Market Risk Management using Stochastic Volatility Models
description
Transcript of Market Risk Management using Stochastic Volatility Models
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Market Risk Managementusing Stochastic Volatility
Models
The Case of European Energy Markets
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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
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Main ObjectivesForecasting Risk Management
MeasuresSV model forecasts of VaR, CVaR and
Greek letter densities
Conditional Moments ForecastsOne-day-ahead densities of VaR, CVaR
and Greek lettersExtreme value theory and VaR, CVaR and
Greek letter densities
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PreliminariesPortfolio Theory Basics for Investors
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2 ,
N
p i ii
p i j i j i j
E R E r
r r
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Kapitalmarkedslinjen
Portefølje Standardavvik
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PreliminariesPortfolio Theory Basics (relevant risk measures):
( , ) ( , )
/ /( )
j M j j Mj F M F j
M M
Cov R R R RE R R E R R
Var R
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Markedsavkastningslinjen
ProsjektA
ProsjektB
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Preliminaries
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.
The Relevant Risk issue:
That is: stotal versus bi = (si * sM) / sM
theTraynor index versus the Sharpe index
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PreliminariesFinancial Products and Markets
Financial Products / “Plain Vanilla” products
Long and Short positions in Assets
Forward Contracts / Future Contracts
Swaps
Options
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PreliminariesEuropean Energy Markets and Activity/Liquidity for 2008 -2009 (annual reports)
Power Futures (TWh) Carbon Trading (tonnes) Spot Power (TWh) Cleared OTC power (TWh)2008 2009 2008 2009 2008 2009 2008 2009
Nord Pool Volume (TWh) 1437 1220 121731 45765 298 286 1140 942Transactions 158815 136030 6685 3792 70 % 72 % 51575 40328
EEX Volume (TWh) 1165 1025 80084 23642 154 203 n/a n/aTransactions 128750 114250 4398 1959 54 % 56 % n/a n/a
Powernext Volume (TWh) 79 87 n/a n/a 203.7 196.3 n/a n/aTransactions n/a n/a 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 n/a n/aTransactions 36150 45900 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.
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PreliminariesFinancial Products and Positions
Hedging Positions for plain Assets
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Profit
Loss
Underlying asset (St)
S0
Long position Asset
Short position Asset
Long Positions Payoff:
St – S0
Short Positions Payoff:
S0 – St
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PreliminariesFinancial Products and Positions
Hedging Positions for plain Forward/Future Products
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Profit
Loss
Underlying asset (St)
K
Long position Forward/Future
Short position Forward/Future
Long Positions Payoff:
St – K
Short Positions Payoff:
K – St
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PreliminariesFinancial Products and Positions
Hedging Positions for plain buying (long) positions in Call/Put options
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Profit
Loss
Underlying asset (St)
K
Buying aCall position
Buying a put position Call position
Payoff:
Max(0;St – K)-c
Put Positions Payoff:
Max(0;K – St)-p
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PreliminariesFinancial Products and Positions
Hedging Positions for plain selling (short) positions in Call/Put options
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Profit
Loss
Underlying asset (St)
K
Selling aCall positionSelling a put
position
Call positionPayoff:
-Max(0;St – K)+c
Put Positions Payoff:
-Max(0;K – St)+p
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PreliminariesManagement of Portfolio Exposures: Greek Letters
DeltaS
The sensitivity of the portfolios value to the price of the underlying asset:
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2( )Gamma
S
The rate of change of the portfolio’s delta with respect to the price of the underlying asset:
( )Vega
The rate of change of the value of the portfolio with respect to the volatility of the underlying asset:
( )ThetaT
The rate of change of the value of the portfolio with respect to the passage of time (time decay):
( )Rhoi
The rate of change of the value of the portfolio with respect to the level of interest rates:
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PreliminariesCalculation of the GREEK LETTERS
Taylor Series Expansion on a single market variable S (volatility and interest rates are assumed constant)
2 2 22 2
2 2
1 1......
2 2S t S t S t
S t S 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 Dt) (quadratic relationship between S and P):
When volatility is uncertain:
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.
2t S
2 22 2
2 2
2 22
2
1 1
2 2
1......
2
S t SS t S
t S tt S t
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Stylized facts about volatility
Definition of volatility (s)The standard deviation of the return (rt) provided by the variable per unit
time when the return is expressed using continuous compounding.
