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1. 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
2. Chapter 3
3. Chapter 3 (cntd)
4. Outline Intro
Estimating Volatility
Stochastic Volatility Models
5. 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
6. 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
7. 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
8. 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
9. 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
10. 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:
11. EV for a Mean-Reverting Process II Recovering of the initial parameters from discrete version:
12. EV: Special Issues The choice of the annualisation factor and use of intra-period data (intraday prices)
Posibilities: sqrt(266)=52x(4+1.107)
Sqrt(273)=52x(4+1.25)
13. EV: Intraday Price Variability
14. EV: Basket Options (Sum of 2 (weighted) or more prices) The Call Option Payoff:
The Put Option Payoff:
15. EV: Basket Options (Sum of 2 weighted or more prices) II Two Commodities (GBM):
PDE:
Volatility:
16. 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)
17. Stochastic Volatility Models (SVM) Ornstein-Uhlenbeck
Vasicek
Ho & Lee
Hull-White
Cox-Ingersoll-Ross
Heath-Jarrow-Morton
Continuous-time above
18. Stochastic Volatility Models (SVM) II Engle (1982): ARCH(q)
Price returns
Variance
Bollerslev (1986): GARCH(p,q)
GARCH(1,1):
19. Stochastic Volatility Models (SVM) IV
20. Stochastic Volatility Models (SVM) III
21. 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
22. Estimation and Testing: Least Squares Stochastic equation:
Minimization:
23. Estimation and Testing: Least Squares II Example I:
Estimation of Mean
24. Estimation and Testing: Least Squares II Example II:
Estimation of Standard Deviation
Unbiased, consistent, efficient
25. Maximum Likelihood Estimation (MLE) Equation:
Probability density function:
Joint distribution:
Likelihood function:
26. MLE I Maximising Equations are:
27. MLE II MLE for St. Dev.:
Consistent
But biased
Unbiased (LSE)
28. Testing
29. Testing II Skewness
Kurtosis
Jarque-Bera Statistic
Goldfeld-Quandt test
30. Testing (Example from Energy Commodity Markets)
31. Testing (Example from Energy Commodity Markets I)
32. Testing (Goodness of Fit) Likelihood Ratio Test:
Schwartz Criterion (SC)
(the most probable model-with the smallest SC):
33. Testing (Goodness of Fit)
34. Testing (Goodness of Fit)
35. Figures (Simulated vs. Actual Data): PD
36. Figures (Simulated vs. Actual Data): JD
37. Figures (Simulated vs. Actual Data): JD+GARCH
38. The End Thank You