ps erc oren -november 19, 20081 hedging fixed price load following obligations in a competitive w...

Oren -November 19, 2008 1

PSERC

Hedging Fixed Price Load Following Obligations in a

Competitive Wholesale Electricity Market

Shmuel [email protected]

IEOR Dept. , University of California at Berkeleyand Power Systems Engineering Research Center (PSerc)

(Based on joint work with Yumi Oum and Shijie Deng)

Presented atThe Electric Power Optimization Center

University of Auckland, New ZealandNovember 19, 2008

mailto:[email protected]


Typical Electricity Supply and Demand Functions

$0

$10

$20

$30

$40

$50

$60

$70

$80

$90

$100

0 20000 40000 60000 80000 100000

Demand (MW)

Marginal Cost ($/MWh)

Marginal Cost

Resource stack includes:Hydro units;Nuclear plants;Coal units;Natural gas units;Misc.


ERCOT Energy Price – On peakBalancing Market vs. Seller’s Choice

January 2002 thru July 2007

$0

$50

$100

$150

$200

$250

$300

$350

$400Ja

n-0

2M

ar-0

2M

ay-0

2Ju

l-02

Sep

-02

No

v-0

2Ja

n-0

3M

ar-0

3M

ay-0

3Ju

l-03

Sep

-03

No

v-0

3Ja

n-0

4M

ar-0

4M

ay-0

4Ju

l-04

Sep

-04

No

v-0

4Ja

n-0

5M

ar-0

5M

ay-0

5Ju

l-05

Sep

-05

No

v-0

5Ja

n-0

6M

ar-0

6M

ay-0

6Ju

l-06

Sep

-06

No

v-0

6Ja

n-0

7M

ar-0

7M

ay-0

7Ju

l-07

En

erg

y P

ric

e (

$/M

Wh

)

Balancing Energy (UBES) Price

Spot Price Energy- ERCOT SC

UBES 2/25/03 $523.45

Shmuel Oren, UC Berkeley 06-19-2008


Electricity Supply Chain

Wholesale electricity market

(spot market)

LSE(load serving

entity

Customers(end users servedat fixed regulated

rate)

Generators

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)


0

200

400

600

800

1000

1200

0 10000 20000 30000 40000 50000 60000

Actual load (MW)

Re

al T

ime

Pri

ce

($

/MW

h)

Period: 4/1998 – 12/2001

Both Price and Quantity are Volatile (PJM Market Price-Load Pattern)


Price and Demand CorrelationElectricity Demand and Price in California

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

9, July ~ 16, July 1998

Dem

and

(M

W)

0

20

40

60

80

100

120

140

160

180

Electricity P

rice ($/MW

h)

Load

Price

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


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


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 (HDD), Cooling Degree Days (CDD)

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


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)

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


Mathematical Formulation

Objective function

Constraint: zero-cost constraint

dqdpqpfpxqprU

pxqprUE

px

px

),()]()[(max

)]()[(max

)(

)(

Utility function over profit

Joint distribution of p and q

0)]([1

pxEB

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


Optimality Condition

)(

)(*|)(*)('

pf

pgppxqprUE

p

)(2

1][)]([ YaVarYEYUE

The Lagrange multiplier is determined so that the constraint is satisfied

Mean-variance utility function:

]|),([

)()(

)()(

]]|),([[

)()(

)()(

11

)(* pqpyE

pfpg

E

pfpg

pqpyEE

pfpg

E

pfpg

apx

p

Q

pQ

p

Q

p


-5 0 5

x 104

0

1

2

3

4

5

6

7

8x 10

-5

profit: y = (r-p)q + x(p)

Pro

babi

lity

= 0 = 0.3 = 0.5 = 0.7 = 0.8

Illustrations of Optimal Exotic Payoffs Under Mean-Var Criterion

rate) retail(flat /120$

60)(,300][

/56$)(,/70$][

Q&Punder ),2.0,69.5,7.0,4(~)log,(log 22

MWhr

MWhqMWhqE

MWhpMWhpE

Nqp

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,

0 50 100 150 200 250

-2

0

2

4

6

8x 10

4

Spot price p

x*(p

)

=0 =0.3 =0.5 =0.7 =0.8


-1 -0.5 0 0.5 1 1.261.5 2 2.5 3

x 104

0

0.5

1

1.5

2

2.5

3x 10

-4

Profit

Pro

babi

lity

After price & quantity hedge

After price hedge

Before hedge

Volumetric-hedging effect on profit

Bivariate lognormal for (p,q)

Comparison of profit distribution for mean-variance utility (ρ=0.8) Price hedge: optimal forward hedge Price and quantity hedge: optimal exotic hedge


