Understanding Volatility

Sheldon NatenbergChicago Trading Co.

440 S. LaSalle St.Chicago, IL 60605

(312) 863-8004shellynat@aol.com

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