View
101
Download
1
Category
Tags:
Preview:
DESCRIPTION
Hedging Fixed Price Load Following Obligations in a Competitive W holesale Electricity Market. Shmuel Oren oren@ieor.berkeley.edu IEOR Dept. , University of California at Berkeley and Power Systems Engineering Research Center (PSerc) (Based on joint work with Yumi Oum and Shijie Deng) - PowerPoint PPT Presentation
Citation preview
Oren -November 19, 2008 1
PSERC
Hedging Fixed Price Load Following Obligations in a
Competitive Wholesale Electricity Market
Shmuel Orenoren@ieor.berkeley.edu
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
Oren -November 19, 2008 2
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.
Oren -November 19, 2008 33
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 $523.45
Shmuel Oren, UC Berkeley 06-19-2008
Oren -November 19, 2008 4
Electricity Supply Chain
Wholesale electricity market
(spot market)
LSE(load serving
entity
Customers(end users servedat fixed regulated
rate)
Generators
Similar exposure is faced by a trader with a fixed price load following obligation(such contracts were auctioned off in New Jersey and Montana to cover default service needs)
Oren -November 19, 2008 5
0
200
400
600
800
1000
1200
0 10000 20000 30000 40000 50000 60000
Actual load (MW)
Re
al T
ime
Pri
ce
($
/MW
h)
Period: 4/1998 – 12/2001
Both Price and Quantity are Volatile (PJM Market Price-Load Pattern)
Oren -November 19, 2008 6
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
Oren -November 19, 2008 7
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 8
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
Oren -November 19, 2008 9
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
Oren -November 19, 2008 10
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 $1 at time 1
! A contract is priced as an expected discounted payoff under risk-neutral measure
Oren -November 19, 2008 11
Optimality Condition
)(
)(*|)(*)('
pf
pgppxqprUE
p
)(2
1][)]([ YaVarYEYUE
The Lagrange multiplier is determined so that the constraint is satisfied
Mean-variance utility function:
]|),([
)()(
)()(
]]|),([[
)()(
)()(
11
)(* pqpyE
pfpg
E
pfpg
pqpyEE
pfpg
E
pfpg
apx
p
Q
pQ
p
Q
p
Oren -November 19, 2008 12
-5 0 5
x 104
0
1
2
3
4
5
6
7
8x 10
-5
profit: y = (r-p)q + x(p)
Pro
babi
lity
= 0 = 0.3 = 0.5 = 0.7 = 0.8
Illustrations of Optimal Exotic Payoffs Under Mean-Var Criterion
rate) retail(flat /120$
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
Oren -November 19, 2008 13
-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
Oren -November 19, 2008 14
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
Oren -November 19, 2008 15
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.
Oren -November 19, 2008 16
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
Oren -November 19, 2008 17
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
Oren -November 19, 2008 18
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)
Oren -November 19, 2008 19
Example:Price and quantity dynamics
Oren -November 19, 2008 20
Standard deviations vs. hedging time
Oren -November 19, 2008 21
Dependency of hedged profit distribution on timing
Optimal
Oren -November 19, 2008 22
Standard deviation vs. hedging time
Oren -November 19, 2008 23
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 24
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
Oren -November 19, 2008 25
Mean Variance ProblemMean Variance Problem
Oum and Oren, July 10, 2008 25
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
Oren -November 19, 2008 26
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
Oum and Oren, July 10, 2008 26
γ)std(Y(x)), , ))h(mean(Y(x(Y(x))VaR γ
Oren -November 19, 2008 27
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:
Oum and Oren, July 10, 2008 27
( ) ( ) ( )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
Oren -November 19, 2008 28
Approximation algorithm to Find Approximation algorithm to Find x*(p)x*(p)
Oum and Oren, July 10, 2008 28
Oren -November 19, 2008 29
ExampleExample
Oum and Oren, July 10, 2008 29
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
Oren -November 19, 2008 30
SolutionSolution xk(p) =(1- ApB)/2k-(r-p)(m + E log p – D) + CpB
Oum and Oren, July 10, 2008 30
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*
Oren -November 19, 2008 32
Efficient FrontiersEfficient Frontiers
Oum and Oren, July 10, 2008 32
Oren -November 19, 2008 33
Impact on Profit Distributions and VaRs Impact on Profit Distributions and VaRs
Oum and Oren, July 10, 2008 33
Oren -November 19, 2008 34
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.
Recommended