Calculation and Application of Historical Volatility: Models and Comparisons

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1. Introduction

Historical Volatility (HistVol) is a crucial metric in financial markets for risk management, option pricing, and asset analysis. It quantifies market uncertainty by analyzing past price fluctuations of an asset. Unlike implied volatility, historical volatility is entirely based on historical price data, making it a backward-looking measure.

This article explores the definition of historical volatility and its calculation methods, introducing several commonly used models: Close-to-Close, EWMA (Exponentially Weighted Moving Average), LinXiao Model, and RiskMetrics Model. By analyzing and comparing these models, we examine their respective applications, strengths, and weaknesses.

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2. Definition of Historical Volatility

Historical volatility is a statistical measure based on an asset's past prices, typically used to describe the magnitude or speed of price changes. Specifically, historical volatility is the standard deviation of asset returns, calculated as:

$$\sigma = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (r_i - \bar{r})^2}$$

where:

• $\sigma$: Historical volatility.
• $r_i = \ln(P_i / P_{i-1})$: Logarithmic return of the asset.
• $\bar{r}$: Average return.
• $N$: Total number of days in the observation period.

Historical volatility can be calculated using different models, each assigning different weights to returns and handling volatility dynamics differently. Below, we introduce four commonly used models.

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3. Models for Calculating Historical Volatility

3.1 Close-to-Close Model

Model Overview

The Close-to-Close model is the simplest method for calculating historical volatility. It uses daily closing prices over a specified time window to compute the standard deviation of logarithmic returns:

$$\sigma_{\text{Close-to-Close}} = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (r_i - \bar{r})^2}$$

Advantages

• Simple and easy to use, with high computational efficiency.
• No need for complex weight assignments or dynamic adjustments.

Disadvantages

• Ignores intraday price fluctuations (e.g., high-low price movements).
• Sensitive to extreme events (e.g., jumps) and struggles to capture dynamic changes in volatility.

Use Cases

Suitable for scenarios requiring quick volatility estimates or when intraday data is unavailable.

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3.2 EWMA Model

Model Overview

The EWMA (Exponentially Weighted Moving Average) model assigns greater weight to more recent returns, reflecting the time-series nature of volatility. Its formula is:

$$\sigma^2_{\text{t}} = (1 - \lambda) r_t^2 + \lambda \sigma^2_{\text{t-1}}$$

where:

• $\sigma_{\text{t}}^2$: Volatility squared at time $t$.
• $r_t$: Return at time $t$.
• $\lambda$: Smoothing parameter, typically set to $0.94$ (as per RiskMetrics recommendations).
• $\sigma_{\text{t-1}}^2$: Volatility squared at time $t-1$.

Advantages

• Captures recent volatility changes through exponential decay, making it more sensitive to new information.
• Well-suited for analyzing dynamic changes in volatility.

Disadvantages

• The choice of $\lambda$ significantly impacts results and requires adjustment based on market characteristics.
• Struggles to capture long-term volatility trends.

Use Cases

Ideal for analyzing assets with strong volatility momentum, particularly in high-frequency trading or short-term risk management.

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3.3 RiskMetrics Model

Model Overview

The RiskMetrics model is a standardized tool based on EWMA, widely used in risk management (e.g., VaR calculations). Its formula is similar to EWMA but uses a fixed smoothing parameter $\lambda = 0.94$:

$$\sigma^2_{\text{t}} = (1 - 0.94) r_t^2 + 0.94 \sigma^2_{\text{t-1}}$$

Advantages

• Simple and easy to implement, with fixed parameters reducing the need for tuning.
• Widely accepted and suitable for analyzing volatility across various asset classes.

Disadvantages

• The fixed smoothing parameter $\lambda$ may not be suitable for all markets.
• Relies heavily on historical data and may not accurately predict future volatility.

Use Cases

Suitable for risk management, portfolio analysis, and other scenarios requiring standardized volatility tools.