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ln Tt
Sr
S
= return in time T expressed with continuous compounding
When T is small it follow that is approximately equal to the standard deviation of the percentage change in the market variable in time T.
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
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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).
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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
u of the variable has the property that, when x is large:
where K and a are constants.
Rewriting using the natural logarithm:
A quick test can now be done for weekly and yearly price changes at NASDAQ OMX energy market. We plot
against ln x.
Prob( )x Kx
ln Prob( ) ln lnx K x
ln Prob( )x
The logarithm of the probability is approx. linearly dependent on ln x for x >3 showing that the power law holds. -8
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ln(p
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x))
Power Law for Nord Pool/EEX Front Week/Month Swap Contracts
NP-Front-Week NP-Front-Month EEX Front Month (base load) EEX Front Month (peak load)
Stylized facts about volatility
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Extreme Value Theory* Application for a Forward contract at NASDAQ OMXEquivalence 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.
We maximize the log-likelihood function:
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( )1ln 1 ix u
Using the estimates (optimized): 0.2013 2.9096and
Calculation of VaR:
The probability that x will be less than 15% is:
The value of one-day 99% VaR for a portfolio where NOK 1 million is invested in the contract is NOK 1 million times:
That is, VaR = 1 million NOK * 0.102987= NOK 102,897
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31 0.15 0,101 0.0025
2809
280931 1 0.99 1 0.102897
31VaR
Stylized facts about volatility
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Stylized facts about volatility
Volatility ClusteringRefer to the observation of large movements of price changes are being followed by large movements. That is, persistence of shocks.
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NP Front Week: Projected and Moving Average Squared Residuals AR(1)-m=4 (15)
SIG-11118000 (Front Week Projection) Moving Average (m=4) Moving Average (m=15)
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Stylized facts about volatility
Asymmetric Volatility (called leverage in equity markets)Refer to the idea that price movements are negatively (positively) correlated with volatility
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Percentage Growth (d)
Nord Pool Front Week: Conditional Variance Function for the "Assymmetry effect"
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Percentage Growth (d)
EEX Base Month: Conditional Variance Function for the "Assymmetry effect"
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Stylized facts about volatility
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.
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.
To get reliable forecasts of future volatilities it is crucial to account for the observed stylised facts.
Implications for reliable future volatilities
11t
d
tu L z 1
1
L
t j t j tj
z a z z
| | 1/ 2d
, valid for ,
and defined for SV model definition:
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Stylized facts about volatility
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One-step-ahead density fK(yt|xt-1,q): xi,t-1 = unconditional mean of the data
Mean(Week) =-0.163487 Stdev(Week) =1.873265 Mean(Month) =-0.054836 Stdev(Month) =1.937181 Covariance =3.11526
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1Nord Pool: Correlation Week - Month Contracts
Co-movements in volatility / Correlations
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Models for volatility estimations/forecasts
Time series models Options-based forecasts
Calculations and Predictions based on past
Standard deviations
Conditional volatility models
Stochastic volatility models
Use the historical information only. Not based on theoretical foundations, but to capture the main features.
From traded option prices and with the help of the Black-Scholes model.
Microsoft Office Excel-regneark (kode)
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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.
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Stochastic volatility Models
Risk management is largely based on historical volatilities. Procedures for using historical data to monitor volatility.
Define n + 1 : number of observationsSi : value of variable at end of i th interval, where i = 0, 1, …, nt : length of time interval
for i = 1, 2, … , n.1
ln ii
i
Su
S
2
1
1
1
n
ii
s u un
The standard deviation of the , where s is the volatility of the variable. The variable s is, therefore an estimate of .
It follows that s itself can be estimated as , where
The standard error of this estimate is approximately:
iu is
ˆs
ˆ
2n
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Correlations /Co-movements in Volatility
Define r : correlation between two variables V1 and V2
1 2 1 2
1 2
( ) ( ) ( )
( ) ( )
E VV E V E V
SD V SD V
An analogy for covariance is the pervious variance/volatility.
1 2 1 2 1 2cov( , ) ( ) ( ) ( )V V E VV E V E V
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.
where E() denotes expected value and SD() denotes standard deviation. The covariance between V1 and V2 is
and the correlation can therefore be written as:
1 2
1 2
cov( , )
( ) ( )
V V
SD V SD V
Stochastic Volatility Models
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Correlations /Co-movements in Volatility and COPULAS
1 1 1 2 2 2F N u and F N u
1 11 1 1 2 2 2
1 11 1 1 2 2 2
,
,
u N F u N F
and
F N u F N u
and N is the cumulative normal distribution function. This means that
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.
