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Transcript of Volatility
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Volatility
Chapter 9
Risk Management and Financial Institutions 2e, Chapter 9, Copyright © John C. Hull 2009 1
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Autocorrelations of Daily S&P 500 Returns for Lags 1 through 100
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Stylized Factzero autocorrelation
Are returns predictable on a daily frequency?
In major markets, daily returns have little autocorrelation. We can write
Returns are almost impossible to predict from their own past.
1,2,3,...for 0, 11 tt RRCorr
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Autocorrelation of Squared Daily S&P500 Returns for Lags 1 through 100
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Stylized FactSD dominates mean
The standard deviation of returns completely dominates the mean of returns at short horizons such as daily.
It is typically not possible to statistically reject a zero mean return.
Our S&P500 data has a daily mean of 0.035% and a daily standard deviation of 1.27% (or ~20% annualized).
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Stylized FactVariance dependence
Variance measured for example by squared returns, displays positive correlation with its own past.
This is most evident at short horizons such as daily or weekly.
Figure shows the autocorrelation in squared returns for the S&P500 data, that is
Models that can capture this variance dependence will be presented!
smallfor ,0),( 21
21 tt RRCorr
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Stylized Factschanging correlations
Correlation between assets appears to be time varying.
Importantly, the correlation between assets appear to increase in highly volatile down-markets and extremely so during market crashes.
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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 9.1, page 177)
The volatility per year is times the daily volatility
Variance rate is the square of volatility
Risk Management and Financial Institutions 2e, Chapter 9, Copyright © John C. Hull 2009 8
252
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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 2e, Chapter 9, Copyright © John C. Hull 2009 9
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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 10
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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 2e, Chapter 9, Copyright © John C. Hull 2009 11
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Normal and Heavy-Tailed Distribution
Risk Management and Financial Institutions 2e, Chapter 9, Copyright © John C. Hull 2009 12
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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 2e, Chapter 9, Copyright © John C. Hull 2009 13
n n ii
m
n ii
m
mu u
um
u
2 2
1
1
1
1
1
( )
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Simplifications Usually Made in Risk Management
Define ui as (Si−Si-1)/Si-1
Assume 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 14
n n ii
m
mu2 2
1
1
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Weighting Scheme
Instead of assigning equal weights to the observations we can set
Risk Management and Financial Institutions 2e, Chapter 9, Copyright © John C. Hull 2009 15
n i n ii
m
ii
m
u2 2
1
1
1
where
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EWMA Model (page 186)
In an exponentially weighted moving average model, the weights assigned to the u2 decline exponentially as we move back through time
This leads to
Risk Management and Financial Institutions 2e, Chapter 9, Copyright © John C. Hull 2009 16
21
21
2 )1( nnn u
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Attractions of EWMA
Relatively little data needs to be storedWe need only remember the current
estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changesRiskMetrics uses l = 0.94 for daily volatility
forecasting
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GARCH (1,1), page 188
In GARCH (1,1) we assign some weight to the long-run average variance rate
Since weights must sum to 1
+ + =1g a b
Risk Management and Financial Institutions 2e, Chapter 9, Copyright © John C. Hull 2009 18
21
21
2 nnLn uV
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GARCH (1,1) continued
Setting = w gVL the GARCH (1,1) model is
and
Risk Management and Financial Institutions 2e, Chapter 9, Copyright © John C. Hull 2009 19
1LV
21
21
2 nnn u
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Example
Suppose
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 20
n n nu21
21
20 000002 013 086 . . .
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Example continued
Suppose 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 day
Risk Management and Financial Institutions 2e, Chapter 9, Copyright © John C. Hull 2009 21
0 000002 013 0 0001 0 86 0 000256 0 00023336. . . . . .