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3.4 LinXiao Model

The LinXiao Model is an improved method for calculating historical volatility, proposed by Lin Xiao in "Volatility Pricing Models for RMB Interest Rate Options, 2019". It addresses the limitations of traditional methods (e.g., RiskMetrics) in handling sudden market events. Below are its key features:

Core Idea

• The LinXiao Model assigns dynamic weights $w_i$ to historical data points:
  • Recent data points (closer to the present) are assigned higher weights.
  • Weights decrease gradually for older data points but at a slower rate.
  • Outliers (e.g., sudden market events) are assigned moderate weights to avoid excessive influence on volatility calculations.

Model Formula

1. Weight distribution for data points:

   $$w_i = \frac{1}{A} \left[1 - 3^{-(n-i+1)}\right], \quad (i = 1, 2, \dots, n)$$

   • $A$: Normalization coefficient ensuring $\sum_{i=1}^n w_i = 1$.
   • $n$: Total number of historical data points.
   • $i$: The $i$-th data point, with more recent data points having higher weights.

2. Calculation of average change:

   $$\bar{x} = \sum_{i=1}^n w_i \cdot x_i$$

   • $x_i = S_i - S_{i-1}$, representing daily price changes.

3. Volatility calculation:

   $$\sigma = \sqrt{250 \cdot \sum_{i=1}^n w_i \cdot (x_i - \bar{x})^2}$$

   • The volatility is annualized (factor of 250 represents trading days in a year).

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Model Features

1. Flexible Weight Distribution:

   • Weights $w_i$ are controlled by an exponential function, ensuring high sensitivity to recent data while avoiding rapid decay for older data.
   • The weight distribution is smoother, capturing long-term trends.

2. Handling Sudden Events:

   • When a data point $x_i$ is an outlier, its weight $w_i$ is not excessively high, preventing undue influence on volatility calculations.
   • The model is more stable and suitable for market data with extreme events.

3. Improving Traditional Methods:

   • RiskMetrics uses a fixed $\lambda$, leading to constant decay rates for historical data and limited flexibility.
   • The LinXiao Model overcomes this limitation through dynamic weight adjustments, making it more suitable for complex market environments.

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Use Cases

1. High-Volatility Markets:

   • Such as FX or commodity markets, where sudden events (e.g., central bank policies, geopolitical risks) are common.

2. Risk Management:

   • Provides more stable historical volatility estimates, avoiding overestimation of risk due to extreme events.

3. Asset Pricing and Hedging:

   • Offers more reasonable historical volatility calculations, aiding in option pricing and hedging strategies.

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3.5 GARCH Model

The GARCH (Generalized Autoregressive Conditional Heteroskedasticity) model is a statistical tool for analyzing volatility in time series data, particularly financial time series like stock returns or exchange rates. Proposed by Bollerslev in 1986, it extends the ARCH (Autoregressive Conditional Heteroskedasticity) model.

3.5.1 Core Idea

The GARCH model assumes that volatility (variance) is not constant but varies over time and exhibits autocorrelation. Specifically, it models current volatility as dependent on past volatility and past random shocks (residuals).

3.5.2 Model Structure

The general form of a GARCH(p, q) model is:

1. Mean Equation:

   $$r_t = \mu_t + \epsilon_t$$

   where $r_t$ is the return at time $t$, $\mu_t$ is the conditional mean (often assumed constant or estimated via other models), and $\epsilon_t$ is the residual term.

2. Variance Equation:

   $$\sigma_t^2 = \alpha_0 + \sum_{i=1}^q \alpha_i \epsilon_{t-i}^2 + \sum_{j=1}^p \beta_j \sigma_{t-j}^2$$

   where $\sigma_t^2$ is the conditional variance at time $t$, $\alpha_0$ is a constant term, $\alpha_i$ are ARCH term coefficients, $\beta_j$ are GARCH term coefficients, $\epsilon_{t-i}^2$ are past squared residuals, and $\sigma_{t-j}^2$ are past conditional variances.

