1. Oum, Oren and Deng, March 24, 20061 Hedging Quantity Risks with Standard Power Options in a Competitive Electricity Market Yumi Oum, University of California at Berkeley Shmuel Oren, University of California at Berkeley Shijie Deng, Georgia Institute of Technology Eleventh Annual POWER Research Conference Berkeley, CA, March 24, 2006

2. Oum, Oren and Deng, March 24, 2006 2 Electricity Supply Chain Wholesale electricity market (spot market) LSE (load serving entity) Customers (end users served at fixed regulated rate) Generators

3. Oum, Oren and Deng, March 24, 2006 3 Volumetric Risk for Load-Serving Entities Properties of electricity demand (load)  Uncertain and unpredictable  Weather-driven  volatile Sources of LSEs exposure  Highly volatile spot price  Flat (regulated) retail rates & limited demand response  Electricity is non-storable (no inventory)  Electricity demand has to be served (no “busy signal”)  Adversely correlated wholesale price and demand Covering expected load with forward contracts will result in a contract deficit when prices are high and contract excess when prices are low so the LSE faces net revenue exposure due to demand fluctuations

4. Oum, Oren and Deng, March 24, 2006 4 Price and Demand Correlation Correlation coefficients: 0.539 for hourly price and load from 4/1998 to 3/2000 at Cal PX 0.7, 0.58, 0.53 for normalized average weekday price and load in Spain, Britain, and Scandinavia, respectively

5. Oum, Oren and Deng, March 24, 2006 5 Tools for Volumetric Risk Management Electricity derivatives  Forward or futures  Plain-Vanilla options (puts and calls)  Swing options (options with flexible exercise rate) Temperature-based weather derivatives  Heating Degree Days, Cooling Degree Days Power-weather Cross Commodity derivatives  Payouts when two conditions are met (e.g. both high temperature & high spot price) Demand response Programs  Interruptible Service Contracts  Real Time Pricing

6. Oum, Oren and Deng, March 24, 2006 6 Scope of This Work Finding an optimal net revenue-hedging strategy for an LSE using plain electricity derivatives and forward contracts  Exploit the correlation between electricity spot price and power demand in volumetric risk management  Characterize an “optimal” exotic hedging instrument  Demonstrate the exposure reduction with an optimal exotic hedge  Replication of the optimal hedge with forward contracts and standard European call and put options  Sensitivity analysis with respect to model parameters.

7. Oum, Oren and Deng, March 24, 2006 7 Model Setup One-period model  At time 0: construct a portfolio with payoff x(p)  At time 1: hedged profit Y(p,q,x(p)) = (r-p)q+x(p) Objective  Find a zero cost portfolio with exotic payoff which maximizes expected utility of hedged profit under no credit restrictions. Spot market LSE Load (q) p r Portfolio x(p) for a delivery at time 1

8. Oum, Oren and Deng, March 24, 2006 8 Mathematical Formulation Objective function Constraint: zero-cost constraint Utility function over profit Joint distribution of p and q Q: risk-neutral probability measure B : price of a bond paying $1 at time 1 ! A contract is priced as an expected discounted payoff under risk-neutral measure

9. Oum, Oren and Deng, March 24, 2006 9 Optimality Condition The Lagrangian multiplier is determined so that the constraint is satisfied Utility functions we examine  CARA (constant absolute risk aversion) utility function:  Mean-variance utility function:

10. Oum, Oren and Deng, March 24, 2006 10 Optimal Payoff Functions Obtained With a CARA utility function With a Mean-variance utility function:

11. Oum, Oren and Deng, March 24, 2006 11 Assumptions on distributions We consider two different joint distributions for price and demand:  Bivariate lognormal-normal distribution:  Bivariate lognormal distribution:

12. Oum, Oren and Deng, March 24, 2006 12 Optimal Exotic Payoff Bivariate lognormal-normal distribution:  For a CARA utility  For a mean-variance utility