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GARCH (p,q)
Risk Management and Financial Institutions 2e, Chapter 9, Copyright © John C. Hull 2009 22
n i n i jj
q
i
p
n ju2 2
11
2
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Other Models
Many other GARCH models have been proposed
For example, we can design a GARCH models so that the weight given to ui
2 depends on whether ui is positive or negative
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Variance Targeting
One way of implementing GARCH(1,1) that increases stability is by using variance targeting
We set the long-run average volatility equal to the sample variance
Only two other parameters then have to be estimated
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Maximum Likelihood Methods
In maximum likelihood methods we choose parameters that maximize the likelihood of the observations occurring
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Calculations for Yen Exchange Rate Data (Table 9.4, page 192)
Risk Management and Financial Institutions 2e, Chapter 9, Copyright © John C. Hull 2009 26
Day Si ui vi =si2 -ln vi-ui
2/vi
1 0.007728
2 0.007779 0.006599
3 0.007746 -0.004242 0.00004355 9.6283
4 0.007816 0.009037 0.00004198 8.1329
5 0.007837 0.002687 0.00004455 9.8568
….
2423 0.008495 0.000144 0.00008417 9.3824
22063.5833
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Daily Volatility of Yen: 1988-1997
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Correlations
Chapter 10
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Monitoring Correlation Between Two Variables X and Y
Define xi=(Xi−Xi-1)/Xi-1 and yi=(Yi−Yi-1)/Yi-1
Also
varx,n: daily variance of X calculated on day n-1
vary,n: daily variance of Y calculated on day n-1
covn: covariance calculated on day n-1
The correlation is
nynx
n
,, varvar
cov
Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 29
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Covariance
The covariance on day n is
E(xnyn)−E(xn)E(yn)
It is usually approximated as E(xnyn)
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Monitoring Correlation continued
EWMA:
GARCH(1,1)
111 )1(covcov nnnn yx
111 covcov nnnn yx
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Multivariate Normal Distribution
Fairly easy to handleA variance-covariance matrix defines
the variances of and correlations between variables
To be internally consistent a variance-covariance matrix must be positive semidefinite
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Chapter 8
Risk Management and Financial Institutions 2e, Chapter 8, Copyright © John C. Hull 2009
The VaR Measure
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The Question Being Asked in VaR
“What loss level is such that we are X% confident it will not be exceeded in N business days?”
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VaR and Regulatory Capital
Regulators base the capital they require banks to keep on VaR
The market-risk capital is k times the 10-day 99% VaR where k is at least 3.0
Under Basel II, capital for credit risk and operational risk is based on a one-year 99.9% VaR
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Advantages of VaR
It captures an important aspect of risk
in a single number It is easy to understand It asks the simple question: “How bad can
things get?”
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VaR vs. Expected Shortfall
VaR is the loss level that will not be exceeded with a specified probability
Expected shortfall is the expected loss given that the loss is greater than the VaR level (also called C-VaR and Tail Loss)
Two portfolios with the same VaR can have very different expected shortfalls
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Distributions with the Same VaR but Different Expected Shortfalls
Risk Management and Financial Institutions 2e, Chapter 8, Copyright © John C. Hull 2009
VaR
VaR
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Normal Distribution Assumption
The simplest assumption is that daily gains/losses are normally distributed and independent with mean zero
It is then easy to calculate VaR from the standard deviation (1-day VaR=2.33s)
The T-day VaR equals times the one-day VaR
Regulators allow banks to calculate the 10 day VaR as times the one-day VaR
T
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Choice of VaR Parameters
Time horizon should depend on how quickly portfolio can be unwound. Bank regulators in effect use 1-day for market risk and 1-year for credit/operational risk. Fund managers often use one month
Confidence level depends on objectives. Regulators use 99% for market risk and 99.9% for credit/operational risk.
A bank wanting to maintain a AA credit rating will often use confidence levels as high as 99.97% for internal calculations.
Quiz 8.12.Risk Management and Financial Institutions 2e, Chapter 8, Copyright © John C. Hull 2009 40
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Back-testing (page 169-171)
Back-testing a VaR calculation methodology involves looking at how often exceptions (loss > VaR) occur
Alternatives: a) compare VaR with actual change in portfolio value and b) compare VaR with change in portfolio value assuming no change in portfolio composition
Suppose that the theoretical probability of an exception is p (=1−X). The probability of m or more exceptions in n days is
knkn
mk
ppknk
n
)1(
)!(!
!
Risk Management and Financial Institutions 2e, Chapter 8, Copyright © John C. Hull 2009 41