Download Forecasting Volatility in Financial Markets

Market Risk Management
using Stochastic Volatility
Models
The Case of European Energy Markets
1
Outline
 Preliminaries, markets, instruments and hedging
– Relevant risk, std, volatility + + + …..
– Markets, instruments and models +++
 Value at Risk, Expected Shortfall, Volatility and Covariances
 Stochastic Volatility Models
– Definition and Motivation
– Projection, estimation and re-projection
 The Nordpool and EEX Energy Markets
 SV model q parameters
 Assessment and empirical findings
 Market Risk Management
 SV-model forecasts and Risk Management
 One-day-ahead forecasts and Risk Management
 Summaries and Conclusions
2
Main Objectives
Forecasting Risk Management Measures
SV model forecasts of VaR, CVaR and Greek
letter densities
Conditional Moments Forecasts
One-day-ahead densities of VaR, CVaR and
Greek letters
Extreme value theory and VaR, CVaR and Greek
letter densities
3
Preliminaries
Portfolio Theory Basics for Investors
E  R p    i  E  ri 
N
i 1
 p2   i j i j   ri , rj 
Kapitalmarkedslinjen
0.13
0.12
0.11
0.1
0.09
0.08
0.07
0.06
0.05
0
0.02
0.04
0.06
Portefølje Standardavvik
0.08
0.1
0.12
4
Preliminaries
Portfolio Theory Basics (relevant risk measures):

Cov( R j , RM )   j   ( R j , RM )
E  R j   RF   E  RM   RF  
/j
/
Var ( RM ) 
M

Markedsavkastningslinjen
11.0000 %
10.0000 %
9.0000 %
Prosjekt
B
8.0000 %
Prosjekt
A
7.0000 %
6.0000 %
5.0000 %
4.0000 %
3.0000 %
2.0000 %
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
5
Preliminaries
The Relevant Risk issue:
The observed municipal and state ownerships often coupled with scale
ownership of many European energy corporations induce greater portion of
wealth invested and less diversification.
Risk adverse managers, stringent actions from regulators and diversification
issues, relative to a perfect world, risk assessment and management
methodologies as well as risk aggregation may be challenging and potentially of
great value to shareholders in the European energy markets.
That is: total versus i = (i * M) / M
theTraynor index versus the Sharpe index
6
Preliminaries
Financial Products and Markets
Financial Products / “Plain Vanilla” products
 Long and Short positions in Assets
 Forward Contracts / Future Contracts
 Swaps
 Options
7
Preliminaries
European Energy Markets and Activity/Liquidity for 2008 -2009 (annual reports)
Power Futures (TWh) Carbon Trading (tonnes) Spot Power (TWh)
2008
2009
2008
2009
2008
2009
Nord Pool Volume (TWh) 1437
1220
121731
45765
298
286
Transactions 158815
136030
6685
3792
70 %
72 %
EEX
Volume (TWh) 1165
1025
80084
23642
154
203
Transactions 128750
114250
4398
1959
54 %
56 %
Powernext Volume (TWh)
79
87
n/a
n/a
203.7
196.3
Transactions
n/a
n/a
n/a
n/a
n/a
n/a
APX/Endex Volume (TWh)
327
412
n/a
n/a
n/a
n/a
Transactions
36150
45900
n/a
n/a
n/a
n/a
Cleared OTC power (TWh)
2008
2009
1140
942
51575
40328
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
* On 1st January 2009, Powernext SA transferred its electricity spot market to EPEX Spot SE and
on 1st September 2009 EEX Power Spot merged with EPEX Spot.
* On 1st April 2009, the Powernext SA futures activity was entrusted to EEX Power Derivatives AG.
8
Preliminaries
Financial Products and Positions
Hedging Positions for plain Assets
Profit
Long position
Asset
Long Positions
Payoff:
St – S0
0
S0
Loss
Underlying
asset (St)
Short Positions
Payoff:
S0 – St
Short position
Asset
9
Preliminaries
Financial Products and Positions
Hedging Positions for plain Forward/Future Products
Profit
Long position
Forward/Future
Long Positions
Payoff:
St – K
0
K
Loss
Underlying
asset (St)
Short Positions
Payoff:
K – St
Short position
Forward/Future
10
Preliminaries
Financial Products and Positions
Hedging Positions for plain buying (long) positions in Call/Put options
Profit
Buying a put
position
Buying a
Call position
Call position
Payoff:
Max(0;St – K)-c
0
K
Underlying
asset (St)
Put Positions
Payoff:
Max(0;K – St)-p
Loss
11
Preliminaries
Financial Products and Positions
Hedging Positions for plain selling (short) positions in Call/Put options
Profit
Call position
Payoff:
-Max(0;St – K)+c
0
K
Loss
Selling a put
position
Underlying
asset (St)
Put Positions
Payoff:
-Max(0;K – St)+p
Selling a
Call position
12
Preliminaries
Management of Portfolio Exposures: Greek Letters

S
The sensitivity of the portfolios value to the price of the
underlying asset:
Delta 
The rate of change of the portfolio’s delta with respect to
the price of the underlying asset:
 2
Gamma()  2
S
The rate of change of the value of the portfolio with
respect to the volatility of the underlying asset:
Vega( ) 
The rate of change of the value of the portfolio with
respect to the passage of time (time decay):
Theta() 
The rate of change of the value of the portfolio with
respect to the level of interest rates:
Rho( ) 



T

i
13
Preliminaries
Calculation of the GREEK LETTERS
Taylor Series Expansion on a single market variable S (volatility and interest rates
are assumed constant)


