This document discusses the concept of volatility in options pricing. It defines volatility as the standard deviation of percentage changes in an underlying asset's price over a period of time. Higher volatility means prices are likely to fluctuate more widely. The document explains how volatility affects options prices and describes methods for estimating historical volatility, implied volatility from current market prices, and forecasting future volatility.

1. Understanding Volatility
Sheldon Natenberg
Chicago Trading Co.
440 S. LaSalle St.
Chicago, IL 60605
(312) 863-8004
shellynat@aol.com
Options Trading Forum
October 2nd, 2002

2. theoretical
value
theoretical
value
pricing
model
exercise price
time to expiration
underlying price
interest rate
volatility
(dividends)
pricing
model
exercise price
time to expiration
underlying price
interest rate
volatility
(dividends)
exercise price
time to expiration
underlying price
interest rate
volatility
(dividends)
-1-

3. long an underlying contract
10%*90 + ……. + 10%*110
long a 100 call
20%*5 + 10%*10 = 2.00
= 100
90 95 105 110100
20% 20% 20% 20% 20%10% 20% 40% 20% 10%
90 95 105 110100
20% 20% 20% 20% 20%
90 95 105 11010090 95 105 11090 95 105 110100
20% 20% 20% 20% 20%20% 20% 20% 20% 20%10% 20% 40% 20% 10%
Expected Return
-2-

4. If the expected return of the 100 call
is 2.00, what is its theoretical value?
The theoretical value is the price
you would be willing to pay today
in order to just break even.
interest rates = 12%
2 months to expiration
2.00 - (2.00 x 2%) = 1.96
-3-

5. underlying prices
probabilities
normal
distribution
normal
distribution
-4-

6. All normal distributions
are defined by their mean
and their standard deviation.
Mean – where the
peak of the curve
is located
Standard deviation –
how fast the curve
spreads out.
-5-

7. 100100
120 call120 call
90 days to
expiration
.25 each day+–.25 each day+–.25 each day+–+–
2.00 each day+–2.00 each day+–2.00 each day+–+–
10.00 each day+–10.00 each day+–10.00 each day+–+–
value =.05
value =.75
value = 8.00
80 put80 put
option valueoption value
-6-

8. +1 S.D.+1 S.D.
+1 S.D. ˜ 34%
-1 S.D.-1 S.D.
-1 S.D. ˜ 34%
+2 S.D.+2 S.D.-2 S.D.-2 S.D.
+2 S.D. ˜ 47.5%
-2 S.D. ˜ 47.5%
±1 S.D. ˜
68% (2/3)
±1 S.D. ˜
68% (2/3)
±2 S.D. ˜
95% (19/20)
±2 S.D. ˜
95% (19/20)
meanmean
-7-

9. Mean
Standard deviation
Volatility: one standard deviation,
in percent, over a one year period.
– the break even price at
expiration for a trade made at
today’s price (forward price)
– volatility
-8-

10. 1-year forward price = 100.00
volatility = 20%
One year from now:
• 2/3 chance the contract will be
between 80 and 120 (100 ± 20%)
• 19/20 chance the contract will be
between 60 to 140 (100 ± 2 x 20%)
• 1/20 chance the contract will be
less than 60 or more than 140
-9-

11. -10-
What does an annual volatility tell
us about movement over some other
time period?
monthly price movement?
weeky price movement?
daily price movement?
volatilityt = volatilityannual x tvvolatilityannual x tvtv

12. -11-
Daily volatility (standard deviation)
Trading days in a year? 250 – 260
Assume 256 trading days
volatilitydaily ˜ volatilityannual / 16
t = 1/256 =tv v1/256=tvtv v1/256 = 1/16

13. -12-
volatilitydaily = 20% / 16 = 1¼%
One trading day from now:
• 2/3 chance the contract will be
between 98.75 and 101.25
(100 ± 1¼%)
• 19/20 chance the contract will be
between 97.50 and 102.50
(100 ± 2 x 1¼%)
16
2/3
19/20

14. -13-
Weekly volatility:
volatilityweekly = volatilityannual / 7.2
t = 1/52 =tv v1/52=tvtv v1/52 ˜ 1/7.2
volatilitymonthly = volatilityannual / 3.5
t = 1/12 =tv v1/12=tvtv v1/12 ˜ 1/3.5
Monthly volatility:

