Page: 1 Carbon Phase II Density Applications using General Scientific Stochastic Volatility Models by Professor Per B Solibakke a Nord Pool Application:

Page: 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

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


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

Page: 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).


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 z1t , z2t and (z3t ) are iid Gaussian random variables. The parameter vector is:

Page: 5

The General Scientific Model methodology (GSM):

2

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 2 2

3 2 2 1 3 2 1 1 2 3 2 1 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 u

b b b u

c c c u

u z

u s r z r z

u s r z r r r r z r r r r r z

0 1 2 0 1 0 1 1 2 1 2 3, , , , , , , , , , ,a a a b b c c s s r r r

1 10 10 12 2 13 3 1

2 22 2 2

3 33 3 3

exp( )t t t t

t t t

t t t

dU dt U U dW

dU U dt dW

dU U dt dW

SDE:

A vector SDE with two stochastic volatility factors.


Page: 6

Application references:

Andersen and Lund (1997): Short rate volatilitySolibakke, P.B (2001): SV model for Thinly Traded Equity MarketsChernov and Ghysel (2002): Option pricing under Stochastic VolatilityDai & Singleton (2000) and Ahn et al. (2002): Affine and quadratic term structure modelsAndersen et al. (2002): SV jump diffusions for equity returnsBansal and Zhou (2002): Term structure models with regime-shiftsGallant & 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):

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.


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.

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 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.

1( | , )t tf y x

n 1 1

,n

t t ty x

1

1

1/ / log | , 0

n

t t n

t

n f y x

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).

1( | , )t tp y x


The scientific model is now viewed as a sharp prior thatrestricts 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 suitablemeasures of the location and scale of the posterior distribution of against , or, better sequential density plots.

.M H

| ,p y x

Page: 10

1,...

, minM j j jj G

d A

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.

Simulated Score Methods and Indirect Inference for Continuous-time Models (some details): Assessment

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:

0 1 0 1ˆˆ | ,..., | ,..., ,L L np y y y p y y y

The unconditional expectations can be generated by a simulation:

0ˆ 0 0

ˆ... ,..., ,..., , ...Ln

L L n y yE g g y y p y y d d

ˆN

t t Ly

Now define: ˆ 0 1

arg maxˆ log | ,..., ,K n

K K LE f y y y

where is the score (SNP) density. 0 1| ,..., ,K Lf y y y

Let . Theorem 1 of Gallant and Long

(1997) states:

0 1 0 1ˆ ˆ| ,..., | ,..., ,K L K L Kf y y y f y y y

0 1 0 1ˆ ˆlim | ,..., | ,...,K L L

Kf y y y p y y y

We study the dynamics of by using as an approximation. p̂ ˆKf


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, simulate from the optimal structural

model and re-project to get

The price of an option (re-projected volatility) can now be calculated as:

* *ˆ | .Kf v y

where and . *1, 1,,...,t t L ty y y

* *,t tv y

* * * * * *ˆˆ ( ) |t Kv y v f v y dv


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


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


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)


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):


Page: 17

Other motivational factors:

4. A model of futures markets directly, without considering spot prices, usingHJM-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):


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

Back to Overview

Page: 20

Carbon front December Contracts:

Mean / Median / Max. Moment Quantile Quantile K-S RESET Serial dependence

Mode std.dev Min. Kurt/Skew Kurt/Skew Normal Z-test (12;6) Q(12) Q2(12)

-0.04364 0.0000 11.5196 2.84118 0.29749 4.2512 4.59075 70.5138 55.7488 1946.270.00000 2.43729 -10.0083 -0.13418 0.03835 {0.1194} {0.0000} {0.0000} {0.0000} {0.0000}

BDS-statistic (e=1) KPSS (Stationary) Augmented ARCH VaR CVaRm=2 m=3 m=4 m=5 Level Trend DF-test (12) 2.5/0.5% 2.5/0.5%

16.6788 23.5820 30.1427 38.4401 0.14330 0.14340 -56.0675 594.675 -5.247 -7.178{0.0000} {0.0000} {0.0000} {0.0000} {0.4121} {0.0568} {0.0000} {0.0000} -8.311 -9.694

Return

Carbon Density Applications for Front December Futures Contracts

Back to Overview

Scientific Models: Stochastic Volatility Model /Parameters (q)

Bayesian Estimation Results1. 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

