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Volatility and Correlation Measures

Introduction

In general, volatility provides a measure of the variability or dispersion of price data per unit of time. This number is important for analyzing many different types of instruments to determine the probability of a certain price being attained on a certain date in the future. Obviously, volatility is a crucial input for any options pricing model, as it will determine the probability of the option expiring in or out of the money.

Historical Volatility

As a tool to estimate future volatility, one can look at historical changes in price. In order to do this, you must first decide how much historical data to analyze and at what intervals to take these measurements. The more data you use, the better your results with respect to obtaining historical norms, minimums and maximums. Using recent historical data provides better information on the current level of volatility. As a general rule, services which supply historical volatilities base their calculations on daily settlement prices, or more specifically the standard deviation of the daily returns (relative changes in price).

Implied Volatility

Implied volatility is the result obtained from a theoretical option pricing model given the market price of the option. When you solve for the implied volatility of an option you are assuming that the option price is known and that the theoretical volatility is unknown. Implied volatility can be thought of as the current market consensus of volatility for the underlying instrument assuming that everyone is using the same theoretical option pricing model, i.e, assuming normal distribution of returns, and therefore log-normal distribution of price (see below).


Technical Details

Calculating Historical Volatility

Calculating historical volatilities requires an assumption about the shape of the distribution of returns for the instrument being analyzed. For stocks, it is assumed that prices are lognormally distributed. This means that the natural logarithm of a price has a normal distribution. This is one of the assumptions used in the Black-Scholes option pricing model. The historical volatility of the Black-Scholes model is simply the standard deviation of the daily change in the natural logarithms of the prices on consecutive trading dates. Such a volatility is often called a log volatility.

There are other types of historical volatilities depending on the data modeling, weighting and convention of mean adjustment. If the data are modeled with a simple normal distribution, the volatility is simply the standard deviation of the changes. Such a volatility is called a normal volatility in this document. Sometimes it is beneficial to put more weight on more recent data. A typical example is the RiskMetrics volatility data set, which is used in many organizations worldwide to estimate value at risk. For this data set exponential weighting is used. The commonly used types of volatility are:

  1. Equally weighted log volatility
  2. Equally weighted normal volatility
  3. Exponentially weighted log volatility
  4. Exponentially weighted normal volatility

Most of the time volatilities are centered on the sample mean. In other words, the volatilities are "adjusted with mean". Sometimes, however, volatilities are not centered. Non-centered volatility time series appear to be more stable than centered volatility series.

Calculating Historical Correlation

If multiple assets or factors are involved in modeling, correlations between assets or factors must be estimated. Oftentimes historical correlations are used. Corresponding to each type of historical volatility defined above there is a type of historical correlation.


Analysis Supported

FINCAD volatility functions can be used for the following:

  • Calculate the historical lognormal and percentage volatility of a range of price or rate data.
  • Calculate various types of historical volatilities and prices adjusted for dividend and splitting/consolidation.
  • Calculate various types of historical correlations and prices adjusted for stock dividends and splitting/consolidation.
  • Calculate various types of historical correlations and prices adjusted for stock dividends and splitting/consolidation, for an arbitrary number of assets.
  • Calculate the correlation matrix of an array of N data points for each of K variables.

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