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1 Market Risk Management using Stochastic Volatility Models The Case of European Energy Markets

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2 Outline Preliminaries, markets, instruments and hedging Preliminaries, markets, instruments and hedging –Relevant risk, std, volatility + + + ….. – Markets, instruments and models +++ Value at Risk, Expected Shortfall, Volatility and Covariances Value at Risk, Expected Shortfall, Volatility and Covariances Stochastic Volatility Models Stochastic Volatility Models –Definition and Motivation –Projection, estimation and re-projection The Nordpool and EEX Energy Markets The Nordpool and EEX Energy Markets SV model parameters SV model parameters Assessment and empirical findings Assessment and empirical findings Market Risk Management Market Risk Management SV-model forecasts and Risk Management SV-model forecasts and Risk Management One-day-ahead forecasts and Risk Management One-day-ahead forecasts and Risk Management Summaries and Conclusions Summaries and Conclusions

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3 Main Objectives Forecasting Risk Management Measures Forecasting Risk Management Measures SV model forecasts of VaR, CVaR and Greek letter densities SV model forecasts of VaR, CVaR and Greek letter densities Conditional Moments Forecasts Conditional Moments Forecasts One-day-ahead densities of VaR, CVaR and Greek letters One-day-ahead densities of VaR, CVaR and Greek letters Extreme value theory and VaR, CVaR and Greek letter densities Extreme value theory and VaR, CVaR and Greek letter densities

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4 Preliminaries Portfolio Theory Basics for Investors

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5 Preliminaries Portfolio Theory Basics (relevant risk measures):

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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: total versus i = ( i * M ) / M theTraynor index versus the Sharpe index

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7 Preliminaries Financial Products and Markets Financial Products / Plain Vanilla products Long and Short positions in Assets Forward Contracts / Future Contracts Swaps Options

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8 Preliminaries European Energy Markets and Activity/Liquidity for 2008 -2009 (annual reports)

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9 Preliminaries Financial Products and Positions Hedging Positions for plain Assets 0 Profit Loss Underlying asset (S t ) S0S0 Long position Asset Short position Asset Long Positions Payoff: S t – S 0 Short Positions Payoff: S 0 – S t

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10 Preliminaries Financial Products and Positions Hedging Positions for plain Forward/Future Products 0 Profit Loss Underlying asset (S t ) K Long position Forward/Future Short position Forward/Future Long Positions Payoff: S t – K Short Positions Payoff: K – S t

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11 Preliminaries Financial Products and Positions Hedging Positions for plain buying (long) positions in Call/Put options 0 Profit Loss Underlying asset (S t ) K Buying a Call position Buying a put position Call position Payoff: Max(0;S t – K)-c Put Positions Payoff: Max(0;K – S t )-p

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12 Preliminaries Financial Products and Positions Hedging Positions for plain selling (short) positions in Call/Put options 0 Profit Loss Underlying asset (S t ) K Selling a Call position Selling a put position Call position Payoff: -Max(0;S t – K)+c Put Positions Payoff: -Max(0;K – S t )+p

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13 Preliminaries Management of Portfolio Exposures: Greek Letters The sensitivity of the portfolios value to the price of the underlying asset: The rate of change of the portfolios delta with respect to the price of the underlying asset: The rate of change of the value of the portfolio with respect to the volatility of the underlying asset: The rate of change of the value of the portfolio with respect to the passage of time (time decay): The rate of change of the value of the portfolio with respect to the level of interest rates:

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14 Preliminaries Calculation of the GREEK LETTERS Taylor Series Expansion on a single market variable S (volatility and interest rates are assumed constant) 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 ): 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.

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15 Why model & forecast volatility? Hard to model and forecast return (price) in energy markets (ACF…) Hard to model and forecast return (price) in energy markets (ACF…) Energy market uncertainty & public confidence Energy market uncertainty & public confidence – determine option price, barometer for vulnerability Stylized facts about volatility Stylized facts about volatility – Volatility clustering, asymmetry and mean reversion… Correlations and Copulas Correlations and Copulas – Increase and decrease in energy market correlation Volatility forecasts Volatility forecasts – In-sample forecast & out-of-sample forecast

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16 ACF of return, squared return, and absolute return NP OYF_Squared Returns NP OYF_Returns NP OYF Absolute Returns

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17 Stylized facts about volatility Definition of volatility ( ) The standard deviation of the return (r t ) provided by the variable per unit time when the return is expressed using continuous compounding. = 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. 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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18 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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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 of the variable has the property that, when x is large: where K and 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. The logarithm of the probability is approx. linearly dependent on ln x for x >3 showing that the power law holds. Stylized facts about volatility

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20 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 n u =31 observations less than u. We maximize the log-likelihood function: Using the estimates (optimized): 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 Stylized facts about volatility

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21 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.

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22 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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23 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, valid for, and defined forSV model definition:

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24 Stylized facts about volatility Co-movements in volatility / Correlations

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25 Models for volatility estimations/forecasts Time series modelsOptions-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.

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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.