1  2
1  2
 2
2
2
 
 S 
 t   2  S   2  t 
 S  t  ......
S
t
2 S
2 t
S t
For a delta neutral portfolio, the first term on the RHS of the equation is zero
(ignoring terms of higher order than t) (quadratic relationship between S and ):
    t    S 2
When volatility is uncertain:



1  2
1  2
2
 
 S 
  
 t   2  S   2   2 
S

t
2 S
2 
1  2
 2
2
 2  t 
 S  t  ......
2 t
S t
Delta hedging eliminates the first term. Second term is eliminated making the
portfolio Vega neutral. Third term is non-stochastic. Fourth term is eliminated by
making the portfolio Gamma neutral.
14
Stylized facts about volatility
Definition of volatility ()
The standard deviation of the return (rt) provided by the variable per unit time when
the return is expressed using continuous compounding.
S 
rt  ln  T 
 S0 
= return in time T expressed with continuous compounding
When T is small it follow that  T is approximately equal to the standard deviation of
the percentage change in the market variable in time T.
Based on Fama (1965); French (1980) & French and Roll (1986) show that volatility is
caused by trading itself  using trading days ignoring days when the exchange is closed.
 T = 1 ~ 252 trading days per year
17
Stylized facts about volatility
Fat tails of asset returns (leptokurtosis)
When the distribution of energy market series are compared with the normal distribution,
fatter tails are observed. Moreover, we also observe too many observations around the
mean. Too little at one std dev. Third moment (≠0) and fourth moment (≠3).
18
Stylized facts about volatility
An alternative to Normal Price Change distributions in Energy Markets
The power law asserts that, for may variables that are encountered in practice, it is approximately true that the value
 of the variable has the property that, when x is large:
Prob(  x)  Kxa
where K and a are constants.
Rewriting using the natural logarithm:
ln Prob(  x)  ln K  a ln x
A quick test can now be done for weekly and yearly price changes at NASDAQ OMX energy market. We plot
ln Prob(  x)against ln x.
Power Law for Nord Pool/EEX Front Week/Month Swap Contracts
-2
-3
-4
ln(prob(v < x))

The logarithm of the
probability is approx.
linearly dependent
on ln x for x >3
showing that the
power law holds.
-5
-6
-7
-8
NP-Front-Week
NP-Front-Month
EEX Front Month (base load)
EEX Front Month (peak load)
19
Stylized facts about volatility
Extreme Value Theory* Application for a Forward contract at NASDAQ OMX
Equivalence to the Power Law (next slide)
Total number of daily price change observations n = 2809, ranging from -12.62% to 16.35%. For the extreme value
theory we consider the left tail of the distribution of returns.
u = -4 % (a value close to the 95% percentile of the distribution). This means that we have nu=31 observations less
than u.
1
We maximize the log-likelihood function:
Using the estimates (optimized):

1
ln 


  ( xi  u ) 
 1 




 1

  0.2013 and   2.9096




Calculation of VaR:
The probability that x will be less than 15% is:
31 
0.15  0,10 
 1  

2809 



1

 0.0025
The value of one-day 99% VaR for a portfolio where NOK 1 million is invested in the contract is NOK 1 million times:
VaR  31



 2809

1

0.99

1


 31
  0.102897
That is, VaR = 1 million NOK * 0.102987= NOK 102,897
20
Stylized facts about volatility
Volatility Clustering
Refer to the observation of large movements of price changes are being followed by
large movements. That is, persistence of shocks.
NP Front Week: Projected and Moving Average Squared Residuals AR(1) -m=4 (15)
22
20
18
16
14
12
10
8
6
4
2
0
SIG-11118000 (Front Week Projection)
Moving Average (m=4)
Moving Average (m=15)
21
Stylized facts about volatility
Asymmetric Volatility (called leverage in equity markets)
Refer to the idea that price movements are negatively (positively) correlated with volatility
Nord Pool Front Week: Conditional Variance Function for the "Assymmetry effect"
50
EEX Base Month: Conditional Variance Function for the "Assymmetry effect"
35
45
30
40
25
35
20
30
15
25
10
20
5
15
10
0
Percentage Growth (d)
Percentage Growth (d)
22
Stylized facts about volatility
, valid for
,
Long Memory (highly persistent volatility)
Especially for high-frequency price series volatility is highly persistent. Therefore, there
are evidence of near unit root behaviour of the conditional variance process and high
persistence in the stochastic volatility process.
SV model definition:
u 1t  1  L   zt
d
L
and
zt   a j  zt  j  z1t
defined for
| d |  1/ 2
j 1
Co-movements in volatility / Correlations
Looking at time series within and across different markets, we observe big movements
in one currency being matched by big movements in another. These observations
suggest importance of multivariate models in modelling cross-correlation in different
products as well as markets.
Implications for reliable future volatilities
To get reliable forecasts of future volatilities it is crucial to account for the observed
stylised facts.
23
Stylized facts about volatility
Co-movements in volatility / Correlations
One-step-ahead density fK(yt|xt-1,q ): xi,t-1 = unconditional mean of the data
Nord Pool: Correlation Week - Month Contracts
1
0.95
0.9
0.14
Mean(Week)
=-0.163487
Stdev(Week)
=1.873265
Mean(Month)
=-0.054836
Stdev(Month)
=1.937181
Covariance
=3.11526
0.12
0.1
0.08
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.06
0.5
5.81
4.65
3.49
2.32
1.16
0.00
-1.16
0.04
0.02
0.35
0.3
0.2
5.62
4.72
0.25
-5.81
5.17
3.82
4.27
2.92
3.37
2.02
-4.65
2.47
1.12
1.57
0.22
0.67
-0.22
-1.12
-3.49
-0.67
-2.02
-1.57
-2.92
-2.47
-3.82
-3.37
-4.72
0.4
-2.32
-4.27
-5.62
-5.17
0
0.45
24
Models for volatility estimations/forecasts
Time series models
Use the historical
information only.
Not based on
theoretical
foundations, but
to capture the
main features.
Options-based forecasts
Calculations and Predictions
based on past
Standard deviations
From traded
option prices and
with the help of
the Black-Scholes
model.
Conditional volatility
models
Stochastic volatility
models
25
Stochastic volatility Models
Value at Risk (VaR):
The gain during time T at the (100 – X)th percentile of the probability
distribution.
Conditional Value at Risk (VaR) (expected shortfall):
The expected loss during time T, conditional on the loss being greater
than the Xth percentile of the probability distribution.
26
Stochastic volatility Models
Risk management is largely based on historical volatilities. Procedures for using
historical data to monitor volatility.
Define
n+1:
Si :
:
number of observations
value of variable at end of i th interval, where i = 0, 1, …, n
length of time interval
 S 
ui  ln  i  for i = 1, 2, … , n.
 Si 1 
2
1 n
s
  ui  u 
n  1 i 1
The standard deviation of the ui is  
variable s is, therefore an estimate of
, where  is the volatility of the variable. The
  .
It follows that s itself can be estimated as ̂
, where ˆ 
The standard error of this estimate is approximately:
s

