1. Page: 1 Carbon Phase II Density Applications using General Scientific Stochastic Volatility Models by Professor Per B Solibakke a Nord Pool Application: Carbon Front December Future Contracts a) Department of Economics, Molde University College Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

2. Background and Outline 1.The Front December Future Contracts NASDAQ OMX: phase II 2008-2012  No EUAs  the theoretical spot-forward relationship does not exist  Price dynamics are depending on total emissions (extreme-dynamics)  EUA options have carbon December futures as underlying instrument Page: 2 2. The dynamics of the forward rates are directly specified  The HJM-approach adopted to modelling forward- and futures prices in commodity markets  Alternatively, we model only those contracts that are traded, resembling swap and LIBOR models in the interest rate market ( also known as market models). Construct the dynamics of traded contracts matching the observed volatility term structure  The EUA options market on carbon contract are rather thin (daily prices are however reported), we estimate the model on the future prices themselves. Black-76 / MCMC simulations Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

3. Background and Outline (cont.) 3. Stochastic Model Specification: Estimation, Assessment and Inference 4. Forecasting unconditional Futures Moments, and Risk Management and Asset Allocation measures 5. Forecasting conditional Futures Moments i.One-step-ahead Conditional Mean (expectations) and Standard Deviation ii.Conditional one-step-ahead Risk Management and Asset Allocation measures iii. Volatility/Particle filtering for Option pricing iv. Multi-step-ahead Mean and Volatility Dynamics v. Mean Reversion and Volatility Persistence measures 6. The EMH case of CARBON Futures/Options for Commodity Markets Page: 3

4. Page: 4 The Carbon NASDAQ OMX commodity market NASDAQ OMXcommodities provide market access to one of Europe’s leading carbon markets. 350 market members from 18 countries covering a wide range of energy producers, consumers and financial institutions. Members can trade cash-settlement derivatives contracts in the Nordic, German, Dutch and UK power markets with futures, forward, option and CfD contracts up to six years’ duration including contracts for days, weeks, months, quarters and years. The reference price for the power derivatives is the underlying day-ahead price as published by Nord Pool spot (Nordics), the EEX (Germany), APX ENDEX (the Netherlands), and N2EX (UK). Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

5. Indirect Estimation and Inference: 1.Projection: The Score generator (A Statistical Model) establish moments:  the Mean (AR-model)  the Latent Volatility ((G)ARCH-model)  Hermite Polynomials for non-normal distribution features 2.Estimation: The Scientific Model – A Stochastic Volatility Model where z 1t, z 2t and (z 3t ) are iid Gaussian random variables. The parameter vector is: Page: 5 The General Scientific Model methodology (GSM): SDE: A vector SDE with two stochastic volatility factors. Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

6. Page: 6 Application references: Andersen and Lund (1997): Short rate volatility Solibakke, P.B (2001): SV model for Thinly Traded Equity Markets Chernov and Ghysel (2002): Option pricing under Stochastic Volatility Dai & Singleton (2000) and Ahn et al. (2002): Affine and quadratic term structure models Andersen et al. (2002): SV jump diffusions for equity returns Bansal and Zhou (2002): Term structure models with regime-shifts Gallant & Tauchen (2010):Simulated Score Methods and Indirect Inference for Continuous-time Models 3.Re-projection and Post-estimation analysis:  MCMC simulation for Risk Management and Asset allocation  Conditional one-step-ahead mean and volatility densities.  Forecasting volatility conditional on the past observed data; and/or extracting volatility given the full data series (particle filtering/option pricing)  The conditional volatility function, multi-step-ahead mean and volatility and mean/volatility persistence. Other extensions for specific applications. The General Scientific Model methodology (GSM):

7. Page: 7 Stochastic Volatility Models: Simulation-based Inference Early references are: Kim et al. (1998), Jones (2001), Eraker (2001), Elerian et al. (2001), Roberts & Stamer (2001) and Durham (2003). A successful approach for diffusion estimation was developed via a novel extension to the Simulated Method of Moments of Duffie & Singleton (1993). Gouriéroux et al. (1993) and Gallant & Tauchen (1996) propose to fit the moments of a discrete-time auxiliary model via simulations from the underlying continuous-time model of interest  EMM/GSM First, use an auxiliary (statistical) model with a tractable likelihood function and generous parameterization to ensure a good fit to all significant features of the time series. Second, a very long sample is simulated from the continuous time model. The underlying parameters are varied in order to produce the best possible fit to the quasi-score moment functions evaluated on the simulated data. Under appropriate regularity, the method provides asymptotically efficient inference for the continuous time parameter vector. Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