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 = u1 to U1 = u1 and V2 = u2 to U2 = u2, where
Stochastic Volatility Models
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A Scientific Stochastic volatility modelLet 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:
where z1t and z2t are iid Gaussian random variables. The parameter vector is:
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.
0 1 1 0 1 2 1
1 0 1 1, 1 0 2
2 0 1 2, 1 0 3
1 1
22 1 1 1 1 2
22 1 3 2 1 2
3 2 22 2
2 3 2 1 3
exp( )
1
( ( )) / 1
1 ( ( )) / 1
t t t t t
t t t
t t t
t t
t t t
t t
t
t
y a a y a v u
b b b u
c c c u
u z
u s r z r z
r z r r r r z
u sr r r r r z
0 1 0 1 1 0 1 2 1 2 3( , , , , , , , , , , )a a b b s c c s r r r
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GSM estimated SV-models for NordPool and EEX European Energy Markets
Stochastic Volatility Models
NP Front Week General Scientific Model. Parallell RunParameter values Scientific Model. Standard
q Mode Mean deviationa0 -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 0.0212070r 0.0321930 0.0346580 0.0233110
log sci_mod_prior 3.5624832
log stat_mod_prior 0 c2(4) =
log stat_mod_likelihood -4397.58339 -3.2525log sci_mod_posterior -4394.02091 {0.516493}
EEX Front Month (base load) General Scientific Model.Parameter values Scientific Model. Standard
q Mode Mean deviationa0 -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 0.0289920r 0.6267800 0.4634600 0.2567000
log sci_mod_prior 4.5115377
log stat_mod_prior 0 c2(2) =
log stat_mod_likelihood -1597.22335 -5.0098log sci_mod_posterior -1592.71181 {0.081684}
NP Front Month General Scientific Model. Parallell RunParameter values Scientific Model. Standard
q Mode Mean deviation
a0 -0.11421 -0.10159 0.030944
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 0 0 0
r 0.032193 -0.0081275 0.022407
log sci_mod_prior 4.78473466
log stat_mod_prior 0 c2(5) =log stat_mod_likelihood -4488.3985 -2.8748log sci_mod_posterior -4483.61377 {0.719281}
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 5.1621327
log stat_mod_prior 0 c2(2) =
log stat_mod_likelihood -1673.34285 -10.257log sci_mod_posterior -1668.18071 {0.005925}
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GSM Assessment of SV Model Simulation fit:
Stochastic Volatility Models
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GSM Assessment of SV Model Simulation fit:
Stochastic Volatility Models
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Stochastic Volatility ModelsSV-model Features (2 markets and 4 contracts): NASDAQ OMX Front Week 100 k
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SV-model Features (2 markets and 4 contracts): NASDAQ OMX Front Week 100 k
Stochastic Volatility Models
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Stochastic Volatility ModelsSV-model Features (2 markets and 4 contracts): EEX Front Month (peak load) 100 k
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SV-model Features (2 markets and 4 contracts): EEX Front Month (peak load) 100 k
Stochastic Volatility Models
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Stochastic Volatility ModelsSV-model Features (2 markets and 4 contracts): Correlation Week/Month 100 k
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1EEX: Correlation Front Month - Base and Peak Load
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1Nord Pool: Correlation Week - Month Contracts
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SV-Models: Risk ManagementDensities Percentiles: VaR, CVaR positions for 4 contracts 100 k
Excel