0 50 100 150 200 250-2

-1

0

1

2

3

4

5

6

7x 10

4

Spot price p

x*(p

)

m2 = m1-0.1m2 = m1-0.05m2 = m1m2 = m1+0.05m2 = m1+0.1

Sensitivity to market risk premium

With Mean-variance utility (a =0.0001)

1.0,1.77

05.0,3.73

0,8.69

05.0,4.66

1.0,1.63

][

12

12

12

12

12

mmif

mmif

mmif

mmif

mmif

pEQ][log2 pEm Q


0 50 100 150 200 250-5

0

5

10

15

20x 10

4

Spot price p

x*(p

)

a =5e-006a =1e-005a =5e-005a =0.0001a =0.001

Sensitivity to risk-aversion

with mean-variance utility (m2 = m1+0.1)

)(2

1][)]([ YaVarYEYUE

(Bigger ‘a’ = more risk-averse)

Note: if m1 = m2 (i.e., P=Q), ‘a’ doesn’t matter for the mean-variance utility.


Replication of Exotic Payoffs

dKKpKxdKpKKxFpFxFxpxF

F

))((''))((''))(('1)()(

0

Forward payoff

Put option payoff

Call option payoff

Bond payoff

Strike < F Strike > F

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)

Payoff

p

Forward price

F

Payoff

K

Payoff

Strike price

K

Strike price


0 50 100 150 200 250-2

-1

0

1

2

3

4

5

6

7x 10

4

Spot price p

payo

ff

x*(p)Replicated when K = $1Replicated when K = $5Replicated when K = $10

0 50 100 150 200 250

-2

0

2

4

6

8x 10

4

Spot price p

x*(p

)

=0 =0.3 =0.5 =0.7 =0.8

0 50 100 150 200 250-500

0

500

Spot price p

x*'(p

)

=0 =0.3 =0.5 =0.7 =0.8

0 50 100 150 200 2500

10

20

30

40

50

60

Spot price p

x*''(

p)

=0 =0.3 =0.5 =0.7 =0.8

Replicating portfolio and discretization)( px )( px )( px

puts calls

F

Payoffs from discretized portfolio


Timing of Optimal Static Hedge

Objective functionUtility function over profit

(A contract is priced as an expected discounted payoff under risk-neutral measure)

Q: risk-neutral probability measure

zero-cost constraint

(Same as Nasakkala and Keppo)


Example:Price and quantity dynamics


Standard deviations vs. hedging time


Dependency of hedged profit distribution on timing

Optimal


Standard deviation vs. hedging time


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.


Extension to VaR Contrained Extension to VaR Contrained Portfolio Portfolio

Oum and Oren, July 10, 2008 24

• Let’s call the Optimal Solution: x*(p)

• Q: a pricing measure• Y(x(p)) = (r-p)q+x(p) includes multiplicative term of

two risk factors• x(p) is unknown nonlinear function of a risk factor• A closed form of VaR(Y(x)) cannot be obtained

5%

VaR


Mean Variance ProblemMean Variance Problem


0)]([E s.t.

kV(Y(x))2

1-E[Y(x)]maxarg(p)x

Q

x(p)

k

px

• For a risk aversion parameter k

• We will show how the solution to the mean-variance problem can be used to approximate the solution to the VaR-constrained problem


Proposition 1Proposition 1 Suppose..

There exists a continuous function h such that

with h increasing in standard deviation (std(Y(x))) and non-increasing in mean (mean(Y(x)))

Then x*(p) is on the efficient frontier of (Mean-VaR) plane and (Mean-

Variance) plane


γ)std(Y(x)), , ))h(mean(Y(x(Y(x))VaR γ


Justification of the Monotonicity Justification of the Monotonicity AssumptionAssumption

The property holds for distributions such as normal, student-t, and Weibull For example, for normally distributed X

The property is always met by Chebyshev’s upper bound on the VaR:


( ) ( ) ( )VaR x z std x E x

12

2

2

1

{ ( ) } ( ) /

{ ( ) ( )} 1/

( ) ( ) ( )t

P x E x k var x k x

P x E x t std x t

VaR x t std x E x


Approximation algorithm to Find Approximation algorithm to Find x*(p)x*(p)



ExampleExample


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)

We assume a bivariate normal dist’n for log p and q

Distribution of Unhedged Profit (120-p)q


SolutionSolution xk(p) =(1- ApB)/2k-(r-p)(m + E log p – D) + CpB


Oren -November 19, 2008 31Oum and Oren, July 10, 2008 31

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)

Finding the Optimal k*


Efficient FrontiersEfficient Frontiers



Impact on Profit Distributions and VaRs Impact on Profit Distributions and VaRs



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.

ps erc oren -november 19, 20081 hedging fixed price load following obligations in a competitive w...

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