Carbon Front December General Scientific Model. Statistical Model SNP-11116000 - fit modelParameter values Scientific Model. Standard Parameters Non-linear-GARCH. Standard

Mode Mean error Mode Mean error

a0 0.026974 0.033957 0.041845 n1 a0[1] 0.010997 0.017017 0.012873

a1 0.053948 0.045583 0.021425 n2 a0[3] 0.009816 -0.027176 0.015376

b0 0.630520 0.624160 0.078653 n3 a0[4] -0.007590 -0.005462 0.003885

b1 0.985140 0.947710 0.038068 n4 a0[5] 0.071771 0.104291 0.017859

c1 0.577490 0.663590 0.080555 n5 a0[6] 0.001586 0.002598 0.003412

s1 0.062399 0.068591 0.016147 n6 A(1,1) 0.004190 -0.000290 0.005238

s2 0.226330 0.196810 0.032872

r1 -0.432440 -0.385280 0.113010 n7 B(1,1) 0.072114 0.043127 0.046907n8 R0[1] 0.151411 0.265661 0.062968

log sci_mod_prior 5.797190 n9 P(1,1) 0.326412 0.430157 0.089448

log stat_mod_prior 0.000000 c2(3) = n10 Q(1,1) 0.926579 0.860786 0.041011

log stat_mod_likelihood -1515.8624 -0.94841 n11 V(1,1) -0.116156 0.037407 0.139785log sci_mod_posterior -1510.0652 {0.81373}

Return



Back to Overview

Page: 24

Scientific Model: Model Assessment – the model concert test

Carbon front December k = 1, 10, 20 and 100 densities – reported.


Page: 25

Scientific Model: The Stochastic Volatility Model: log-sci-mod-posterior

Log sci-mod-posterior (every 25th observation reported): Optimum is along this path!

c2(4)

-0.94841{0.81373}

Optimum:


Page: 26

Scientific Model: Carbon q-paths and densities; 240.000 simulations


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 k increases. The result indicates that the model fits and that the scale measure increases, indicating that the scientific model has empirical content.

Reported Proportion Number of Proportion%-rejected Moved Rejects Accepted

theta1 ( 1) 0.49051424 0.1255875 60.525 125.5875

theta2 ( 2) 0.47925517 0.1248125 59.7375 124.8125

theta3 ( 3) 0.47381869 0.12480417 59.1625 124.804167

theta4 ( 4) 0.4807526 0.12526667 60.2333333 125.266667

theta5 ( 5) 0.47864768 0.1262625 60.4583333 126.2625

theta6 ( 6) 0.47833745 0.12455 59.5333333 124.55

theta7 ( 7) 0.48576032 0.12464583 60.5958333 124.645833

theta8 ( 8) 0.48436667 0.12407083 64.1208333 124.070833

Sum 0.48436667 1 484.366667 1000

Return



Back to Overview

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

Carbon Density Applications for Front December Futures ContractsReturn

Back to Overview

Scientific Model: The Stochastic Volatility Model.

Page: 31

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.



Page: 32

Risk assessment and management: CARBON VaR / CVaR



Page: 33

Asset Allocation/Dynamic Hedging: CARBON GREEK Letters


The Scientific Model: Simulations from the optimal SV-model:

Realized Volatility and continuous / jump volatility (5 minutes simulations):

Page: 34

0

0.00002

0.00004

0.00006

0.00008

0.0001

0.00012

0.00014

0.00016

0.00018

0.0002

Realized Volatility

0

0.00002

0.00004

0.00006

0.00008

0.0001

0.00012

0.00014

0.00016

0.00018

0.0002Continuous Volatility

-0.00003

-0.00002

-0.00001

0

0.00001

0.00002

0.00003

Jump Volatility

Return



Back to Overview

Page: 36

Scientific Model: Re-projections / Nonlinear Kalman filtering

Of immediate interest of eliciting the dynamics of observables:

0 1 0 0 1 0( | ) ( | , )k ky x y f y x dy One-step ahead conditional mean:

One-step ahead conditional volatility:

'

0 1 0 0 1 0 0 1 0 1 0( | ) ( | ) ( | ) ( | , )k kVar y x y y x y y x f y x dy



Page: 37

SV Model: One-step-ahead conditional moments

0 1 0 0 1 0( | ) ( | , )k ky x y f y x dy '

0 1 0 0 1 0 0 1 0 1 0( | ) ( | ) ( | ) ( | , )k kVar y x y y x y y x f y x dy


Scientific Model Re-projections: Conditional SV-model moments:

Conditional VaR/CVaR for Risk Management and Greeks for Asset allocation

Page: 38

Return


Back to Overview

Page: 40


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.