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SV-Models: Risk ManagementEVT: VaR, CVaR Positions for 4 contracts 100 k
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SV-Models: Risk ManagementEVT densities: VaR, CVaR Positions for 4 contracts 100 k
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SV-Models: Risk ManagementGreek Letter densities (delta reported) for NASDAQ Week and Month 100 k
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SV-Models: Risk ManagementGreek Letter densities (delta reported) for EEX Base and Peak Load Month Futures 100 k
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SV-Models: Risk Management
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MU Week + Month
Frequency Week Frequency Month Week Kernel Week Normal distribution Month Kernel Month Normal distribution
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SIG Week + Month
Frequency Week Frequency Month Week Kernel Month Kernel Week Normal distribution Month Normal distribution
Bivariate Estimations: NASDAQ OMS Front Week – Front Month
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SV-Models: Risk ManagementBivariate Estimations: EEX Front Base Month – Front Peak Month
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MU Month Base & Peak
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
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Frequency Month (base load) Frequency Month (peak load) Month (base load) Kernel
Month (peak load) Kernel Month (base load) Normal distribution Month (peak load) Normal distribution
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SV-Models: Risk ManagementBivariate Estimations: NASDAQ OMX Front Month – EEX Front Base Month
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MU NP - EEX Month
Frequency Month NP Frequency Month EEX Month NP Kernel
Month NP Normal distribution Month EEX Kernel Month EEX Normal distribution
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Frequency Month NP Frequency Month EEX Month NP Kernel
Month EEX Kernel Month NP Normal distribution Month EEX Normal distribution
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SV-Models: Risk ManagementForecast unconditional First Moment: VaR/CVaR measures from Uni- and Bivariate Estimations (precentiles)
Univariate (long positions)Nord Pool EEX
Confidence Front Week Front Month Base Month Peak Monthlevels: VaR CVaR VaR CVaR VaR CVaR VaR CVaR
99.90 % 0.0333 0.0410 0.0240 0.0287 0.0195 0.0245 0.0246 0.030299.50 % 0.0237 0.0298 0.0176 0.0216 0.0129 0.0171 0.0171 0.021899.00 % 0.0198 0.0256 0.0152 0.0189 0.0107 0.0144 0.0140 0.018697.50 % 0.0155 0.0206 0.0122 0.0156 0.0079 0.0111 0.0104 0.014595.00 % 0.0124 0.0172 0.0102 0.0134 0.0060 0.0090 0.0080 0.011890.00 % 0.0096 0.0140 0.0082 0.0112 0.0043 0.0070 0.0059 0.0093
Bivariate (long positions)Nord Pool EEX Nord-Pool & EEX
Confidence Front Week Front Month Base Month Peak Month Front Month Base Monthlevels: VaR CVaR VaR CVaR VaR CVaR VaR CVaR VaR CVaR VaR CVaR
99.90 % 0.0378 0.0464 0.0343 0.0416 0.0228 0.0285 0.0307 0.0379 0.0150 0.0178 0.0220 0.027599.50 % 0.0266 0.0338 0.0240 0.0303 0.0148 0.0197 0.0210 0.0272 0.0114 0.0138 0.0144 0.019199.00 % 0.0220 0.0289 0.0201 0.0261 0.0123 0.0166 0.0171 0.0230 0.0099 0.0121 0.0119 0.016097.50 % 0.0170 0.0230 0.0155 0.0209 0.0090 0.0128 0.0125 0.0178 0.0079 0.0101 0.0087 0.012495.00 % 0.0133 0.0190 0.0122 0.0173 0.0068 0.0103 0.0094 0.0143 0.0064 0.0086 0.0066 0.009990.00 % 0.0098 0.0152 0.0092 0.0139 0.0048 0.0080 0.0067 0.0111 0.0048 0.0070 0.0047 0.0077
