Transcript Document

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
CFE-2011, Parallel Sessions, Monday 19/12/2011
Page: 1
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
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
Page: 2
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
CFE-2011, Parallel Sessions, Monday 19/12/2011
Page: 3
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).
CFE-2011, Parallel Sessions, Monday 19/12/2011
Page: 4
The General Scientific Model methodology (GSM):
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
ySDE:
t  a0  a1  yt 1  a0   exp(1,t   2,t )u1t
 10bdt exp(
   u12U 2t  13U 3t )dW1t
1t b
dU
1,t 
0
1
1,t 1  b010
2t
dU 2t   22U 2t dt  dW2t
2,t 
c0  c1 2,t 1  c0   u3t
dU 3t   33U 3t dt  dW3t
u1t  z1t
where z1t , z2t and (z3t ) are iid Gaussian
random variables. The parameter vector
is:
  a0 , a1, a2 , b0 , b1, c0 , c1, s1, s2 , r1, r2 , r3 
A vector SDE with 2two stochastic volatilityfactors.

 1 r   z 
  r   r  r  /
u2t  s1 r1  z1t 

u3t  s2  r2  z1t


1
3
2t
2
1
1 r
2
1



 z2t  1  r22   r3   r2  r1   / 1  r12

CFE-2011, Parallel Sessions, Monday 19/12/2011

2


  z3t 


Page: 5
The General Scientific Model methodology (GSM):
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.
Applications:
Andersen and Lund (1997):
Solibakke, P.B (2001):
Chernov and Ghysel (2002):
Dai & Singleton (2000) and
Ahn et al. (2002):
Andersen et al. (2002):
Bansal and Zhou (2002):
Gallant & Tauchen (2010):
Short rate volatility
SV model for Thinly Traded Equity Markets
Option pricing under Stochastic Volatility
Affine and quadratic term structure models
SV jump diffusions for equity returns
Term structure models with regime-shifts
Simulated Score Methods and Indirect
Inference for Continuous-time Models
CFE-2011, Parallel Sessions, Monday 19/12/2011
Page: 6
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.
CFE-2011, Parallel Sessions, Monday 19/12/2011
Page: 7
Simulated Score Methods and Indirect Inference for Continuous-time
Models (some details): Estimation
Simulated Score Estimation:
Suppose that: f ( yt | xt-1, r ) 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 r n
n
the average of the score of the data  yt , xt 1t 1
satisfies:
n
å¶ / ¶q (log f ( y | x
1/ n
t
t-1, rn
)) = 0
t=1
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 p( yt | xt-1,q ) , 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
Page: 8
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)
Details for parameter  estimation:
n
(
)
Compute: L (q ) = Õ f yt | xt-1, g (q ) , where  yt , xt 1 denotes the observed data
and n is the samplet=1size. Given a current  0 and the corresponding  0  g  0 we
obtain the pair  ' , '  as follows (the M-H algorithm):
I. Draw
q
II. Simulate
*

0 *
according to q  ,
 yt , xt 1tN1
 
 

according to p( yt | xt-1,q )
*
*
N
*
III. Compute   g  and y (parameter functionals) from simulation  yt , xt 1t 1
IV. Define
( ) ( ) (
( ) ( ) (
ì
*
*
* 0
ï L q × p q × q q ,q
a = min í1,
ï L q 0 × p q * × q q 0 ,q *
î
( ) (
)
) üïý
) ïþ
I. With probability a , q ' ,h ' = q * ,h* , otherwise
 ,    , .
'
'
0
0
Page: 9
Simulated Score Methods and Indirect Inference for Continuous-time
Models (some details): Assessment
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.
(
) (
¢
d h, M = min h - h j A j h - h j
(
)
j=1,...G
)
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.
Page: 10
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:

