1. Historical VolatilityDerivative Classroom -Series IV
ssss
Introduction
Volatility is a statistical measure of the amount of fluctuation in a stock’s price
within a given period of time. A highly volatile stock would have rapid up and
down movements in its stock price. A stock with very little movement in its price
would constitute low volatility. There are two main measures of Volatility:
Historical Volatility & Implied Volatility. Historical volatility will be discussed
Historical Volatility
in this report whereas implied volatility will be explained in the subsequent report
&
Implied Volatility Historical Volatility is a statistical measure of the volatility of a futures contract,
security, or other instrument over a specified number of past trading days
Historical volatility is the measure of a stock’s price movement based on its
historical prices. It measures how active a stock price typically is over a certain
period of time. Usually, historical volatility is measured by taking the daily (close-
to-close) percentage price changes in a stock and calculating the average over a
given time period. The average is then expressed as an annualized percentage.
Short-term or more active traders tend to use shorter time periods for measuring
historical volatility, the most common being 5-day, 10-day, 20-day and 30-day.
Intermediate-term and long-term investors tend to use longer time periods, most
commonly 60-day, 180-day and 360-days
Calculation
⎛ S ⎞ Step I: Measure the day-to-day price changes in the market
R t = LN ⎜ t
⎜S ⎟
⎟
⎝ t −1 ⎠ Calculate the natural log of the ratio (Rt) of a stock’s price (S) from the current
day (t) to the previous day (t-1)
∑ R t Step II: Calculate the average day-to-day price changes over a
R m = n certain period
n
Calculate the Average daily price change (Rm) for a period of time (n)
Step III: Find out how far prices vary from the average calculated
in Step II
Karun Mutha
The historical volatility (HV) is the “average variance” from the mean (the
Sr. Vice President& Head Derivatives “standard deviation”), this is the average deviation of the values from the mean,
Tel +91-22-67897833
Email: Karun.mutha@hsbcinv.com and is estimated as
Tina Khetan
Analyst - Derivatives
Tel +91-22-67897828
Email: Tina.khetan@hsbcinv.com HV = ∑ Rt
2
HV =
∑ (R t − Rm ) 2
n OR n −1
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2. Historical VolatilityDerivative Classroom -Series IV
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Sometimes historical volatility is estimated by “ditching the mean” and using the
2nd formula mentioned. The latter formula for HV is statistically called a non-
centered approach. Traders commonly use it because it is closer to what would