1 10 1 ,..., )( | ) |t L tk x y yVar y x



Page: 41

SV Model: filtered volatility / particle filtering

1 10 1 ( ,..., )| | 0,...,t L tk x y yy x t n

0

0.05

0.1

0.15

0.2

Conditonal

Mean

Density

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

Frequency xt-1=-10% Frequency xt-1=-5% Frequency xt-1=-3% Frequency xt-1= Mean (-0.032)

Frequency xt-1=0% Frequency xt-1=+3% Frequency x-1=+5% Frequency x-1=+10%

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

GAUSS-hermite Quadrature Density Distribution


Page: 42


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).


*1 2 1exp

t T

t t t t

t

v v v u dt



Page: 43

The NASDAQ OMX market: The Re-projected Volatility Model Option Prices

Unconditional and Conditional Mean Volatility Densities


0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016Simulated Sigma UN-Conditional Mean Densities

Frequency Normal Log-Normal

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0.011

0.012

0.013Simulated Sigma Conditional Mean Densities

Frequency Normal Log-Normal

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Option

Prices

(

€)

Strike Prices (€)

Carbon Option Market and SV-Model Maturity December-11 for 2011/09/05Maturity Option Contract September 19th 2011

Market closing prices call Sep-21 put Sep-21 SV-Model prices call Sep-21 put Sep-21

Page: 44


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)).


it

ititit volatilitydreprojecte

volatilitydreprojectevolatilityimpliedR

,1

,1,1,1



Page: 45


Market Implied Risk Premiums (re-projected model versus market prices (st-1))

<1m

<3m

<6m

<9m

<12

m

>12

m

0

0.1

0.2

0.3

0.4

0.5

0.6

ITM Call ITM PUT ATM Call ATM Put OTM Call OTM Put

02.09.2011 Call/Put Contracts

0.6

0

0.5

9 0.3

7

0.3

7

0.5

8

0.5

8

0.2

9

0.2

2

0.2

3

0.2

3

0.2

3

0.2

9

0.1

5

0.1

3

0.1

2

0.1

2

0.1

2

0.1

4

0.1

7

0.1

4 0.1

4

0.1

4

0.1

4

0.1

6

0.1

7 0.1

4

0.1

4

0.1

4

0.1

3

0.1

6

0.1

5

0.1

4

0.1

2

0.1

1

0.1

1

0.1

2

Market ImpliedYearly Volatility

(non-diversifiable risk)


Series1

Series2

Series3

Series4

Series5

Series60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

02.09.201107.10.201104.11.201102.12.201106.01.201203.02.201202.03.201204.04.201204.05.2012

0.37

0.440.681.

82

1.53

1.21

1.11

1.191.

04

0.23

0.130.

47

0.98

1.37

1.16

0.991.

090.88

0.120.140.160.

84

1.23

1.07

0.95

1.020.88

0.14

0.140.15

0.78

1.04

0.89

0.82

1.01

0.82

0.140.

24

0.29

0.710.

81

0.71

0.98

0.8

0.12

0.8

0.7

1.01

0.83

ATMMarket ImpliedYearly Volatility

(non-diversifiable risk)

Re-Projected Model Estimation dates

Page: 46


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.




Page: 47


Market versus Model Option Prices ((1) September 2011, (2) December 2011, (3) March 2012, (4) June 2012, (5) September 2012 and (6) December 2012):

Option Prices Market closing prices SV-Model prices Black-76 FormulaNECDEC1 Strike Price call Sep-11 put Sep-11 call Sep-11 put Sep-11 call Sep-11 put Sep-1102.09.2011 9.0 3.5856 0.0023 3.5853 0.0001

9.5 3.09 0.01 3.0915 0.0054 3.0866 0.000610.0 2.60 0.01 2.5983 0.0127 2.5900 0.003410.5 2.12 0.03 2.1151 0.0284 2.1012 0.014011.0 1.66 0.07 1.6472 0.0616 1.6321 0.044211.5 1.25 0.16 1.2143 0.1284 1.2014 0.112812.0 0.87 0.28 0.8384 0.2495 0.8305 0.241312.5 0.54 0.45 0.5382 0.4479 0.5355 0.445613.0 0.35 0.76 0.3271 0.7343 0.3207 0.730113.5 0.22 1.12 0.1902 1.0980 0.1781 1.086914.0 0.13 1.54 0.1091 1.5192 0.0918 1.500014.5 0.08 1.99 0.0638 1.9722 0.0440 1.951515.0 0.05 2.46 0.0373 2.4451 0.0197 2.426515.5 0.03 2.94 0.0225 2.9293 0.0083 2.914416.0 0.02 3.43 0.0136 3.4201 0.0033 3.4088


