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Understanding Volatility
Sheldon NatenbergChicago Trading Co.
440 S. LaSalle St.Chicago, IL 60605
(312) [email protected]
Options Trading Forum
October 2nd, 2002
theoreticalvalue
theoreticalvalue
pricingmodel
exercise price
time to expiration
underlying price
interest rate
volatility
(dividends)
pricingmodel
exercise price
time to expiration
underlying price
interest rate
volatility
(dividends)
exercise price
time to expiration
underlying price
interest rate
volatility
(dividends)
-1-
long an underlying contract10%*90 + ……. + 10%*110
long a 100 call20%*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-
If the expected return of the 100 callis 2.00, what is its theoretical value?
The theoretical value is the priceyou would be willing to pay todayin order to just break even.
interest rates = 12%2 months to expiration
2.00 - (2.00 x 2%) = 1.96
-3-
underlying prices
probabilities
normaldistribution
normaldistribution
-4-
All normal distributionsare defined by their mean
and their standard deviation.
Mean – where thepeak of the curveis located
Standard deviation –how fast the curvespreads out.
-5-
100100
120 call120 call
90 days toexpiration
.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-
+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-
Mean
Standard deviation
Volatility: one standard deviation,in percent, over a one year period.
– the break even price atexpiration for a trade made attoday’s price (forward price)
– volatility
-8-
1-year forward price = 100.00volatility = 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-
-10-
What does an annual volatility tellus about movement over some othertime period?
monthly price movement?weeky price movement?daily price movement?
volatilityt = volatilityannual x tvvolatilityannual x tv tv
-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=tv tv v1/256 = 1/16
-12-
volatilitydaily = 20% / 16 = 1¼%One trading day from now:
• 2/3 chance the contract will bebetween 98.75 and 101.25(100 ± 1¼%)
• 19/20 chance the contract will bebetween 97.50 and 102.50(100 ± 2 x 1¼%)
16
2/3
19/20
-13-
Weekly volatility:
volatilityweekly = volatilityannual / 7.2t = 1/52 =tv v1/52=tv tv v1/52 ˜ 1/7.2
volatilitymonthly = volatilityannual / 3.5t = 1/12 =tv v1/12=tv tv v1/12 ˜ 1/3.5
Monthly volatility:
-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
-15-
daily standard deviation = 1.80stock = 68.50; volatility = 42.0%
+1.25 -.95 +.35+.70 -1.60
Is 42% a reasonable volatilityestimate?How often do you expect to see an occurrence greater than onestandard deviation?
-16-
8+8+8–8–
00
normaldistribution
normaldistribution
lognormaldistributionlognormaldistribution
-17-
normaldistribution
110 call
lognormaldistribution
underlying price = 100
3.0090 put 3.00
3.002.50
110 call = 2.75 90 put = 3.00Are the options mispriced?Could there is something wrongwith the model?
-18-
The volatility ofthe underlying contract over someperiod in the future
future volatility:
historical volatility:
forecast volatility:
The volatilityof the underlying contract oversome period in the past
Someone’sestimate of future volatility
-19-
derived from the prices of optionsin the marketplace
implied volatility:
the marketplace’s forecast offuture volatility
-20-
exercise price
time to expiration
underlying price
interest rate
volatility
exercise price
time to expiration
underlying price
interest rate
volatility
pricingmodelpricingmodel
theoreticalvalue
theoreticalvalue
2.50
3.25
volatility 27%27%
??????31%
implied volatilityimplied volatility
-21-
future volatility
implied volatility
= value
= price
historical volatilityforecast volatilityhistorical volatilityforecast volatility
Option trading decisions oftenbegin by comparing
to
-22-
Volatility Trading
Initially buy underpriced options or strategies, or selloverpriced options or strategies
Offset the option position by taking an opposing marketposition, delta neutral, in the underlying contract
Periodically buy or sell an appropriate amount of theunderlying contract to remain delta neutral over the lifeof the strategy (dynamic hedging)
At expiration liquidate the entire position
In theory, when the position is closed out the totalprofit (or loss) should be approximately equal to theamount by which the options were originally mispriced.
-23-
Volatility Trading Risks
You may have incorrectlyestimated the future volatility
The model may be wrong
-24-
SPX Historical VolatilityJanuary 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 volatility250-day volatility
-25-
Volatility characteristics
mean reversion – volatility tends toreturn to its historical average
serial correlation – in the absence ofother data, the best volatility guess overthe next time period is the volatility whichoccurred over the previous time period.
momentum – a trend in volatility is likely to continue
-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)
impl
ied
vola
tility
(%
)
-27-
(G)ARCH
Volatility Forecasting Methods
– (generalized) auto-regressive conditionalheteroscedasticity
(V)ARIMA– (vector) auto-regressive integratedmoving average
-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)
num
ber
of o
ccur
renc
es
number of days: 3186biggest 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: -.0263kurtosis: +3.9072
-29-
Volatility Skew:
The tendency of options atdifferent exercise prices to tradeat different implied volatilities
A consequence ofhow people use optionsweaknesses in the pricing model
-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