ˆ
2n
27
Stochastic Volatility Models
Correlations /Co-movements in Volatility
For risk management, if changes in two or more variables have a high positive
correlation, the company’s total exposure is very high; if the variables have a correlation
of zero, the exposure is less, but still quite large; if they have a high negative correlation,
the exposure is quite low because a loss on one of the variables is likely to be offset by a
gain on the other.
Define
:
correlation between two variables V1 and V2

E (V1V2 )  E (V1 ) E (V2 )
SD(V1 ) SD(V2 )
where E() denotes expected value and SD() denotes standard deviation. The covariance between V1 and V2 is
cov(V1 ,V2 )  E(VV
1 2 )  E (V1 ) E (V2 )
and the correlation can therefore be written as:

cov(V1 , V2 )
SD(V1 ) SD(V2 )
An analogy for covariance is the pervious variance/volatility.
28
Stochastic Volatility Models
Correlations /Co-movements in Volatility and COPULAS
Often there is no natural way of defining a correlation structure between two marginal distribution (unconditional
distributions). This is where COPULAS come in. Formally, the Gaussian copula approach is: Suppose that F1 and F2
are the cumulative marginal probability distributions of V1 and V2. We map V1 = 1 to U1 = u1 and V2 = 2 to U2 =
u2, where
F1 1   N u1  and F2 2   N u2 
and N is the cumulative normal distribution function. This means that
u1  N 1  F1 1  ,
u2  N 1  F2 2 
and
1  F11  N  u1  ,
2  F21  N  u2 
The variables U1 and U2 are then assumed to be bivariate normal.
The key property of a copula model is that it preserves the marginal distribution of V1 and V2 (however unusual they
may be) while defining a correlation structure between them.
Other copulas is the Student-t copula
Multivariate copulas exists and Factor models can be used.
29
A Scientific Stochastic volatility model
Let yt denote the percent change in the price of security/portfolio. A stochastic volatility
model in the form used by Gallant, Hsieh and Tauchen (1997) with a slight modification
to produce leverage (asymmetry) effects is:
yt  a0  a1  yt 1  a0   exp(1t  v2t )  u1t
1t  b0  b1  1,t 1  b0   u2t
 2t  c0  c1   2,t 1  c0   u3t
u1t  z1t

