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PSERC Hedging Fixed Price Load Following Obligations in a Competitive Wholesale Electricity Market Shmuel Oren [email protected] IEOR Dept. , University of California at Berkeley and Power Systems Engineering Research Center (PSerc) (Based on joint work with Yumi Oum and Shijie Deng) Presented at The Electric Power Optimization Center University of Auckland, New Zealand November 19, 2008 Oren -November 19, 2008 1 Typical Electricity Supply and Demand Functions $100 Marginal Cost ($/MWh) $90 Marginal Cost Resource stack includes: Hydro units; Nuclear plants; Coal units; Natural gas units; Misc. $80 $70 $60 $50 $40 $30 $20 $10 $0 0 20000 40000 60000 80000 100000 Demand (MW) Oren -November 19, 2008 2 ERCOT Energy Price – On peak Balancing Market vs. Seller’s Choice January 2002 thru July 2007 $400 $350 UBES 2/25/03 $523.45 Balancing Energy (UBES) Price Spot Price Energy- ERCOT SC $250 $200 $150 $100 $50 $0 Ja n02 M ar02 M ay -02 Ju l-0 2 Sep 02 Nov02 Ja n03 M ar03 M ay -03 Ju l-0 3 Sep 03 Nov03 Ja n04 M ar04 M ay -04 Ju l-0 4 Sep 04 Nov04 Ja n05 M ar05 M ay -05 Ju l-0 5 Sep 05 Nov05 Ja n06 M ar06 M ay -06 Ju l-0 6 Sep 06 Nov06 Ja n07 M ar07 M ay -07 Ju l-0 7 Energy Price ($/MWh) $300 Oren -November 19,06-19-2008 2008 Shmuel Oren, UC Berkeley 33 Electricity Supply Chain Generators Wholesale electricity market (spot market) Customers (end users served at fixed regulated rate) LSE (load serving entity Similar exposure is faced by a trader with a fixed price load following obligation (such contracts were auctioned off in New Jersey and Montana to cover default service needs) Oren -November 19, 2008 4 Both Price and Quantity are Volatile (PJM Market Price-Load Pattern) Real Time Price ($/MWh) 1200 1000 Period: 4/1998 – 12/2001 800 600 400 200 0 0 10000 20000 30000 40000 50000 60000 Actual load (MW) Oren -November 19, 2008 5 Price and Demand Correlation Electricity Demand and Price in California 18000 180 Load 160 Price 14000 140 12000 120 10000 100 8000 80 6000 60 4000 40 2000 20 0 Electricity Price ($/MWh) Demand (MW) 16000 0 9, July ~ 16, July 1998 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 Oren -November 19, 2008 6 Volumetric Risk for Load Following Obligation Properties of electricity demand (load) Uncertain and unpredictable Weather-driven volatile Sources of exposure Highly volatile wholesale spot price Flat (regulated or contracted) 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 load Covering expected load with forward contracts will result in a contract deficit when prices are high and contract excess when prices are low resulting in a net revenue exposure due to load fluctuations Oren -November 19, 2008 7 Tools for Volumetric Risk Management Electricity derivatives Temperature-based weather derivatives Heating Degree Days (HDD), Cooling Degree Days (CDD) Power-weather Cross Commodity derivatives Forward or futures Plain-Vanilla options (puts and calls) Swing options (options with flexible exercise rate) Payouts when two conditions are met (e.g. both high temperature & high spot price) Demand response Programs Interruptible Service Contracts Real Time Pricing Oren -November 19, 2008 8 Volumetric Static Hedging 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) Spot market r p Load (q) LSE x(p) Portfolio for a delivery at time 1 Objective Find a zero cost portfolio with exotic payoff which maximizes expected utility of hedged profit under no credit restrictions. Oren -November 19, 2008 9 Mathematical Formulation Objective function Utility function