VolatilityChapter 10RiskManagementandFinanciallnstitutions3e,Chapter10,CopyrightJohnC.Hull2012
Volatility Chapter 10 Risk Management and Financial Institutions 3e, Chapter 10, Copyright © John C. Hull 2012 1
Definition of VolatilitySuppose that S, is the value of a variable onday i. The volatility per day is the standarddeviation of ln(S, /Si-1) Normally days when markets are closed areignored in volatility calculations (see BusinessSnapshot 10.1, page 207) The volatility per year is /252 times the dailyvolatilityVariance rate is the square of volatility2RiskManagementandFinancialInstitutions3e,Chapter10,CopyrightJohnC.Hull 2012
Definition of Volatility ⚫ Suppose that Si is the value of a variable on day i. The volatility per day is the standard deviation of ln(Si /Si-1 ) ⚫ Normally days when markets are closed are ignored in volatility calculations (see Business Snapshot 10.1, page 207) ⚫ The volatility per year is times the daily volatility ⚫ Variance rate is the square of volatility Risk Management and Financial Institutions 3e, Chapter 10, Copyright © John C. Hull 2012 2 252
ImpliedVolatilities Of the variables needed to price an optionthe one that cannot be observed directly isvolatility We can therefore imply volatilities frommarket prices and viceversa3RiskManagementandFinancialInstitutions3e,Chapter10,CopyrightJohnC.Hull2012
Implied Volatilities ⚫ Of the variables needed to price an option the one that cannot be observed directly is volatility ⚫ We can therefore imply volatilities from market prices and vice versa Risk Management and Financial Institutions 3e, Chapter 10, Copyright © John C. Hull 2012 3
ViXIndex:AMeasureoftheImpliedVolatility of the S&P500 (Figure10.1, page208)9080706050403020W100Jan-2005Jan-2008Jan-2010Jan-2004Jan-2006Jan-2007Jan-2009Jan-2011RiskManagementandFinancialInstitutions3e,Chapter10,CopyrightJohnC.Hull20124
VIX Index: A Measure of the Implied Volatility of the S&P 500 (Figure 10.1, page 208) Risk Management and Financial Institutions 3e, Chapter 10, Copyright © John C. Hull 2012 4
AreDailyChangesinExchangeRatesNormally Distributed? Table 10.1, page 209Real World (%)Normal Model (%)>1 SD31.7325.045.274.55>2SD1.340.27>3SD0.290.01>4SD>5SD0.080.000.030.00>6SD5RiskManagementandFinancial Institutions3eChapter10,CopyrightJohnC.Hull2012
Are Daily Changes in Exchange Rates Normally Distributed? Table 10.1, page 209 Real World (%) Normal Model (%) >1 SD 25.04 31.73 >2SD 5.27 4.55 >3SD 1.34 0.27 >4SD 0.29 0.01 >5SD 0.08 0.00 >6SD 0.03 0.00 Risk Management and Financial Institutions 3e, Chapter 10, Copyright © John C. Hull 2012 5
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 = − − = − = − = ( )