1. 1 Lecture Plan 9 00 -10 30 : Modelling volatility and correlation in energy markets Time Varying Volatility and Correlation in Energy Markets Models for Time Varying Volatility Exponential Weighted Moving Average of Volatility Univariate GARCH Models Models for Time Varying Correlation Exponential Weighted Moving Average of Correlation Multivariate GARCH Models Realized Volatility and Correlation Using Intra Daily Data Future Options Pricing & Implied Volatility Excel Case: Modeling time varying volatility in the ICE Oil Futures Market Modeling time varying correlation ICE Oil and Gas Oil Futures Market Modeling realized volatility and correlation in the Nord Pool Electricity Futures Market Implied volatility in the Nord Pool Electricity Futures Option Markets

2. 2 Why modelling variance/volatility covariance/correlation? Pricing options with one or more underlying Estimating the value at risk of a position or a portfolios Determining optimal allocations between a set of risky assets Finding hedge ratios

3. 3 Datasets used Crude Oil ICE Gas Oil ICE Carbon ICE

4. 4 Why non-linear models for energy futures data? Motivation: the linear structural (and time series) models cannot explain a number of important features common to much financial energy market data - leptokurtosis - volatility clustering or volatility pooling - leverage effects

5. 5 Time Varying Volatility An example of a structural model is with u t  N(0, ). The assumption that the variance of the errors is constant is known as homoscedasticity, i.e. Var (u t ) =. What if the variance of the errors is not constant? - heteroscedasticity Is the variance of the errors likely to be constant over time? Not for financial futures data

6. 6 Volatility Clustering Volatility clustering

7. 7 Volatility Clustering Volatility clustering

8. 8 Non-linear Models: A Definition A non-linear data generating process as one that can be written y t = f(u t, u t-1, u t-2, …) where u t is an iid error term and f is a non-linear function. A slightly more specific definition as y t = g(u t-1, u t-2, …)+ u t  2 (u t-1, u t-2, …) where g is a function of past error terms only and  2 is a variance term. Models with nonlinear g() are “non-linear in mean”, while those with nonlinear  2 () are “non-linear in variance”.

9. 9 Types of non-linear models There are many types of non-linear models, e.g. –ARCH / GARCH –Regime Switching Models –Non-Linear Quantile Regression Models –Discrete Choice Models –Copula Functions

10. 10 Moving Average (MA) of Volatility One simple alternative to calculate time varying volatility is simply to calculate the standard deviation of returns using a “window” and roll this window forward using T observations A typical window could be 100 days Note that we here put equal weights on the observations

11. 11 Moving Average (MA) of Volatility

12. 12 The exponentially weighted moving average (EWMA) estimate the conditional volatility of a given return r t and is more flexible than the MA model as this model weights all previous volatilities equal In EWMA, the lower the λ, the less weight will be given to older observations relative to recent observations, which helps to capture changes in σ t It can be shown that σ 2 t can be expressed like (assuming no mean return): σ 2 t = λr 2 t-1 + (1-λ)σ 2 t-1 Exponential Weighted Moving Average (EWMA) of Volatility

13. 13 Exponential Weighted Moving Average (EWMA) of Volatility σ 2 t = λr 2 t-1 + (1-λ)σ 2 t-1 σ t = √σ 2 t λ is usually set to 0.94 for daily data

14. 14 Autoregressive Conditionally heteroscedastic (ARCH) Models A model which does not assume that the variance is constant What could the current value of the variance of the errors plausibly depend upon? –Previous squared error terms. This leads to the autoregressive conditionally heteroscedastic model for the variance of the errors: =  0 +  1 This is known as an ARCH(1) model (Engle (1982))

15. 15 Autoregressive Conditionally heteroscedastic (ARCH) Models

16. 16 Autoregressive Conditionally Heteroscedastic (ARCH) Models (cont’d) The full model would be y t =  1 +  2 x 2t +... +  k x kt + u t, u t  N(0, ) where =  0 +  1 We can easily extend this to the general case where the error variance depends on q lags of squared errors: =  0 +  1 +  2 +...+  q This is an ARCH(q) model.

