trondheim, 29-30 th june 2011page: 1 stochastic volatility models the nord pool energy market by per...

Trondheim, 29-30th June 2011 Page: 1

Stochastic Volatility Models

The Nord Pool Energy Market

by

Per Bjarte Solibakke

Department of Economics, Molde University College

Page: 2

Stochastic Volatility: Origins and Overview

Stochastic Volatility (SV) models are used to capture the impact of time-varying volatility on financial markets and decision making (endemic in markets) .

The success of SV models is multidisciplinary: financial economics, probability theory and econometrics are blended to produce methods that aid our understanding of option pricing, efficient portfolio allocation and accurate risk assessment and management.

Heterogeneity has implications for the theory and practice of economics and econometrics. Heterogeneity for asset pricing theory means higher rewards are required as an asset is exposed to more systematic risk.

SV models bring us closer to reality, allowing us to make better decisions, inspire new theory and improve model building.

Page: 3


The SV approach indirectly specifies the predictive distribution of returns via the structure of the scientific model (need numerical computations in most cases). The advantage is that it is more convenient and perhaps more natural to model the volatility as having its own stochastic process. The disadvantage is that the likelihood function is not directly available.

From the late 1990s the SV models have taken centre stage in the econometric analysis of volatility forecasting using high-frequency data based on realized volatility (RV) and related concepts. This is mainly due to the fact that the econometric analysis of realized volatility is tied to continuous time processes.

The close connection between SV and RV models allows econometricians to harness the enriched information set available through high frequency data to improve, by an order of magnitude, the accuracy of their volatility forecasts.

Page: 4


The central intuition in the SV literature is that asset returns are well approximated by a mixture distribution where the mixture reflects the level of activity or news arrivals. Clark 1973 originates this approach by specifying asset prices as subordinated stochastic processes directed by the increments to an underlying activity variable. Clark (1973) stipulates:

where Yi denotes the logarithmic asset price at time i and yi = Yi – Yi-1 the corresponding continuously compounded return over [i-1, i].The is a normally distributed random variable with mean zero, variance , and independent increments, and is a real-valued process initiated at with non-negative and non-decreasing sample paths (time change).

, 0,1,2,...iiY X i

2X i

iX

i

0 0

21 1( ) ~ (0, ( ))i i i X i iy N

The Mixture of Distributions Hypotheses (MDH) inducing heteroskedastic return volatility and, if the time-change process is positively serially correlated, also volatility clustering.

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Stochastic Volatility: Continuous Time Model

Emphasising that the log-price process is a martingale, we can write:

; , 0,i t ti t tY M and X W M W t

where W is BM and W and t are independent processes. However, due to the fact that asset pricing theory asserts (systematic risk) positive excess returns relative to the risk-free interest rate, asset prices are not martingales. Instead, assuming frictionless markets, weak no-arbitrage condition the asset prices will be a special semi-martingale, leading to the general formulation:

0Y Y A M

where the finite variation process, A, constitutes the expected mean return. An specification is: with rf denoting the risk-free interest rate and b representing a risk premium due to the non-diversifiable variance risk. The distributional MDH result generalizes to:

t f tA r t

| ~ ( , )t t f t tY N r t

Note that the persistence in return volatility is not represented in the model.

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Stochastic Volatility: Continuous Time Model

A decade later Taylor (1982), accommodates volatility clustering. Taylor models the risky part of returns as a product process:

1i i i i im M M

where hi is a non-zero mean Gaussian process. A first order auto-regression is (hi is a zero mean, Gaussian white noise process):

1 ( )i i ih h

exp( / 2)i ih

and in continuous time, using the Itô stochastic integral representation (and where jumps are allowed):

e is assumed to follow an auto-regression with zero mean and unit variance, while s is some non-negative process. The model is completed by assuming e is orthogonal to s and

0

t

t s sM dW

Page: 7

Stochastic Volatility and Realized Variance

Assuming M is a process with continuous martingale sample paths then the celebrated Dambis-Dubins-Schwartz theorem, ensures that M can be written as a time changed BM with the time-change being the quadratic variation (QV) process:

1sup 0 .j j jt t for n

As M has continuous sample paths, so must [M]. If [M] is absolutely continuous (stronger condition), M can be written as a SV process (Doob, 1953).Together this implies that a time-changed BMs are canonical in continuous sample path price processes and SV models arise as special case.