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SV-Models: Risk ManagementForecast Second Moment: Uni- and Bivariate Estimations NASDAQ OMX Front Week
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Mean
Density
One-step-ahead density fK(yt|xt-1,q) conditional on data value y = -0.347
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Mean
Density
One-step-ahead density fK(yt|xt-1,q) xt-1= -10,-5,-3,-1,-0.347, 0,+1,+3,+5,+10%
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%
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GAUSS-Hermite Quadrature: Conditional Mean Density Distribution
Reprojected Quadrature
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Percentage Growth (d)
The Conditional Variance Function for the "Assymmetry effect"
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SV-Models: Risk ManagementForecast Second Moment: Uni- and Bivariate Estimations NASDAQ OMX Front Month
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One-step-ahead density fK(yt|xt-1,q) conditional on data value y = -0.137
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One-step-ahead density fK(yt|xt-1,q) xt-1=-10,-5,-3,-1, -0.137, 0,+1,+3,+5,+10%
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%
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
GAUSS-Hermite Quadrature: Conditional Mean Density Distribution
Reprojected Quadrature
0
5
10
15
20
25
30
35
40
45
50
Percentage Growth (d)
The Conditional Variance Function for the "Assymmetry effect"
50
SV-Models: Risk ManagementForecast Second Moment: Uni- and Bivariate Estimations EEX Front Month (base load)
0
0.1
0.2
0.3
0.4
0.5
Conditonal
Mean
Density
One-step-ahead density fK(yt|xt-1,q) conditional on data value y = -0.044
0
0.025
0.05
0.075
0.1
0.125
0.15
0.175
0.2
0.225
0.25
0.275
0.3
0.325
0.35
0.375
0.4
0.425
0.45
0.475
0.5
0.525
Conditonal
Mean
Density
One-step-ahead density fK(yt|xt-1,q) xt-1= -10,-5,-3,-1,0,mean,+1,+3,+5,+10%
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%
0
0.1
0.2
0.3
0.4
GAUSS-Hermite Quadrature: Conditional Mean Density Distribution
Reprojected Quadrature
4
5
6
7
8
9
10
11
Percentage Growth (d)
The Conditional Variance Function for the "Assymmetry effect"
51
SV-Models: Risk ManagementForecast Second Moment: Uni- and Bivariate Estimations EEX Front Month (peak load)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Conditonal
Mean
Density
One-step-ahead density fK(yt|xt-1,q) conditional on data value y = -0.117
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Conditonal
Mean
Density
One-step-ahead density fK(yt|xt-1,q) xt-1= -10,-5,-3,-1,-0.12,0,+1,+3,+5,+10%
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%
0
0.1
0.2
0.3
0.4
GAUSS-Hermite Quadrature: Conditional Mean Density Distribution
Reprojected Quadrature
0
5
10
15
20
25
30
35
40
45
Percentage Growth (d)
The Conditional Variance Function for the "Assymmetry effect"
52
SV-Models: Risk ManagementForecast Second Moment: Uni- and Bivariate Estimations NASDAQ OMX Front Week/Month
-6.27
-5.02
-3.76
-2.51-1.25
0.001.25
2.513.76
5.026.27
0
0.05
0.1
0.15
0.2
0.25
0.3
-6.5
1
-5.9
9
-5.4
7
-4.9
5
-4.4
3
-3.9
1
-3.3
9
-2.8
6
-2.3
4
-1.8
2
-1.3
0
-0.7
8
-0.2
6
0.26
0.78
1.30
1.82
2.34
2.86
3.39
3.91
4.43
4.95
5.47
5.99
6.51
Front Month Contracts
Conditional
Density
Front Week Contracts
One-step-ahead density fK(yt|xt-1,q): xi,t-1 = unconditional mean (-0.32 /-0.12)
Mean(Week) =-0.129685 Stdev(Week) =2.170127 Mean(Month) =-0.146504 Stdev(Month) =2.090196 Covariance =4.41692 Correlation =0.97375
-10.58
-8.47
-6.35
-4.23
-2.120.00
2.124.23
6.358.4710.58
0
0.02
0.04
0.06
0.08
0.1
