Volatility

Volatility

Chapter 9

Risk Management and Financial Institutions 2e, Chapter 9, Copyright © John C. Hull 2009 1

Autocorrelations of Daily S&P 500 Returns for Lags 1 through 100

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

Autocorrelation of Squared Daily S&P500 Returns for Lags 1 through 100

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).

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

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.

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


252

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


VIX Index: A Measure of the Implied Volatility of the S&P 500 (Figure 9.1, page 178)


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


Normal and Heavy-Tailed Distribution


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)


n n ii

m

n ii

m

mu u

um

u

2 2

1

1

1

1

1

( )

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


n n ii

m

mu2 2

1

1

Weighting Scheme

Instead of assigning equal weights to the observations we can set


n i n ii

m

ii

m

u2 2

1

1

1

where

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


21

21

2 )1( nnn u

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


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


21

21

2 nnLn uV

GARCH (1,1) continued

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

and


1LV

21

21

2 nnn u

Example

Suppose

The long-run variance rate is 0.0002 so that the long-run volatility per day is 1.4%


n n nu21

21

20 000002 013 086 . . .

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


0 000002 013 0 0001 0 86 0 000256 0 00023336. . . . . .

GARCH (p,q)


n i n i jj

q

i

p

n ju2 2

11

2

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


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


Maximum Likelihood Methods

In maximum likelihood methods we choose parameters that maximize the likelihood of the observations occurring


Calculations for Yen Exchange Rate Data (Table 9.4, page 192)


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

Daily Volatility of Yen: 1988-1997


Correlations

Chapter 10


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


Covariance

The covariance on day n is

E(xnyn)−E(xn)E(yn)

It is usually approximated as E(xnyn)


Monitoring Correlation continued

EWMA:

GARCH(1,1)

111 )1(covcov nnnn yx

111 covcov nnnn yx


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


Chapter 8

Risk Management and Financial Institutions 2e, Chapter 8, Copyright © John C. Hull 2009

The VaR Measure

33

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?”


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


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?”


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


Distributions with the Same VaR but Different Expected Shortfalls


VaR

VaR

38

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


10

39

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

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(

)!(!

!


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