Parameter Interpretation:

• ARCH Terms ($\alpha_i \epsilon_{t-i}^2$): Represent the impact of past squared residuals on current volatility.
• GARCH Terms ($\beta_j \sigma_{t-j}^2$): Represent the impact of past volatility on current volatility.
• Constant Term ($\alpha_0$): Represents the long-term average level of volatility.

GARCH model parameters are typically estimated using Maximum Likelihood Estimation (MLE). Steps include:

1. Assuming residuals $\epsilon_t$ follow a normal or other distribution (e.g., t-distribution).
2. Constructing the likelihood function.
3. Maximizing the likelihood function using numerical optimization methods (e.g., Newton-Raphson).

GARCH models are widely used in finance for:

• Volatility Forecasting: Predicting future asset price volatility for risk management.
• Option Pricing: Volatility is a key input for option pricing models (e.g., Black-Scholes).
• Risk Management: Calculating risk metrics like VaR (Value at Risk).

Extensions of GARCH, such as EGARCH (Exponential GARCH) and TGARCH (Threshold GARCH), capture more complex features like asymmetry (e.g., leverage effects).

The GARCH model is a powerful tool for analyzing and forecasting financial time series volatility. By incorporating ARCH and GARCH terms, it captures volatility clustering and persistence, providing critical support for risk management and asset pricing in financial markets.

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4. Model Comparison

Below is a summary of the strengths and weaknesses of the Close-to-Close, EWMA (RiskMetrics), and LinXiao Models, along with their applicability in different scenarios:

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Comparison Table

Model	Advantages	Disadvantages	Use Cases
Close-to-Close	1. Simple and easy to use; 2. High computational efficiency; 3. Low data requirements.	1. Ignores intraday volatility; 2. Sensitive to extreme events; 3. Cannot capture dynamic volatility changes.	Scenarios with limited data or stable markets; Quick volatility estimates.
EWMA (RiskMetrics)	1. Dynamic weighting, with higher weights for recent data; 2. Reflects short-term volatility changes; 3. Simple and widely used.	1. Fixed parameters limit flexibility; 2. Weak at capturing long-term trends; 3. Sensitive to jumps.	Short-term volatility analysis; High-frequency trading or intraday risk management; Risk management (e.g., VaR).
LinXiao	1. Flexible dynamic weight distribution, capturing long-term trends; 2. Stable handling of extreme events; 3. Suitable for complex markets.	1. Higher computational complexity; 2. Complex parameter tuning; 3. High data requirements.	High-volatility markets (e.g., FX, commodities); Time series with jump risks; Risk management and asset pricing (e.g., option pricing).

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Use Cases

• Close-to-Close Model:

  • The most basic and simple volatility calculation method, suitable for scenarios with limited data or quick volatility estimates. However, it may perform poorly in dynamic volatility analysis or extreme market conditions.

• EWMA Model (RiskMetrics):

  • A more advanced dynamic volatility model, suitable for short-term volatility analysis and risk management, especially in high-frequency trading and VaR calculations.
  • Its fixed parameters limit flexibility, and it struggles to handle jump events effectively.

• LinXiao Model:

  • A newer, improved method that balances short-term volatility and long-term trends through flexible weight distribution. It handles extreme events stably.
  • Suitable for complex market environments (e.g., high volatility or jump risks) but requires higher computational effort and parameter tuning.

Recommendation: In practice, choose the model based on market characteristics and analysis needs. For example:

• Quick Estimates: Use the Close-to-Close model.
• Short-Term Dynamic Volatility Analysis: Use the EWMA model.
• Long-Term Trend Analysis or Complex Markets: Use the LinXiao model.

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References

1. RiskMetrics Group (1996) - RiskMetrics Technical Document.
2. Hull, J. C. (2018) - Options, Futures, and Other Derivatives.
3. JP Morgan (1996) - Introducing RiskMetrics.
4. Engle, R. F. (1982) - Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of UK Inflation.
5. Lin Xiao (2019) - Volatility Pricing Models for RMB Interest Rate Options.