Carbon Phase II Density Applications using General Scientific Stochastic Volatility Models by

1. Carbon Phase II Density Applications using General Scientific Stochastic Volatility Models by Professor Per B Solibakkea 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 • 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 • 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 Page: 2

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 • One-step-ahead Conditional Mean (expectations) and Standard Deviation • 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. The Carbon NASDAQ OMX commodity market NASDAQ OMX commodities 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 Page: 4

5. Indirect Estimation and Inference: 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 Estimation: The Scientific Model – A Stochastic Volatility Model The General Scientific Model methodology (GSM): SDE: where z1t , z2t and (z3t ) are iid Gaussian random variables. The parameter vector is: A vector SDE with two stochastic volatility factors. Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am Page: 5

6. The General Scientific Model methodology (GSM): • 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. 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 Page: 6

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 Page: 7

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. Page: 8

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 xt-1 is the state vector of the observable process at time t-1 and yt is the observable process. Fitted by maximum likelihood we get an estimate of the average of the score of the data satisfies: 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 r 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 Page: 9

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 Aj 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. Simulated Score Methods and Indirect Inference for Continuous-time Models (some details): Assessment For a well fitting scientific model: The location measure should not move by a scientifically meaningful amount as k increases. The result indicates that the model fits and that the scale measure increases, indicating that the scientific model has empirical content. Page: 10

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: where is 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 Page: 11

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, simulate from the optimal structural model and re-project to get The price of an option (re-projected volatility) can now be calculated as: where and . Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am Page: 12

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. 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 Page: 14

15. Objectives (purpose): (cont) • 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 FutureMarkets (EMH) • Serial correlation in Mean and Volatility • Price-Trend-Forecasting models and Risk premiums • Predictability and Efficient use of Available Information Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am Page: 15

16. Objectives (why): 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) Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am Page: 16

17. Objectives (why): 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. Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am Page: 17

18. Carbon Density Applications MCMC estimation/inference: • 1. NASDAQ OMX Carbon front December contracts • 2. The Statistical model and the Stochastic Volatility Model • 3. Model assessment (relaxing the prior): model appropriate? • 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 Data Characteristics Estimation Results Model Assessment SV Model Findings Risk Man./Asset Alloc Re-projection/Post-Est Filtering/Option Prices EMH/Model Summary Page: 18

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

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22. Return Carbon Density Applications for Front December Futures Contracts Scientific Models: Stochastic Volatility Model /Parameters (q) 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) Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am Page: 22

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

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

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

27. Return Carbon Density Applications for Front December Futures Contracts • 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 k increases. The result indicates that the model fits and that the scale measure increases, indicating that the scientific model has empirical content. Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am Page: 27

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29. Return Carbon Density Applications for Front December Futures Contracts Empirical Model Findings: • For the mean stochastic equation: • Positive mean drift (a0 = 0.026; s.e. = 0.03) and serial correlation (a1 = 0.054; s.e. 0.021) for the CARBON contracts • For the latent volatility: two stochastic volatility equations: • Positive constant parameter (e0.6305 >> 1) • Two volatility factors (s1 = 0.0624, s.e.=0.0161; s2 = 0.2263, s.e.=0.0329) • persistence is high for s1 with associated (b1 = 0.985, s.e. = 0.0381) ; persistence is lower for s2 with associated (b2 = 0.5775, s.e.=0.0806) • Asymmetry is strong and negative (r1 = -0.4324, s.e.=0.1130) Page: 29

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31. Carbon Density Applications for Front December Futures Contracts Scientific Model: The Stochastic Volatility Model. Risk assessment and management: CARBON VaR / CVaR 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, nu 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 Page: 31

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

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

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

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36. Carbon Density Applications for Front December Futures Contracts 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: Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am Page: 36

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

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

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40. Carbon Density Applications for Front December Futures Contracts Scientific Model: Re-projections / Nonlinear Kalman filtering Filtered volatility is the one-step ahead conditional standard deviation evaluated at data values: where yt denotes the data and yk0 denotes the kth element of the vector y0, k = 1,…M. Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am Page: 40

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

42. Carbon Density Applications for Front December Futures Contracts 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 fK 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 fK : The conditional volatility mean can be used to obtain an estimate of: for the purpose of pricing an option (particle filtering). Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am Page: 42

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

44. Carbon Density Applications for Front December Futures Contracts 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 Ri 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)). Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am Page: 44

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

46. Carbon Density Applications for Front December Futures Contracts 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. Workshop Molde, Session Monday 21/May/2012: 14:00am – 15:30am Page: 46

47. Carbon Density Applications for Front December Futures Contracts Scientific Model: The Stochastic Volatility Model. 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): Page: 47

48. Carbon Density Applications for Front December Futures Contracts Scientific Model: The Stochastic Volatility Model. 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) Page: 48

49. Carbon Density Applications for Front December Futures Contracts 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, CMi is the model call price and Ciis 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. Page: 49

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