In the SV case:

21

1

limj j

n

t ttj

M p M M

for any sequence of partitions t0 = 0 <t1 < … < tn = t with

That is, the increments to the quadratic variation (QV) process are identical to the corresponding integrated return variance generated by the SV model.

2

0

t

stM ds

Page: 8

Stochastic Volatility: Extensions

1. Jumps:

Eraker et al. (2003) deem this extension critical for adequate model fit.

2 2 , 0t td dt dz

where z is a subordinator with independent, stationary and non-negative increments. The unusual timing convention for zlt ensures that the stationary distribution of s2 does not depend on l. These OU processes are analytical tractable (affine model class).

Barndorf-Nielsen and Shephard (2001): Pure jump processes

2. Long Memory

Barndorff-Nilsen (2001): infinite superposition of non-negative OU processes. The process can be used for option pricing without excessive computational effort.

Page: 9

Stochastic Volatility: Extensions (cont.)

3. Multivariate models

1

J

j jj

M F G

where the factors F(1), F(2),…, F(J) are independent univariate SV models, J<N, and G is correlated (Nx1) BM, and the (Nx1) vector of factor loadings, b(j), remains constant through time.

Diebold and Nerlove (1989) cast a multivariate SV model within the factor structure used in many areas of asset prticing:

Harvey (1994) introduced a more limited multivariate discrete time model. Harvey suggest having the martingale components be given as a direct rotation of a p-dimensinal vector of univariate SV processes (implemented in OX 6.0).

Recently, the area has seen a dramatic increase in activity (Chib et al., 2008).

Page: 10

Stochastic Volatility: 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 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: 11

Stochastic Volatility: Simulation-based inference (EMM)

Applications:

Andersen and Lund (1997): Short rate volatilityChernov and Ghysel (2002): Option pricing under SVDai & 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-shiftsSolibakke, P.B (2001): SV model for Thinly Traded Equity Markets

Gallant, A.R. and R.E. McCulloch, 2009, On the determination of general statistical models with application to asset pricing, Journal of The American Statistical Association, 104, 117-131.

Third, the re-projection step obtains:

Forecasting volatility conditional on the past observed data; and/or extracting volatility given the full data series. Conditional one-step-ahead mean and volatility densities. The conditional volatility function is available.

Page: 12

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: 13

Simulated Score Methods and Indirect Inference for Continuous-time Models

Three basic steps:

1. Projection step: project the data into the reduced-form auxiliary-model.

2. Estimation step: parameters for the structural model (i.e. SV) is estimated by GMM using an appropriate weighting matrix.

3. Reprojection step: entails post-estimation analysis of simulations for the purposes of prediction, filtering and model assessment.

Page: 14

Application Stochastic Volatility (SV):

NORD POOL energy market

FRONT Product Contracts

See also the PHELIX and CARBON applications (working papers):

Solibakke, P.B., S. Westgaard, S., and G. Lien,2010, Stochastic Volatility Models for EEX Base and Peak Load Forward Contracts using GSM

Solibakke, P.B., S. Westgaard, S., and G. Lien,2010, Stochastic Volatility Models for Carbon Front December Contracts

Front Day/Week/Month/Quarter/Year Contracts: Stochastic Volatility

Page: 15

Higher Understanding of the Market in general Serial correlation and non-normality in the Mean equations Volatility clustering and persistence in the volatility equations

Models derived from scientific considerations is always preferable Likelihood is not observable because of latent variables (volatility) The model’s output 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 is credible

Financial Contracts Characteristics for Hedging (derivatives based on) General Forward Contracts

Research Objectives (purpose):

Page: 16

Research Objectives (purpose):