15. -14-
daily standard deviation?
stock = 68.50; volatility = 42.0%
˜ 68.50 x 42% / 16
= 68.50 x 2.625% ˜ 1.80
weekly standard deviation?
˜ 68.50 x 42% / 7.2
= 68.50 x 5.83% ˜ 4.00

16. -15-
daily standard deviation = 1.80
stock = 68.50; volatility = 42.0%
+1.25 -.95 +.35+.70 -1.60
Is 42% a reasonable volatility
estimate?
How often do you expect to see
an occurrence greater than one
standard deviation?

17. -16-
8+8+8–8–
00
normal
distribution
normal
distribution
lognormal
distribution
lognormal
distribution

18. -17-
normal
distribution
110 call
lognormal
distribution
underlying price = 100
3.00
90 put 3.00
3.00
2.50
110 call = 2.75 90 put = 3.00
Are the options mispriced?
Could there is something wrong
with the model?

19. -18-
The volatility of
the underlying contract over some
period in the future
future volatility:
historical volatility:
forecast volatility:
The volatility
of the underlying contract over
some period in the past
Someone’s
estimate of future volatility

20. -19-
derived from the prices of options
in the marketplace
implied volatility:
the marketplace’s forecast of
future volatility

21. -20-
exercise price
time to expiration
underlying price
interest rate
volatility
exercise price
time to expiration
underlying price
interest rate
volatility
pricing
model
pricing
model
theoretical
value
theoretical
value
2.50
3.25
volatility 27%27%
??????31%
implied volatilityimplied volatility

22. -21-
future volatility
implied volatility
= value
= price
historical volatility
forecast volatility
historical volatility
forecast volatility
Option trading decisions often
begin by comparing
to

23. -22-
Volatility Trading
Initially buy underpriced options or strategies, or sell
overpriced options or strategies
Offset the option position by taking an opposing market
position, delta neutral, in the underlying contract
Periodically buy or sell an appropriate amount of the
underlying contract to remain delta neutral over the life
of the strategy (dynamic hedging)
At expiration liquidate the entire position
In theory, when the position is closed out the total
profit (or loss) should be approximately equal to the
amount by which the options were originally mispriced.

24. -23-
Volatility Trading Risks
You may have incorrectly
estimated the future volatility
The model may be wrong

25. -24-
SPX Historical Volatility
January 1990 - August 2002
5%
10%
15%
20%
25%
30%
35%
Jan-90 Jan-91 Jan-92 Jan-93 Jan-94 Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Jan-00 Jan-01 Jan-02
50-day volatility
250-day volatility

26. -25-
Volatility characteristics
mean reversion – volatility tends to
return to its historical average
serial correlation – in the absence of
other data, the best volatility guess over
the next time period is the volatility which
occurred over the previous time period.
momentum – a trend in volatility is
likely to continue

27. -26-
Volatility Cones
20
22
24
26
28
30
32
34
36
38
40
0 3 6 9 12 15 18 21 24 27 30 33 36
time to expiration (months)
impliedvolatility(%)

28. -27-
(G)ARCH
Volatility Forecasting Methods
– (generalized) auto-
regressive conditional
heteroscedasticity
(V)ARIMA– (vector) auto-
regressive integrated
moving average

29. -28-
SPX Daily Price Changes: January 1990 - August 2002
0
25
50
75
100
125
150
175
200
225
250
-7% -6% -5% -4% -3% -2% -1% 0% 1% 2% 3% 4% 5%
daily price change (nearest 1/8 percent)
numberofoccurrences
number of days: 3186
biggest up move: +5.73% (24 July 2002)
biggest down move: -6.87% (27 October 1997)
mean: +.0364%
standard deviation: 1.0217%
volatility: 16.24%
skewness: -.0263
kurtosis: +3.9072

30. -29-
Volatility Skew:
The tendency of options at
different exercise prices to trade
at different implied volatilities
A consequence of
how people use options
weaknesses in the pricing model

31. -30-
SPX June Implied Volatilities - 22 February 2002
14
16
18
20
22
24
26
28
30
32
34
36
38
750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300 1350 1400