Estimating Volatilities and Correlations. Following Options, Futures, and Other Derivatives, 5th edition by John C. Hull. Chapter 17. m 2 2.

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1 Estiatig Volatilities ad Correlatios Followig Optios, Futures, ad Other Derivatives, 5th editio by Joh C. Hull Chapter 17 Stadard Approach to Estiatig Volatility Defie as the volatility per day betwee day -1 ad day, as estiated at ed of day -1 Defie S i as the value of arket variable at ed of day i Defie u i = l(s i /S i-1 ) = ( u i u ) 1 i= 1 1 u = 1 i = 1 u i Page 1 1

2 Siplificatios Usually Made Defie u i as (S i -S i-1 )/S i-1 Assue that the ea value of u i is zero Replace -1 by This gives 1 = u i = 1 i Statistical properties ^ 4 ERQM = E ( )^ = If prices follow a logoral process Exaple: with 50 historical price, the relative error o variace estiate is still about 0% Page

the 0% error Volatility based o Ope/Close/High/Low Source: Malik

3 Efficiet estiators with Highs ad Lows Parkiso Rogers Satchell Gara Klass Exeple: with =50, relative error is about 10%, to be copared with the 0% error Volatility based o Ope/Close/High/Low Source: Malik Magdo-Isail, Air F. Atiya Volatility Estiatio Usig High, Low, ad Close Data Page 3 3

4 Difficulty i evaluatig ad coparig volatility odels is due to the fact that volatility is ot directly observable. A approach to solve this particular proble is to copare the volatility forecasts, with squared returs Weightig Schee Istead of assigig equal weights to the observatios we ca set i = 1 = α i iu = 1 i α i where = 1 Page 4 4

5 ARCH() Model I a ARCH() odel we also assig soe weight to the log-ru variace rate, V L : γ + = γv where i = 1 α L i + = 1 α i = iu 1 i EWMA Model I a expoetially weighted ovig average odel, the weights assiged to the u declie expoetially as we ove back through tie This leads to = λ 1 + ( 1 λ) u 1 Page 5 5

6 Attractios of EWMA Relatively little data eeds to be stored We eed oly reeber the curret estiate of the variace rate ad the ost recet observatio o the arket variable Tracks volatility chages RiskMetrics uses λ = 0.94 for daily volatility forecastig GARCH (1,1) I GARCH (1,1) we assig soe weight to the log-ru average variace rate = γv L + αu 1 + β 1 Sice weights ust su to 1 γ + α + β =1 Page 6 6

7 GARCH (1,1) cotiued Settig ω = γv the GARCH (1,1) odel is = ω + α u 1 + β 1 ad V L ω = 1 α β Exaple Suppose = u The log-ru variace rate is so that the logru volatility per day is 1.4% Page 7 7

8 Exaple cotiued Suppose that the curret estiate of the volatility is 1.6% per day ad the ost recet percetage chage i the arket variable is 1%. The ew variace rate is = The ew volatility is 1.53% per day GARCH (p,q) p = ω + α u + β i i j i = 1 j = 1 q j Page 8 8

9 Other Models We ca desig GARCH odels so that the weight give to u i depeds o whether u i is positive or egative We do ot have to assue that the coditioal distributio is oral (e.g. Studet residuals) Variace Targetig Oe way of ipleetig GARCH(1,1) that icreases stability is by usig variace targetig We set the log-ru average volatility equal to the saple variace Oly two other paraeters the have to be estiated Page 9 9

10 Maxiu Likelihood Methods I axiu likelihood ethods we choose paraeters that axiize the likelihood of the observatios occurrig Exaple 1 We observe that a certai evet happes oe tie i te trials. What is our estiate of the proportio of the tie, p, that it happes? The probability of the outcoe is 10p(1 p) We axiize this to obtai a axiu likelihood estiate: p=0.1 9 Page 10 10

11 Exaple Estiate the variace of observatios fro a oral distributio with ea zero Maxiize: or: This gives: i = 1 1 u i exp πv v u i l( v) i 1 v = 1 v = ui i = 1 Applicatio to GARCH We choose paraeters that axiize ui l( vi ) i= 1 vi Page 11 11

12 How Good is the Model? The Ljug-Box statistic tests for autocorrelatio We copare the autocorrelatio of the u i with the autocorrelatio of the u i / i Is i a predictor of u i+1? Perfor a regressio of i versus u i+1 Forecastig Future Volatility A few lies of algebra shows that E[ + k ] = V L + ( α + β) k ( V L ) The variace rate for a optio expirig o day is 1 1 E k k = 0 [ + ] Page 1 1

13 Correlatios Defie u i =(U i -U i-1 )/U i-1 ad v i =(V i -V i-1 )/V i-1 Also u, : daily vol of U calculated o day -1 v, : daily vol of V calculated o day -1 cov : covariace calculated o day -1 The correlatio is cov /( u, v, ) Correlatios cotiued Uder EWMA cov = (1 β) u -1 v -1 +β cov -1 Uder GARCH (1,1) cov = ω + α u -1 v -1 +β cov -1 Page 13 13

A random variable is a variable whose value is a numerical outcome of a random phenomenon.

A random variable is a variable whose value is a numerical outcome of a random phenomenon. The Practice of Statistics, d ed ates, Moore, ad Stares Itroductio We are ofte more iterested i the umber of times a give outcome ca occur tha i the possible outcomes themselves For example, if we toss