u2t  s1 r1  z1t  1  r12  z2t



 r  z  (r  (r  r )) / 1  r 2  z  
3
2 1
2t
 2 1t

u3t  s2 

2
 1  r 2  (r  (r  r )) / 1  r 2  z 
2
3
2 1
3t




where z1t and z2t are iid Gaussian random variables. The parameter vector is:
  (a0 , a1 , b0 , b1 , s1 , c0 , c1 , s2 , r1 , r2 , r3 )
REF: Clark (1973), Tauchen & Pitts (1983), Gallant, Hsieh, and Tauchen (1991, 1997), Andersen (1994), and Durham
(2003). See Shephard (2004) and Taylor (2005) for more background and references.
30
Stochastic Volatility Models
GSM estimated SV-models for NordPool and EEX European Energy Markets
NP Front Week General Scientific Model. Parallell Run
Parameter values Scientific Model.
Standard
q
Mode
Mean
deviation
a0
-0.3455100
-0.3439200
0.0362740
a1
0.1603800
0.1616200
0.0117630
b0
0.9504900
0.9464300
0.0460600
b1
0.2660200
0.1751400
0.3753100
c1
0.9697400
0.9687400
0.0089971
s1
0.3300100
0.3235100
0.0232200
s2
0.1034300
0.1042300
0.0212070
r
0.0321930
0.0346580
0.0233110
EEX Front Month (base load) General Scientific Model.
Parameter values Scientific Model.
Standard
q
Mode
Mean
deviation
a0
-0.1005800
-0.1036800
0.0290180
a1
0.1531800
0.1524500
0.0163440
b0
0.5415200
0.5171100
0.0907430
b1
0.9930500
0.9868200
0.0085251
c1
0.8915800
0.7486800
0.2359500
s1
0.0541580
0.0791910
0.0355660
s2
0.1535500
0.1526100
0.0289920
r
0.6267800
0.4634600
0.2567000
log sci_mod_prior
log sci_mod_prior
3.5624832
log stat_mod_prior
log stat_mod_likelihood
log sci_mod_posterior
0
-4397.58339
-4394.02091
c2 (4) =
-3.2525
{0.516493}
NP Front Month General Scientific Model. Parallell Run
Parameter values Scientific Model.
Standard
q
Mode
Mean
deviation
a0
-0.11421
-0.10159
0.030944
log
log
log
log
a1
0.10047
0.11203
0.016788
b0
0.80606
0.82536
0.042584
b1
0.79323
0.79608
0.013226
c1
0
0
0
s1
0.23126
0.23091
0.0048139
s2
r
0
0.032193
0
-0.0081275
0
0.022407
sci_mod_prior
stat_mod_prior
stat_mod_likelihood
sci_mod_posterior
4.78473466
0
-4488.3985
-4483.61377
log stat_mod_prior
log stat_mod_likelihood
log sci_mod_posterior
0
-1597.22335
-1592.71181
c2 (2) =
-5.0098
{0.081684}
EEX Front Month (peak load) General Scientific Model.
Parameter values Scientific Model.
Standard
q
Mode
Mean
deviation
a0
-0.1836000
-0.1822500
0.0354710
a1
0.1604000
0.1618700
0.0159110
b0
0.6935800
0.6792700
0.0872020
b1
0.9798300
0.9791800
0.0043944
c1
0.2208400
0.2795600
0.3138000
s1
0.1122400
0.1105900
0.0138830
s2
0.2606700
0.2483200
0.0379230
r
0.3446600
0.3399800
0.0818190
log sci_mod_prior
c2(5) =
-2.8748
{0.719281}
4.5115377
log stat_mod_prior
log stat_mod_likelihood
log sci_mod_posterior
5.1621327
0
-1673.34285
-1668.18071
c2 (2) =
-10.257
{0.005925}
31
Stochastic Volatility Models
GSM Assessment of SV Model Simulation fit:
32
Stochastic Volatility Models
GSM Assessment of SV Model Simulation fit:
33
Stochastic Volatility Models
SV-model Features (2 markets and 4 contracts): NASDAQ OMX Front Week 100 k
34
Stochastic Volatility Models
SV-model Features (2 markets and 4 contracts): NASDAQ OMX Front Week 100 k
35
Stochastic Volatility Models
SV-model Features (2 markets and 4 contracts): EEX Front Month (peak load) 100 k
36
Stochastic Volatility Models
SV-model Features (2 markets and 4 contracts): EEX Front Month (peak load) 100 k
37
Stochastic Volatility Models
SV-model Features (2 markets and 4 contracts): Correlation Week/Month 100 k
Nord Pool: Correlation Week - Month Contracts
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
EEX: Correlation Front Month - Base and Peak Load
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
38
SV-Models: Risk Management
Excel
Densities Percentiles: VaR, CVaR positions for 4 contracts 100 k
39
SV-Models: Risk Management
EVT: VaR, CVaR Positions for 4 contracts 100 k
40
SV-Models: Risk Management
EVT densities: VaR, CVaR Positions for 4 contracts 100 k
41
SV-Models: Risk Management
Greek Letter densities (delta reported) for NASDAQ Week and Month 100 k
42
SV-Models: Risk Management
Greek Letter densities (delta reported) for EEX Base and Peak Load Month Futures 100 k
