3_volatility - VOLATILITY definition and other basics...

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VOLATILITY J. Wei, Department of Management, U of T 1 MGTD78 definition and other basics weighted moving average model GARCH(1, 1) maximum likelihood estimation of GARCH volatility forecasting and volatility term structure
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VOLATILITY J. Wei, Department of Management, U of T 2 MGTD78 Definition of volatility : standard deviation of the variable Variance rate (or variance) : square of volatility Calendar days or trading days (see Business Snapshot 9.1)? volatility seems to be generated by trading; we therefore count only trading days – 252 days in a year. Example: annual variance is 0.0625. Then annual volatility is 0.25; daily variance is 0.0625 / 252 = 0.000248; daily volatility is 0.25/sqrt(252) = 0.01575. Implied volatility: volatility implied from option prices (C71) VIX (Fig 9.1)
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VOLATILITY IN FINANCIAL VARIABLES J. Wei, Department of Management, U of T 3 MGTD78 Black-Scholes model assumes normal distribution of returns Financial returns are usually not normal. Example in Table 9.2 (10-year period, 12 exchange rates): % days on which realized FX is n standard deviations away from mean: real world(%) Normal distribution(%) > 1 SD 25.04 31.73 > 2 SD 5.27 4.55 > 3 SD 1.34 0.27 > 4 SD 0.29 0.01 > 5 SD 0.08 0.00 > 6 SD 0.03 0.00 large / extreme changes are more likely than predicted by normal: heavy-tail or fat-tail distributions (see Figure 9.1)
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POWER LAW J. Wei, Department of Management, U of T 4 MGTD78 Alternative to normal distribution when extreme values are considered It is an approximation, applicable to many situations The Power Law: when x is large, the value of variable v approximately satisfies Prob(v > x) = Kx α , where K and α are constants Application: use not-so-extreme values of x to estimate K and α , then calculate probability for extreme realizations Example: the probability that v > 6 is 0.004 when K = 10.368 and = 4. Then we can calculate the α following:
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STANDARD METHOD OF ESTIMATION J. Wei, Department of Management, U of T 5 MGTD78 Review of C71 (on day n, going back m days): where In risk management, we make three changes use percentage change (reflecting reality): assuming to be zero (for simplicity) replace m -1 by m (conforming to ML estimate) ( 29 u u 1 - m 1 σ m 1 i 2 i - n 2 n = - = u m 1 u , S S ln u m 1 i i n 1 - i i i = - = = and 1 - i 1 - i i i S S S u - = u u m 1 σ m 1 i 2 i - n 2 n = =
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WEIGHTING SCHEME J. Wei, Department of Management, U of T 6 MGTD78 Standard method assigns equal weight (1/m) to all observations Make sense to assign more weights to recent observations: The weighting scheme ( α i ) varies. The most common is to
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3_volatility - VOLATILITY definition and other basics...

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