8. Page: 8 Simulated Score Methods and Indirect Inference for Continuous-time Models (some details): The idea (Bansal et al., 1993, 1995 and Gallant & Lang, 1997; Gallant & Tauchen, 1997): Use the expectation with respect to the structural model of the score function of an auxiliary model as the vector of moment conditions for GMM estimation. The score function is the derivative of the logarithm of the density of the auxiliary model with respect to the parameters of the auxiliary model. The moment conditions which are obtained by taking the expectations of the score depends directly upon the parameters of the auxiliary model and indirectly upon the parameters of the structural model through the dependence of expectation operator on the parameters of the structural model. Replacing the parameters from the auxiliary model with their quasi-maximum likelihood estimates, leaves a random vector of moment conditions that depends only on the parameters of the structural model.

9. Page: 9 Simulated Score Methods and Indirect Inference for Continuous-time Models (some details): Estimation Simulated Score Estimation: Suppose that: is a reduced form model for observed time series, where x t-1 is the state vector of the observable process at time t-1 and y t is the observable process. Fitted by maximum likelihood we get an estimate of the average of the score of the datasatisfies: That is, the first-order condition of the optimization problem. Having a structural model (i.e. SV) we wish to estimate, we express the structural model as the transition density, where  is the parameter vector. It can be relatively easy to simulate the structural model and is the basic setup of simulated method of moments (Duffie and Singleton, 1993; Ingram and Lee, 1991). Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

10. The scientific model is now viewed as a sharp prior that restricts the -parameters to lie on the manifold Main question: How do the results change as this prior is relaxed? That is: How does the marginal posterior distribution of a parameter or functional of the statistical model change? Distance from the manifold: where A j is the scaling matrices. Hence, we propose that the scientific model be assessed by plotting suitable measures of the location and scale of the posterior distribution of against, or, better sequential density plots. Page: 10 For a well fitting scientific model: The location measure should not move by a scientifically meaningful amount as  increases. The result indicates that the model fits and that the scale measure increases, indicating that the scientific model has empirical content. Simulated Score Methods and Indirect Inference for Continuous-time Models (some details): Assessment

11. Page: 11 Simulated Score Methods and Indirect Inference for Continuous-time Models (some details): Re-projection / Post-Estimation Analysis Elicit the dynamics of the implied conditional density for observables: The unconditional expectations can be generated by a simulation: Now define: whereis the score (SNP) density. Let. Theorem 1 of Gallant and Long (1997) states: We study the dynamics of by using as an approximation. Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

12. Page: 12 Simulated Score Methods and Indirect Inference for Continuous-time Models (some details): Re-projection / Post-Estimation Analysis Filtered volatility involves estimating an unobserved state variable conditional upon all past and present values. Denote v (unobserved) and y (contemporaneous and lagged observed variables) thus modified by v* and y*, respectively. Hence, simulatefrom the optimal structural model and re-project to get The price of an option (re-projected volatility) can now be calculated as: whereand. Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

13. Application: Financial CARBON Futures Contracts NORD POOL (Phase II: 2008-2012) Front December Futures Contracts (EUA options will have the December futures as the underlying instrument) NEDEC (-X) specification at NASDAQ OMX FTP-server Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

14. Page: 14 Objectives (purpose of the working paper): Higher Understanding of the Carbon Futures Commodity Markets  the Mean equations (drift, serial correlation, mean reversion)  the Volatility equations (constant, persistence, asymmetry, multiple factors) Models derived from scientific considerations and theory is always preferable  Fundamentals of Stochastic Volatility Models  Likelihood is not observable due to latent variables (volatility)  The model is continuous but observed discretely (closing prices) Bayesian Estimation Approach is credible  Accepts prior information  No growth conditions on model output or data  Estimates of parameter uncertainty (distributions) is credible Establish Financial Contracts Characteristics Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

15. Page: 15 Value-at-Risk / Expected Shortfall for Risk Management  Stochastic Volatility models are well suited for simulation  Using Simulation and Extreme Value Theory for VaR-/CVaR-Densities Simulations and Greek Letters Calculations for Asset Allocation  Direct path wise hedge parameter estimates  MCMC superior to finite difference, which is biased and time-consuming Re-projection for Simulations and Forecasting (conditional moments)  Conditional Mean and Volatility forecasting  Volatility Filtering and Pricing of Commodity Options  Consequences of Shocks, Multiple-step ahead Dynamics and Persistence. The Case against the Efficiency of Future Markets (EMH)  Serial correlation in Mean and Volatility  Price-Trend-Forecasting models and Risk premiums  Predictability and Efficient use of Available Information Objectives (purpose): (cont) Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