13. Oum, Oren and Deng, March 24, 2006 13 Optimal Exotic Payoff Bivariate lognormal distribution:  For a mean-variance utility

14. Oum, Oren and Deng, March 24, 2006 14 Illustrations of Optimal Exotic Payoffs Bivariate lognormal-normal distribution Dist’n of p Dist’n of profit r Optimal exotic payoff CARAMean-Var

15. Oum, Oren and Deng, March 24, 2006 15 Illustrations of Optimal Exotic Payoffs Bivariate lognormal distribution: Dist’n of profit Optimal exotic payoff Mean-Var E[p] Note: For the mean-variance utility, the optimal payoff is linear in p when correlation is 0, Mean-Var

16. Oum, Oren and Deng, March 24, 2006 16 Volumetric-hedging effect on profit dist’n Bivariate lognormal for (p,q) Bivariate lognormal-normal for (p,q) Comparison of profit distribution for an LSE with mean-variance utility (ρ=0.8)  Price hedge: optimal forward hedge  Price and quantity hedge: optimal exotic hedge

17. Oum, Oren and Deng, March 24, 2006 17 Sensitivity to market risk premium With Mean-variance utility (a =0.0001) With CARA utility

18. Oum, Oren and Deng, March 24, 2006 18 Sensitivity to risk-aversion with mean-variance utility (m2 = m1+0.1) with CARA utility Optimal payoffs by risk aversion (Bigger ‘a’ = more risk-averse) Note: if m1 = m2 (i.e., P=Q), ‘a’ doesn’t matter for the mean-variance utility.

19. Oum, Oren and Deng, March 24, 2006 19 We’ve obtained an exotic payoff function that we like to replicate with standard forward contracts plus a portfolio of call and put options. Carr and Madan(2001) showed any twice continuously differentiable function can be written as: Replication ( s  forward price F) Replication with Standard Instruments Forward payoff Put option payoff Call option payoff Bond payoff Strike < F Strike > F Payoff p Forward price F Payoff K Strike price K

20. Oum, Oren and Deng, March 24, 2006 20 Replication of Exotic Payoffs Forward payoff Put option payoff Call option payoff Bond payoff Strike < FStrike > F Thus, exact replication can be obtained from  a long cash position of size x(F)  a long forward position of size x’(F)  long positions of size x’’(K) in puts struck at K, for a continuum of K which is less than F (i.e., out-of-money puts)  long positions of size x’’(K) in calls struck at K, for a continuum of K which is larger than F (i.e., out-of-money calls) Q: How to determine the options position given a limited number of available strike prices?

21. Oum, Oren and Deng, March 24, 2006 21 For put options with available strikes K 1 <K 2 <…<K n Thus, use size of long put position for strike price K i Similarly, for call options with available strikes k 1 <k 1 <…<k n, Thus, use size of long call position for strike price k i Discretization Payoff of put option struck at K i

22. Oum, Oren and Deng, March 24, 2006 22 Replicating Portfolio (CARA/lognormal-normal) putscalls Payoffs from discretized portfolio F

23. Oum, Oren and Deng, March 24, 2006 23 Replicating Portfolio (MV/lognormal-normal) putscalls F Payoffs from discretized portfolio

24. Oum, Oren and Deng, March 24, 2006 24 Replicating Portfolio (MV/lognormal-lognormal) putscalls F Payoffs from discretized portfolio

25. Oum, Oren and Deng, March 24, 2006 25 Conclusion We have constructed an optimal portfolio for hedging the increased exposure due to correlated price and volumetric risk for LSEs  Forward contracts plus a spectrum of call and put options We have shown a good way of replicating exotic options with available call and put options given discrete strike prices. The results obtained with an optimal hedging portfolio provide a useful benchmark for simpler hedging strategies. Extensions  Decide the best hedging timing  Use Value-at-Risk measure  Optimal hedging under credit limit constraints