market risk management using stochastic volatility models

1

Market Risk Managementusing Stochastic Volatility

Models

The Case of European Energy Markets

2

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

3

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

4

PreliminariesPortfolio Theory Basics for Investors

1

2 ,

N

p i ii

p i j i j i j

E R E r

r r

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0 0.02 0.04 0.06 0.08 0.1 0.12

Kapitalmarkedslinjen

Portefølje Standardavvik

5

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

2.0000 %

3.0000 %

4.0000 %

5.0000 %

6.0000 %

7.0000 %

8.0000 %

9.0000 %

10.0000 %

11.0000 %

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Markedsavkastningslinjen

ProsjektA

ProsjektB

6

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

7

PreliminariesFinancial Products and Markets

Financial Products / “Plain Vanilla” products

Long and Short positions in Assets

Forward Contracts / Future Contracts

Swaps

Options

8

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.

9

PreliminariesFinancial Products and Positions

Hedging Positions for plain Assets

0

Profit

Loss

Underlying asset (St)

S0

Long position Asset

Short position Asset

Long Positions Payoff:

St – S0

Short Positions Payoff:

S0 – St

10


Hedging Positions for plain Forward/Future Products

0

Profit

Loss


K

Long position Forward/Future

Short position Forward/Future

Long Positions Payoff:

St – K

Short Positions Payoff:

K – St

11


Hedging Positions for plain buying (long) positions in Call/Put options

0

Profit

Loss


K

Buying aCall position

Buying a put position Call position

Payoff:

Max(0;St – K)-c

Put Positions Payoff:

Max(0;K – St)-p

12


Hedging Positions for plain selling (short) positions in Call/Put options

0

Profit

Loss


K

Selling aCall positionSelling a put

position

Call positionPayoff:

-Max(0;St – K)+c

Put Positions Payoff:

-Max(0;K – St)+p

13

PreliminariesManagement of Portfolio Exposures: Greek Letters

DeltaS

The sensitivity of the portfolios value to the price of the underlying asset:

2

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:

14

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

17

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.

0

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

18


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).

19

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

-7

-6

-5

-4

-3

-2

ln(p

rob

(v <

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)


20

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:

11

( )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

1

31 0.15 0,101 0.0025

2809

280931 1 0.99 1 0.102897

31VaR


21


Volatility ClusteringRefer to the observation of large movements of price changes are being followed by large movements. That is, persistence of shocks.

0

2

4

6

8

10

12

14

16

18

20

22

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)

22


Asymmetric Volatility (called leverage in equity markets)Refer to the idea that price movements are negatively (positively) correlated with volatility

10

15

20

25

30

35

40

45

50

Percentage Growth (d)

Nord Pool Front Week: Conditional Variance Function for the "Assymmetry effect"

0

5

10

15

20

25

30

35


EEX Base Month: Conditional Variance Function for the "Assymmetry effect"

23


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:

24


-5.81

-4.65

-3.49

-2.32-1.16

0.001.16

2.323.49

4.655.81

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

-5.62

-5.17

-4.72

-4.27

-3.82

-3.37

-2.92

-2.47

-2.02

-1.57

-1.12

-0.67

-0.22

0.22

0.67

1.12

1.57

2.02

2.47

2.92

3.37

3.82

4.27

4.72

5.17

5.62

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

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1Nord Pool: Correlation Week - Month Contracts

Co-movements in volatility / Correlations

25

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)

26

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.

27

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

28

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

29

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


30

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

31

GSM estimated SV-models for NordPool and EEX European Energy Markets


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




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 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




32

GSM Assessment of SV Model Simulation fit:


33

GSM Assessment of SV Model Simulation fit:


34

Stochastic Volatility ModelsSV-model Features (2 markets and 4 contracts): NASDAQ OMX Front Week 100 k

35

SV-model Features (2 markets and 4 contracts): NASDAQ OMX Front Week 100 k


36

Stochastic Volatility ModelsSV-model Features (2 markets and 4 contracts): EEX Front Month (peak load) 100 k

37

SV-model Features (2 markets and 4 contracts): EEX Front Month (peak load) 100 k


38

Stochastic Volatility ModelsSV-model Features (2 markets and 4 contracts): Correlation Week/Month 100 k

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1EEX: Correlation Front Month - Base and Peak Load

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1Nord Pool: Correlation Week - Month Contracts

39

SV-Models: Risk ManagementDensities Percentiles: VaR, CVaR positions for 4 contracts 100 k

Excel

40

SV-Models: Risk ManagementEVT: VaR, CVaR Positions for 4 contracts 100 k

41

SV-Models: Risk ManagementEVT densities: VaR, CVaR Positions for 4 contracts 100 k

42

SV-Models: Risk ManagementGreek Letter densities (delta reported) for NASDAQ Week and Month 100 k

43

SV-Models: Risk ManagementGreek Letter densities (delta reported) for EEX Base and Peak Load Month Futures 100 k

44

SV-Models: Risk Management

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

MU Week + Month

Frequency Week Frequency Month Week Kernel Week Normal distribution Month Kernel Month Normal distribution

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

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

45

SV-Models: Risk ManagementBivariate Estimations: EEX Front Base Month – Front Peak Month

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

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

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05 SIG Week + Month

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

46

SV-Models: Risk ManagementBivariate Estimations: NASDAQ OMX Front Month – EEX Front Base Month

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

MU NP - EEX Month

Frequency Month NP Frequency Month EEX Month NP Kernel

Month NP Normal distribution Month EEX Kernel Month EEX Normal distribution

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05 SIG Month NP + EEX

Frequency Month NP Frequency Month EEX Month NP Kernel

Month EEX Kernel Month NP Normal distribution Month EEX Normal distribution

47

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

48

SV-Models: Risk ManagementForecast Second Moment: Uni- and Bivariate Estimations NASDAQ OMX Front Week

0

0.05

0.1

0.15

0.2

0.25

Conditonal

Mean

Density

One-step-ahead density fK(yt|xt-1,q) conditional on data value y = -0.347

0

0.05

0.1

0.15

0.2

0.25

Conditonal

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%

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


The Conditional Variance Function for the "Assymmetry effect"

49

SV-Models: Risk ManagementForecast Second Moment: Uni- and Bivariate Estimations NASDAQ OMX Front Month

0

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

0.225

Conditonal

Mean

Density


0

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

0.225

Conditonal

Mean

Density

One-step-ahead density fK(yt|xt-1,q) xt-1=-10,-5,-3,-1, -0.137, 0,+1,+3,+5,+10%



0

0.05

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SV-Models: Risk ManagementForecast Second Moment: Uni- and Bivariate Estimations EEX Front Month (base load)

0

0.1

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0.5

Conditonal

Mean

Density


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%



0

0.1

0.2

0.3

0.4



4

5

6

7

8

9

10

11



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


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= "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



0

5

10

15

20

25

30

35

40

45



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)


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)


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


Conditional

Density

EEX Front Month (base 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

-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


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


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


Conditional

Density

NP Front Month (base load)


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)


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


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


57


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.

market risk management using stochastic volatility models

Documents

positionshedging positions

cput positions payoff

risk managementsummaries

hedgingrelevant risk

risk assessment

risk aggregation

st s0short positions

st kshort positions