Option Prices Market closing prices SV-Model prices Black-76 FormulaNECDEC1 Strike Price call Dec-11 put Dec-11 call Dec-14 put Dec-14 call Dec-14 put Dec-1402.09.2011 7.0 5.55 0.01 5.5506 0.0084 5.5403 0.0014

7.5 5.0606 0.0154 5.0478 0.00438.0 4.5787 0.0273 4.5591 0.01118.5 4.0976 0.0469 4.0776 0.02509.0 3.6367 0.0771 3.6076 0.05059.5 3.18 0.11 3.1930 0.1233 3.1548 0.093010.0 2.76 0.19 2.7538 0.1904 2.7249 0.158610.5 2.36 0.29 2.3552 0.2838 2.3237 0.252811.0 2.00 0.42 1.9856 0.4091 1.9560 0.380511.5 1.67 0.59 1.6491 0.5700 1.6253 0.545312.0 1.36 0.78 1.3568 0.7719 1.3334 0.748812.5 1.09 1.00 1.0989 1.0161 1.0804 0.991213.0 0.87 1.28 0.8865 1.2917 0.8650 1.271313.5 0.69 1.59 0.7105 1.6119 0.6848 1.586414.0 0.54 1.94 0.5635 1.9646 0.5362 1.9334

Risk Premium 14.5 0.42 2.31 0.4484 2.3433 0.4157 2.308215.0 0.32 2.71 0.3546 2.7434 0.3192 2.707215.5 0.24 3.13 0.2823 3.1631 0.2430 3.126416.0 0.18 3.57 0.2250 3.6061 0.1834 3.562316.5 0.14 4.01 0.1796 4.0530 0.1374 4.011717.0 0.10 4.48 0.1425 4.5097 0.1022 4.471917.5 0.07 4.94 0.1149 4.9804 0.0756 4.940718.0 0.05 5.42 0.0926 5.4508 0.0555 5.416118.5 0.04 5.90 0.0744 5.9354 0.0406 5.896619.0 0.03 6.39 0.0603 6.4122 0.0295 6.380919.5 0.02 6.88 0.0494 6.8888 0.0214 6.868220.0 0.01 7.37 0.0401 7.3812 0.0154 7.3577

Option Prices Market closing pricesSV-Model prices Black-76 PricesENOYR-12 Strike Price call Mar-12 put Mar-12call Mar-12 put Mar-12 call Mar-12 put Mar-1202.09.2011 5.0 8.0048 0.0001

5.5 7.5145 0.00036.0 7.0248 0.00106.5 6.5362 0.00307.0 6.0501 0.00737.5 5.5683 0.01608.0 5.1176 0.0549 5.0935 0.03178.5 4.6570 0.0873 4.6288 0.05759.0 4.2128 0.1325 4.1780 0.09719.5 3.77 0.17 3.7827 0.1928 3.7448 0.154510.0 3.36 0.25 3.3781 0.2736 3.3329 0.233110.5 2.98 0.36 2.9830 0.3795 2.9455 0.336111.0 2.62 0.49 2.6292 0.5102 2.5852 0.466311.5 2.29 0.66 2.2992 0.6679 2.2536 0.625212.0 1.99 0.85 1.9978 0.8556 1.9519 0.813912.5 1.71 1.06 1.7216 1.0758 1.6800 1.032513.0 1.45 1.29 1.4841 1.3234 1.4374 1.280513.5 1.23 1.57 1.2642 1.6042 1.2230 1.556614.0 1.04 1.87 1.0827 1.9079 1.0351 1.859214.5 0.88 2.20 0.9185 2.2311 0.8718 2.186315.0 0.74 2.55 0.7791 2.5843 0.7309 2.535915.5 0.61 2.92 0.6605 2.9533 0.6101 2.905616.0 0.51 3.31 0.5625 3.3405 0.5073 3.293316.5 0.42 3.71 0.4758 3.7477 0.4203 3.696817.0 0.35 4.13 0.4039 4.1671 0.3471 4.114017.5 0.28 4.56 0.3407 4.5945 0.2857 4.543118.0 0.23 5.00 0.2909 5.0342 0.2345 4.982418.5 0.19 5.45 0.2451 5.4789 0.1920 5.430419.0 0.15 5.90 0.2074 5.9354 0.1568 5.885719.5 0.12 6.37 0.1767 6.3967 0.1278 6.347220.0 0.10 6.84 0.1506 6.8603 0.1040 6.813820.5 0.08 7.31 0.1299 7.3302 0.0845 7.2848