16. Page: 16 SV models has a simple structure and explain the major stylized facts. Moreover, market volatilities change so frequent that it is appropriate to model the volatility process by a random variable. Note, that all model estimates are imperfect and we therefore has to interpret volatility as a latent variable (not traded) that can be modelled and predicted through its direct influence on the magnitude of returns. Mainly three motivational factors: 1. Unpredictable event on day t; proportional to the number of events per day. (Taylor, 86) 2. Time deformation, trading clock runs at a different rate on different days; the clock often represented by transaction/trading volume (Clark, 73). 3. Approximation to diffusion process for a continuous time volatility variable; (Hull & White (1987) Objectives (why): Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

17. Page: 17 Other motivational factors: 4. A model of futures markets directly, without considering spot prices, using HJM-type models. A general summary of the modelling approaches for forward curves can be found in Eydeland and Wolyniec (2003). Matching the volatility term structure. 5. In order to obtain an option pricing formula the futures are modelled directly. Mean and volatility functions deriving prices of futures as portfolios. Such models can price standardized options in the market. Moreover, the models can provide consistent prices for non-standard options. 6. Enhance market risk management, improve dynamic asset/portfolio pricing, improve market insights and credibility, making a variety of market forecasts available, and improve scientific model building for commodity markets. Objectives (why): Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

18. Page: 18 1. NASDAQ OMX Carbon front December contracts 2. The Statistical model and the Stochastic Volatility Model 3. Model assessment (relaxing the prior): model appropriate? 4.Empirical Findings in the mean and latent volatility. Unconditional mean and latent volatility paths/distributions Carbon Post-Estimation Analysis: 1. SV-model simulations: i) Mean, Risk Management and Asset Allocation, ii) Realized Volatility (continuous versus jumps). 2 Conditional Mean and Volatility: i) Risk management and Asset Allocation ii) Volatility (particle) Filtering and the Pricing of Options iii) Variance functions, multi-step ahead dynamics and persistence 3. EMH and Model Summary/Conclusion Carbon Density Applications MCMC estimation/inference: Model Assessment SV Model Findings Risk Man./Asset Alloc EMH/Model Summary Data Characteristics Estimation Results Re-projection/Post-Est Filtering/Option Prices

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20. Page: 20 Carbon front December Contracts: Return Carbon Density Applications for Front December Futures Contracts

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22. Scientific Models: Stochastic Volatility Model /Parameters (  ) Bayesian Estimation Results 1. Several serial Bayesian runs establish the mode We tune the scientific model until the posterior quits climbing and it looks like the mode has been reached: Then: 2. A final parallel run with 32 (8 cores *4 CPUs) CPUs and 240.000 MCMC simulations (Linux/Ubuntu 12.04 LTS & OPEN_MPI (Indiana University) parallell computing) Page: 22 Return Carbon Density Applications for Front December Futures Contracts Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

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24. Page: 24 Scientific Model: Model Assessment – the model concert test Carbon front December k = 1, 10, 20 and 100 densities – reported. Carbon Density Applications for Front December Futures Contracts

25. Page: 25 Scientific Model: The Stochastic Volatility Model: log-sci-mod-posterior Log sci-mod-posterior (every 25 th observation reported): Optimum is along this path! Optimum: Carbon Density Applications for Front December Futures Contracts

26. Page: 26 Scientific Model: Carbon  -paths and densities; 240.000 simulations Carbon Density Applications for Front December Futures Contracts

27. Page: 27 Scientific Model: Stochastic Volatility The chains look good. Rejection rates are:  The MCMC chain has found its mode.  A well fitted scientific SV model: The location measure is not moved by a scientifically meaningful amount as  increases. The result indicates that the model fits and that the scale measure increases, indicating that the scientific model has empirical content. Return Carbon Density Applications for Front December Futures Contracts Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

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29. Empirical Model Findings: For the mean stochastic equation:  Positive mean drift (a 0 = 0.026; s.e. = 0.03) and serial correlation (a 1 = 0.054; s.e. 0.021) for the CARBON contracts For the latent volatility: two stochastic volatility equations:  Positive constant parameter (e 0.6305 >> 1)  Two volatility factors (s 1 = 0.0624, s.e.=0.0161; s 2 = 0.2263, s.e.=0.0329)  persistence is high for s 1 with associated (b 1 = 0.985, s.e. = 0.0381) ; persistence is lower for s 2 with associated (b 2 = 0.5775, s.e.=0.0806)  Asymmetry is strong and negative (   = -0.4324, s.e.=0.1130) Page: 29 Carbon Density Applications for Front December Futures Contracts Return

