The weights on past estimates of the volatility decrease exponentially as we
move back through time.
Example 10.5
(Exponentially decreasing weights)
.
The EWMA model gives
σ
2
n
-
1
=
λσ
2
n
-
2
+ (1
-
λ
)
r
2
n
-
2
4
MS4226 TST16/17B
Chapter 10
5
MS4226 TST16/17B
Chapter 10
City University of Hong Kong
Department of Management Sciences
MS4226 Risk Management Models
Semester B 2016/2017
Chapter 10 Summary
(a) If the returns per day arei.i.d.,V ar(RT) =TV ar(R)⇒σT=√Tσ(b) The law asserts thatP(V > x) =Kx-α(c) The estimate for the volatility per day is the sample standard deviation ofreturnsˆσ=sr,wheres2r=1n-1nsummationdisplayi=1(ri-¯r)2.(d) The EWMA model isσ2n=λσ2n-1+ (1-λ)r2n-1.
6
City University of Hong Kong
Department of Management Sciences
MS4226 Risk Management Models
Semester B 2016/2017
Chapter 10 Question1. The volatility of independent and identically distributed returns of an asset is2% per day. What is the standard deviation of the return in three days?2. The volatility of an asset is 25% per annum. Suppose that daily returns areindependent and identically distributed.(a) What is the standard deviation of the return in one trading day?(b) Assume that the daily returns have a normal distribution with zero mean.Find the 95% confidence limits for the return in one day.3. Why do traders assume 252 rather than 365 days in a year when using volatil-ities?4. The number of visitors to websites follows the power law such thatP(V > x) =Kx-2.Suppose that 1% of sites get 500 or more visitors per day.
