[펌] Volatility Measures - Historical, ARCH,EWMA and GARCH

Volatility Measures - Historical, ARCH,EWMA and GARCH

Volatility of the underlying asset is a key input in theoretical option pricing models. The most popular volatility measures used by market practitioners are:

Historical Volatility
ARCH (Auto Regressive Conditional Heteroskedasticity)
EWMA (Exponential Weighted Moving Average)
GARCH (Generalized Autoregressive Conditional Heteroskedasticity)

Each of these volatility measures has its own pros and cons. These models are neither fool proof nor deterministic but sadly these are the ones we know of currently. In this series of tutorials, we will focus on the derivation of different volatility measures, their pros and cons and practical considerations of each of these measures.

To begin with, let us look at historical volatility. Understanding this measure is crucial before we explore ARCH, EWMA and GARCH. Mathematically, historical volatility can be defined as the standard deviation of an asset’s returns over a year. Conceptually, one can talk about it as a measurement of change (both up and down movements) in asset’s price over a given period. To begin with, let us take a hypothetical asset X. Let the closing price of the asset X on any given day i to be p_i. Then simple return of asset X is given by

r_i - Return of asset X on any given day i
p_i - Closing price of asset X on any given day i
p_i-1 - Closing price of asset X on the previous day i-1

But generally, we don’t use the simple return for the calculation of historical volatility. Instead we use the continuous compounding return. The continuous compounding return of asset X is given by

ln - natural logarithm

Before proceeding further, we need to understand the role of time intervals and the duration of intervals in calculating the historical volatility. One can use daily, weekly or monthly intervals. In this tutorial, we will be working on daily intervals (using closing price of asset X on each day for a series of trading days). When working on daily data, the most commonly used durations are 20/50 trading days.

Steps to calculate historical volatility
1. Calculate the average of returns of the hypothetical asset X over recent “m” time intervals (μ).

2. Calculate the volatility of the asset returns using the formula given below. Keep in mind that the volatility given by this formula represents the volatility for the chosen time interval. Here, we chose daily intervals. Hence, σ represents one day volatility of the asset X.

3. Next how do we get the annualized volatility of the asset from its one day volatility obtained above? Simple! For example, let the one day volatility of the asset X be 1%. The formula for annualized historical volatility (which utilizes the “square root of time” rule) is given by

Where t = number of trading days in a year = 252

Note: Historical volatility over any given time period, say 30 days or 180 days, can also be calculated using the above formula. In this case, t = 30 and 180 respectively.


Pros of Historical Volatility Measure

• HV is computationally inexpensive.
• HV is easy to understand.
• HV is simple measure compared to other volatility measures

Cons of Historical Volatility Measure

• HV is a backward measure. The assumption here is that it is a reasonable forecast of future volatility.
• HV assigns equal weights to all observations during the sampling period.
• HV assumes the price movements are random (Gaussian) and stable. Often, volatility does not obey a Gaussian distribution law .

References

1. http://www.riskglossary.com/
2. http://currencies.thefinancials.com/FAQs1b.html

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