Option Prices Market closing prices SV-Model prices Black-76 PricesENOYR-12 Strike Price call Jun-12 put Jun-12 call Jun-12 put Jun-12 call Jun-12 put Jun-1202.09.2011 5.0 7.9319 0.0009

5.5 7.4477 0.00276.0 6.9656 0.00666.5 6.4874 0.01447.0 6.0151 0.02817.5 5.5514 0.05038.0 5.1301 0.1190 5.0991 0.08398.5 4.7018 0.1724 4.6612 0.13209.0 4.2841 0.2420 4.2406 0.19739.5 3.88 0.30 3.8859 0.3318 3.8398 0.282610.0 3.50 0.41 3.5100 0.4413 3.4611 0.389810.5 3.15 0.55 3.1616 0.5713 3.1060 0.520711.0 2.82 0.71 2.8244 0.7328 2.7757 0.676311.5 2.51 0.89 2.5275 0.9128 2.4706 0.857212.0 2.23 1.10 2.2423 1.1208 2.1908 1.063412.5 1.96 1.32 1.9930 1.3516 1.9359 1.294413.0 1.72 1.56 1.7662 1.6059 1.7050 1.549513.5 1.51 1.84 1.5554 1.8915 1.4971 1.827514.0 1.32 2.14 1.3783 2.1861 1.3108 2.127214.5 1.15 2.46 1.2057 2.5089 1.1447 2.447115.0 1.00 2.80 1.0578 2.8498 0.9973 2.785715.5 0.86 3.15 0.9314 3.2063 0.8670 3.141316.0 0.75 3.52 0.8216 3.5782 0.7522 3.512516.5 0.64 3.91 0.7152 3.9637 0.6514 3.897717.0 0.55 4.31 0.6306 4.3657 0.5632 4.295417.5 0.48 4.72 0.5562 4.7745 0.4863 4.704418.0 0.41 5.14 0.4873 5.1904 0.4192 5.123418.5 0.35 5.57 0.4265 5.6180 0.3610 5.551119.0 0.30 6.01 0.3755 6.0503 0.3106 5.986619.5 0.25 6.46 0.3334 6.4905 0.2669 6.428920.0 0.22 6.91 0.2919 6.9399 0.2292 6.877220.5 0.18 7.36 0.2600 7.3918 0.1967 7.3306

Option Prices Market closing prices SV-Model prices Black-76 PricesENOYR-12 Strike Price call Sep-12 put Sep-12 call Sep-12 put Sep-12 call Sep-12 put Sep-1202.09.2011 5.0 7.8448 0.0040

5.5 7.3696 0.00926.0 6.8989 0.01906.5 6.4348 0.03537.0 5.9798 0.06087.5 5.5365 0.09798.0 5.1566 0.1918 5.1073 0.14918.5 4.7452 0.2654 4.6946 0.21689.0 4.3440 0.3590 4.3004 0.30319.5 3.99 0.43 3.9822 0.4659 3.9263 0.409510.0 3.63 0.56 3.6344 0.5996 3.5737 0.537310.5 3.30 0.72 3.3102 0.7511 3.2431 0.687211.0 2.99 0.89 3.0045 0.9291 2.9351 0.859611.5 2.71 1.09 2.7155 1.1226 2.6495 1.054512.0 2.44 1.31 2.4621 1.3429 2.3861 1.271512.5 2.18 1.54 2.2144 1.5811 2.1442 1.510013.0 1.94 1.78 2.0011 1.8429 1.9230 1.769313.5 1.73 2.06 1.8012 2.1201 1.7215 2.048214.0 1.55 2.36 1.6150 2.4305 1.5385 2.345614.5 1.38 2.68 1.4517 2.7408 1.3729 2.660515.0 1.22 3.01 1.3074 3.0691 1.2235 2.991515.5 1.08 3.36 1.1725 3.4164 1.0890 3.337516.0 0.96 3.72 1.0565 3.7726 0.9682 3.697116.5 0.85 4.09 0.9410 4.1493 0.8600 4.069417.0 0.75 4.48 0.8505 4.5327 0.7632 4.453017.5 0.66 4.88 0.7616 4.9259 0.6768 4.847118.0 0.58 5.28 0.6806 5.3285 0.5998 5.250518.5 0.51 5.70 0.6125 5.7414 0.5312 5.662419.0 0.45 6.12 0.5495 6.1594 0.4703 6.081819.5 0.39 6.55 0.4965 6.5851 0.4161 6.508120.0 0.34 6.99 0.4453 7.0147 0.3681 6.940520.5 0.30 7.43 0.4014 7.4547 0.3255 7.3784

