Download Risk Management in Competitive Electricity Markets

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
EU ' (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