Book Review: Energy Derivatives: Pricing and Risk Management by Clewlow and Strickland, 2000 Chapter

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Book Review Energy Derivatives Pricing and
Risk Management by Clewlow and Strickland,
2000Chapter 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

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Chapter 3
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Chapter 3 (cntd)
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Outline

• Intro
• Estimating Volatility
• Stochastic Volatility Models

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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

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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

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Estimating Volatility (EV)

• EV From Historical Data
• EV For a Mean-Reverting Process
• EV Special Issues
• Intraday Price Variability
• EV for a Basket
• Implied Volatility

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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

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EV from Historical Data II

• Step 1 log price returns (lpr)-log(1r)
• Step 2 st. dev. of lpr
• Step 3 annualization
• \sigmasqrt(n)\sigma(lpr)
• Standard usage
• Seasonality effect

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EV for a Mean-Reverting Process

• Ornstein-Uhlenbeck process (OU)
• Solution
• Discrete analogue (autoregressive process)
• OU is the limiting case for
• (dt-gt0)
• \nu_t-zero mean and variance

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EV for a Mean-Reverting Process II

• Recovering of the initial parameters from
  discrete version

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EV Special Issues

• The choice of the annualisation factor and use of
  intra-period data (intraday prices)
• Posibilities sqrt(266)52x(41.107)
• Sqrt(273)52x(41.25)

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EV Intraday Price Variability
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EV Basket Options (Sum of 2 (weighted) or more
prices)

• The Call Option Payoff
• The Put Option Payoff

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EV Basket Options (Sum of 2 weighted or more
prices) II

• Two Commodities (GBM)
• PDE
• Volatility

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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) its
  described by the kurtosis (4th moment around the
  mean) (for normal 3)

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Stochastic Volatility Models (SVM)

• Ornstein-Uhlenbeck
• Vasicek
• Ho Lee
• Hull-White
• Cox-Ingersoll-Ross
• Heath-Jarrow-Morton
• Continuous-time above

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Stochastic Volatility Models (SVM) II

• Engle (1982) ARCH(q)
• Price returns
• Variance
• Bollerslev (1986) GARCH(p,q)
• GARCH(1,1)

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Stochastic Volatility Models (SVM) IV
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Stochastic Volatility Models (SVM) III
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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

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Estimation and Testing Least Squares

• Stochastic equation
• Minimization

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Estimation and Testing Least Squares II

• Example I
• Estimation of Mean

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Estimation and Testing Least Squares II

• Example II
• Estimation of Standard Deviation
• Unbiased, consistent, efficient

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Maximum Likelihood Estimation (MLE)

• Equation
• Probability density function
• Joint distribution
• Likelihood function

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MLE I

• Maximising Equations are

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MLE II

• MLE for St. Dev.
• Consistent
• But biased
• Unbiased (LSE)

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Testing
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Testing II

• Skewness
• Kurtosis
• Jarque-Bera Statistic
• Goldfeld-Quandt test

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Testing (Example from Energy Commodity Markets)
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Testing (Example from Energy Commodity Markets I)
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Testing (Goodness of Fit)

• Likelihood Ratio Test
• Schwartz Criterion (SC)
• (the most probable model-with the smallest SC)

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Testing (Goodness of Fit)
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Testing (Goodness of Fit)
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Figures (Simulated vs. Actual Data) PD
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Figures (Simulated vs. Actual Data) JD
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Figures (Simulated vs. Actual Data) JDGARCH
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The End

• Thank You