0.12
-11
.43
-10
.52
-9.6
0
-8.6
9
-7.7
7
-6.8
6
-5.9
4
-5.0
3
-4.1
1
-3.2
0
-2.2
9
-1.3
7
-0.4
6
0.4
6
1.3
7
2.2
9
3.2
0
4.1
1
5.0
3
5.9
4
6.8
6
7.7
7
8.6
9
9.6
0
10
.52
11
.43 NP Front Month
Conditional
Density
NP Front Week
One-step-ahead density fK(yt|xt-1,q): xi,t-1=-5%; -5%
Mean(Week) =-1.13343 Stdev(Week) =3.527478 Mean(Month) =-0.955496 Stdev(Month) =3.81 Covariance =13.1947 Correlation =0.98177
-10.84
-8.67
-6.50
-4.34-2.17
0.002.17
4.346.50
8.6710.84
0
0.02
0.04
0.06
0.08
0.1
0.12
-12
.24
-11
.26
-10
.28
-9.3
0
-8.3
2
-7.3
4
-6.3
7
-5.3
9
-4.4
1
-3.4
3
-2.4
5
-1.4
7
-0.4
9
0.4
9
1.4
7
2.4
5
3.4
3
4.4
1
5.3
9
6.3
7
7.3
4
8.3
2
9.3
0
10
.28
11
.26
12
.24
NP Front Month
Conditional
Density
NP Front Week
One-step-ahead density fK(yt|xt-1,q): xi,t-1=+5%; +5%
Mean(Week) =0.690381 Stdev(Week) =3.612949 Mean(Month) =0.402447 Stdev(Month) =4.080466 Covariance =14.5089 Correlation =0.98415
-15
-10
-5
0
5
10
15
-8
-4
0
4
8
0
0.01
0.02
0.03
0.04
0.05
0.06
X Y
ZGauss-Hermite Quadrature Nord Pool Front Week - Month
Frame 001 11 Mar 2011 cartesianplt
0.7
0.75
0.8
0.85
0.9
0.95
1
8
9
10
11
12
13
14
15
16
17
18
GrowthVariance NP Week Variance NP Month NP Covariance NP Correlation
Variance-Co-Variance/Correlation
53
SV-Models: Risk ManagementForecast Second Moment: Uni- and Bivariate Estimations EEX Front Month Base and Peak
-3.94
-3.15
-2.36
-1.58-0.79
0.000.79
1.582.36
3.153.94
0
0.2
0.4
0.6
0.8
1
1.2
-2.6
9
-2.4
8
-2.2
6
-2.0
5
-1.8
3
-1.6
2
-1.4
0
-1.1
9
-0.9
7
-0.7
5
-0.5
4
-0.3
2
-0.1
1
0.11
0.32
0.54
0.75
0.97
1.19
1.40
1.62
1.83
2.05
2.26
2.48
2.69
Front Month (peak load)
Conditional
Density
Front Month (base load)
One-step-ahead density fK(yt|xt-1,q): xi,t-1 = unconditional mean (-0.04 /-0.11)
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
-11.20
-8.96
-6.72
-4.48
-2.240.00
2.244.48
6.728.9611.20
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
-11
.22
-10
.32
-9.4
2
-8.5
3
-7.6
3
-6.7
3
-5.8
3
-4.9
4
-4.0
4
-3.1
4
-2.2
4
-1.3
5
-0.4
5
0.4
5
1.3
5
2.2
4
3.1
4
4.0
4
4.9
4
5.8
3
6.7
3
7.6
3
8.5
3
9.4
2
10
.32
11
.22 EEX Front Month (peak load)
Conditional
Density
EEX Front Month (base load)
One-step-ahead density fK(yt|xt-1,q): xi,t-1=-5%; -5%
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
-11.76
-9.41
-7.05
-4.70-2.35
0.002.35
4.707.05
9.4111.76
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
-11
.63
-10
.70
-9.7
7
-8.8
4
-7.9
1
-6.9
8
-6.0
5
-5.1
2
-4.1
9
-3.2
6
-2.3
3
-1.4
0
-0.4
7
0.4
7
1.4
0
2.3
3
3.2
6
4.1
9
5.1
2
6.0
5
6.9
8
7.9
1
8.8
4
9.7
7
10
.70
11
.63 EEX Front Month (peak load)
Conditional
Density
EEX Front Month (base load)
One-step-ahead density fK(yt|xt-1,q): xi,t-1=+5%; +5%
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
-4-3
-2
-1
0
1
2
3
4
5
-5-4
-3-2
-1
0
1
2
3
4
5
0
0.02
0.04
0.06
0.08
0.1
X Y
ZGauss-Hermite Quadrature EEX Front Months - Base and peak Load
Frame 001 11 Mar 2011 cartesianplt
0.5
0.6
0.7
0.8
0.9
1
4
5
6
7
8
9
10
11
12
13
14
15
16
Growth
EEX Front Month (base) Variance EEX Front Month (peak) Variance EEX Front Month Covariance EEX Front Month Correlation
EEX Variance, Co-Variance and Correlation
54
SV-Models: Risk ManagementForecast Second Moment: Uni- and Bivariate Estimations NASDAQ and EEX Front Month (base)
-2.64
-2.12
-1.59
-1.06
-0.530.00
0.531.06
1.592.122.64
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-4.8
2-4
.43
-4.0
5
-3.6
6
-3.2
8
-2.8
9
-2.5
0
-2.1
2
-1.7
3
-1.3
5
-0.9
6
-0.5
8
-0.1
9
0.19
0.58
0.96
1.35
1.73
2.12
2.50
2.89
3.28
3.66
4.05
4.43
4.82
EEX Front Month
Conditional
Density
NP Front Month