over profit max E U [( r p)q x( p)] x( p) max x( p) Joint distribution of p and q U [( r p)q x( p)] f ( p, q)dqdp Constraint: zero-cost constraint 1 Q E [ x( p )] 0 B ! A contract is priced as an expected discounted payoff under risk-neutral measure Q: risk-neutral probability measure B : price of a bond paying $1 at time 1 Oren -November 19, 2008 10 Optimality Condition The Lagrange multiplier is determined so that the constraint is satisfied EU ' (r p)q x * ( p) | p * g ( p) f p ( p) Mean-variance utility function: 1 E[U (Y )] E[Y ] aVar (Y ) 2 g ( p ) g ( p) f p ( p) f p ( p) 1 Q x * ( p) 1 E[ y ( p, q) | p] E [ E[ y ( p, q) | p]] a g ( p) g ( p) Q Q E E f ( p) p f p ( p) Oren -November 19, 2008 11 Illustrations of Optimal Exotic Payoffs Under Mean-Var Criterion Bivariate lognormal distribution: 8 (log p, log q) ~ N (4, 0.7 2 , 5.69, 0.2 2 , ) under P & Q 7 E[ p] $70 / MWh, ( p) $56 / MWh E[q] 300MWh, (q) 60MWh r $120 / MWh (flat retail rate) Probability 6 x*(p) 4 5 =0 = 0.3 = 0.5 = 0.7 = 0.8 4 3 1 4 x 10 6 Dist’n of profit 2 Optimal exotic payoff 8 -5 x 10 0 -5 Mean-Var =0 =0.3 =0.5 =0.7 =0.8 5 4 x 10 Note: For the mean-variance utility, the optimal payoff is linear in p when correlation is 0, E[p] 2 0 profit: y = (r-p)q + x(p) 0 -2 0 50 100 150 Spot price p 200 250 Oren -November 19, 2008 12 Volumetric-hedging effect on profit Comparison of profit distribution for mean-variance utility (ρ=0.8) Price hedge: optimal forward hedge Price and quantity hedge: optimal exotic hedge Bivariate lognormal for (p,q) -4 3 x 10 2.5 After price & quantity hedge 2 Probability 1.5 After price hedge 1 Before hedge 0.5 0 -1 -0.5 0 0.5 1 1.261.5 Profit 2 Oren -November 19, 2008 2.5 3 4 x 10 13 Sensitivity to market risk premium 63.1, 66.4, Q E [ p] 69.8, 73.3, 77.1, m2 E [log p] Q if m2 m1 0.1 if m2 m1 0.05 if m2 m1 0 if m2 m1 0.05 if m2 m1 0.1 With Mean-variance utility (a =0.0001) 4 7 x 10 6 5 x*(p) 4 m2 = m1-0.1 m2 = m1-0.05 m2 = m1 m2 = m1+0.05 m2 = m1+0.1 3 2 1 0 -1 -2 0 50 100 150 Spot price p Oren -November 19, 2008 200 250 14 Sensitivity to risk-aversion (Bigger ‘a’ = more risk-averse) 1 E[U (Y )] E[Y ] aVar (Y ) 2 with mean-variance utility (m2 = m1+0.1) 4 20 x 10 a =5e-006 a =1e-005 a =5e-005 a =0.0001 a =0.001 15 x*(p) 10 5 0 -5 0 50 100 150 Spot price p 200 250 Note: if m1 = m2 (i.e., P=Q), ‘a’ doesn’t matter for the mean-variance utility. Oren -November 19, 2008 15 Replication of Exotic Payoffs Strike < F Strike > F F x( p) x( F ) 1 x' ( F )( p F ) x' ' ( K )( K p) dK x' ' ( K )( p K ) dK 0 Bond payoff Forward payoff F Put option payoff Payoff Payoff Payoff F K Forward price Call option payoff Strike price K p Strike price 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) Oren -November 19, 2008 16 Replicating portfolio and discretization x 10 60 =0 =0.3 =0.5 =0.7 =0.8 50 0 x*''(p) 2 puts =0 =0.3 =0.5 =0.7 =0.8 calls 40 =0 =0.3 =0.5 =0.7 =0.8 30 20 0 10 -2 0 50 100 150 Spot price p 200 250 -500 0 50 100 150 Spot price p 200 250 0 0 50 F 100 150 Spot price p 200 250 4 7 x 10 6 Payoffs from discretized portfolio 5 4 payoff x*(p) 4 x( p ) 500 x*'(p) 6 x( p ) x( p) 4 8 x*(p) Replicated when K = $1 Replicated when K = $5 Replicated when K = $10 3 2 1 0 -1 -2 0 Oren -November 19, 2008 50 100 150 Spot price p 200 250 17 Timing of Optimal Static Hedge Objective function Utility function over profit Q: risk-neutral probability measure zero-cost constraint (A contract is priced as an expected discounted payoff under risk-neutral measure) (Same as Nasakkala and Keppo) Oren -November 19, 2008 18 Example: Price and quantity dynamics