Option Prices Market closing prices SV-Model prices Black-76 PricesENOYR-12 Strike Price call Dec-12 put Dec-12 call Dec-12 put Dec-12 call Dec-12 put Dec-1202.09.2011 5.0 7.7784 0.0078

5.5 7.3109 0.0164Valgt strike 6.0 6.8495 0.0312

6.5 6.3964 0.05437.0 5.9542 0.08817.5 5.5250 0.13518.0 5.1582 0.2444 5.1111 0.19748.5 4.7749 0.3306 4.7144 0.27689.0 4.3971 0.4354 4.3364 0.3749

Ant_Runder 9.5 4.05 0.52 4.0314 0.5565 3.9783 0.493010.0 3.72 0.66 3.7038 0.7010 3.6409 0.631710.5 3.40 0.83 3.3978 0.8621 3.3245 0.791411.0 3.10 1.02 3.1083 1.0503 3.0292 0.972311.5 2.82 1.22 2.8365 1.2517 2.7549 1.174112.0 2.57 1.45 2.5846 1.4795 2.5010 1.396312.5 2.31 1.67 2.3507 1.7207 2.2668 1.638313.0 2.07 1.91 2.1393 1.9818 2.0516 1.899313.5 1.87 2.19 1.9473 2.2643 1.8544 2.178214.0 1.68 2.49 1.7585 2.5618 1.6742 2.474114.5 1.52 2.81 1.5976 2.8799 1.5099 2.786015.0 1.36 3.14 1.4478 3.2003 1.3605 3.112615.5 1.23 3.49 1.3155 3.5454 1.2248 3.453116.0 1.10 3.84 1.2002 3.8956 1.1018 3.806316.5 0.99 4.21 1.0831 4.2590 0.9906 4.171217.0 0.89 4.60 0.9790 4.6407 0.8901 4.546917.5 0.80 4.99 0.8917 5.0183 0.7994 4.932318.0 0.71 5.39 0.8107 5.4112 0.7177 5.326718.5 0.64 5.80 0.7243 5.8281 0.6441 5.729319.0 0.57 6.21 0.6675 6.2253 0.5779 6.139219.5 0.51 6.63 0.5994 6.6389 0.5184 6.555920.0 0.46 7.06 0.5526 7.0715 0.4650 6.978620.5 0.41 7.50 0.5017 7.4879 0.4170 7.4067


Page: 48


Market Prices versus Model Prices (for t 2011/09/02): (1 to 6 Sep-11 to Dec12)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Option

Prices

(

€)

Strike Prices (€)

Carbon Option Market and SV-Model Maturity December-11 for 2011/09/02Maturity Option Contract September 21st 2011



0

1

2

3

4

5

Option

Prices

(

€)

Strike Prices (€)

Carbon Option Market and SV-Model Maturity December-11 for 2011/09/05Maturity Option Contract December 14th 2011.

Market closing prices call Dec-14 put Dec-14 SV-Model prices call Dec-14 put Dec-14

0

1

2

3

4

5

Option

Prices

(

€)

Strike Prices (€)

Carbon Option Market and SV-Model Maturity December-12 for 2011/09/02Maturity Option Contract March 21st 2012

Market closing prices call Mar-21 put Mar-21 SV-Model prices call Mar-21 put Mar-21

0

1

2

3

4

5

Option

Prices

(

€)

Strike Prices (€)

Carbon Option Market and SV-Model Maturity December-12 for 2011/09/05Maturity Option Contract June 20th 2012 (201 days)

Market closing prices call Jun-20 put Jun-20 SV-Model prices call Jun-20 put Jun-20

0

1

2

3

4

5

6

Option

Prices

(

€)

Strike Prices (€)

Electricity Option Market and SV-Model Maturity Dec-12 for 2011/09/02Maturity Option Contract September 19th 2012


0

1

2

3

4

5

6

Option

Prices

(

€)

Strike Prices (€)

Electricity Option Market and SV-Model Maturity Dec-12 for 2011/09/02Maturity Option Contract December 12th 2012

Market closing prices call Dec-12 put Dec-12 SV-Model prices call Dec-12 put Dec-12

Page: 49


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 Ci 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.



