Download Extreme Value Theory

Extreme Value Theory for
High Frequency Financial Data
Abhinay Sawant
April 20, 2009
Economics 201FS
Background: Value at Risk

An important topic in financial risk management is the measurement
of Value at Risk (VAR). For example, if the 10-Day VAR of a $10
million portfolio is estimated to be $1 million, then we would interpret
this is as: “We are 99% certain that we will not lose more than $1
million in the next 10 days.”

VAR is typically used as a gauge of exposure to financial risk for
firms and regulators to determine the amount of capital to set aside
to cover unexpected losses. VAR is also used to set risk limits by
firms on individual traders.

Although practitioners typically calculate VAR based on the normal
distribution, experience has shown that asset returns have fatter
tails and asymmetry in tail distributions. An alternative approach that
takes these issues into consideration is Extreme Value Theory.
Background: Extreme Value Theory

Under order statistics, one normally imagines that the distribution of
the maximum return max{r1, …, rn} converges to either 0 or 1:
Fr max ( x )  [ F ( x )] n

Given independent and identically distributed log returns {r1, …, rn}
the maximum follows the Generalized Extreme Value Distribution:
F ( x) 
Motivation: High-Frequency Data

There is limited literature on the use of the high-frequency data for
Extreme Value Theory VAR estimation. This paper aims to
implement EVT with actual stock price data and realize the benefits
of using data at high-frequency.

Potential Benefits to high-frequency data is:

Better Estimation: More high-frequency returns allow for better chance
of getting extreme returns.

Intraday VAR: Rather than setting aside capital once a day, adjusting
the VAR depending on time of day may minimize capital.

Use of Recent Data: Some EVT methods such as Block Maxima
require large amounts of data. High-frequency data allows more financial
data from more recent years.
Methods: Block Maxima Estimation

Divide the time series {r1, …, rn} into blocks of size n and pull out the
maximum of each block:
{r1, …, rn}
r1 *
{r1, …, rn}
{r1, …, rn}
r2 *
…
r3 *

Use the maximums of each block to estimate the parameters of the
GEV distribution (maximum likelihood).

In order to determine the value at risk:
P(rn *  VAR)  P(rt  VAR)
n
Methods: Standardization of Data

One of the assumptions behind GEV estimation is that the data is
independent and identically distributed. In order to meet this
assumption, we divide the log return by the realized variance
(measured by high-frequency and sampling) over the same time
period:
One-Day
Return
Half-Day
Return
RV (10 min)
Half-Day
Return
RV (10 min)
RV (10 min)
Methods: Predicting VAR

GEV distribution is fitted to standardized data, and VAR is
determined in standardized terms. The VAR in real terms is then
determined by un-standardizing the data by multiplying by
tomorrow’s realized volatility:

Tomorrow’s realized volatility is either:

Known: Tests the validity of the extreme value distribution for VAR.

Forecasted: Using HAR-RV regression:
RVt 1  0   D RVt  W RVt 5   M RVt 22
Methods: Testing the VAR Model

Binomial Distribution: Given a VAR model with probability p of
exceedance of n trials, then the number of exceedances should be
binomially distributed as Bin(n,p). Therefore, a p-value is calculated
for a two-sided test exceedances compared to expected value np.

Kupiec Statistic: The p-value for a powerful two-tailed test
proposed by Kupiec is also calculated.

 2  ln (1  p )n m  p m   2  ln 1  m / n 
m = exceedances
n = trials
n m

 m / n  ~ 12
m
p = probability of exceedance
Model: Testing the VAR Model

Bunching: The following is the test statistic proposed by
Christofferson. A low p-value suggests bunching is likely.
u01
 2  ln[( 1   ) u00  u10   u01  u11 ]  2  ln[( 1   01 ) u00  01
(1   11 ) u10  11u11 ~  12


u 01  u11
u 00  u 01  u10  u11
 01 
u 01
u 00  u 01
 11
u11

u10  u11
Mean VAR: Since the level of VAR sets the level of capital allocated
for unexpected loss, a good VAR test would minimize the amount of
capital allocated.
Results: EVT from 1-Day Returns
(1912 Trials, Known Future RV)
Left Tail
VAR
Breaks
Pct% Binomial
Kupiec
Bunch AvgVAR
2.5%
64
3.33%
0.0208
0.0262
0.3731
-2.97%
1.0%
20
1.04%
0.7415
0.8572
0.5164
-3.42%
0.5%
12
0.62%
0.3441
0.4559
0.6977
-3.69%
0.1%
2
0.10%
0.6039
0.9548
0.0042
-4.17%
Pct% Binomial
Kupiec
Bunch AvgVAR
Right Tail
VAR
Breaks
2.5%
51
2.65%
0.5996
0.6668
0.0952
2.87%
1.0%
19
0.99%
0.9172
0.9615
0.5377
3.45%
0.5%
11
0.57%
0.5178
0.6592
0.7218
3.81%
0.1%
1
0.05%
0.8554
0.4638
0.9742
4.50%
Results: EVT from 1-Day Returns
(1912 Trials, Forecasted RV)
Left Tail
VAR
Breaks
Pct% Binomial
Kupiec
Bunch AvgVAR
2.5%
49
2.55%
0.8122
0.8871
0.0002
-3.34%
1.0%
23
1.20%
0.3237
0.3993
0.0292
-3.84%
0.5%
19
0.99%
0.0043
0.0074
0.0128
-4.14%
0.1%
12
0.62%
0.0000
0.0000
0.0622
-4.68%
Pct% Binomial
Kupiec
Bunch AvgVAR
Right Tail
VAR
Breaks
2.5%
37
1.93%
0.1154
0.0934
0.1997
3.22%
1.0%
17
0.88%
0.7190
0.6052
0.5816
3.88%
0.5%
13
0.68%
0.2158
0.2975
0.6738
4.29%
0.1%
2
0.10%
0.6039
0.9548
0.9485
5.06%
HAR-RV Estimation
Bunching?
Results: Intraday VAR
(Known Future RV, Right Tail, 0.5% VAR)
DailyRet
Pct% Binomial
Kupiec
Bunch AvgVAR
1
0.57%
0.5178
0.6592
0.7218
3.81%
2
0.60%
0.3248
0.4005
0.9742
2.53%
4
0.56%
0.4061
0.4674
0.3731
1.65%
8
0.60%
0.2278
0.2741
0.516
1.02%
VAR Estimation, 1x Day
VAR Estimation, 2x Day
VAR Estimation, 16x Day
Parameter Estimation, 1x Day
Parameter Estimation, 2x Day
Parameter Estimation, 16x Day
Results: Parameter Estimation
(Right Tail, 0.5% VAR)
DailyRet
VAR
ξ
α
β
1
2.30
-0.20
0.35
2.40
2
2.20
0.03
0.25
2.20
4
2.10
-0.23
0.20
2.10
8
1.95
-0.23
0.13
1.90
16
1.78
-0.30
0.90
1.80
Conclusions

Given sufficient knowledge about future realized volatility, extreme
value theory appears to provide a good measurement of VAR.

Volatility forecasting problems appear to cause more issues with the
left tail than the right.

Given sufficient knowledge about future realized volatility, intraday
VAR seems plausible and a good method to minimize the use of
capital.
Final Computations

Forecasting daily VAR from intraday VAR using the horizon rule:
VARL  L  VAR1
ˆ

Realized volatility forecasting for intraday volatility

Testing the other two EVT methods: (1) Peaks Over Thresholds, (2)
Nonparametric Hill Estimation