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31. Scientific Model: The Stochastic Volatility Model. Page: 31 Risk assessment and management: CARBON VaR / CVaR Carbon Density Applications for Front December Futures Contracts Extreme Value Theory (Gnedenko, 1943) is used for VaR and CVAR calculations. VaR is calculated as: CVaR is calculated from VaR as: where (for power law implementation) is the 95% percentile of the empirical distribution, n is the total number of observations, n u is the number of observations exceeding u, and q = VaR confidence level. The empirical distribution is from a 250 k long unconditional SV optimal simulation. Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

32. Scientific Model: The Stochastic Volatility Model. Page: 32 Risk assessment and management: CARBON VaR / CVaR Carbon Density Applications for Front December Futures Contracts

33. Scientific Model: The Stochastic Volatility Model. Page: 33 Asset Allocation/Dynamic Hedging: CARBON GREEK Letters Carbon Density Applications for Front December Futures Contracts

34. The Scientific Model: Simulations from the optimal SV-model: Realized Volatility and continuous / jump volatility (5 minutes simulations): Page: 34 Return Carbon Density Applications for Front December Futures Contracts Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

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36. Page: 36 Scientific Model: Re-projections / Nonlinear Kalman filtering Of immediate interest of eliciting the dynamics of observables: One-step ahead conditional mean: One-step ahead conditional volatility: Carbon Density Applications for Front December Futures Contracts Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

37. Page: 37 SV Model: One-step-ahead conditional moments Carbon Density Applications for Front December Futures Contracts

38. Scientific Model Re-projections: Conditional SV-model moments: Conditional VaR/CVaR for Risk Management and Greeks for Asset allocation Page: 38 Return Carbon Density Applications for Front December Futures Contracts

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40. Page: 40 Scientific Model: Re-projections / Nonlinear Kalman filtering Filtered volatility is the one-step ahead conditional standard deviation evaluated at data values: where y t denotes the data and y k0 denotes the k th element of the vector y 0, k = 1,…M. Carbon Density Applications for Front December Futures Contracts Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

41. Page: 41 SV Model: filtered volatility / particle filtering Carbon Density Applications for Front December Futures Contracts

42. Page: 42 Scientific Model: Re-projections / Nonlinear Kalman filtering The most predominant application is Option Pricing. Step 1: A long simulation(i.e. 250 k): where v * is the unobserved volatility and y * the observed returns (incl. lags). Step 2: Make a new projection to get the BIC optimal f K density with lags set generously long and non-linearity is available. The result is the unconditional mean from the raw the simulations and the conditional volatility is the conditional mean from f K : The conditional volatility mean can be used to obtain an estimate of: for the purpose of pricing an option (particle filtering). Carbon Density Applications for Front December Futures Contracts Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

43. Scientific Model: The Stochastic Volatility Model. Page: 43 The NASDAQ OMX market: The Re-projected Volatility Model Option Prices Unconditional and Conditional Mean Volatility Densities Carbon Density Applications for Front December Futures Contracts

44. Page: 44 Scientific Model: Re-projections / Nonlinear Kalman filtering Risk Premium (non-diversifiable risk) calculations are based on information at t-1. For contract i we obtain the risk premium R i at t-1 as: The risk premium is interpreted as non-diversifiable risk in the commodity market and is added as a constant to the Re-projected and the Black-76 average volatility for market comparisons. Note that the risk premium does not imply arbitrage opportunities within a market if risk is treated consistently. However, the risk premium may induce arbitrage opportunities between carbon markets (i.e. NASDAQ OMX and ECX (theice.com)). Carbon Density Applications for Front December Futures Contracts Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

45. Scientific Model: The Stochastic Volatility Model. Page: 45 The NASDAQ OMX market: The Re-projected Volatility Model Option Prices Market Implied Risk Premiums (re-projected model versus market prices (  t-1 )) Carbon Density Applications for Front December Futures Contracts

46. Page: 46 Scientific Model: Re-projections / Nonlinear Kalman filtering Price observations (for the next table): Market prices are raw data at close for a specific date. Re-projected Volatility and option prices: (or any other more complex function) N >= 250.000 conditional moment observations from re-projection step (see page 43 in this presentation). The density is adjusted for the risk premium for contract i. The Black´76 prices are calculated from an average of the re-projected volatility adjusted for the risk premium at t-1. Carbon Density Applications for Front December Futures Contracts Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