43
SV-Models: Risk Management
Bivariate Estimations: NASDAQ OMS Front Week – Front Month
0.08
0.045
MU Week + Month
SIG Week + Month
0.07
0.04
0.06
0.035
0.03
0.05
0.025
0.04
0.02
0.03
0.015
0.02
0.01
0.01
0.005
0
Frequency Week
Frequency Month
0
Week Kernel
Week Normal distribution
Month Kernel
Month Normal distribution
Frequency Week
Frequency Month
Week Kernel
Month Kernel
Week Normal distribution
Month Normal distribution
44
SV-Models: Risk Management
Bivariate Estimations: EEX Front Base Month – Front Peak Month
0.1
MU Month Base & Peak
SIG Week + Month
0.05
0.09
0.045
0.08
0.04
0.07
0.035
0.06
0.03
0.05
0.025
0.04
0.02
0.03
0.015
0.02
0.01
0.01
0.005
0
0
Frequency Month (base load)
Frequency Month (peak load)
Month (base load) Kernel
Frequency Month (base load)
Frequency Month (peak load)
Month (base load) Kernel
Month (base load) Normal distribution
Month (peak load) Kernel
Month (peak load) Normal distribution
Month (peak load) Kernel
Month (base load) Normal distribution
Month (peak load) Normal distribution
45
SV-Models: Risk Management
Bivariate Estimations: NASDAQ OMX Front Month – EEX Front Base Month
0.09
MU NP - EEX Month
SIG Month NP + EEX
0.05
0.08
0.045
0.07
0.04
0.06
0.035
0.03
0.05
0.025
0.04
0.02
0.03
0.015
0.02
0.01
0.01
0.005
0
0
Frequency Month NP
Frequency Month EEX
Month NP Kernel
Month NP Normal distribution
Month EEX Kernel
Month EEX Normal distribution
Frequency Month NP
Frequency Month EEX
Month NP Kernel
Month EEX Kernel
Month NP Normal distribution
Month EEX Normal distribution
46
SV-Models: Risk Management
Forecast unconditional First Moment: VaR/CVaR measures from Uni- and Bivariate
Estimations (precentiles)
Univariate (long positions)
Nord Pool
Front Month
Confidence Front Week
CVaR
CVaR VaR
VaR
levels:
0.0333 0.0410 0.0240 0.0287
99.90 %
0.0237 0.0298 0.0176 0.0216
99.50 %
0.0198 0.0256 0.0152 0.0189
99.00 %
0.0155 0.0206 0.0122 0.0156
97.50 %
0.0124 0.0172 0.0102 0.0134
95.00 %
0.0096 0.0140 0.0082 0.0112
90.00 %
EEX
Base Month
CVaR
VaR
0.0195 0.0245
0.0129 0.0171
0.0107 0.0144
0.0079 0.0111
0.0060 0.0090
0.0043 0.0070
Bivariate (long positions)
Nord Pool
Front Month
Confidence Front Week
CVaR
CVaR VaR
VaR
levels:
0.0378 0.0464 0.0343 0.0416
99.90 %
0.0266 0.0338 0.0240 0.0303
99.50 %
0.0220 0.0289 0.0201 0.0261
99.00 %
0.0170 0.0230 0.0155 0.0209
97.50 %
0.0133 0.0190 0.0122 0.0173
95.00 %
0.0098 0.0152 0.0092 0.0139
90.00 %
EEX
Base Month
CVaR
VaR
0.0228 0.0285
0.0148 0.0197
0.0123 0.0166
0.0090 0.0128
0.0068 0.0103
0.0048 0.0080
Peak Month
CVaR
VaR
0.0246 0.0302
0.0171 0.0218
0.0140 0.0186
0.0104 0.0145
0.0080 0.0118
0.0059 0.0093
Peak Month
CVaR
VaR
0.0307 0.0379
0.0210 0.0272
0.0171 0.0230
0.0125 0.0178
0.0094 0.0143
0.0067 0.0111
Nord-Pool & EEX
Base Month
Front Month
CVaR
CVaR VaR
VaR
0.0150 0.0178 0.0220 0.0275
0.0114 0.0138 0.0144 0.0191
0.0099 0.0121 0.0119 0.0160
0.0079 0.0101 0.0087 0.0124
0.0064 0.0086 0.0066 0.0099
0.0048 0.0070 0.0047 0.0077
47
SV-Models: Risk Management
Forecast Second Moment: Uni- and Bivariate Estimations NASDAQ OMX Front Week
One-step-ahead density fK(yt|xt-1,q) conditional on data value y = -0.347
One-step-ahead density fK(yt|xt-1,q) xt-1= -10,-5,-3,-1,-0.347, 0,+1,+3,+5,+10%
0.25
C
o
n
d
i
t
o
n
a
l
0.25
C
o
n
d
i
t
o
n
a
l
0.2
0.15
M
e
a
n
D
e
n
s
i
t
y
0.2
0.15
M
e
a
n
0.1
0.1
D
e
n
s
i
t
y
0.05
0.05
0
0
Frequency xt-1=-10%
Frequency xt-1=-5%
Frequency xt-1=-3%
Frequency xt-1=-1%
Frequency xt-1=0%
Frequency xt-1= "Mean (0.037)"
Frequency xt-1=+1%
Frequency xt-1=+3%
Frequency xt-1=+5%
Frequency xt-1=+10%
GAUSS-Hermite Quadrature: Conditional Mean Density Distribution
The Conditional Variance Function for the "Assymmetry effect"
0.4
50
0.35
45
40
0.3
35
0.25
30
0.2
25
20
0.15
15
0.1
10
0.05
5
0
0
Reprojected Quadrature
48
Percentage Growth (d)
SV-Models: Risk Management
Forecast Second Moment: Uni- and Bivariate Estimations NASDAQ OMX Front Month
One-step-ahead density fK(yt|xt-1,q) conditional on data value y = -0.137
