scientific carbon stochastic volatility model estimation and inference:

CFE-2011, Parallel Sessions, Monday 19/12/2011 Page: 1

Scientific Carbon Stochastic Volatility Model Estimation and Inference:

Forecasting (Un-)Conditional Moments for Options Applications

by

Per Bjarte Solibakkea

a) Department of Economics, Molde University College

Background and Outline

1. The Front December Future Contracts NASDAQ OMX: phase II 2008-2012

No existence of EUAs spot-forward relationship does not exist

EUA options have carbon December futures as underlying instrument

Price dynamics are depending on total emissions

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, we will therefore estimate the option prices on the future prices themselves. Black-76 / MCMC simulations.

CFE-2011, Parallel Sessions, Monday 19/12/2011

Background and Outline (cont.)

3. Stochastic Model Specification, Estimation, Assessment and Inference

4. Forecasting unconditional Futures and Options Moments,

and measures for risk management and asset allocation

5. Forecasting conditional Futures and Options Moments

i. One-step-ahead Conditional Mean (expectations)

ii. One-step-ahead Standard deviation / Particle filtering

iii. Multi-step-ahead Mean and Volatility Dynamics

iv. Mean / Volatility Persistence

6. Conditional Risk Management and Asset Allocation Measures

7. The EMH case for CARBON commodity markets

Page: 3CFE-2011, Parallel Sessions, Monday 19/12/2011

Page: 4

The Carbon NASDAQ OMX commodity market

NASDAQ OMX commodities provide access to one of Europe’s leading carbon markets.

350 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

Applications:

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 Option pricing, 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) The conditional volatility function, Multi-step-ahead mean and volatility and mean/volatility persistence. Other extensions.

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.


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.

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.

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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 1,

nt t t

y x

Having a structural model (i.e. SV) we wish to estimate, we express the structural model as the transition density , where q 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).


Details for parameter q estimation:

Compute: where denotes the observed data and n is the sample size. Given a current and the corresponding we obtain the pair as follows (the M-H algorithm):

I. Draw according to

II. Simulate according to

III. Compute and (parameter functionals) from simulation

IV. Define

V. With probability otherwise Page: 9

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

The scientific model is built using financial market insight/knowledge

Stochastic volatility model computable from a simulation

Metropolis-Hastings algorithm to compute the posterior (only need of a function proportional to the prior)

1,t ty x 0 0 0g

' ',

0 *,q

1 1,

Nt t t

y x

* *g 1 1,

Nt t t

y x

' ' 0 0, , .

Main question: How do the results change as the prior is relaxed?

That is: How does the marginal posterior distribution of a parameter or functional of the statistical model change?

Distance measurement: where Aj is the scaling matrices.

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For a well fitting scientific model:The location measure should not move by a scientifically meaningful amount as k increases. However, the scale measure can increase.

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

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


Application:

Financial CARBON Contracts (EUA)NORD POOL (Phase II: 2008-2012)

Front December Futures Contracts(EUA options will have the December futures as the underlying instrument)


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Objectives (purpose):

Higher Understanding of the Carbon Futures Commodity Markets the Mean equations the Volatility equations

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 (densities) Accepts prior information No growth conditions on model output or data Estimates of parameter uncertainty (distributions) is credible

Financial Contracts Characteristics and Risk Assessment & Management The Financial Contracts Characteristics


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Value-at-Risk / Expected Shortfall for Risk Management Stochastic Volatility models are well suited 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

The Case against the Efficiency of Future Markets (EMH) Serial correlation in Mean and Volatility Price-Trend-Forecasting models and Risk premiums Predictability

Objectives (purpose): (cont)


Page: 15

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: 16

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: 17

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: Option prices, Risk management and Asset Allocation (unconditional).

2. Conditional mean and volatility, particle filtering, variance functions,

multi-step ahead dynamics and persistence.