Page: 50


MRE/MARE relative pricing errors (no. of contracts Sep-11 to Dec-12).

Moneyness Volatility Days to Maturityx=ln (S/K) Models < 30 dager 30 - 180 days 180 - 360 days > 360 days Overall

MRE MARE MRE MARE MRE MARE MRE MARE MRE MAREReprojected -0.2631 0.2631 -0.2020 0.2020 -0.1417 0.1417 -0.2023 0.2023

Call x < -0.3Black'76 0.0000 0.0000 -0.0346 0.0346 -0.0555 0.0555 -0.0300 0.0300

OTMReprojected 0.1818 0.1818 -0.1067 0.1067 -0.0942 0.0942 -0.0778 0.0778 -0.0242 0.1151

-0.3 < x < -0.1Black'76 0.3000 0.3000 0.0000 0.0000 -0.0051 0.0051 -0.0298 0.0298 0.0663 0.0837

Reprojected 0.0803 0.0803 -0.0267 0.0267 -0.0381 0.0381 -0.0400 0.0400 -0.0061 0.0463-0.1 < x < 0

Black'76 0.1414 0.1414 0.0000 0.0000 0.0022 0.0022 -0.0137 0.0137 0.0325 0.0393ATM

Reprojected 0.0201 0.0201 -0.0023 0.0064 -0.0157 0.0157 -0.0164 0.0164 -0.0036 0.01460 < x < 0.1

Black'76 0.0229 0.0229 0.0161 0.0161 0.0116 0.0116 0.0007 0.0071 0.0128 0.0144

Reprojected 0.0015 0.0015 -0.0001 0.0014 -0.0020 0.0020 -0.0002 0.0016 -0.0002 0.0016Call 0.1 < x < 0.3

Black'76 0.0042 0.0042 0.0115 0.0115 0.0146 0.0146 0.0115 0.0115 0.0105 0.0105ITM

Reprojected -0.0026 0.0026 0.0000 0.0000 0.0025 0.0025 -0.0001 0.0017x > 0.3

Black'76 0.0053 0.0053 0.0128 0.0128 0.0100 0.0100 0.0094 0.0094

Moneyness Volatility Days to Maturityx=ln (S/K) Models < 30 dager 30 - 180 days 180 - 360 days > 360 days Overall

MRE MARE MRE MARE MRE MARE MRE MARE MRE MAREReprojected 0.0000 0.0000 -0.0039 0.0039 -0.0054 0.0054 -0.0015 0.0019 -0.0027 0.0028

Put x < -0.3Black'76 0.0000 0.0000 0.0013 0.0013 0.0043 0.0043 0.0058 0.0058 0.0029 0.0029

ITMReprojected 0.0043 0.0043 -0.0095 0.0095 -0.0128 0.0128 -0.0113 0.0113 -0.0073 0.0095

-0.3 < x < -0.1Black'76 0.0116 0.0116 0.0021 0.0021 0.0034 0.0034 0.0005 0.0016 0.0044 0.0047

Reprojected 0.0226 0.0226 -0.0136 0.0136 -0.0233 0.0233 -0.0264 0.0264 -0.0102 0.0215-0.1 < x < 0

Black'76 0.0273 0.0273 0.0048 0.0048 0.0045 0.0045 -0.0069 0.0069 0.0074 0.0109ATM

Reprojected 0.1169 0.1169 -0.0011 0.0128 -0.0242 0.0242 -0.0260 0.0260 0.0164 0.04500 < x < 0.1

Black'76 0.1511 0.1511 0.0298 0.0298 0.0154 0.0154 -0.0011 0.0058 0.0488 0.0505

Reprojected -0.0249 0.0310 -0.0379 0.0379 -0.0361 0.0361 -0.0330 0.0350Put 0.1 < x < 0.3