Oren -November 19, 2008 19 Standard deviations vs. hedging time Oren -November 19, 2008 20 Dependency of hedged profit distribution on timing Optimal Oren -November 19, 2008 21 Standard deviation vs. hedging time Oren -November 19, 2008 22 Optimal hedge and replicating portfolio at optimal time *In reality hedging portfolio will be determined at hedging time based on realized quantities and prices at that time. Oren -November 19, 2008 23 Extension to VaR Contrained Portfolio max E[Y(x)] x(p) s.t. E Q [x(p)] 0 VaR γ (Y(x)) V0 5% where VaR γ (X) ν such that P{X ν} 1 γ • Let’s call the Optimal Solution: x*(p) • Q: a pricing measure • Y(x(p)) = (r-p)q+x(p) • x(p) is unknown nonlinear function of a risk factor • A closed form of VaR(Y(x)) cannot be obtained VaR includes multiplicative term of two risk factors Oren -November 19, 2008 Oum and Oren, July 10, 2008 24 24 Mean Variance Problem • For a risk aversion parameter k 1 x (p) arg max E[Y(x)] - kV(Y(x)) x(p) 2 s.t. E Q [ x( p)] 0 k • We will show how the solution to the mean-variance problem can be used to approximate the solution to the VaR-constrained problem Oren -November 19, 2008 Oum and Oren, July 10, 2008 25 25 Proposition 1 Suppose.. There exists a continuous function h such that VaR γ (Y(x)) h(mean(Y(x )) , std(Y(x)), γ) with h increasing in standard deviation (std(Y(x))) and nonincreasing in mean (mean(Y(x))) Then x*(p) is on the efficient frontier of (Mean-VaR) plane and (MeanVariance) plane Oren -November 19, 2008 Oum and Oren, July 10, 2008 26 26 Justification of the Monotonicity Assumption The property holds for distributions such as normal, student-t, and Weibull For example, for normally distributed X VaR ( x) z std ( x) E ( x) The property is always met by Chebyshev’s upper bound on the VaR: P{ x E ( x) k} var ( x) / k 2 x P{x E ( x) t std ( x)} 1/ t 2 VaR1 1 2 ( x) t std ( x) E ( x) t Oren -November 19, 2008 Oum and Oren, July 10, 2008 27 27 Approximation algorithm to Find x*(p) Obtain xk(p) that maximizes E[Y(x)]-kV(Y(x))/2 Calculate associated VaR(k) Find smallest k such that VaR(k) ≤ V0 Oren -November 19, 2008 Oum and Oren, July 10, 2008 28 28 Example We assume a bivariate normal dist’n for log p and q Under P log p ~ N(4, 0.72) q~N(3000,6002) corr(log p, q) = 0.8 Under Q: log p ~ N(4.1, 0.72) Distribution of Unhedged Profit (120-p)q Oren -November 19, 2008 Oum and Oren, July 10, 2008 29 29 Solution xk(p) =(1- ApB)/2k-(r-p)(m + E log p – D) + CpB Oren -November 19, 2008 Oum and Oren, July 10, 2008 30 30 Finding the Optimal k* Once we have optimal xk(p), we can calculate associated VaR by simulating p and q and Find smallest k such that VaR(k) ≤ V0 (Smallest k => Largest Mean) Oren -November 19, 2008 Oum and Oren, July 10, 2008 31 31 Efficient Frontiers Oren -November 19, 2008 Oum and Oren, July 10, 2008 32 32 Impact on Profit Distributions and VaRs Oren -November 19, 2008 Oum and Oren, July 10, 2008 33 33 Conclusion Risk management is an essential element of competitive electricity markets. The study and development of financial instruments can facilitate structuring and pricing of contracts. Better tools for pricing financial instruments and development of hedging strategy will increase the liquidity and efficiency of risk markets and enable replication of contracts through standardized and easily tradable instruments Financial instruments can facilitate market design objectives such as mitigating risk exposure created by functional unbundling, containing market power, promoting demand response and ensuring generation adequacy. Oren -November 19, 2008 34