5. Conditional Risk Management and Asset Allocation

6. EMH and Model Summary/Conclusion

Carbon Application MCMC estimation/inference:

Assessment

Model Findings

Risk M/Asset Alloc

Conditional Moments

EMH/ Model Summary

Data Characteristics

Estimation Results

Re-projection/Post-Est


Back to Overview


Page: 19

Application Carbon Front December Contracts

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

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Scientific Models: Stochastic Volatility Model /Parameters (q)

Bayesian Estimation Results1. Several serial Bayesian runs establishing the mode

We tune the scientific model until the posterior quits climbing and it looks like the mode has been reached:

2. A final parallel run with 24 (8 *3) CPUs and 240.000 MCMC simulations(OPEN_MPI (Indiana University) parallell computing)

Page: 21


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}

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Page: 23

Scientific Model: Model Assessment – the model concert test

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


Page: 24

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

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


Page: 25

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


Page: 26

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 result indicates that the model fits and that location measure is stable and 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

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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: 28

Application Carbon Front December ContractsReturn

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Scientific Model: The Stochastic Volatility Model.

Page: 30

The Option market versus SV-model prices 05.09.2011


Option Prices 05.09.2011 Market closing prices SV-Model pricesDEC-11 Strike Price call Dec-11 put Dec-11 call Dec-11 put Dec-11 Volume 0 6.0 2.59 0.06 2.54 0.04

0 6.5 2.15 0.12 2.09 0.080 7.0 1.72 0.19 1.66 0.130 7.5 1.33 0.29 1.30 0.230 8.0 0.97 0.43 0.92 0.390 8.5 0.67 0.62 0.62 0.610 9.0 0.46 0.91 0.41 0.890 9.5 0.31 1.25 0.26 1.230 10.0 0.2 1.64 0.17 1.620 10.5 0.12 2.06 0.11 2.110 11.0 0.07 2.51 0.06 2.520 11.5 0.04 2.97 0.04 3.040 12.0 0.03 3.45 0.03 3.510 12.5 0.01 3.93 0.01 3.960 13.0 0.01 4.42 0.01 4.47

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0

Option

Prices

Strike Prices

Carbon Option Market and SV-Model prices Maturity 2011 for 2011/09/05

Market closing prices call Dec-11 put Dec-11 SV-Model prices call Dec-11 put Dec-11


Page: 31

Risk assessment and management: CARBON VaR / CVaR



Page: 32

Asset Allocation/Dynamic Hedging: CARBON GREEK Letters

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Page: 34

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

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.

For instance, one might wish to obtain an estimate of:

for the purpose of pricing an option (from the re-projection step).

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


*1 2exp( )

t T

t

t

v v dt


Page: 35

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


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SV Model: filtered volatility /particle filtering for Option pricing

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: 37

SV Model: Conditional variance functions (asymmetry)

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


Page: 38

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: 39

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

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

Conditional VaR/CVaR for RM and Greeks for Asset allocation

Page: 41



Scientific Model Reprojections: Extensions using SV-model simulations:

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

Page: 42


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


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Scientific Model Re-projections: Post Estimation Analysis

Post estimation analysis add new information to market participants:

Option prices for any derivative for any maturity. Credible densities are available for all call/put prices.

Credible densities for VaR/CVaR and Greek letters are available for risk management and asset allocation

Conditional mean (expectations) is narrow information from the history?

The filtered volatility (particle filter) add information for the one-day-ahead conditional volatility. Conditional return densities for obs. Xt-1. Gauss quadrature densities are available.

Conditional variance functions evaluates the surprise to economic agents from market shocks.

Multi-step-ahead dynamics for the mean and volatility are available

Conditional Risk management and asset allocation measures available

Realized Volatility can be obtained from simulation step change (96 steps per day = 5 minutes data). Page: 43

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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

EMH (weak form/semi-strong form) seems clearly acceptable.

Page: 45



Main Findings for CARBON 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.

Option Prices can easily be generated for any maturity. We compared two maturities market prices to model prices (mean and distributions).

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.

Imperfect tracking (incomplete markets) suggest that simulation is a well-suited methodology for derivative pricing Page: 46

Application Carbon Front December ContractsReturn

scientific carbon stochastic volatility model estimation and inference:

Documents

carbon phase

future prices

futures prices

carbon commoditymarket

carbon contract

high prices

conditional risk management

eua options market