L M is log-normally distributed, we obtain the following closed-form solution forP : 1
where the measure change is given by
2
! , so that the asset price dynamics becomes
(% & ) &
! is the Radon-Nykodym derivative to further change probability measure from
P to ! Pˆ such that Wt W~t 2ht
According to (21), %
&
( T D*
h T so that the associated probability measure can incur more extremal events.
Secondly, we estimate decay rates of P1 and P under three scaling scenarios in time and space. When 2
e * ( * is obtained. Therefore we get the following decay rates for the first two
moments under a small time scale T:
These results show that the decay rate of the second moment is twice of the decay rate of the first moment, which implies P12 GP2 as E goes to zero, so that an asymptotic zero variance rate for the importance
> ( > is obtained so that an asymptotic zero variance rate is
confirmed.
When maturity is short and default threshold is large, one can expect the increase of decay speed of these moments. Let D( *1 E and T (E for E //1, then one can obtain the following decay rate estimates
2 2
2 1 2
0 0
lim logP 2 lim log =P -1.
E E E E
&
> ( > Note that the scaling order is E2 in this scenario which is faster than E in previous two scenarios.
Table 1: Estimation of Extremal Event Probability and Its CVaR with Two Different Loss Thresholds. sampling, and variance reduction ratio, respectively. VR is defined as
. . of 2
3. Sample means and standard errors shown in parenthesis are reported in columns of BMC and IS.
Table 2: Comparisons of CVaR Approximation Based on Normal Approximation and Importance Sampling.
1 2 3 4 Column 2, two sets of CVaRs are approxmated. The first set of CVaR, reported in the Column 3, is an approximation obtained from the closed-form solution under a normality assumption. N. Approx.
denotes the normal approximation.
2. The second set of CVaR is estimated by the proposed importance sampling estimator, denoted by IS.
Sample means and their standard errors shown in parentheses are reported in the Column 4.
Table 3: Construction of Conditional Exceptions
conditional Day before
Current day No exception exception unconditional no exception T00 (T0
!
1*20"
T10(T1!
1*21"
T!
1*2"
exception T01(T0
! "
20 T11(T1! "
21 T! "
2total T 0 T 1 T T( ) 0 T1
Remark: Tij denotes the number of days in which state j occurred in one day while it was state i the previous day. 2i represents the probability of observing an exceedances conditional on state i the previous day.
Table 4: Descriptive Statistics of the three Foreign Exchange Rate Data Panel 1: Original Daily Data
JPY SGD CAD
Mininum 87.915 1.34665 0.9218 1st Quantile 107.525 1.575763 1.1683 Mean 114.5035 1.653406 1.334586 Median 115.265 1.6875 1.357725 3rd Quantitle 120.35 1.737975 1.506788 Maximum 147.41 1.85325 1.6147 Standard Deviation10.00708 0.119753 0.188313
Skewness 0.254427-0.7232320 0.295178 Kurtosis 0.435041-0.2974507 1.273715
Panel 2: Daily Return Data
JPY SGD CAD
Mininum -0.04565 -0.03523 -0.03737 1st Quantile -0.00384 -0.00184 -0.00301 Mean -0.00012 -5.9E-05 -9.3E-05
Median 0 -7.2E-05 -5E-05
3rd Quantitle 0.0040030.0016810.002774
Standard Deviation0.0072310.0036340.005528 Skewness -0.43005 -0.205780.043318 Kurtosis 3.67931 9.8711634.425934
Table 5: Backtesting Outcomes of Exchange Rate VaR Estimates
Table 6: Descriptive Data for S&P500 & VIX
S&P500 VIX S&P500 Return.
VIX Return.
Mininum 676.53 9.89 -0.0947 -0.29987 1st Quantile 1190.47 12.26 -0.00532 -0.03683 Mean 1248.455 21.26077 -0.00018 0.000435 Median 1275.55 15.59 0.000739 -0.0033 3rd Quantitle 1400.565 25.14 0.005524 0.032046 Maximum 1565.15 80.86 0.102457 0.496008 Standard Deviation 203.4692 13.0195 0.015136 0.066551 Skewness -0.85291 1.859021 -0.10722 0.574778 Kurtosis -0.0239 3.345993 9.331141 4.770059
Table 7: Backtesting Outcomes of S&P 500 VaR Estimates
RiskMetrics
Significance 1% Significance 5%
LRuc Reject VaR Model LRuc Reject VaR Model
LRind Reject VaR Model LRind Don't Reject VaR Model
LRcc Reject VaR Model LRcc Reject VaR Model
Historical Simulation
Significance 1% Significance 5%
LRuc Reject VaR Model LRuc Reject VaR Model
LRind Don't Reject VaR Model LRind Don't Reject VaR Model
LRcc Reject VaR Model LRcc Reject VaR Model
SV
Significance 1% Significance 5%
LRuc Don't Reject VaR Model LRuc Reject VaR Model LRind Don't Reject VaR Model LRind Don't Reject VaR Model
LRcc Don't Reject VaR Model LRcc Reject VaR Model GARCH(1,1)
Significance 1% Significance 5%
LRuc Reject VaR Model LRuc Reject VaR Model
LRind Don't Reject VaR Model LRind Reject VaR Model
LRcc Reject VaR Model LRcc Reject VaR Model
Figure 1: Time Series Plots of Exchange Rate Series Panel 1: Original Daily Data
Panel 2: Daily Return Series
Figure 2: Parameter Estimates of the Improved Estimation Procedure for Exchange Rate Series
Panel 1: JPY
Panel 2: SGD
Panel 3: CAD
Figure 3: Time Series Plot of VaR and CVaR Estimates by the Proposed Procedure for Foreign Exchange Rate
Panel 1: JPY
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009
-0.03-0.02-0.010.000.010.02
JPYVaR95 CVaR95
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009
-0.04-0.03-0.02-0.010.000.010.02
JPY VaR99 CVaR99
Panel 2:
SGD
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009
-0.020-0.015-0.010-0.0050.0000.0050.0100.015
SGD VaR95 CVaR95
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009
-0.025-0.020-0.015-0.010-0.0050.0000.0050.0100.015
SGD VaR99 CVaR99
Panel 3:
CAD
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009
-0.03-0.02-0.010.000.010.020.03
CAD VaR95 CVaR95
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009
-0.04-0.03-0.02-0.010.000.010.020.03
CAD VaR99 CVaR99
Figure 4: Time Series Plots of S&P 500 and Its VIX Panel 1: Original Daily Data
Panel 2: Daily Return Series
Figure 5: Time Series Plot of Rho between S&P500 and VIX
Figure 6: Parameter Estimates of the Improved Estimation Procedure for S&P 500 with VIX Series
Figure 7: Time Series Plot of VaR and CVaR Estimates by the Proposed Procedure for S&P 500
Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3
2005 2006 2007 2008 2009
-0.7-0.6-0.5-0.4-0.3-0.2-0.10.00.1
SP500 VaR95 CVaR95
Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3
2005 2006 2007 2008 2009
-0.4-0.3-0.2-0.10.00.1
SP500 VaR99 CVaR99
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