accurately explains the volatility portion that can be attributed entirely to a trend in the price evolution…[The] Rogers–Satchell volatility estimation, which is simply based on the current day open, high, low, and close prices.”

#### Yang-Zhang

“Yang and Zhang noted that Garman-Klass and Rogers-Satchell estimators are arithmetic averages of their corresponding single-period estimators, whereas the classical close-to-close estimator is a multiperiod-based one. They argued that an unbiased variance estimator, which would be both drift-independent and able to handle opening jumps, must be based on multiple periods. Yang and Zhang proposed a new minimum-variance__, unbiased, multiperiod-based variance estimator__.”

“Yang and Zhang chose the constant k in order to minimize the variance of their estimator…Yang and Zhang commented in that k can never reach zero or one, and this fact proves that __neither the classical close-to-close estimator nor the Rogers–Satchell estimator alone has the property of minimum variance__. The estimator with minimum variance is a linear combination of both the close-to-close volatility and the Rogers-Satchell volatility with positive weights. The weight applied to Rogers-Satchell volatility is always greater than the weight applied to close-to-close volatility, which reflects the fact that the variance of Rogers-Satchell is smaller.”

#### Intrinsic entropy

“Over the past decades, __the use of entropy in modelling various economic phenomena, along with the emergence of econophysics as a scientific discipline, has resulted in rapid progress__ being made in economics outside of the mainstream. Information entropy has been used both to assess the price fluctuations of financial instruments in connection with the maximum entropy distribution [12] or for studying the predictability of stock market returns.”

“__Employing the price of the preceding transaction as a reference price preserves the atomicity of each trade__ within the overall pool of transactions that constitute the trading day on the stock exchange. Consequently, a Markov chain is thereby constructed, in which the price of each individual trade is compared only to the price of the preceding trade. We proposed a new unbiased volatility estimator based on multiperiod data and on the intrinsic entropy model.”

“Diverging from the previously presented volatility models that take into account only the elements related to the traded prices, namely open, high, low, and close prices for a trading day (OHLC), the __intrinsic entropy model takes into account the traded volumes__ during the considered time frame as well.”

“We adjusted the intraday intrinsic entropy model…for exchange-traded securities in order to __connect daily OHLC prices with the ratio of the corresponding daily volume to the overall volume traded in the considered period__. The intrinsic entropy model conceptualizes this ratio as an entropic probability or market credence assigned to the corresponding price level…[It] __follows the Yang and Zhang approach regarding the treatment of the overnight jumps, opening jumps, and the drift__ manifested during the trading day.”

“[Here] qi/Q represents the ratio between daily traded volume qi and the overall traded volume Q [also called the] the degree of credence that the investors and the market provide to the price levels or to the intensity of price changes.”

The intrinsic entropy-based estimator of historical volatility does not produce results in a “comparable range of values with the variance-based estimators…[but] are consistently in a lower range of values compared to the estimates produced by the other volatility estimators, while changing relevantly from one day to another. __The information that is brought in by the daily traded volume and the entropic mechanism through which the intrinsic entropy-based estimations are computed provide for more dynamic changes__, although we note that these estimates can offer a more valuable perspective of the overall market evolution for short time horizons.”

### Empirical findings

“We considered for this comparison the historical daily trading data for S&P 500, Dow 30, NYSE Composite, NASDAQ Composite, Russell 2000, Hang Seng Index, and Nikkei 225 indices. The estimates are computed for the following n-period intervals, going back from 31 January 2021: 5, 10, 15, 20, 30, 60, 90, 150, 260, and 520. The estimates are computed on a daily basis, by rolling back n-period time windows, corresponding to the considered intervals.”

“The volatility estimates provided by the intrinsic entropy consistently show the mean in a lower value range, while the coefficient of variation confirms the earlier observation that the intrinsic __entropy estimates change on a daily basis__. This peculiar characteristic of the volatility estimates produced by the intrinsic entropy estimator suggests that it __may be more useful in estimating the market volatility for short-term trading purposes__ rather than characterizing the evolution of the historical volatility over the long term.”

“Not having access to the true, unobserved volatility of the market we [use] the classical close-to-close volatility estimator, as a benchmark…__In addition to the ____mean squared error ____and proportional bias ____ indicators, we note the volatility estimators’ efficiency__. The efficiency of an estimator is defined as the variance of a benchmark estimator divided by the variance of that particular estimator:”

“[The table below] presents the mean squared error (MSE), proportional bias (PB), and efficiency values for the Parkinson, Garman–Klass, Rogers–Satchell, Yang–Zhang, and intrinsic entropy volatility estimators relative to the classical close-to-close estimator as a benchmark. The computation process uses the S&P 500 stock market index daily trading data for various moving time windows.”

“We computed the volatility estimators’ efficiency for a series of successive time intervals from 5 to 20 days, along with a 30-day window. [The figure below] shows the evolution of the volatility estimators’ efficiency for the S&P 500 (GSPC) index over these time intervals…__The intrinsic entropy-based estimator’s efficiency is…higher than the other volatility estimators…for short time intervals __of between 5 and 11 days, representing roughly one to two weeks of trading.”