17. 17 Testing for “ARCH Effects” First, run any postulated linear regression of the form given in the equation above, e.g. y t =  1 +  2 x 2t +... +  k x kt + u t saving the residuals,. Then square the residuals, and regress them on q own lags to test for ARCH of order q, i.e. run the regression where v t is iid. Perform a standard F test. Under H 0 all the coefficients are 0 and we do not have heteroscedasiticity.

18. 18 The null and alternative hypotheses are H 0 :  1 = 0 and  2 = 0 and  3 = 0 and... and  q = 0 H 1 :  1  0 or  2  0 or  3  0 or... or  q  0. If the value of the test statistic is greater than the critical value from F distribution, then reject the null hypothesis. Testing for “ARCH Effects”

19. 19 Testing for “ARCH Effects”

20. 20 Problems with ARCH(q) Models How do we decide on q? The required value of q might be very large Non-negativity constraints might be violated. –When we estimate an ARCH model, we require  i >0  i=1,2,...,q (since variance cannot be negative) A natural extension of an ARCH(q) model which gets around some of these problems is a GARCH model.

21. 21 Generalised ARCH (GARCH) Models Due to Bollerslev (1986). Allow the conditional variance to be dependent upon previous own lags The variance equation is now This is a GARCH(1,1) model, which is like an ARMA(1,1) model for the variance equation. It can be shown that GARCH(1,1) is equivalent with ARCH(∞). The pros with GARCH is there are much fewer parameters to estimate.

22. 22 The Unconditional Variance under the GARCH Specification The unconditional variance (mean reversion level of variance) of u t is given by when is termed “non-stationarity” in variance is termed integrated GARCH For non-stationarity in variance, the conditional variance forecasts will not converge on their unconditional value as the horizon increases.

23. 23 Estimation of GARCH Models Since the model is no longer of the usual linear form, we cannot use ordinary least square (OLS). We use another technique known as maximum likelihood (MLE) or non-linear least square (NLS). The method works by finding the most likely values of the parameters given the model and the actual data. More specifically, we form a log-likelihood function and maximise it.

24. 24 Estimation of ARCH / GARCH Models (cont’d) The steps involved in actually estimating an ARCH or GARCH model are as follows 1.Specify the appropriate equations for the mean and the variance - e.g. an GARCH(1,1) model: 2.Specify the log-likelihood function to maximise: 3. The computer will maximise the function and give parameter values and their standard errors Max! L = ∑[ -ln(σ t ) - (1/2)*(r 2 t / σ 2 t ) ]

25. 25 Estimation of GARCH(1,1) models

26. 26 Estimation of GARCH(1,1) models Use solver for non-linear optimisation. Obs! Excel can be sensitive to starting values of parameters!

27. 27 Estimation of GARCH(1,1) models

28. 28 Extensions of GARCH type models Mean and volatility specifications GARCH GARCH in Mean A-GARCH GJR-GARCH Exponential GARCH Specifications of the error terms Normal Student-t Normal Mixture GARCH Markow Switching GARCH

29. 29 A great reference on these and other empirical and quantitative methods in finance with 1500+ spreadsheets I use the books from Carol in many courses at master and PhD level

30. 30 Simulation of univarite GARCH models After estimation, we might want to simulate the GARCH models with the given parameters. The applications could be –Out of sample simulation for Value at Risk and Conditional Value at Risk calculations –Financial option pricing with Monte Carlo simulation We will demonstrate a simple Excel example. Large scale simulations should be performed in Ox, Eviews, R, Matlab, Gauss, Stata etc.

31. 31 Simulation of GARCH models

32. 32 Why modelling covariance/correlation? Estimating the VaR and CVaR of portfolios Determining optimal allocations between a set of risky assets Pricing multi-asset options Hedging the risk of portfolios

33. 33 Modelling Dynamic Correlation “Clustering” is also evident in correlation not just volatilities. During time of “crises”, correlation tend to increase as asset prices have greater tendency to move in the same direction (this is what we call correlation clustering). Clustering in correlation can be captured by multivariate GARCH models.

