Download Understanding Volatility

Options Trading Forum
October 2nd, 2002
Understanding Volatility
Sheldon Natenberg
Chicago Trading Co.
440 S. LaSalle St.
Chicago, IL 60605
(312) 863-8004
[email protected]
exercise price
time to expiration
underlying price
-1-
interest rate
volatility
(dividends)
pricing
model
theoretical
value
90
95
100
105
110
20%
10%
20%
20%
40%
20%
10%
20%
long an underlying contract
-2-
10%*90 + ……. + 10%*110 = 100
long a 100 call
20%*5 + 10%*10 = 2.00
Expected Return
-3-
The theoretical value is the price
you would be willing to pay today
in order to just break even.
If the expected return of the 100 call
is 2.00, what is its theoretical value?
interest rates = 12%
2 months to expiration
2.00 - (2.00 x 2%) = 1.96
underlying prices
probabilities
-4-
normal
distribution
Standard deviation –
how fast the curve
spreads out.
Mean – where the
peak of the curve
is located
-5-
All normal distributions
are defined by their mean
and their standard deviation.
100
+
– .25 each day
value =.05
-6-
+
– 2.00 each day
value =.75
80 put
+
– 10.00 each day
value = 8.00
90 days to
expiration
120 call
option value
+1 S.D. ˜ 34%
-1 S.D. ˜ 34%
±1 S.D. ˜
68% (2/3)
mean
±2 S.D. ˜
95% (19/20)
+2 S.D. ˜ 47.5%
-2 S.D. ˜ 47.5%
-7-
-1 S.D. +1 S.D.
-2 S.D.
+2 S.D.
Mean – the break even price at
expiration for a trade made at
today’s price (forward price)
-8-
Standard deviation – volatility
Volatility: one standard deviation,
in percent, over a one year period.
-9-
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
What does an annual volatility tell
us about movement over some other
time period?
-10-
monthly price movement?
weeky price movement?
daily price movement?
volatilityt = volatilityannual x v t
Daily volatility (standard deviation)
Trading days in a year? 250 – 260
-11-
Assume 256 trading days
t = 1/256
v t = v 1/256 = 1/16
volatilitydaily ˜ volatilityannual / 16
volatilitydaily = 20% / 16 = 1¼%
One trading day from now:
-12-
• 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%)
Weekly volatility:
t = 1/52 v t = v 1/52 ˜ 1/7.2
volatilityweekly = volatilityannual / 7.2
-13-
Monthly volatility:
t = 1/12 v t = v 1/12 ˜ 1/3.5
volatilitymonthly = volatilityannual / 3.5
stock = 68.50; volatility = 42.0%
-14-
daily standard deviation?
˜ 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
stock = 68.50; volatility = 42.0%
daily standard deviation = 1.80
+.70 +1.25 -.95 -1.60
+.35
-15-
Is 42% a reasonable volatility
estimate?
How often do you expect to see
an occurrence greater than one
standard deviation?
8
–
normal
distribution
0
8
+
lognormal
distribution
-16-
-17-
underlying price = 100
normal
lognormal
distribution distribution
3.00
3.00
110 call
90 put
3.00
2.50
110 call = 2.75 90 put = 3.00
Are the options mispriced?
Could there is something wrong
with the model?
future volatility: The volatility of
the underlying contract over some
period in the future
-18-
historical volatility: The volatility
of the underlying contract over
some period in the past
forecast volatility: Someone’s
estimate of future volatility
implied volatility:
-19-
derived from the prices of options
in the marketplace
the marketplace’s forecast of
future volatility
implied volatility
exercise price
time to expiration
-20-
pricing
model
underlying price
interest rate
volatility
3.25
31%
???
theoretical
value
2.50
27%
Option trading decisions often
begin by comparing
-21-
implied volatility = price
to
future volatility = value
historical volatility
forecast volatility
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
-22-
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.
Volatility Trading Risks
-23-
You may have incorrectly
estimated the future volatility
The model may be wrong
SPX Historical Volatility
January 1990 - August 2002
35%
50-day volatility
250-day volatility
30%
25%
-24-
20%
15%
10%
5%
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
Volatility characteristics
-25-
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.
mean reversion – volatility tends to
return to its historical average
momentum – a trend in volatility is
likely to continue
Volatility Cones
40
38
36
-26-
implied volatility (%)
34
32
30
28
26
24
22
20
0
3
6
9
12
15
18
21
time to expiration (months)
24
27
30
33
36
Volatility Forecasting Methods
-27-
(G)ARCH – (generalized) autoregressive conditional
heteroscedasticity
(V)ARIMA– (vector) autoregressive integrated
moving average
SPX Daily Price Changes: January 1990 - August 2002
250
225
200
175
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
-28-
number of occurrences
150
125
100
75
50
25
0
-7%
-6%
-5%
-4%
-3%
-2%
-1%
0%
daily price change (nearest 1/8 percent)
1%
2%
3%
4%
5%
Volatility Skew:
The tendency of options at
different exercise prices to trade
at different implied volatilities
-29-
A consequence of
how people use options
weaknesses in the pricing model
SPX June Implied Volatilities - 22 February 2002
38
36
34
32
30
28
-30-
26
24
22
20
18
16
14
750
800
850
900
950
1000
1050
1100
1150
1200
1250
1300
1350
1400