Options Trading Forum October 2 nd, 2002 Understanding Volatility Sheldon Natenberg Chicago Trading Co. 440 S. LaSalle St. Chicago, IL 60605 (312) 863-8004 shellynat@aol.com
exercise price time to expiration -1- underlying price interest rate pricing model theoretical value volatility (dividends)
90 95 100 105 110 20% 10% 20% 20% 40% 20% 10% 20% long an underlying contract -2-10%*90 +. + 10%*110 long a 100 call = 100 20%*5 + 10%*10 = 2.00 Expected Return
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 -3- 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.
-6- +.25 each day value =.05 + 2.00 each day value =.75 80 put + 10.00 each day value = 8.00 100 90 days to expiration 120 call option value
+1 S.D. 34% -1 S.D. 34% +2 S.D. 47.5% -2 S.D. 47.5% mean ±1 S.D. 68% (2/3) ±2 S.D. 95% (19/20) -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.
1-year forward price = 100.00 volatility = 20% One year from now: -9-2/3 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? monthly price movement? -10- weeky price movement? daily price movement? volatility t = volatility annual x v t
Daily volatility (standard deviation) Trading days in a year? 250 260-11- Assume 256 trading days t = 1/256 v t = v1/256 = 1/16 volatility daily volatility annual / 16
volatility daily = 20% / 16 = 1¼% One trading day from now: 2/3 chance the contract will be between 98.75 and 101.25-12- (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 = v1/52 1/7.2 volatility weekly = volatility annual / 7.2-13- Monthly volatility: t = 1/12 v t = v1/12 1/3.5 volatility monthly = volatility annual / 3.5
stock = 68.50; volatility = 42.0% daily standard deviation? 68.50 x 42% / 16-14- = 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 8 normal distribution +8+ 8 lognormal distribution 0-16-
underlying price = 100 110 call normal distribution 3.00 lognormal distribution 3.00-17- 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 historical volatility: The volatility -18- of the underlying contract over some period in the past forecast volatility: Someone s estimate of future volatility
implied volatility: derived from the prices of options -19- in the marketplace the marketplace s forecast of future volatility
-20- exercise price time to expiration underlying price interest rate 31%??? volatility 27% implied volatility pricing model 3.25 theoretical value 2.50
Option trading decisions often begin by comparing implied volatility = price -21- 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 You may have incorrectly -23- 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
40 Volatility Cones 38 36 34-26- implied volatility (%) 32 30 28 26 24 22 20 0 3 6 9 12 15 18 21 24 27 30 33 36 time to expiration (months)
Volatility Forecasting Methods (G)ARCH (generalized) autoregressive conditional -27- 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 150-28- number of occurrences 125 100 75 50 25 0-7% -6% -5% -4% -3% -2% -1% 0% 1% 2% 3% 4% 5% daily price change (nearest 1/8 percent)
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
38 SPX June Implied Volatilities - 22 February 2002 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