34. 34 Rolling correlation Crude Oil – Gas Oil (equal weight moving average) Modelling Dynamic Correlation

35. 35 Exponential weighted correlation Crude Oil – Gas Oil Modelling Dynamic Correlation

36. 36 Multivariate GARCH Models Multivariate GARCH models are used to estimate and to forecast covariances and correlations. The basic formulation is similar to that of the GARCH model, but where the covariances as well as the variances are permitted to be time-varying There are many classes of multivariate GARCH formulation that are widely used: VECH, diagonal VECH, BEKK, CCC, DCC, Factor GARCH, Orthogonal GARCH, RiskMetrics MGARCH. One problem is to estimate these models for many futures positions. DCC is the most used alternative. Simulation of MGARCH is also possible with the use of Cholesky Factorization

37. 37 Realized Volatility

38. 38 Realized Volatility Quarter NP

39. 39 Realized Volatility Quarter NP

40. 40 Realized Correlation

41. 41 Realized Volatility Quarter NP

42. 42 Implied Volatility An alternative estimate of volatility is the implied volatility Implied volatility Backing out implied volatility from B&S 76 prices Volatility curves / smiles Term structure of volatilities Implied volatility surfaces

43. 43 Implied volatility is a transformation of a standard European option price It is the volatility that, when input into the Black-Scholes-Merton (BSM), yields the price of the option (or volatility “backed out” from market prices) Implied Volatility

44. 44 Implied volatility surface/smile: Since market participants (unlike the BSM assumption) do not believe in constant volatilities, there will be a surface of market implied volatilities over strikes/moneyness Term structure of implied volatility: The market implied volatility of all options of the same strike/moneyness but different maturities also forms a function that converges to the long term implied volatility. Implied Volatility

45. 45 A dynamic model of implied volatility (unlike BSM) is also a dynamic model of the option Price Hedge ratios If we can forecast market implied volatility successfully then we can also forecast the market price of the option and hedge the option accurately We might also get better measures of risk using implied volatility as input Implied Volatility

46. 46 Volatility smiles and term structure of volatilites varies in the following markets Equities Currencies Bonds/Interest rates Commodities (Energy, Metals, Aggriculturals) Equity options have negative volatility skew, especially for short term options Currency options have fairly symmetrical smiles for all maturities Interest options have negative skews Commodity options sometimes have negative skews and sometimes positive skews over time/maturities Implied Volatility

47. 47 Implied Volatility “ Backing out” Implied Volatility from a Market Price: Set the market price f m (K,T) of a standard European call or put option with strike K and maturity T equal to the BSM model price. That is, assuming the option is priced from the market price of the future F: where ω=1 for a call and ω=-1 for a put, and Φ(.) is the standard normal distribution function

48. 48 Implied Volatility “Backing out” Implied Volatility from a Market Price: We know K and T, and we can observe the risk free interest rate r and the market price of the option f m Hence we can ask, what is the value of σ that satisfies the equation above?

49. 49 Implied Volatility

50. 50 Implied Volatility When implied volatilities are “back out” this way for a given maturity, we will observe that they do not form a flat line (all implied volatilities are the same for all strikes) but rather form a smile or smirk That is market prices of different options on the same underlying have different implied volatilties

51. 51 Implied Volatility

52. 52 Implied Volatility Nordic Q3 2012 Expiry 21.06.2012

53. 53 Publications on analysis of energy markets volatility and correlation

54. 54 Publications on analysis of energy markets volatility and correlation

55. 55 Publications on analysis of energy markets volatility and correlation

56. 56 Publications on analysis of energy markets volatility and correlation

57. 57 Publications on analysis of energy markets volatility and correlation

58. 58 Publications on analysis of energy markets volatility and correlation

59. 59 Publications on analysis of energy markets volatility and correlation

60. 60 Publications on analysis of energy markets volatility and correlation

61. 61 Publications on analysis of energy markets volatility and correlation

62. 62 Publications on analysis of energy markets volatility and correlation

63. 63 Excel Exercise Use exercise data for all energy commodities and: –Graph returns and squared returns and discuss –Calculate the Moving Average Volatility (30 versus 100 days) and Exponential Weighted Moving Average Volatility with λ=0.94. What are the findings? Calculate the rolling correlation (30, 100 days) and Exponential Weighted Moving Average Correlation between Nordpool 1M versus Nordpool 1Q…. and so on. What are the findings? Use the Volatilities Nordic Option Market 27 apr 2012 spreadsheet. Calculate the implied volatility for the different options (check with ICAP calculations). Draw the volatility smile for the different options.