Heavy TailsDaily exchange rate changes are not normallydistributedThe distribution has heavier tails than the normaldistributionIt is more peaked than the normal distributionThis means that small changes and largechanges are more likely than the normaldistribution would suggestMany market variables have this propertyknown as excess kurtosisRiskManagementandFinancialInstitutions3e,Chapter10,CopyrightJohnC.Hull 20126
Heavy Tails ⚫ Daily exchange rate changes are not normally distributed ⚫ The distribution has heavier tails than the normal distribution ⚫ It is more peaked than the normal distribution ⚫ This means that small changes and large changes are more likely than the normal distribution would suggest ⚫ Many market variables have this property, known as excess kurtosis Risk Management and Financial Institutions 3e, Chapter 10, Copyright © John C. Hull 2012 6
Normal and Heavy-TailedDistribution-NormalHeavyTailed-4-20667RiskManagementandFinancialInstitutions3e,Chapter10,CopyrightJohnC.Hull20127
Normal and Heavy-Tailed Distribution Risk Management and Financial Institutions 3e, Chapter 10, Copyright © John C. Hull 2012 7
Alternatives to Normal Distributions:The Power Law (See page 211)Prob(v > x) = Kx-αThis seems to fit the behavior of thereturns on many market variables betterthan the normal distribution8RiskManagementandFinancialInstitutions3e,Chapter10,CopyrightJohnC.Hull2012
Alternatives to Normal Distributions: The Power Law (See page 211) Prob(v > x) = Kx-a This seems to fit the behavior of the returns on many market variables better than the normal distribution Risk Management and Financial Institutions 3e, Chapter 10, Copyright © John C. Hull 2012 8
Log-LogTestforExchangeRateData00.51.520In(x)-1-23[(x<alodu4-5-6-7.-8 -9-10RiskManagementandFinancialInstitutions3e,Chapter10,CopyrightJohnC.Hull20129
Log-Log Test for Exchange Rate Data Risk Management and Financial Institutions 3e, Chapter 10, Copyright © John C. Hull 2012 9
Standard Approach to EstimatingVolatility. Define on as the volatility per day betweenday n-1 and day n, as estimated at end of dayn-1 Define S, as the value of market variable atend of day iDefine u;= ln(S/Si-1)m1Z(n-i -u)2(um-i=1m1ZuUn-mi=110RiskManagementandFinancialInstitutions3e,Chapter10,CopyrightJohnC.Hull2012
Standard Approach to Estimating Volatility ⚫ Define sn as the volatility per day between day n-1 and day n, as estimated at end of day n-1 ⚫ Define Si as the value of market variable at end of day i ⚫ Define ui= ln(Si /Si-1 ) Risk Management and Financial Institutions 3e, Chapter 10, Copyright © John C. Hull 2012 10 s n n i i m n i i m m u u u m u 2 2 1 1 1 1 1 = − − = − = − = ( )