Value-at-Risk / Expected Shortfall for Risk Management Stochastic Volatility models are well suited simulation Using Simulation and Extreme Value Theory for VaR-/ES-Densities

Simulation and Greek Letters Calculations for Portfolio Management Direct path-wise hedge parameter estimates MCMC superior to finite difference, which is biased and time-consuming

The Case against the Efficiency of Future Markets (EMH) Serial correlation in Mean and Volatility Price-Trend-Forecasting models and the Construction of trading rules


Research Design (how):

Time-series of Front price contracts from Mondays to Fridays

Score generator (A Statistical Model) Serial Correlation in the Mean (AR-model) Volatility Clustering in the Latent Volatility ((G)ARCH-model) Hermite Polynomials for higher order features and non-normality

Scientific Model – Stochastic Volatility Models

0 1 1 0 1

0 1 1 0 2

1 1

22 1 2

exp( )

1

t t t t

t t t

t t

t t t

y a a y a u

b b b u

u z

u s r z r z

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

0 1 0 1, , , , ,a a b b s r

Page: 17


The simple two-factor model:

Research Design (how):

Scientific Model – A Stochastic Volatility Model

where z1t , z2t and z3t are iid Gaussian random variables. The parameter vector now becomes:

Page: 18

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


Three-factor models (with Cholesky-decomposition):

Research Design (how): Scientific Model – A Stochastic Volatility Model

where z1t, z2t, z3t and z4t are iid Gaussian random variables. The parameter vector now becomes:

Page: 19

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 2 3 2 1 1 3

4 3 4 1 5 4 1

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

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

u s r z r r r

21 2 6 4 2 3 4/ 1 ... ... / ... ...t t tr z r r r z z

0 1 0 1 0 1 0 1 1 2 3 1 2 3 4 5 6, , , , , , , , , , , , , , , ,a a b b c c d d s s s r r r r r r


Four-factor models (with Cholesky-decomposition):

Front Electricity Market Data contracts NP (Monday-Friday, not holidays):

3911

, 1, / / / /i t t

y i Day Week Month Quarter Year

Page: 22

Raw data series characteristic plots


Page: 23


Three basic steps:




Statistical Model Characteristics (conditional one-step ahead distributions (moments))

Page: 24


Statistical Model Characteristics (conditional variance functions)

Page: 26


Page: 28


Three basic steps:




Scientific Models: The Stochastic Volatility Models


Page: 29

Front Year Contract Scientific ModelParameter values Scientific Model. Standard

Mode Mean error

a0 0.0466920 0.0469680 0.0142350

a1 0.0758150 0.0792210 0.0130730

b0 0.0718260 0.1895900 0.0989570

b1 0.9866100 0.9722500 0.0256910

c1 0.6714900 0.5870600 0.0850520

s1 0.0825190 0.0842100 0.0152590

s2 0.1964400 0.2046700 0.0211520

r1 0.2352600 0.3050900 0.0954660

r2 -0.1460400 -0.1405100 0.0538790

log sci_mod_prior 4.5339809 c2(5)

log stat_mod_prior 0 -2.05220log stat_mod_likelihood -4261.38868 {0.84188}log sci_mod_posterior -4256.85470