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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 observations S i : value of variable at end of i th interval, where i = 0, 1, …, n :length of time interval for i = 1, 2, …, n. The standard deviation of the, where 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:

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28 Correlations /Co-movements in Volatility Define :correlation between two variables V 1 and V 2 An analogy for covariance is the pervious variance/volatility. For risk management, if changes in two or more variables have a high positive correlation, the companys 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 V 1 and V 2 is and the correlation can therefore be written as: Stochastic Volatility Models

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29 Correlations /Co-movements in Volatility and COPULAS and N is the cumulative normal distribution function. This means that The variables U 1 and U 2 are then assumed to be bivariate normal. The key property of a copula model is that it preserves the marginal distribution of V 1 and V 2 (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 F 1 and F 2 are the cumulative marginal probability distributions of V 1 and V 2. We map V 1 = 1 to U 1 = u 1 and V 2 = 2 to U 2 = u 2, where Stochastic Volatility Models

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30 A Scientific Stochastic volatility model Let y t 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 z 1t and z 2t 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.

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31 GSM estimated SV-models for NordPool and EEX European Energy Markets Stochastic Volatility Models

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32 GSM Assessment of SV Model Simulation fit: Stochastic Volatility Models

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33 GSM Assessment of SV Model Simulation fit: Stochastic Volatility Models

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

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

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

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

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

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39 SV-Models: Risk Management Densities Percentiles: VaR, CVaR positions for 4 contracts 100 k Excel

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40 SV-Models: Risk Management EVT: VaR, CVaR Positions for 4 contracts 100 k

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41 SV-Models: Risk Management EVT densities: VaR, CVaR Positions for 4 contracts 100 k

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

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

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44 SV-Models: Risk Management Bivariate Estimations: NASDAQ OMS Front Week – Front Month

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45 SV-Models: Risk Management Bivariate Estimations: EEX Front Base Month – Front Peak Month

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46 SV-Models: Risk Management Bivariate Estimations: NASDAQ OMX Front Month – EEX Front Base Month

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47 SV-Models: Risk Management Forecast unconditional First Moment: VaR/CVaR measures from Uni- and Bivariate Estimations (precentiles)

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48 SV-Models: Risk Management Forecast Second Moment: Uni- and Bivariate Estimations NASDAQ OMX Front Week

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49 SV-Models: Risk Management Forecast Second Moment: Uni- and Bivariate Estimations NASDAQ OMX Front Month

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

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51 SV-Models: Risk Management Forecast Second Moment: Uni- and Bivariate Estimations EEX Front Month (peak load)

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52 SV-Models: Risk Management Forecast Second Moment: Uni- and Bivariate Estimations NASDAQ OMX Front Week/Month

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53 SV-Models: Risk Management Forecast Second Moment: Uni- and Bivariate Estimations EEX Front Month Base and Peak

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54 SV-Models: Risk Management Forecast Second Moment: Uni- and Bivariate Estimations NASDAQ and EEX Front Month (base)

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55 SV-Models: Risk Management Extreme Value Theory for First Conditional Moment 5 k iterations Excel

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56 SV-Models: Risk Management Extreme Value Theory for First Conditional Moment 5 k iterations

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57 SV-Models: Risk Management Extreme Value Theory for First Conditional Moment 5 k iterations

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58 Risk Management (aggregation) Economic Capital and RAROC Panel A: Panel B: Hybrid approach: The market risk economic capital: 233.41 The basis risk economic capital: 159.37 The operational risk economic capital: 98.32 E total : 299.73 Copula approach: MCMC 10 k for well behaved distributions with correlation structures (Cholesky): Normal distribution: E total = 305.06 st.dev =47.5 Student-t (4 df): E total = 304.21 st.dev =51.8 Student-t (2 df): E total = 318.58 st.dev = 222.4

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59 Risk Management (aggregation) Economic Capital and RAROC: Using Copulas and Correlation structures VaR/CVaR for Normal distributions

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60 Risk Management (aggregation) Economic Capital and RAROC: Using Copulas and Correlation structures VaR/CVaR for Student-t distribution 4 degrees of freedom

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61 Risk Management (aggregation) Economic Capital and RAROC: Using Copulas and Correlation structures VaR/CVaR for Student-t distribution 2 degrees of freedom

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62 Future work….? Operational Forecasting (efficient algorithms) Operational Forecasting (efficient algorithms) Higher Conditional Moments (skew/kurtosis) Higher Conditional Moments (skew/kurtosis) Volatility (particle filtering) and pricing exotic options Volatility (particle filtering) and pricing exotic options Multiple-ahead-forecasts for mean and volatility Multiple-ahead-forecasts for mean and volatility Persistence measures Persistence measures New information and the SV models- concept New information and the SV models- concept Multivariate SV models forecasts: market arbitrage Multivariate SV models forecasts: market arbitrage Closed-form solution SV models and energy markets. Closed-form solution SV models and energy markets.

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63 Summary & Conclusions Free methodology Free methodology SV-models for energy, equity, currency markets. Portfolio applications and forecasting. SV-models for energy, equity, currency markets. Portfolio applications and forecasting. The number of CPUs are not important any longer. Apple (linux) 8 core computer with HYPERTHREAD has 16 cores for running OPEN- MPI (downloadable from Indiana Univeristy) The number of CPUs 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. Running every day obtaining one-day-ahead forecasts, induce 30-50% VaR/CVaR reduction and the Greek letters seem to move significantly.

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