One-step-ahead density fK(yt|xt-1,q) xt-1=-10,-5,-3,-1, -0.137, 0,+1,+3,+5,+10%
0.225
0.225
C
o
n
d
i
t
o
n
a
l
C
0.2
o
n
d
0.175
i
t
o
0.15
n
a
l
0.125
0.1
M
e
a
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D 0.075
e
n
s 0.05
i
t
y 0.025
D
e
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s
i
t
y
M
e
a
n
0.2
0.175
0.15
0.125
0.1
0.075
0.05
0.025
0
0
Frequency xt-1=-10%
Frequency xt-1=-5%
Frequency xt-1=-3%
Frequency xt-1=-1%
Frequency xt-1=0%
Frequency xt-1= "Mean (0.037)"
Frequency xt-1=+1%
Frequency xt-1=+3%
Frequency xt-1=+5%
Frequency xt-1=+10%
GAUSS-Hermite Quadrature: Conditional Mean Density Distribution
0.4
The Conditional Variance Function for the "Assymmetry effect"
50
0.35
45
0.3
40
0.25
35
30
0.2
25
0.15
20
0.1
15
10
0.05
5
0
49
0
Reprojected Quadrature
Percentage Growth (d)
SV-Models: Risk Management
Forecast Second Moment: Uni- and Bivariate Estimations EEX Front Month (base load)
One-step-ahead density fK(yt|xt-1,q) conditional on data value y = -0.044
One-step-ahead density fK(yt|xt-1,q) xt-1= -10,-5,-3,-1,0,mean,+1,+3,+5,+10%
0.525
0.5
0.5
C
o
n
d
i
t
o
n
a
l
M
e
a
n
D
e
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s
i
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y
C
o
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d
i
t
o
n
a
l
0.4
0.3
0.475
0.45
0.425
0.4
0.375
0.35
0.325
0.3
0.275
M
e
a
n
0.25
0.225
0.2
0.2
0.175
D
e
n
s
i
t
y
0.1
0.15
0.125
0.1
0.075
0.05
0.025
0
0
Frequency xt-1=-10%
Frequency xt-1=-5%
Frequency xt-1=-3%
Frequency xt-1=-1%
Frequency xt-1=0%
Frequency xt-1= "Mean (0.037)"
Frequency xt-1=+1%
Frequency xt-1=+3%
Frequency xt-1=+5%
Frequency xt-1=+10%
GAUSS-Hermite Quadrature: Conditional Mean Density Distribution
The Conditional Variance Function for the "Assymmetry effect"
0.4
11
10
0.3
9
8
0.2
7
0.1
6
5
0
4
Reprojected Quadrature
50
Percentage Growth (d)
SV-Models: Risk Management
Forecast Second Moment: Uni- and Bivariate Estimations EEX Front Month (peak load)
One-step-ahead density fK(yt|xt-1,q) conditional on data value y = -0.117
One-step-ahead density fK(yt|xt-1,q) xt-1= -10,-5,-3,-1,-0.12,0,+1,+3,+5,+10%
0.4
C
o
n
d
i
t
o
n
a
l
M
e
a
n
D
e
n
s
i
t
y
0.4
C
o
n
d
i
t
o
n
a
l
0.35
0.3
0.25
0.35
0.3
0.25
M
e
a
n
0.2
0.15
0.2
0.15
D
e
n
s
i
t
y
0.1
0.05
0.1
0.05
0
0
Frequency xt-1=-10%
Frequency xt-1=-5%
Frequency xt-1=-3%
Frequency xt-1=-1%
Frequency xt-1=0%
Frequency xt-1= "Mean (-0.117)"
Frequency xt-1=+1%
Frequency xt-1=+3%
Frequency xt-1=+5%
Frequency xt-1=+10%
GAUSS-Hermite Quadrature: Conditional Mean Density Distribution
The Conditional Variance Function for the "Assymmetry effect"
0.4
45
40
35
0.3
30
25
0.2
20
15
0.1
10
5
0
0
Reprojected Quadrature
51
Percentage Growth (d)
SV-Models: Risk Management
Forecast Second Moment: Uni- and Bivariate Estimations NASDAQ OMX Front Week/Month
One-step-ahead density f K(yt |xt-1,q ): xi,t-1=-5%; -5%
One-step-ahead density fK(yt|xt-1,q ): xi,t-1 = unconditional mean (-0.32 /-0.12)
0.06
0.04
10.58
8.47
6.35
4.23
2.12
0.00
-2.12
0.02
0.15
NP Front Month
10.52
-10.58
11.43
8.69
9.60
6.86
7.77
5.03
5.94
3.20
-8.47
4.11
1.37
2.29
-1.37
0.46
-0.46
-3.20
-6.35
-2.29
-5.03
NP Front Week
-4.11
-7.77
-6.86
-5.94
-4.23
-8.69
-9.60
0
-11.43
0.2
D
e
n
s
i
t
y
-10.52
0.25
0.1
0.08
One-step-ahead density f K(yt |xt-1,q ): xi,t-1=+5%; +5%
0.1
0.08
0.06
0.04
10.84
8.67
6.50
4.34
2.17
0.00
-2.17
0.02
0
11.26
-10.84
NP Front Month
12.24
9.30
10.28
7.34
8.32
5.39
6.37
3.43
-8.67
4.41
1.47
2.45
-1.47
0.49
-0.49
-3.43
-6.50
-2.45
-5.39
-4.41
-4.34
NP Front Week
-8.32
-12.24
6.51
5.47
-6.27
5.99
4.43
4.95
3.39
3.91
2.34
D
e
n
s
i
t
y
-5.02 Front Month Contracts
2.86
1.30
1.82
0.26
Front Week Contracts
0.78
-1.30
-0.78
-3.76
-0.26
-2.34
-1.82
-3.91
-3.39
-2.86
-5.47
-4.95
-2.51
-4.43
-6.51
-5.99
0
Mean(Week)
=0.690381
Stdev(Week)
=3.612949
Mean(Month)
=0.402447
Stdev(Month)
=4.080466
Covariance
=14.5089
Correlation
=0.98415
0.12
-7.34
0.05
C
o
n
d
i
t
i
o
n
a
l
-6.37
6.27
5.02
3.76
2.51
1.25
0.00
-1.25
-9.30
0.1
-11.26
D
e
n
s
i
t
y