Black'76 0.0929 0.0929 0.0390 0.0390 0.0045 0.0045 0.0455 0.0455OTM

Reprojected -0.1053 0.1053 -0.0781 0.0781 -0.0545 0.0545 -0.0793 0.0793x > 0.3

Black'76 0.0625 0.0625 0.0416 0.0416 0.0000 0.0000 0.0347 0.0347


Call-Contracts:Put Contracts:


35%

40%

45%

50%

55%

60%

65%

70%

75%

80%

Market Implied Yearly Volatility

Moneyness (ln(S/K))

Market, Reprojected and Black'76 Implied VolatilitySeptember 2011: Time to Maturity: < 1 Month

Market impl. Volatility (Call) Reprojected impl. Volatility (Call)Market impl. Volatility (Put) Reprojected impl. Volatility (Put) Black'76 Implied Volatility (Call/Put)

0.34

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5Market Implied Yearly Volatility

Moneyness (ln(S/K))

Market, Reprojected and Black'76 Implied VolatilitySeptember-2012: Time to Maturity: 1 - 4 Months

Market Implied Volatility (Call) Reprojected Implied Volatility (Call)Market Implied Volatility (Put) Reprojected Implied Volatility (Put) Black'76 Implied Volatility


Page: 51


Risk Adjusted Implied Volatilities:


0.33

0.34

0.35

0.36

0.37

0.38

0.39

0.4

0.41

Market Implied Yearly Volatility

Moneyness (ln(S/K))

Market, Reprojected and Black'76 Implied VolatilitySeptember 2011: Maturity Option Contract: 1-6 Months

Market impl. Volatility (Call) Reprojected impl. Volatility (Call)

Market impl. Volatility (Put) Reprojected impl. Volatility (Put) Black'76 Implied Volatility (Call/Put)

0.34

0.35

0.36

0.37

0.38

0.39

0.4Market Implied Yearly Volatility

Moneyness (ln(K/S))

Market, Reprojected and Black'76 Implied VolatilitySeptember 2011: Maturity Option Contract: 6 - 12 months

Market Implied Volatility (Call) Reprojected Implied Volatility (Call)Market Implied Volatility (Put) Reprojected Implied Volatility (Put) Black'76 Implied Volatility (Call/Put)

0.34

0.35

0.36

0.37

0.38

0.39

0.4


Moneyness (ln(S/K))

Market, Reprojected and Black'76 Implied VolatilitySeptember 2011: Maturity Option Contracts: > 12 Months

Market Implied Volatility (Call) Reprojected Implied Volatility (Call)Market Implied Volatility (Put) Reprojected Implied Volatility (Put) Black'76 Implied Volatility (Call/Put)

0.34

0.345

0.35

0.355

0.36

0.365

0.37

0.375

0.38

0.385

0.39


Moneyness (ln(S/K))

Market, Reprojected and Black'76 Implied VolatilitySeptember 2011: Maturity Option Contracts: > 12 Months

Market impl. Volatility (Call) Reprojected impl. Volatility (Call)Market impl. Volatility (Put) Reprojected impl. Volatility (Put) Black'76 Implied Volatility (Call/Put)

Page: 52

SV Model: Conditional variance functions (asymmetry)

(shocks to a system that comes as a surprise to the economic agents)


Page: 53

SV Model: Multistep-ahead volatility dynamics

(volatility impulse-response profiles)

0

0.5

1

1.5

2

2.5

3

Var

ian

ce E

[Var

(yk

,j|x

-1)

DAYS

Multistep Ahead Dynamics s2j

dy0 dy-1 (low) dy+1 (high) dy-3 (low) dy+3 (high) dy-6 (low) dy+6 (high) dy-10 (low) dy+10 (high)



Page: 54

SV Model: Mean and Volatility Persistence (half-lives = –ln2 / b)

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Mean

Days

CARBON Profile Bundles for the MEAN (overplots of profiles)

0

3

6

9

12

15

18

21

24

27

30

Vol

atil

ity

Days

CARBON Profile Bundles for the VOLATILITY (overplots of profiles)

Halflives:

28.238149SE=1.324



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. Xt-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



Back to Overview

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


Main Findings for CARBON Applications and Front December contracts:

Stochastic Volatility models give a deeper insight of price processes andthe 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 functionsinterpret 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

Page: 1 Carbon Phase II Density Applications using General Scientific Stochastic Volatility Models by Professor Per B Solibakke a Nord Pool Application:

Documents

future prices

futures prices

high prices

carbon phase

daily prices

market models

option market

derivative prices