pˆ  y0 | y L ,..., y1   p y0 | y L ,..., y1 ,ˆn

The unconditional expectations can be generated by a simulation:  yˆt t  L
N


Eˆ  g    ... g  y L ,..., y0  p y L ,..., y0 , ˆn d y L ...d y0
n
Let fˆK  y0 | y L ,..., y1   f K  y0 | y L ,..., y1 , ˆK  . Theorem 1 of Gallant and Long
(1997) states:
lim fˆK  y0 | y L ,..., y1   pˆ  y0 | y L ,..., y1 
K 
We study the dynamics of pˆ by using fˆK as an approximation.
CFE-2011, Parallel Sessions, Monday 19/12/2011
Page: 11
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
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
Page: 13
Objectives (purpose): (cont)
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
CFE-2011, Parallel Sessions, Monday 19/12/2011
Page: 14
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)
CFE-2011, Parallel Sessions, Monday 19/12/2011
Page: 15
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.
CFE-2011, Parallel Sessions, Monday 19/12/2011
Page: 16
Carbon Application MCMC estimation/inference:
1. NASDAQ OMX Carbon front December contracts
Data Characteristics
2. The Statistical model and the Stochastic Volatility Model
Estimation Results
3. Model assessment (relaxing the prior): model appropriate?
Assessment
4.
Empirical Findings in the mean and latent volatility.
Unconditional mean and latent volatility paths/distributions
Model Findings
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.
Risk M/Asset Alloc
Re-projection/Post-Est
3.
Conditional Risk Management and Asset Allocation
Conditional Moments
4.
EMH and Model Summary/Conclusion
EMH/ Model Summary
CFE-2011, Parallel Sessions, Monday 19/12/2011
Page: 17
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Overview
CFE-2011, Parallel Sessions, Monday 19/12/2011
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Application Carbon Front December Contracts
Carbon front December Contracts:
Mean /
Median /
Max.
Moment
Quantile
{y }
1191
i,t t=1
Quantile
, i = Carbon Front December
K-S
RESET Serial dependence
Mode
std.dev
Min. Kurt/Skew Kurt/Skew Normal
Z-test
(12;6)
-0.04364
0.0000
11.5196 2.84118
0.29749
4.2512
4.59075
70.5138
0.00000
2.43729 -10.0083 -0.13418
0.03835 {0.1194} {0.0000} {0.0000}
BDS-statistic (e =1)
KPSS (Stationary)
Augmented ARCH
m=2
m=3
m=4
m=5
Level
Trend
DF-test
(12)
16.6788
23.5820 30.1427 38.4401
0.14330
0.14340
-56.0675
594.675
{0.0000} {0.0000} {0.0000} {0.0000} {0.4121} {0.0568} {0.0000} {0.0000}
Q(12)
55.7488
{0.0000}
VaR
2.5/0.5%
-5.247
-8.311
Q2 (12)
1946.27
{0.0000}
CVaR
2.5/0.5%
-7.178
-9.694
Page: 19
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CFE-2011, Parallel Sessions, Monday 19/12/2011
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Application Carbon Front December Contracts
Scientific Models: Stochastic Volatility Model /Parameters ()
1. Several serial
BayesianResults
runs establishing the mode
Bayesian
Estimation
Carbon
Front
Scientific
Model. quits
Statistical
Model
- fitmode
modelhas
We tune
the December
scientific General
model until
the posterior
climbing
and SNP-11116000
it looks like the
Parameter
values Scientific Model.
Standard
Parameters Non-linear-GARCH.
Standard
been reached:

Mode
Mean
error

Mode
Mean
error
0.033957
n1 a0[1]
0.010997
2.a0A final 0.026974
parallel run with
24 (8 *3)0.041845
CPUs and 240.000
MCMC
simulations0.017017 0.012873
a1
0.053948
0.045583
0.021425 University)
n2 a0[3]parallell
0.009816
-0.027176 0.015376
(OPEN_MPI
(Indiana
computing)
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
0.046907
n8 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
log stat_mod_likelihood -1515.8624
log sci_mod_posterior
-1510.0652
c2 (3) =
-0.94841
{0.81373}
n10 Q(1,1)
n11 V(1,1)
0.926579
-0.116156
CFE-2011, Parallel Sessions, Monday 19/12/2011
0.860786
0.037407
0.041011
0.139785
Page: 21
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CFE-2011, Parallel Sessions, Monday 19/12/2011
Application Carbon Front December Contracts
Scientific Model: Model Assessment – the model concert test
Carbon front December k = 1, 10, 20 and 100 densities – reported.
Page: 23
Application Carbon Front December Contracts
Scientific Model: The Stochastic Volatility Model: log-sci-mod-posterior
Log sci-mod-posterior (every 25th observation reported): Optimum is along this path!
Page: 24
Application Carbon Front December Contracts
Scientific Model: Carbon -paths and densities; 240.000 simulations
Page: 25
Return
Application Carbon Front December Contracts
Scientific Model: Stochastic Volatility
The chains look good. Rejection rates are:
Reported
theta1 ( 1)
%-rejected
0.49051424
Proportion
Moved
0.1255875
Number of
Rejects
60.525
Proportion
Accepted
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
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
0.12526667 60.2333333 125.266667
1
484.366667
1000
 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.
CFE-2011, Parallel Sessions, Monday 19/12/2011
Page: 26
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Application Carbon Front December 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: 28
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CFE-2011, Parallel Sessions, Monday 19/12/2011
Application Carbon Front December Contracts
Scientific Model: The Stochastic Volatility Model.
The Option market versus SV-model prices 05.09.2011
Carbon Option Market and SV-Model prices Maturity 2011 for 2011/09/05
Option Prices
05.09.2011 Market closing prices
DEC-11
Strike Price call Dec-11 put Dec-11
Volume 0
6.0
2.59
0.06
0
6.5
2.15
0.12
0
7.0
1.72
0.19
0
7.5
1.33
0.29
0
8.0
0.97
0.43
0
8.5
0.67
0.62
0
9.0
0.46
0.91
0
9.5
0.31
1.25
0
10.0
0.2
1.64
0
10.5
0.12
2.06
0
11.0
0.07
2.51
0
11.5
0.04
2.97
0
12.0
0.03
3.45
0
12.5
0.01
3.93
0
13.0
0.01
4.42
4.5
4
O 3.5
p
t 3
i
o
n 2.5
P 2
r
i
1.5
c
e
s 1
0.5
SV-Model prices
call Dec-11 put Dec-11
2.54
0.04
2.09
0.08
1.66
0.13
1.30
0.23
0.92
0.39
0.62
0.61
0.41
0.89
0.26
1.23
0.17
1.62
0.11
2.11
0.06
2.52
0.04
3.04
0.03
3.51
0.01
3.96
0.01
4.47
0
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
Strike Prices
Market closing prices call Dec-11
put Dec-11
SV-Model prices call Dec-11
put Dec-11
Page: 30
Application Carbon Front December Contracts
Scientific Model: The Stochastic Volatility Model.
Risk assessment and management: CARBON VaR / CVaR
Page: 31
Return
Application Carbon Front December Contracts
Scientific Model: The Stochastic Volatility Model.
Asset Allocation/Dynamic Hedging: CARBON GREEK Letters
Page: 32
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CFE-2011, Parallel Sessions, Monday 19/12/2011
Application Carbon Front December Contracts
Scientific Model: Re-projections / nonlinear Kalman filtering
Of immediate interest of eliciting the dynamics of observables:
One-step ahead conditional mean: ( y0 | x1 )   y0 f k ( y0 | x1 ,  k ) dy0
One-step ahead conditional volatility:
Var ( y0 | x1 )    y0  ( y0 | x1 )  y0  ( y0 | x1 )  f k ( y0 | x1 ,  k )dy0
'
Filtered volatility is the one-step ahead conditional standard deviation
evaluated at data values:
Var ( yk 0 | x1 ) |x1  yt L ,..., yt 1 )
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:
 