Mean(Week)
=-0.129685
Stdev(Week)
=2.170127
Mean(Month)
=-0.146504
Stdev(Month)
=2.090196
Covariance
=4.41692
Correlation
=0.97375
0.3
Mean(Week)
=-1.13343
Stdev(Week)
=3.527478
Mean(Month)
=-0.955496
Stdev(Month)
=3.81
Covariance
=13.1947
Correlation
=0.98177
0.12
-10.28
C
o
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C
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Frame 001  11 Mar 2011  cartesianplt
Z
Gauss-Hermite Quadrature Nord Pool Front Week - Month
18
1
Variance-Co-Variance/Correlation
X
Y
17
0.95
16
15
0.9
14
0.06
0.05
0.04
0.03
0.02
0.01
0
-8
13
-15
Z
NP
0.85
12
-10
-5
0
-4
Fro
nt 0
Mo
nth
5
10
4
8
15
W
nt
Fro
P
N
e
0.8
11
ek
10
0.75
9
8
0.7
Variance NP Week
Growth
Variance NP Month
NP Covariance
NP Correlation
52
SV-Models: Risk Management
Forecast Second Moment: Uni- and Bivariate Estimations EEX Front Month Base and Peak
One-step-ahead density f K(yt |xt-1,q ): xi,t-1=-5%; -5%
One-step-ahead density fK(yt|xt-1,q ): xi,t-1 = unconditional mean (-0.04 /-0.11)
11.22
9.42
10.32
7.63
8.53
5.83
6.73
4.04
EEX Front Month (peak load)
Mean(EEX
Month(base))
=0.801637
Stdev(Month(
base))
=3.918852
Mean(Month(
peak))
=0.843551
Stdev(Month(
peak))
=3.877847
Covariance
=14.9603
Correlation
=0.98444
0.12
0.1
0.08
0.06
11.76
9.41
7.05
4.70
2.35
0.00
-2.35
0.04
0.02
0
11.63
9.77
7.91
8.84
6.05
-11.76 EEX Front Month (peak load)
10.70
Frame 001  11 Mar 2011  cartesianplt
6.98
4.19
-9.41
5.12
2.33
3.26
1.40
0.47
-0.47
-3.26
-2.33
-7.05
-1.40
-5.12
-4.19
-6.98
-4.70
EEX Front Month (base load)
Gauss-Hermite Quadrature EEX Front Months - Base and peak Load
4.94
2.24
3.14
0.45
1.35
-0.45
-3.14
-2.24
-1.35
-4.94
-4.04
-6.73
-5.83
-9.42
-8.53
-7.63
-4.48
-6.72
-8.96
-11.20
0.14
-11.63
2.48
Front Month (peak load)
2.69
2.05
2.26
1.62
1.83
1.19
1.40
0.75
0.97
0.11
0.32
0.54
-0.32
-0.11
-0.97
-0.75
D
e
n
s
i
t
y
-3.15
-3.94
0
-6.05
C
o
n
d
i
t
i
o
n
a
l
-2.36
-0.54
-1.40
-1.19
-2.05
-1.83
Front Month (base load)
0.02
One-step-ahead density f K(yt |xt-1,q ): xi,t-1=+5%; +5%
-1.58
-1.62
-2.69
-2.48
0
11.20
8.96
6.72
4.48
2.24
0.00
-2.24
0.04
EEX Front Month (base load)
3.94
3.15
2.36
1.58
0.79
0.00
-0.79
0.2
0.06
-9.77
0.4
0.1
0.08
-8.84
D
e
n
s
i
t
y
0.12
-7.91
0.6
0.14
-11.22
0.8
0.16
-10.70
1
D
e
n
s
i
t
y
Mean(EEX
Month (base))
=-0.949634
Stdev(Month(
base))
=3.732667
Mean(Month(
peak)) =1.05158
Stdev(Month(
peak))
=3.739318
Covariance
=13.8024
Correlation
=0.98888
0.18
-10.32
Mean(Month(
base)) =0.013149
Stdev(Month(
base))
=0.897826
Mean(Month(
peak)) =0.035861
Stdev(Month(
peak))
=1.313046
Covariance
=1.14086
Correlation
=0.96774
1.2
-2.26
C
o
n
d
i
t
i
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a
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C
o
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d
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Z
16
1
EEX Variance, Co-Variance and Correlation
15
X
Y
14
0.9
13
0.1
12
0.08
11
0.06
10
0.04
0.02
0.8
9
0.7
8
-5
0
-4
-4
-3
-3
EE
XF
ron -1
tM
on 0
th
(
7
-2
-2
0
pe 1
ak
loa 2
d)
1
M
nt
Fro
EX
2
3
3
4
4
E
th
on
d)
-1
loa
se
(ba
0.6
6
5
4
0.5
5 5
Growth
EEX Front Month (base) Variance
EEX Front Month (peak) Variance
EEX Front Month Covariance
53
EEX Front Month Correlation
SV-Models: Risk Management
Forecast Second Moment: Uni- and Bivariate Estimations NASDAQ and EEX Front Month (base)
One-step-ahead density f K(yt |xt-1,q ): xi,t-1=-5%; -5%
One-step-ahead density fK(yt|xt-1,q ): xi,t-1 = unconditional mean (-0.13 /-0.04)
C
o
n
d
i
t
i
o
n
a
l
11.22
9.42
10.32
7.63
8.53
5.83
6.73
4.04
4.94
2.24
3.14
0.45
1.35
-0.45
-3.14
-2.24
-1.35
-4.94
-4.04
-6.73
-5.83
-9.42
-8.53
-7.63
Mean(NP
Month)
=0.475304
Stdev(NP
Month)
=4.395987
Mean(EEX
Month)
=0.780463
Stdev(EEX
Month)
=4.395225
Covariance
=18.9263
Correlation
=0.97955
0.1
0.09
0.08
0.07
0.06
0.05
0.02
0.01
0
13.19
11.08
-13.19 EEX Front Month (base load)
12.13
8.97
10.02
6.86
7.91
4.75
-10.55
5.80
2.64
3.69
1.58
0.53
-0.53
-7.91
-2.64
-3.69
-5.28
NP Front Month (base load)
-1.58
EEX Front Month
13.19
