Jumps in High Volatility Environments and Extreme Value Theory

Report
Jumps in High Volatility Environments
and Extreme Value Theory
Abhinay Sawant
March 4, 2009
Economics 201FS
Overview

Jumps in High Volatility Last Environment: Updated method from
previous time

Extreme Value Theory: Read current literature on topic but haven’t
decided how to apply it to data
Set-Up of Test

Pre-Lehman Period: All data through September 12, 2008

Post-Lehman Period: September 15, 2008 – January 4, 2009 (78 days)

Difference of Sample Means t Test:
t

X 2  X1
s12 s22

n1 n2
Assumption: t distribution is approximately normal for high sample size
Results: Financial Stocks
Company Name
zQP
zTP
Bank of America (BAC)
1.631
1.669
Bank of New York (BK)
–0.483
-0.578
0.354
0.459
Capital One Financial (COF)
-0.603
0.426
Goldman Sachs (GS)
-0.633
-0.568
1.727
1.748
-0.255
-0.319
Regions Financial Corp. (RF)
1.650
1.653
U.S. Bancorp (USB)
0.055
0.048
Wells Fargo (WFC)
0.944
1.012
Citigroup (C)
JPMorgan Chase (JPM)
Morgan Stanley (MS)
Results: Non-Financial Stocks
Company Name
zQP
zTP
Cisco (CSCO)
-1.875
-1.797
Intel (INTC)
-1.472
-1.476
Hewlett-Packard (HPQ)
-1.547
-1.477
Pfizer (PFE)
-0.053
0.015
Merck (MRK)
-0.086
-0.065
Johnson & Johnson (JNJ)
-0.161
-0.222
Wal-Mart (WMT)
-2.193
-2.070
Procter & Gamble (PG)
0.734
0.865
PepsiCo (PEP)
0.051
0.096
Lockheed Martin (LMT)
-0.860
-0.720
Caterpillar (CAT)
-4.385
-4.246
Honeywell (HON)
-2.991
-2.760
Jumps in High Volatility Environments

Regression of Realized Volatility on Z-Scores

Comparisons across Industries
Extreme Value Theory
Extreme Value Theory
Extreme Value Theory: Background Theory

General Pareto Distribution (GPD) describes values of x above the
threshold u:
 

F ( x | x  u)  1  1   ( x  u) 
 


1

 0

ξ and β are to be estimated using Maximum Likelihood Estimation

Hill’s Estimator:

1 k 1
 
  ln X i ,n  ln X k ,n
k  1 i 1
Extreme Value Theory: Background Theory

Extreme Value Theory allows for the estimation of risk metrics:

 



  n
VaRp  u    
 p   1

   Nu 



VaRp ˆ    u

ES p 
1
1  ˆ
Extreme Value Theory: Current Literature

High-frequency tail estimation has efficiency benefits since intraday
data allows for observable extremes (Cotter and Longin, 2004)

Margin setting based on closing prices alone underestimates the
risk, when compared with intraday data (Cotter and Longin, 2004)

High-frequency volatility estimator based on EVT provides superior
forecasting abilities when compared to GARCH discrete time
models (Bali and Weinbaum, 2006)
Further Direction

Does the financial crisis period offer extreme values of returns and
can GPD model adequately estimate these values of returns?

At high frequency, do the extreme intraday returns represent jumps
or rapid movement in prices?

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