*
t
t T
 exp(v  v )dt
1
2
t
for the purpose of pricing an option (from the re-projection step).
CFE-2011, Parallel Sessions, Monday 19/12/2011
Page: 34
Application Carbon Front December Contracts
SV Model: One-step-ahead conditional moments
( y(0 y| 0x|1x)1)  y0fyk 0( y0| (xy10,|xk)1dy
Var
) 0 y0  ( y0 | x1 )  f k ( y0 | x1 ,  k )dy0
'
Page: 35
Application Carbon Front December Contracts
SV Model: filtered volatility /particle filtering for Option pricing
yk 0 | x1 |x1 ( ytL ,..., yt1 ) t  0,..., n
0.4
GAUSS-hermite Quadrature Density Distribution
One-step-ahead density
fK(yt|xt-1,) xt-1 =-10,-5, -3, -1, m, 0, +1, +3, +5, +10%
0.2
C
o
0.35
n
d
i
t 0.3
o
n
a
0.25
l
0.15
0.1
M
e 0.2
a
n
0.15
D
e
n
s 0.1
i
t
y
0.05
0.05
0
0
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%
Page: 36
Application Carbon Front December Contracts
SV Model: Conditional variance functions (asymmetry)
(shocks to a system that comes as a surprise to the economic agents)
Page: 37
Application Carbon Front December Contracts
SV Model: Multistep-ahead volatility dynamics
(volatility impulse-response profiles)
Multistep Ahead Dynamics s2j
3
Variance E[Var(yk,j |x-1)
2.5
2
1.5
1
0.5
0
DAYS
dy0
dy-1 (low)
dy+1 (high)
dy-3 (low)
dy+3 (high)
dy-6 (low)
CFE-2011, Parallel Sessions, Monday 19/12/2011
dy+6 (high)
dy-10 (low)
dy+10 (high)
Page: 38
Return
Application Carbon Front December Contracts
SV Model: Mean and Volatility Persistence (half-lives = –ln2 / )
30
4
CARBON
Profile
Bundles
for the
(overplots
of profiles)
CARBON
Profile
Bundles
for VOLATILITY
the MEAN (overplots
of profiles)
Half
lives:
3.5
27
3
28.238149
SE=1.324
24
2.5
2
21
1.5
Mean
Volatility
18
1
0.5
15
0
12
-0.5
-1
9
-1.5
-26
-2.5
3
-3
-3.50
Days
Days
CFE-2011, Parallel Sessions, Monday 19/12/2011
Page: 39
Back to
Overview
CFE-2011, Parallel Sessions, Monday 19/12/2011
Application Carbon Front December Contracts
Scientific Model Re-projections: Conditional SV-model moments:
Conditional VaR/CVaR for RM and Greeks for Asset allocation
CFE-2011, Parallel Sessions, Monday 19/12/2011
Page: 41
Return
Application Carbon Front December Contracts
Scientific Model Reprojections: Extensions using SV-model simulations:
Realized Volatility and continuous / jump volatility (5 minutes simulations):
0.0002
0.00003
Continuous
Jump Volatility
Volatility
Realized Volatility
0.0002
0.00018
0.00018
0.00002
0.00016
0.00016
0.00014
0.00001
0.00014
0.00012
0.00012
0.0001
0
0.0001
0.00008
0.00008
-0.00001
0.00006
0.00006
0.00004
0.00004
-0.00002
0.00002
0.00002
-0.000030
0
CFE-2011, Parallel Sessions, Monday 19/12/2011
Page: 42
Return
Application Carbon Front December Contracts
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-dayahead 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
Back to
Overview
CFE-2011, Parallel Sessions, Monday 19/12/2011
Application Carbon Front December Contracts
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.
CFE-2011, Parallel Sessions, Monday 19/12/2011
Page: 45
Return
Application Carbon Front December Contracts
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