CFE-2011, Parallel Sessions, Monday 19/12/2011Page: 1
Scientific Carbon Stochastic Volatility Model Estimation and
Inference: Forecasting (Un-)Conditional Moments for Options
Applications by Per Bjarte Solibakke a a) Department of Economics,
Molde University College
Slide 2
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
Slide 3
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
Slide 4
Page: 4 The Carbon NASDAQ OMX commodity market NASDAQ
OMXcommodities provide access to one of Europes 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). CFE-2011, Parallel Sessions, Monday
19/12/2011
Slide 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. CFE-2011,
Parallel Sessions, Monday 19/12/2011
Slide 6
Page: 6 Applications: 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 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): CFE-2011, Parallel Sessions, Monday
19/12/2011
Slide 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). Gouriroux 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. CFE-2011, Parallel Sessions,
Monday 19/12/2011 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.
Slide 8
Page: 8 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). CFE-2011, Parallel Sessions, Monday 19/12/2011
Slide 9
Details for parameter 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 I. 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)
Slide 10
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 A j is the scaling matrices. Page: 10 For a well
fitting scientific model: The location measure should not move by a
scientifically meaningful amount as increases. However, the scale
measure can increase. Simulated Score Methods and Indirect
Inference for Continuous-time Models (some details):
Assessment
Slide 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: Let. Theorem 1 of Gallant and
Long (1997) states: We study the dynamics of by using as an
approximation. CFE-2011, Parallel Sessions, Monday 19/12/2011
Slide 12
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) CFE-2011,
Parallel Sessions, Monday 19/12/2011
Slide 13
Page: 13 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 CFE-2011, Parallel Sessions,
Monday 19/12/2011
Slide 14
Page: 14 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) CFE-2011, Parallel
Sessions, Monday 19/12/2011
Slide 15
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): CFE-2011,
Parallel Sessions, Monday 19/12/2011
Slide 16
Page: 16 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): CFE-2011, Parallel Sessions,
Monday 19/12/2011
Slide 17
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. 3.Conditional Risk Management and
Asset Allocation 4.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 CFE-2011,
Parallel Sessions, Monday 19/12/2011
Slide 18
Back to Overview CFE-2011, Parallel Sessions, Monday
19/12/2011
Slide 19
Page: 19 Application Carbon Front December Contracts Carbon
front December Contracts: Return
Slide 20
Back to Overview CFE-2011, Parallel Sessions, Monday
19/12/2011
Slide 21
Scientific Models: Stochastic Volatility Model /Parameters ( )
Bayesian Estimation Results 1. 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 Application Carbon Front December Contracts
Return CFE-2011, Parallel Sessions, Monday 19/12/2011
Slide 22
Back to Overview CFE-2011, Parallel Sessions, Monday
19/12/2011
Slide 23
Page: 23 Scientific Model: Model Assessment the model concert
test Carbon front December k = 1, 10, 20 and 100 densities
reported. Application Carbon Front December Contracts
Slide 24
Page: 24 Scientific Model: The Stochastic Volatility Model:
log-sci-mod-posterior Log sci-mod-posterior (every 25 th
observation reported): Optimum is along this path! Application
Carbon Front December Contracts
Slide 25
Page: 25 Scientific Model: Carbon -paths and densities; 240.000
simulations Application Carbon Front December Contracts
Slide 26
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. Return Application Carbon Front December
Contracts CFE-2011, Parallel Sessions, Monday 19/12/2011
Slide 27
Back to Overview CFE-2011, Parallel Sessions, Monday
19/12/2011
Slide 28
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: 28 Application Carbon Front December Contracts
Return
Slide 29
Back to Overview CFE-2011, Parallel Sessions, Monday
19/12/2011
Slide 30
Scientific Model: The Stochastic Volatility Model. Page: 30 The
Option market versus SV-model prices 05.09.2011 Application Carbon
Front December Contracts
Slide 31
Scientific Model: The Stochastic Volatility Model. Page: 31
Risk assessment and management: CARBON VaR / CVaR Application
Carbon Front December Contracts
Slide 32
Scientific Model: The Stochastic Volatility Model. Page: 32
Asset Allocation/Dynamic Hedging: CARBON GREEK Letters Return
Application Carbon Front December Contracts
Slide 33
Back to Overview CFE-2011, Parallel Sessions, Monday
19/12/2011
Slide 34
Page: 34 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: 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. For instance, one might wish to obtain an estimate of:
for the purpose of pricing an option (from the re-projection step).
Application Carbon Front December Contracts CFE-2011, Parallel
Sessions, Monday 19/12/2011
Slide 35
Page: 35 SV Model: One-step-ahead conditional moments
Application Carbon Front December Contracts
Slide 36
Page: 36 SV Model: filtered volatility /particle filtering for
Option pricing Application Carbon Front December Contracts
Slide 37
Page: 37 SV Model: Conditional variance functions (asymmetry)
(shocks to a system that comes as a surprise to the economic
agents) Application Carbon Front December Contracts
Slide 38
Page: 38 SV Model: Multistep-ahead volatility dynamics
(volatility impulse-response profiles) Application Carbon Front
December Contracts CFE-2011, Parallel Sessions, Monday
19/12/2011
Slide 39
Page: 39 SV Model: Mean and Volatility Persistence (half-lives
= ln2 / ) Application Carbon Front December Contracts Return
CFE-2011, Parallel Sessions, Monday 19/12/2011
Slide 40
Back to Overview CFE-2011, Parallel Sessions, Monday
19/12/2011
Slide 41
Scientific Model Re-projections: Conditional SV-model moments:
Conditional VaR/CVaR for RM and Greeks for Asset allocation Page:
41 Application Carbon Front December Contracts CFE-2011, Parallel
Sessions, Monday 19/12/2011
Slide 42
Scientific Model Reprojections: Extensions using SV-model
simulations: Realized Volatility and continuous / jump volatility
(5 minutes simulations): Page: 42 Application Carbon Front December
Contracts CFE-2011, Parallel Sessions, Monday 19/12/2011
Return
Slide 43
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. X t-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 Return Application Carbon
Front December Contracts
Slide 44
Back to Overview CFE-2011, Parallel Sessions, Monday
19/12/2011
Slide 45
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
Application Carbon Front December Contracts CFE-2011, Parallel
Sessions, Monday 19/12/2011
Slide 46
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
functions interpret 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 Contracts Return