Volatility Chapter 9 Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Definition of VolatilitySuppose 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 9.1, page 177)The volatility per year is times the daily volatilityVariance rate is the square of volatility

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Implied VolatilitiesOf the variables needed to price an option the one that cannot be observed directly is volatilityWe can therefore imply volatilities from market prices and vice versaRisk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

VIX Index: A Measure of the Implied Volatility of the S&P 500 (Figure 9.1, page 178)Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Are Daily Changes in Exchange Rates Normally Distributed? Table 9.2, page 181Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Real World (%)Normal Model (%)>1 SD25.0431.73>2SD5.274.55>3SD1.340.27>4SD0.290.01>5SD0.080.00>6SD0.030.00

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Heavy TailsDaily exchange rate changes are not normally distributedThe distribution has heavier tails than the normal distributionIt is more peaked than the normal distributionThis means that small changes and large changes are more likely than the normal distribution would suggestMany market variables have this property, known as excess kurtosis Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Normal and Heavy-Tailed DistributionRisk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Alternatives to Normal Distributions: The Power Law (See page 182)

Prob(v > x) = Kx-a

This seems to fit the behavior of the returns on many market variables better than the normal distributionRisk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Log-Log Test for Exchange Rate Data Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Standard Approach to Estimating VolatilityDefine sn as the volatility per day between day n-1 and day n, as estimated at end of day n-1Define Si as the value of market variable at end of day iDefine ui= ln(Si/Si-1)

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Simplifications Usually Made in Risk ManagementDefine ui as (SiSi-1)/Si-1Assume that the mean value of ui is zeroReplace m-1 by m

This gives

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Weighting SchemeInstead of assigning equal weights to the observations we can set

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

ARCH(m) ModelIn an ARCH(m) model we also assign some weight to the long-run variance rate, VL:

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

EWMA Model (page 186)In an exponentially weighted moving average model, the weights assigned to the u2 decline exponentially as we move back through timeThis leads to

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Attractions of EWMARelatively little data needs to be storedWe need only remember the current estimate of the variance rate and the most recent observation on the market variableTracks volatility changesRiskMetrics uses l = 0.94 for daily volatility forecastingRisk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

GARCH (1,1), page 188In GARCH (1,1) we assign some weight to the long-run average variance rate

Since weights must sum to 1g + a + b =1

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

GARCH (1,1) continuedSetting w = gVL the GARCH (1,1) model is

and

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

ExampleSuppose

The long-run variance rate is 0.0002 so that the long-run volatility per day is 1.4%Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Example continuedSuppose that the current estimate of the volatility is 1.6% per day and the most recent percentage change in the market variable is 1%.The new variance rate is

The new volatility is 1.53% per dayRisk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

GARCH (p,q) Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Other ModelsMany other GARCH models have been proposedFor example, we can design a GARCH models so that the weight given to ui2 depends on whether ui is positive or negativeRisk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Variance TargetingOne way of implementing GARCH(1,1) that increases stability is by using variance targetingWe set the long-run average volatility equal to the sample varianceOnly two other parameters then have to be estimatedRisk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Maximum Likelihood MethodsIn maximum likelihood methods we choose parameters that maximize the likelihood of the observations occurringRisk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Example 1 (page 190)We observe that a certain event happens one time in ten trials. What is our estimate of the proportion of the time, p, that it happens?The probability of the outcome is

We maximize this to obtain a maximum likelihood estimate: p=0.1

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Example 2 (page 190-191)Estimate the variance of observations from a normal distribution with mean zeroRisk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Application to GARCH (1,1)We choose parameters that maximize

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Calculations for Yen Exchange Rate Data (Table 9.4, page 192) Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

DaySiuivi =si2-ln vi-ui2/vi10.00772820.0077790.00659930.007746-0.0042420.000043559.628340.0078160.0090370.000041988.132950.0078370.0026870.000044559.8568.24230.0084950.0001440.000084179.382422063.5833

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Daily Volatility of Yen: 1988-1997Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Forecasting Future Volatility (Equation 9.14, page 195)A few lines of algebra shows that

To estimate the volatility for an option lasting T days we must integrate this from 0 to T

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Forecasting Future Volatility contThe volatility per year for an option lasting T days is

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

Volatility Term Structures (Equation 9.16, page 197)The GARCH (1,1) model allows us to predict volatility term structures changesWhen s(0) changes by Ds(0), GARCH (1,1) predicts that s(T) changes by

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009*

Risk Management and Financial Institutions 2e, Chapter 9, Copyright John C. Hull 2009

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