1992 Volatility Sides

8/12/2019 1992 Volatility Sides

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Definition:

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Historical Volatility

Ilistorical volatilityis a formula that looks at setUement priceshlfi-for a number of previous trading days, say 20 for 2o-day \Nt

historical volatility. Tb calculate, take the standard deviation offr# ;'#;; J;; ;"*, artze In m ath em atic aI no t ati 04 ht,

where NlnP t

Pr-,D

ialrl IIL| Ua, Uf Uf fr;4f

z d' ' (to"'^ - .

r\*ry@ )=61

number of days of observation (zo) \(3.)lt" @natural logarithmsettlement price for each of the 20 dayssettlement price on the day before P*number of trading days in a year (252)

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Implied VolatilityDefinition:

Implied volatility is calculated using the BlackModel in reverse. Instead of plugging in the fivevariables (futures price, strike price, interest rate,volatility, days to expiration) and calculating atheoretical premium, what if we use the currentoption premium on the exchange floor? We couldthen work backward a.nd find the volatility that themarket is implying when trading the option at thatprice. There is no simple formula for calculatingimplied volatility. Look back to the "Black Modelufor the theoretical options formula used by O&A.To solve for volatility you must use a techniquecalled reiteration. That is, plug in an estimate ofvolatility; calculate the premium; then compare thetheoretical and the actual premiums. If they aredifferent, adjust your volatility value and re-calcu-late the premium. Do this repeatedly until the theeretical premium equals the actual premium. Whenthey are the sa.me, the final inputted volatility isthe implied volatility. To have OSzA calculate im-plied volatility, chooserrTable: Calls and Puts" fromthe main menu.

C=where:

dl =d2=C=r=t-

F=S=ln=

V=

Blaek Model

.''t[FN(d,)-SN(dr)]

tln (F/S) + (V2t) t2) t N {t)Iln (F/s) - (v'?t) A tN {\Call premium"Risk-Free" interest rate (i.e., T-bill rate)Time to expiration (in fractions of a year;i.e.,'90 days = t = 90/365 = .25\

Price of underlying futures contract

Strike price of option contractnatural logarithmbase of the natural logarithmAnnualized estimated volatility of theunderlying futures PricesArea under the curve of a nonnal distri -butionito the left of (d,) or (dr) respectively

Similarly, the formula for a put option is:

NO=

P=where:P=

-e-n [F N (-dr) - S N (-dr)]

Put premium

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\IolatilityTtre volatility figure assigned to a futures contract represents a onestandard deviation price change (in percent terms) at the end of aone year time period. In other words, if S&P 5OO stock indexfutures are trading at 3OO.OO and has a volatility of 1Oo/o, then wecould expect to see the futures trade between 27O and 33O (3OO tLOo/o) 680/o of the time at the end of one year. Is this to say that wecould never trade outside this range? The answer is a clear-cut no2 times out of 3 we will be within one standard deviation up ordown. 1 time out of 3 we will trade outside a one standard deviationmove at the end of one year. More advanced level trad.ing requiressome study of elementary probability.

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Volatility (cont'd)Given that most options have expirations of less than one year itwould be helpful if we could take the volatility value (remembervolatility and Standard Deviation are synonJrrnous) and convert it toa more usable form. In our example above, if volatility is LOo/o whatis a one standard deviation price change for a week? 40 days? (N)number of days?

Tlo convert a yearly standard deviation to a weekly stand.ard devia-tion, divide the volatility by the square root of the number of weeksin a trading year:

10 10 = 1.38o/o7.2

(3oo) (t1.38%) - 4.14 ptsfor one week.

\J '-k\.-'o I

4 {.tb/\), \' - 'N= one standard deviation price move

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\Iolatility (cont'd)TIo convert (n) number of days (such as days to expiration) from ayearly standard deviation to a one standard deviation move duringthat period:

number of days365

= volatility for (n) nu:rrber of days

Using our S&P example, what would be a one standard. deviationprice move over a BO-day time period?

Thus, at the end of a 3o-day period with volatility @ LOo/o, we wouldexpect S&P futures to trade in a range of 308.6O to 291 .4O two-thirds of the time.

)*"atirity)

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Price Distribution at ExpirationPresentPrice

Notes:

LowVolatility

ModerateVolatility

HighVolatility

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Volat1trty and ProbabilityCIearIy it is useful to convert annualized volatility values intovolatilities more appropriate to the time horizon of the optiontrader. of even more use to the option trader though, is thelikelihood of a particular strategy turning out p"olitaUe. Usingvolatility and the probabilities associated with it, one could forexarnple calculate the probabitity of a futures price attaining agiven strike price - i.e., what are the chances of a particular movegiven currelLt futures prices, volatility and a particular strikeprice?

The D-Mark is trading at around .5OOO with volatility around LOo/o.There are 1BO days until expiration. What is the probability of a D-Mark 43 put being in-the-money by expiration?

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Volatility and Probability (cont'd)First convert the annual volatility to a 18O-day volatility:

)

(.10) = .O7O or 7o/o .5OOO (t .O7) = * .O35 or 35O ticks

Ttrerefore, a one standard deviation price change up anddown for 18O days is .5OOO + .O35O (or a range of .5350 to .4680).

For the 43 put to be in-ttre-money by expiration, the D-Mark wou1d.have to d.rop .O7OO pts. (or 7OO tieks). If .O35 pts. = 1 standard devia-tion, then .O7OO pts. = 2 standard deviations (.O7OO/.O35O = 2.OO). Tbfind the probability of a 2 standard deviation move, consult a table of

normal probability distributions. Ttre probability = .OZZ8 or Z.Z8o/o otabout 1 time out of 44.

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\Iolatility SkewsAs you plot actual option premiums versus their strike prices (use theOptions and Alternatives software to study this effect), you'I1 noticethat not all implied volatilities are identical. The implied volatilities ofthe at-the-money options wiII differ from the implied volatilities of theout-of-the-money options. Traders and representatives from academiahave offered several reasons for the existence of volatility skews:

1. Risk/Reward ratio of buying vs. selling out-of-the-money options(as well as the margin requirements for selling options).

2. Sentiment or preponderance of bullishness or bearishness.

3.Assumptions made by option pricing models

vs. real life(theoretical vs. actual).

There is no right answer. However, the answer is not as important asthe fact that a skilled trader can take advantage of volatility skews.

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Volatility Skew inOptions on Yen Futures

Volatility vs. OTM Strike JYM = 70.45, JYH - 70.22 (1126190 close)

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67p 69p 70p 71c 72c 74c 75c 76cE March YENffi June YEN

Percent

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Volatility Skew inOptions on S&P 5OO Futures

Volatility vs. OTM Strike SPM = BBO.G5 (Llz6,lgo close)

Percent

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1B

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305 p 310p 315p 320p 325p 330p 335c 340c 345c 350c 355c 360c

- June S&P

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1992 Volatility Sides

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