One-step-ahead density fK(yt|xt-1,q): xi,t-1 = unconditional mean (-0.13 /-0.04)
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
-11.20
-8.96
-6.72
-4.48-2.24
0.002.24
4.486.72
8.9611.20
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
-11
.22
-10
.32
-9.4
2
-8.5
3
-7.6
3
-6.7
3
-5.8
3
-4.9
4
-4.0
4
-3.1
4
-2.2
4
-1.3
5
-0.4
5
0.4
5
1.3
5
2.2
4
3.1
4
4.0
4
4.9
4
5.8
3
6.7
3
7.6
3
8.5
3
9.4
2
10
.32
11
.22 EEX Front Month (peak load)
Conditional
Density
NP Front Month (base load)
One-step-ahead density fK(yt|xt-1,q): xi,t-1=-5%; -5%
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
-13.19
-10.55
-7.91
-5.28-2.64
0.002.64
5.287.91
10.5513.19
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
-13
.19
-12
.13
-11
.08
-10
.02
-8.9
7
-7.9
1
-6.8
6
-5.8
0
-4.7
5
-3.6
9
-2.6
4
-1.5
8
-0.5
3
0.5
3
1.5
8
2.6
4
3.6
9
4.7
5
5.8
0
6.8
6
7.9
1
8.9
7
10
.02
11
.08
12
.13
13
.19 EEX Front Month (base load)
Conditional
Density
NP Front Month (base load)
One-step-ahead density fK(yt|xt-1,q): xi,t-1=+5%; +5%
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
-8
-4
0
4
-3-2
-10
12
3
00.010.020.030.040.050.060.070.08
X Y
ZGauss-Hermite Quadrature NP-EEX Front Months Contracts
Frame 001 11 Mar 2011 cartesianplt
0.4
0.5
0.6
0.7
0.8
0.9
1
2
4
6
8
10
12
14
16
18
20
Growth
Var Week Var Month Covariance Correlation
NP and EEX Front Month Variance, Co-Variance, and Correlation
55
SV-Models: Risk ManagementExtreme Value Theory for First Conditional Moment 5 k iterations
Excel
56
SV-Models: Risk ManagementExtreme Value Theory for First Conditional Moment 5 k iterations
57
SV-Models: Risk ManagementExtreme Value Theory for First Conditional Moment 5 k iterations
58
Risk Management (aggregation)Economic Capital and RAROC
Business Units (billion €)Hydro power Network Telephone
Economic Capital generation (B1) operation (B2) communication (B3)Market risk (M) 150 45 82Basis Risk (B) 95 38 50Operational Risk (O) 55 25 34
Panel A:
CorrelationStructure 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 0 0.25 0 0.2 1 -0.1 0.1 0.2 0.05OB2 0.1 0 0.2 0.1 -0.1 1 0 0.1 0MB3 0.3 0.05 0.1 0 0.1 0 1 0.1 0BB3 0 0.1 0.1 0 0.2 0.1 0.1 1 0.05OB3 0.05 0 0 0.1 0.05 0 0 0.05 1
Panel B:
1 1
n n
total i j iji j
E E E
Hybrid approach:
The market risk economic capital: 233.41The basis risk economic capital: 159.37
The operational risk economic capital: 98.32
Etotal: 299.73
Copula approach:
MCMC 10 k for well behaved distributions with correlation structures (Cholesky):
Normal distribution:Etotal = 305.06 st.dev =47.5
Student-t (4 df):Etotal = 304.21 st.dev =51.8
Student-t (2 df):Etotal = 318.58 st.dev = 222.4
59
Risk Management (aggregation)Economic Capital and RAROC: Using Copulas and Correlation structuresVaR/CVaR for Normal distributions
60
Risk Management (aggregation)Economic Capital and RAROC: Using Copulas and Correlation structuresVaR/CVaR for Student-t distribution 4 degrees of freedom
61
Risk Management (aggregation)Economic Capital and RAROC: Using Copulas and Correlation structuresVaR/CVaR for Student-t distribution 2 degrees of freedom
62
Future work….?Operational Forecasting (efficient algorithms)Higher Conditional Moments (skew/kurtosis)Volatility (particle filtering) and pricing exotic
optionsMultiple-ahead-forecasts for mean and
volatilityPersistence measuresNew information and the SV models-
conceptMultivariate SV models forecasts: market
arbitrageClosed-form solution SV models and energy
markets.
63
Summary & ConclusionsFree methodologySV-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 OPEN-MPI (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.