1. 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 28 th, 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 (4 th 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)

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