actually affect their profits and losses. It also performs better when “n” is small or
when there is a strong trend in the particular stock.
Step IV: Express volatility as an annual percentage
To annualize the historical volatility, the above result is multiplied by the square
root of the average number of trading days in a year (approx. 252 days).
Practical Example
N ift y R t= L n (S t/
D a te (R t)^ 2 (R t-R m )^ 2
C lo s in g S (t-1 ))
2 9 - J u l- 0 9 4 5 1 3 .5 0 -0 .0 1 1 1 4 8 0 .0 0 0 1 2 4 0 .0 0 1 4 4 2
3 0 - J u l- 0 9 4 5 7 1 .4 5 0 .0 1 2 7 5 8 0 .0 0 0 1 6 3 0 .0 0 0 1 9 8
3 1 - J u l- 0 9 4 6 3 6 .4 5 0 .0 1 4 1 1 9 0 .0 0 0 1 9 9 0 .0 0 0 1 6 2
3 -A u g -0 9 4 7 1 1 .4 0 0 .0 1 6 0 3 6 0 .0 0 0 2 5 7 0 .0 0 0 1 1 6
4 -A u g -0 9 4 6 8 0 .5 0 -0 .0 0 6 5 8 0 0 .0 0 0 0 4 3 0 .0 0 1 1 1 6
5 -A u g -0 9 4 6 9 4 .1 5 0 .0 0 2 9 1 2 0 .0 0 0 0 0 8 0 .0 0 0 5 7 2
6 -A u g -0 9 4 5 8 5 .5 0 -0 .0 2 3 4 1 8 0 .0 0 0 5 4 8 0 .0 0 2 5 2 5
7 -A u g -0 9 4 4 8 1 .4 0 -0 .0 2 2 9 6 4 0 .0 0 0 5 2 7 0 .0 0 2 4 7 9
1 0 -A u g -0 9 4 4 3 7 .6 5 -0 .0 0 9 8 1 1 0 .0 0 0 0 9 6 0 .0 0 1 3 4 2
1 1 -A u g -0 9 4 4 7 1 .3 5 0 .0 0 7 5 6 5 0 .0 0 0 0 5 7 0 .0 0 0 3 7 1
1 2 -A u g -0 9 4 4 5 7 .5 0 -0 .0 0 3 1 0 2 0 .0 0 0 0 1 0 0 .0 0 0 8 9 6
1 3 -A u g -0 9 4 6 0 5 .0 0 0 .0 3 2 5 5 5 0 .0 0 1 0 6 0 0 .0 0 0 0 3 3
1 4 -A u g -0 9 4 5 8 0 .0 5 -0 .0 0 5 4 3 3 0 .0 0 0 0 3 0 0 .0 0 1 0 4 1
1 7 -A u g -0 9 4 3 8 7 .9 0 -0 .0 4 2 8 5 9 0 .0 0 1 8 3 7 0 .0 0 4 8 5 6
1 8 -A u g -0 9 4 4 5 8 .9 0 0 .0 1 6 0 5 1 0 .0 0 0 2 5 8 0 .0 0 0 1 1 6
1 9 -A u g -0 9 4 3 9 4 .1 0 -0 .0 1 4 6 3 9 0 .0 0 0 2 1 4 0 .0 0 1 7 1 9
2 0 -A u g -0 9 4 4 5 3 .4 5 0 .0 1 3 4 1 6 0 .0 0 0 1 8 0 0 .0 0 0 1 8 0
2 1 -A u g -0 9 4 5 2 8 .8 0 0 .0 1 6 7 7 8 0 .0 0 0 2 8 1 0 .0 0 0 1 0 1
2 4 -A u g -0 9 4 6 4 2 .8 0 0 .0 2 4 8 6 1 0 .0 0 0 6 1 8 0 .0 0 0 0 0 4
2 5 -A u g -0 9 4 6 5 9 .3 5 0 .0 0 3 5 5 8 0 .0 0 0 0 1 3 0 .0 0 0 5 4 1
2 6 -A u g -0 9 4 6 8 0 .8 5 0 .0 0 4 6 0 4 0 .0 0 0 0 2 1 0 .0 0 0 4 9 4
2 7 -A u g -0 9 4 6 8 8 .2 0 0 .0 0 1 5 6 9 0 .0 0 0 0 0 2 0 .0 0 0 6 3 8
S u m m a t io n 0 .0 2 6 8 2 8 0 .0 0 6 5 4 6 0 .0 2 0 9 4 2
N o o f d a y s t o b e u s e d in D e n o m in a t io n n= 22 n = 2 2 -1 = 2 1
T a k in g D iv is io n 0 .0 0 0 2 9 8 0 .0 0 0 9 9 7
F o r m u la (0 .0 0 6 5 4 6 /2 2 ) 0 .0 2 0 9 4 2 /2 1 )
T a k in g S q u a r e R o o t w e g e t D a ily V o la t ilit y
D a ily V o la t ilit y 1 .7 3 % 3 .1 6 %
A n n u a lis e d V o la t ilit y = D a ily V o l* s q r t ( 2 5 2 )
A n n u a lis e d V o la t ilit y 2 7 .3 9 % 5 0 .1 3 %
Inference
When no of days is less its recommended to use first formula. As we consider
more trading days we need to calculate volatility of the price taking deviation from
the mean.
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