MS4226 Chapter 10 - City University of Hong Kong Department of Management Sciences MS4226 Risk Management Models Semester B 2016\/2017 Chapter 10

MS4226 Chapter 10 - City University of Hong Kong Department...

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City University of Hong Kong Department of Management Sciences MS4226 Risk Management Models Semester B 2016/2017 Chapter 10: Volatility 1 Introduction It 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 common to 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 track of variations in volatility through time. 2 Definition of Volatility Volatility is the standard deviation of continuously compounded return R per day (or per year) σ = radicalbig V ar ( R ) . If the returns per day are independent and identically distributed (with the same variance), the variance of returns over T days is T times the variance of returns over one day. V ar ( R T ) = TV ar ( R ) σ T = This means that the standard deviation of returns over T days is T times the standard deviation of returns over one day. Example 10.1. An asset price is $ 60 and its volatility is 2% per day. Suppose returns are independent and identically distributed with zero mean. The volatility of returns over five days is = 5 × 0 . 02 = 0 . 04472 . If the returns have a normal distribution, we can be 95% certain that the five-day return is bracketleftBig 0 ± 1 . 96 × parenleftBig 5 × 0 . 02 parenrightBigbracketrightBig . We can also be 95% certain that the asset price is bracketleftBig 60 e - 1 . 96 × ( 5 × 0 . 02 ) , 60 e 1 . 96 × ( 5 × 0 . 02 ) bracketrightBig = [54 . 96 , 65 . 50] . 1
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MS4226 TST16/17B Chapter 10 2.1 Business Days versus Calendar Days Volatility is much higher on business days than on non-business days. We may ignore weekends and public holidays when calculating and using volatilities. The usual assumption is that there are 252 days per year. Example 10.2. If the returns per day are independent and identically distributed, then σ year = 252 σ day 3 Do Returns Have a Normal Distribution In practice, most financial variables are more likely to experience big moves than the normal distribution would suggest. Table 1 shows the results of a test of normality using daily movements in 12 different exchange rates over a 10-year period.
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