Front Month Contract Scientific ModelParameter values Scientific Model. Standard

Mode Mean error

a0 -0.0823820 -0.0943910 0.0260010

a1 0.1126500 0.1151500 0.0109900

b0 0.8207500 0.8230300 0.0448190

b1 0.9700400 0.9684000 0.0063873

c1 0.1538600 0.0799180 0.1052300

s1 0.1108000 0.1088900 0.0110800

s2 0.2325200 0.2258900 0.0223330

r1 0.1115800 0.1277400 0.0597970

r2 -0.0645280 -0.0513250 0.0343710



Front Week Contract Scientific ModelParameter values Scientific Model. Standard

Mode Mean error

a0 -0.3535500 -0.3386800 0.0343690

a1 0.1649900 0.1630200 0.0101890

b0 0.9282400 0.9097100 0.0377290

b1 0.9629300 0.9586200 0.0067092

c1 -0.7046900 -0.4795800 0.1723800

s1 0.1255600 0.1241200 0.0093417

s2 0.2272300 0.2689200 0.0380850

r1 -0.0524850 -0.0135910 0.0696300

r2 0.0490910 0.0292610 0.0329160



Front Day Contract Scientific ModelParameter values Scientific Model. Standard

Mode Mean errora0 -0.3231800 -0.3082100 0.0208610

a1 0.0115830 0.0063304 0.0056443

b0 1.4237000 1.4043000 0.0078410

b1 0.8811800 0.8808500 0.0011206

c1 -0.0257450 -0.0650240 0.0229130

d1 -0.1821900 -0.1342300 0.0267100

s1 0.2844400 0.2938600 0.0044842

s2 0.0867860 0.0802750 0.0029063

s3 0.0039528 0.0011121 0.0014336

r1 0.0904810 0.0833680 0.0048483

r2 0.1366100 0.1627700 0.0201910

r4 -0.4880100 -0.4680400 0.0397570

log sci_mod_prior -9.2086914 c2(6)


Scientific Models: The Stochastic Volatility Models


Page: 30

Mean Equation Volatility Equation Volatility serial correlationDrift Serial Correlation Constant Sigma_1 Sigma_2 Sigma_3 1st factor 2nd factor 3rd factor

Year 0.046968 0.079221 0.18959 0.08421 0.20467 0.97225 0.58706Contracts {0.014235} {0.013073} {0.098957} {0.015259} {0.021152} {0.025691} {0.085052}

Quarter 0.0044977 0.083847 0.73992 0.13175 0.12685 0.94659 0.24882Contracts {0.026073} {0.011955} {0.160100} {0.029349} {0.052855} {0.035534} {0.210450}

Month -0.094391 0.11515 0.82303 0.10889 0.22589 0.9684 0.079918Contracts {0.026001} {0.010990} {0.044819} {0.011080} {0.022333} {0.006387} {0.105230}

Week -0.33868 0.16302 0.90971 0.12412 0.26892 0.95862 -0.47958Contracts {0.034369} {0.010189} {0.037729} {0.009342} {0.038085} {0.006709} {0.172380}

Day -0.32318 0.0063304 1.4043 0.29386 0.080275 0.0011121 0.88085 -0.065024 -0.13423Contracts {0.020861} {0.005644} {0.007841} {0.004484} {0.002906} {0.001434} {0.001121} {0.022913} {0.026710}

Empirical Findings:

Three factor models for week, month, quarter and year financial contracts. Four factor model for one-day forward (spot) product.

For the mean stochastic equations: Changing drift and serial correlation for the five series Negative drift for the shortest contracts. Are there risk premiums in the market for

the shortest products? Positive drift for the longest contracts (quarter and year).

Scientific Model: The Stochastic Volatility Model.


Page: 34

Empirical Findings:

For the latent volatility stochastic equation: The size and complexity of volatility seem to increase the shorter the life of the

contracts. Positive constant parameter for the contracts (b0). However, for the year and quarter

is coefficients are close to zero. For the longest contracts (quarter and year) the persistence (b1) is quite high.

For the shortest contracts the persistence are much lower. The volatility parameters show higher volatility for the shortest contracts. The factors show quite different characteristics

Asymmetry is highest for longest contracts ( and positive).



Page: 35

The Stochastic Volatility Model: Empirical Findings (densities)


Page: 36

The Stochastic Volatility Model: Empirical Findings (volatility factors)


Page: 37

Page: 39


Three basic steps:



3. Reprojection step: entails post-estimation analysis of simulations for the purposes of prediction, filtering and (model assessment).