10.55
7.91
5.28
2.64
0.00
-2.64
0.03
-5.80
4.43
-2.64
4.82
3.66
4.05
2.89
3.28
2.12
2.50
1.35
-2.12
1.73
0.19
0.58
0.96
-0.58
-0.19
-1.73
-1.35
-1.59
-0.96
-2.50
-2.12
-3.66
-3.28
NP Front Month
EEX Front Month (peak load)
0.04
D
e
n
s
i
t
y
-1.06
-2.89
-4.82
-4.43
-4.05
0
-6.72
-8.96
-11.20
One-step-ahead density f K(yt |xt-1,q ): xi,t-1=+5%; +5%
-4.75
0.1
-4.48
NP Front Month (base load)
2.64
2.12
1.59
1.06
0.53
0.00
-0.53
0.2
0
-13.19
D
e
n
s
i
t
y
0.02
-7.91
0.3
11.20
8.96
6.72
4.48
2.24
0.00
-2.24
0.04
-6.86
0.4
0.06
-8.97
0.5
0.1
0.08
-11.08
0.6
D
e
n
s
i
t
y
0.12
-10.02
0.7
0.14
-11.22
0.8
0.16
-10.32
Mean(NP
Month) =0.057598
Stdev(NP
Month)
=1.605624
Mean(EEX
Month) =0.015215
Stdev(EEX
Month)
=0.881296
Covariance
=1.32782
Correlation
=0.93837
Mean(NP
Month) =0.949634
Stdev(NP
Month)
=3.732667
Mean(EEX
Month) =1.05158
Stdev(EEX
Month)
=3.739318
Covariance
=13.8024
Correlation
=0.98888
0.18
-12.13
C
o
n
d
i
t
i
o
n
a
l
C
o
n
d
i
t
i
o
n
a
l
Frame 001  11 Mar 2011  cartesianplt
Z
Gauss-Hermite Quadrature NP-EEX Front Months Contracts
20
1
NP and EEX Front Month Variance, Co-Variance, and Correlation
18
X
Y
0.9
16
14
0.8
12
-8
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
-3
EE
0.7
10
-4
0
-2
X F -1 0
ron
tM 1 2
ont
h
4
NP
nt
Fro
nth
Mo
8
0.6
6
0.5
4
3
2
0.4
54
Growth
Var Week
Var Month
Covariance
Correlation
SV-Models: Risk Management
Excel
Extreme Value Theory for First Conditional Moment 5 k iterations
55
SV-Models: Risk Management
Extreme Value Theory for First Conditional Moment 5 k iterations
56
SV-Models: Risk Management
Extreme Value Theory for First Conditional Moment 5 k iterations
57
Risk Management (aggregation)
Economic Capital and RAROC
Panel A:
Business Units (billion €)
Hydro power
Network
Economic Capital
generation (B1)
operation (B2)
Market risk (M)
150
45
Basis Risk (B)
95
38
Operational Risk (O)
55
25
Hybrid
approach:
Copula
approach:
Telephone
communication (B3)
82
50
n
i 1 j 1
behaved distributions
with correlation
structures
(Cholesky)
The market
risk :
34
economic capital: 233.41
Panel B:
Correlation
Structure
n
Etotal 10
Ei  well
E j ij
MCMC
k for
MB1
BB1
OB1
MB2
BB2
OB2
MB3
BB3
OB3
MB1
1
0.35
0.2
0.4
0
0.1
0.3
0
0.05
BB1
0.35
1
0.15
0.15
0.25
0.25
0.05
0.1
0
OB1
0.2
0.15
1
0.15
0
0.2
0.1
0.1
0
MB2
0.4
0.15
0.15
1
0.2
0.1
0
0
0.1
BB2
OB2
MB3
BB3
OB3
0
0.1
0.3
0
0.05
0.25
0
0.05
0.1
0
0
0.2
0.1
0.1
0
0.2
0.1
0
0
0.1
1
-0.1
0.1
0.2
0.05
-0.1
1
0
0.1
0
0.1
0
1
0.1
0
0.2
0.1
0.1
1
0.05
0.05
0
0
0.05
1
Normal distribution:
The basis risk
Etotal = 305.06 st.dev =47.5
economic capital: 159.37
Student-t (4 df):
operational
EtotalThe
= 304.21
st.devrisk
=51.8
economic capital: 98.32
Student-t (2 df):
Etotal = 318.58 st.dev = 222.4
Etotal: 299.73
58
Risk Management (aggregation)
Economic Capital and RAROC: Using Copulas and Correlation structures
VaR/CVaR for Normal distributions
59
Risk Management (aggregation)
Economic Capital and RAROC: Using Copulas and Correlation structures
VaR/CVaR for Student-t distribution 4 degrees of freedom
60
Risk Management (aggregation)
Economic Capital and RAROC: Using Copulas and Correlation structures
VaR/CVaR for Student-t distribution 2 degrees of freedom
61
Future work….?
Operational Forecasting (efficient algorithms)
Higher Conditional Moments (skew/kurtosis)
Volatility (particle filtering) and pricing exotic
options
Multiple-ahead-forecasts for mean and volatility
Persistence measures
New information and the SV models- concept
Multivariate SV models forecasts: market arbitrage
Closed-form solution SV models and energy
markets.
62
Summary & Conclusions
Free methodology
SV-models for energy, equity, currency markets.
Portfolio applications and forecasting.
The number of CPU’s are not important any
longer. Apple (linux) 8 core computer with
HYPERTHREAD has 16 cores for running OPENMPI (downloadable from Indiana Univeristy)
Running every day obtaining one-day-ahead
forecasts, induce 30-50% VaR/CVaR reduction
and the Greek letters seem to move significantly.
63