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City University of Hong KongDepartment of Management SciencesMS4226 Risk Management ModelsSemester B 2016/2017Chapter 10: Volatility1IntroductionIt is important for a financial institution to monitor the volatilities of market vari-ables (such as interest rates, exchange rates, equity prices and commodity prices)on which the value of its portfolio depends. Volatility is first defined. It is commonto assume that percentage returns from market variables are normally distributed.The power law is presented as an alternative.The exponentially weighted moving average model is discussed. It has the dis-tinctive feature that volatility is not constant. This model attempts to keep trackof variations in volatility through time.2Definition of VolatilityVolatility is the standard deviation of continuously compounded returnRper day(or per year)σ=radicalbigV ar(R).If the returns per day are independent and identically distributed (with the samevariance), the variance of returns overTdays isTtimes the variance of returns overone day.V ar(RT) =TV ar(R)⇒σT=√TσThis means that the standard deviation of returns overTdays is√Ttimes thestandard deviation of returns over one day.Example 10.1.An asset price is$60 and its volatility is 2% per day.Supposereturns are independent and identically distributed with zero mean. The volatility ofreturns over five days is√Tσ=√5×0.02 = 0.04472.If the returns have a normal distribution, we can be 95% certain that the five-dayreturn isbracketleftBig0±1.96×parenleftBig√5×0.02parenrightBigbracketrightBig.We can also be 95% certain that the asset price isbracketleftBig60e-1.96×(√5×0.02),60e1.96×(√5×0.02)bracketrightBig= [54.96,65.50].1
MS4226 TST16/17BChapter 102.1Business Days versus Calendar DaysVolatility is much higher on business days than on non-business days.We mayignore weekends and public holidays when calculating and using volatilities.Theusual assumption is that there are 252 days per year.Example 10.2.If the returns per day are independent and identically distributed,thenσyear=√252σday3Do Returns Have a Normal DistributionIn practice, most financial variables are more likely to experience big moves than thenormal distribution would suggest. Table 1 shows the results of a test of normalityusing daily movements in 12 different exchange rates over a 10-year period.