47. Scientific Model: The Stochastic Volatility Model. Page: 47 The NASDAQ OMX market: The Re-projected Volatility Model Option Prices Market versus Model Option Prices ((1) September 2011, (2) December 2011, (3) March 2012, (4) June 2012, (5) September 2012 and (6) December 2012): Carbon Density Applications for Front December Futures Contracts

48. Scientific Model: The Stochastic Volatility Model. Page: 48 The NASDAQ OMX market: The Re-projected Volatility Model Option Prices Market Prices versus Model Prices (for t 2011/09/02): (1 to 6 Sep-11 to Dec12) Carbon Density Applications for Front December Futures Contracts

49. Page: 49 Scientific Model: Re-projections / Nonlinear Kalman filtering Finally, we calculate the mean relative error and mean absolute error (calls): where n is the number of strike contracts at time t, C M i is the model call price and C i is the observed market price (for put options P replaces C). The MRE statistic measure the average relative biases of the model option prices, while the MARE statistic measures the dispersion of the relative biases of the model prices. The difference between MRE and MARE suggests the direction of the bias of the model prices, namely when MRE and MARE are of the same absolute values, it suggests that the model systematically misprices the options to the same direction as the sign of MRE, while when MARE is much larger than MRE in absolute magnitude, it suggests that the model is inaccurate in pricing options but the mispricing is less systematic. Carbon Density Applications for Front December Futures Contracts

50. Scientific Model: The Stochastic Volatility Model. Page: 50 The NASDAQ OMX market: The Re-projected Volatility Model Option Prices MRE/MARE relative pricing errors (no. of contracts Sep-11 to Dec-12). Carbon Density Applications for Front December Futures Contracts Call-Contracts: Put Contracts: Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

51. Scientific Model: The Stochastic Volatility Model. Page: 51 The NASDAQ OMX market: The Re-projected Volatility Model Option Prices Risk Adjusted Implied Volatilities: Carbon Density Applications for Front December Futures Contracts

52. Page: 52 SV Model: Conditional variance functions (asymmetry) (shocks to a system that comes as a surprise to the economic agents) Carbon Density Applications for Front December Futures Contracts

53. Page: 53 SV Model: Multistep-ahead volatility dynamics (volatility impulse-response profiles) Carbon Density Applications for Front December Futures Contracts Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

54. Page: 54 SV Model: Mean and Volatility Persistence (half-lives = –ln2 /  ) Carbon Density Applications for Front December Futures Contracts Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

55. Scientific Model Re-projections: Post Estimation Analysis Post estimation analysis add new information to market participants:  Conditional mean (expectations) densities  information from past?  Conditional Risk management and asset allocation measures (VaR/ CVaR) are available  The filtered volatility (particle filter) add information for the one-day- ahead conditional volatility. Conditional return densities for obs. X t-1.  The re-projection methodology prices any option over the whole strike interval for any complexity. Identify market mispricing/arbitrage opportunities  Conditional variance functions evaluates the surprise to economic agents from market shocks.  Multi-step-ahead dynamics for the mean and volatility are available  Identify volatility persistence, serial correlation and mean reversion Page: 55 Return Carbon Density Applications for Front December Futures Contracts Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am

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57. CARBON front December contracts and EMH:  Drift in the mean (risk premium) is positive but negligible (insignificant)  The positive serial correlation in the mean (0.054) is probable not tradable  The volatility clustering is strong (0.985) but probably not tradable  Asymmetry is strong (-0.432) but not tradable  The mean and volatility is stochastic and not predictable FUTURES:  EMH (weak form/semi-strong form) seems clearly acceptable. OPTIONS:  Option Prices are strongly influenced by risk premiums. The NASDAQ OMX market and the re-projected model show systematic option (mis-)pricing. Page: 57 Carbon Density Applications for Front December Futures Contracts

58. Main Findings for CARBON Applications and Front December contracts:  Stochastic Volatility models give a deeper insight of price processes and the stochastic flow of information interpretation  The Stochastic Volatility model and the statistical model seem to work well in concert (indirect estimation)  The MC chains look good and rejection is acceptable giving a reliable and viable stochastic volatility model  The SV-model results induce serial correlation in mean and volatility, persistence and negative asymmetry. One volatility factor is slowly moving while the second is quite choppy.  Risk management procedures are available from Stochastic Volatility models and Extreme Value Theory (VaR/CVaR and Greek letters)  Conditional moments, particle filtering and volatility variance functions interpret asymmetry, pricing options and evaluates shocks.  Option Prices can be generated from re-projected volatility for any maturity/ complexity. Non-diversifiable risk implies volatility risk premiums.  Imperfect tracking (incomplete markets) suggest that simulation is a well-suited methodology for derivative pricing. Page: 58 Return