# Book Review â€˜ Energy Derivatives Pricing and Risk Management

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					       Book Review: ‘Energy Derivatives: Pricing and
Risk Management’ by Clewlow and
Strickland, 2000
Chapter 3: Volatility Estimation in Energy
Markets

Anatoliy Swishchuk
Math & Comp Lab
Dept of Math & Stat, U of C
‘Lunch at the Lab’ Talk
November 28th, 2006
Chapter 3
Chapter 3 (cntd)
Outline

 Intro
 Estimating Volatility
 Stochastic Volatility Models
Intro

   Volatility can be defined and estimated in the
context of a specific stochastic process for
the price returns
   Volatility definition and measure should
capture the key features of energy markets,
such as the seasonal dependence
Intro II (most important issues)

   Investment Assets vs. Consumption Goods
(Commodities cannot be treated as purely
financial assets)
   Prices of Energy Commodities Display
Seasonality
   Commodity Prices Often Display Jump
Behaviour
   Prices Gravitate to the Cost of Production
Estimating Volatility (EV)

   EV From Historical Data
   EV For a Mean-Reverting Process
   EV: Special Issues
   Implied Volatility
EV from Historical Data

   Step 1: Calculate Logarithmic Price Returns
   Step 2: Calculate Standard Deviations of the
Logarithmic Price Returns
   Step 3: Annualize the St. Dev. By Multiplying
it by the Correct Factor
EV from Historical Data II

  Step 1: log price returns
(lpr)-log(1+r)
 Step 2: st. dev. of lpr
 Step 3: annualization
\sigma=sqrt(n)\sigma(lpr)
Standard usage

Seasonality effect
EV for a Mean-Reverting Process

   Ornstein-Uhlenbeck process
(OU)
 Solution
 Discrete analogue
(autoregressive process)
 OU is the limiting case for
(dt->0):
\nu_t-zero mean and variance:
EV for a Mean-Reverting Process II

   Recovering of the initial
parameters from
discrete version:
EV: Special Issues

   The choice of the
annualisation factor and
use of intra-period data
   Posibilities:
sqrt(266)=52x(4+1.107)
   Sqrt(273)=52x(4+1.25)
EV: Basket Options (Sum of 2 (weighted) or
more prices)

   The Call Option Payoff:

   The Put Option Payoff:
EV: Basket Options (Sum of 2
weighted or more prices) II

   Two Commodities
(GBM):

   PDE:

   Volatility:
Implied Volatility (IV)

   IV: Vol. that is used as an input to an option pricing
formula that equates the model price with the
market price
   Existence of fat tails (leptokurtic): it’s described by
the kurtosis (4th moment around the mean) (for
normal 3)
Stochastic Volatility Models (SVM)

   Ornstein-Uhlenbeck
   Vasicek
   Ho & Lee
   Hull-White
   Cox-Ingersoll-Ross
   Heath-Jarrow-Morton
   Continuous-time above
Stochastic Volatility Models (SVM) II

  Engle (1982): ARCH(q)
Price returns
Variance

   Bollerslev (1986):
GARCH(p,q)

   GARCH(1,1):
Stochastic Volatility Models (SVM) IV
Stochastic Volatility Models (SVM) III
EV: Estimation and Testing

   Parameters
Estimation
   Usefulness of a
parameter estimator:
   Unbiased and Efficient
   Unbiased is good
   Biased but Efficient
may be preferable to
an unbiased
Estimation and Testing: Least Squares

   Stochastic equation:

   Minimization:
Estimation and Testing: Least Squares II

   Example I:

   Estimation of Mean
Estimation and Testing: Least Squares II

   Example II:

   Estimation of
Standard
Deviation
   Unbiased,
consistent,
efficient
Maximum Likelihood Estimation (MLE)

   Equation:

   Probability density
function:

   Joint distribution:

   Likelihood
function:
MLE I

   Maximising
Equations
are:
MLE II

   MLE for St. Dev.:

   Consistent
   But biased

   Unbiased (LSE)
Testing
Testing II

   Skewness

   Kurtosis

   Jarque-Bera Statistic

   Goldfeld-Quandt test
Testing (Example from Energy
Commodity Markets)
Testing (Example from Energy
Commodity Markets I)
Testing (Goodness of Fit)

   Likelihood Ratio Test:

   Schwartz Criterion (SC)
   (the most probable
model-with the smallest
SC):
Testing (Goodness of Fit)
Testing (Goodness of Fit)
Figures (Simulated vs. Actual Data): PD
Figures (Simulated vs. Actual Data): JD
Figures (Simulated vs. Actual Data):
JD+GARCH
The End

   Thank You


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