Page: 40

Scientific Model: Reprojections (nonlinear Kalman filtering)

Of immediate interest of eliciting the dynamics of the observables:

0 1 0 0 1 0( | ) ( | , )K Ky x y f y x dy 1. One-step ahead conditional mean:

2. 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

3. 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

Scientific Model: Reprojections (one-step-ahead conditional moments)

0 1 0 0 1 0( | ) ( | , )K Ky x y f y x dy


Page: 42

Scientific Model: Reprojections (one-step-ahead conditional moments) '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: 43

Scientific Model: Reprojections (filtered volatility) Front Year


0

0.1

0.2

0.3

0.4

0.5

0.6

Conditonal

Mean

Density

One-step-ahead density fK(yt|xt-1,) xt-1=-10,-5,-3,-2,-1,0,mean,+1,+2,+3,+5,+10%

Frequency xt-1=-10% Frequency xt-1=-5% Frequency xt-1=-3% Frequency xt-1=-2%

Frequency xt-1=-1% Frequency xt-1= "Mean (0.033)" Frequency xt-1=0% Frequency xt-1=+1%

Frequency xt-1=+2% Frequency x-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

0.45

GAUSS-Hermite Quadrature Density Distribution (Particle Filtering)

Reprojected Quadrature Projected Quadrature

Page: 45

Scientific Model: Reprojections (multistep ahead dynamics)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 260

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 Multistep Ahead Dynamics Front Year Contracts (s2j)

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

DAYS

Var

ian

ce E

[Var

(yk,

j|x-

1]

Page: 46

Scientific Model: Reprojections (persistence mean and volatility)


-5

-4

-3

-2

-1

0

1

2

3

4

5

6

Me

an

Days

Reprojection Profile Bundles for the Front Year: THE MEAN

0

2.5

5

7.5

10

12.5

Vol

atil

ity

Days

Reprojection Profile Bundles for Front Year Reprojection Volatility

Half life in number of days

24.54522

SE = 1.944

Risk assessment and management: VaR / Expected Shortfall

Using Extreme Value Theory estimates of VaR and Expected Shortfall can be calculated.

The power law is found to be approx. true and is used to estimate the tails of distributions (EVT).


-25

-20

-15

-10

-5

0

0 0.693147181 1.098612289 1.386294361 1.609437912 1.791759469 1.945910149

Power Law for One-Year December Forward Contracts

K = 4


Page: 47

Risk assessment and management: VaR / Expected Shortfall

Note that setting u = b/ x , the cumulative probability distribution of x when x is large is:

saying that the probability of the variable being greater than x is Kx-a

where and which

implies that the extreme value theory is consistent with the power law and

VaR becomes:


1/( ) 1 / /uF x n n x

1// /uK n n x

1/

/ / 1 1uVaR u n n q

where q is the confidence level, n is the total number of observations and nu is number of observations x > u. (Gnedenko, D.V., 1943)


Page: 48

Front Week Contracts: 6000 Simulations – VaR Densities


Page: 49

Front Week Contracts: 6000 Simulations – CVaR DensitiesFront Week Contracts: 6000 Simulations – Greeks Densities

Main Findings for the Nord Pool Energy Market

Stochastic Volatility models give a deeper insight of price processes and the EMH

The Stochastic Volatility model and the statistical model work well in concert.

The M-H algorithm helps to keep parameter estimates within correct theoretical values (i.e. s – positive and abs(r)<+1)

VaR / CVAR for risk management and Greek letters (portfolio management)

are easily obtainable from the SV models and Extreme Value Theory.

Imperfect tracking (incomplete markets) suggest that simulation is the only available methodology for derivative pricing methodologies.


Page: 50

Scientific Model: Stochastic Volatility models - Summary and Conclusions

Show the use of a Bayesian M-H algorithm application (SV)

Reliable and credible SV-model parameters are obtained

MCMC extends parameter findings from nonlinear optimizers

Mean and Volatility conditional forecasts is available. Preliminary results suggest close to normal densities with much smaller standard deviations.

Volatility clustering and asymmetry suggests non-linear price dynamics for the Nord Pool energy market

SV-models is therefore a fruitful and practical methodology for descriptive statistics, forecasting and predictions, risk and portfolio management.


Page: 51

trondheim, 29-30 th june 2011page: 1 stochastic volatility models the nord pool energy market by per...

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