1. Spot Price Models

2. Spot Price Dynamics An examination of the previous graph shows several items of interest. –Price series exhibit spikes with seasonal component (more in summer than other times). –Prices show seasonality (higher in summer with a secondary peak in winter) –Volatility varies over period –Prices revert to a mean

3. Returns

4. Objective Find model / return generating process that describes the preceding graphs Several models have been used. No clear cut conclusion Trade-off between ease of implementation (use) and descriptive power. Art rather than science.

5. Models Commonly Used General Brownian Motion (GBM) Mean Reversion (Ornstein-Uhlenbeck) Mean Reversion with Jumps Stochastic Volatility GARCH(p,q) Markov switching Model

6. Geometric Brownian Motion GBM model in natural log of price ( x= ln(S)). Drift does not depend upon x as

7. Simulating Brownian Motion

8. Brownian Motion in Trees

9. Mean Reversion Models Continuous time model in natural log of price.   is the mean   the rate of reversion to the mean   the standard deviation

10. Mean Reversion: Simulation Discrete version Issue of step size –Drift term function of log of price

11. Aside: half-life of MR Average time to return to one-half a deviation from the average price The larger , the shorter the time to revert.

12. MR Parameter Estimation Historical Basis: regression of change in log of price on log of price. We now have the slope and intercept and can compute the mean reversion parameters.

13. Scatter Plot Day Ahead

14. Parameter Estimation From the OLS estimation, we can recover the parameter values of the process.

15. MR Jump Diffusion Process Electricity Prices exhibit jumps. The jumps do not persist, but are more like spikes, i.e. quickly return to a ‘base’ level. The last term represents the jump component.

16. Estimation of Jump Parameters The MR jump process can be estimated via a number of techniques. These range from the heuristic through formal statistical methods such as maximum likelihood. ML tends to overestimate jump intensity hence we will simply use the heuristic approach. Estimation via recursive filter Need the intensity, size and variance of jumps.

17. Estimation Procedure Determine some (arbitrary) level at which a return is a jump. Count number of ‘jumps’ and divide by time in years to obtain  (frequency). Compute jump return and standard deviation. Repeat until estimates converge.

18. Stochastic Volatility This is an extension to GBM wherein the volatility is no longer constant, but random. Two factors: Price and Volatility. Mean reverting in volatility.

19. GARCH Generalized Autoregressive Conditional Heteroskedacity Bollerslev (1986) Process for variance.

20. Markov Switching These models are based upon the idea that the returns will be due to multiple regimes. Simplest model is two states, one a regular state and the other a spike. Inputs: –Transition matrix – two stochastic equations.

21. Markov Switching Transition matrix p(t) is the probability of a spike on day t. q(t) is the probability that a spike ends at t.

22. Comparison of Models Issues in simulation Terminal values more widely dispersed in GBM. Mean expressed in terms of ln(S).