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## Monday, July 19, 2010

### what 80% volatility means

( http://lvb.wiwi.hu-berlin.de/members/personalpages/wh/talks/DetHaeSlidesVarianceSwapForecast060531.pdf has the precise math definition of Realized Variance )

We need to differentiate between i-vol vs h-vol .....

Q: what does 24% volatility mean for an option expiring in 3 months?
A: it means the stdev for an observable is ----> 24% * √ 3m/12m = 24% * 1/2 = 12%.
See P277 [[Complete guide ]]

Now let's define that observable. If today's closing price is \$100, and closing price in 3 months is X, then (X-100)/100 is the observable.

Therefore, 3-month forward price is likely (two-thirds likelihood) to fall between ±12% of current price, or between \$88 and \$112. Here we ignore interest rate and dividend.

Now forget about options. Consider a stock like IBM.

Q: what does a vol of 80% means for IBM?
A: see P 76 [[Options Vol trading]]. Average day will see a 5% move (80%/√ 252. More precisely 66.666% of the days will see moves under 5%; 33.333% of the days will see wider/wilder moves.

Q: a longer example -- what does 25% volatility mean for IBM's closing prices tomorrow vs today?
A: it means the stdev for IBM daily percentage return is 25% * √ 1 / 252days = 25% / 15.9 = 1.57% 

A longer answer: Take the underlier's daily closing price for today (\$100) vs tomorrow (X). The daily percentage return (X-100)/100  could be 1%, -2%, 0.5% .., but for clarity I'd rather compute PriceRelative to get 1.01, 0.98, 1.005...

Now if we simulate 1000 times and plot a histogram of daily returns defined as log(PR), then we get a mean. The mean is most likely very close to 0. The histogram is likely to resemble a normal curve. If today closes at \$100, and if IBM does follow an annualized vol of 25%, then tomorrow's close is likely (two-thirds likelihood) to fall within \$98.43 to \$101.57. Note the numbers are imprecise because we are assuming IBM is equally like to lose 1.57% or gain 1.57%, but this is the naive model. It also predicts "equally likely to gain 101% or lose 101%" to become NEGATIVE. BS model assumes log(PR) is normal so 1.57% gain is as likely as a decline to 98.454%.

IF (big IF) IBM continues to *follow* the 25% annualized vol, and if we observe its daily PR for 5 years, we will see that most of the time (two-third of the time), daily PR falls somewhere between ±1.57% using the naive model. See P34 [[Trading Options in Turbulent Markets ]]

 There are 252 trading days in a year. Our 25% vol is an annualized vol.