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Oum, Oren and Deng, March 24, 2006
1
Hedging Quantity Risks with Standard Power Options in a Competitive Electricity Market
Yumi Oum, University of California at BerkeleyShmuel Oren, University of California at BerkeleyShijie Deng, Georgia Institute of Technology
Eleventh Annual POWER Research ConferenceBerkeley, CA, March 24, 2006
Oum, Oren and Deng, March 24, 2006
2
Electricity Supply Chain
Wholesale electricity market
(spot market)
LSE(load serving
entity)
Customers(end users servedat fixed regulated
rate)
Generators
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
Oum, Oren and Deng, March 24, 2006
4
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
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
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.
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
Oum, Oren and Deng, March 24, 2006
8
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
Oum, Oren and Deng, March 24, 2006
9
Optimality Condition
)(
)(*|)(*)('
pf
pgppxqprUE
p
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:
aYea
YU 1
)(
)(2
1][)]([ YaVarYEYUE
Oum, Oren and Deng, March 24, 2006
10
Optimal Payoff Functions Obtained
]|[ln]|[ln
)(
)(ln
)(
)(ln
1)(* ),(),( peEEpeE
pg
pfE
pg
pf
apx qpayQqpaypQp
With a CARA utility function
With a Mean-variance utility function:
]|),([
)()(
)()(
]]|),([[
)()(
)()(
11
)(* pqpyE
pfpg
E
pfpg
pqpyEE
pfpg
E
pfpg
apx
p
Q
pQ
p
Q
p
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:
),(~log:
),(log),,(~),,(~log:2
2
221
smNpQUnder
qpCorrumNqsmNpPUnder
),(~log:
)log,(log),,(~log),,(~log:2
2
221
smNpQUnder
qpCorrumNqsmNpPUnder qq
Oum, Oren and Deng, March 24, 2006
12
Optimal Exotic Payoff Bivariate lognormal-normal distribution:
For a CARA utility
For a mean-variance utility
)1()(22
1)(log
))(()(log)(log1
)(*
222222
1
2
12
2
2
1
122212
22
22
22
22
uepepraesmpps
u
epms
ummp
s
urmp
s
mm
apx
smsmsm
sm
22
222
122
2121
212
22121
2
12
2212
12
)3)((
12
)3)((
)(
)(log)(11
)(*
smsms
mm
s
mmmm
s
mm
s
mmmm
esmrms
um
s
umerpe
mps
umprpe
apx
),(~log:
),(log),,(~),,(~log:2
2
221
smNpQUnder
qpCorrumNqsmNpPUnder
Oum, Oren and Deng, March 24, 2006
13
Optimal Exotic Payoff Bivariate lognormal distribution:
For a mean-variance utility
22
221222222
12212
22121
2212
122
2121
12
1)1(
2
1)(
2
1)1(
2
1)(
2
)3)((
)1(2
1)(log
2
)3)((
)(11
)(*
ss
uumm
s
ummuumm
s
um
s
mm
s
mmmm
umps
um
s
mm
s
mmmm
qqq
q
q
erepe
eprpea
px
),(~log:
)log,(log),,(~log),,(~log:2
2
221
smNpQUnder
qpCorrumNqsmNpPUnder qq
Oum, Oren and Deng, March 24, 2006
14
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 50 100 150 200 250-2
-1
0
1
2
3
4x 10
7
Spot price p
x*(p
)
=0 =0.3 =0.5 =0.7 =0.8
Illustrations of Optimal Exotic Payoffs
0 50 100 150 200 250 3000
0.002
0.004
0.006
0.008
0.01
0.012
0.014
Price p
Pro
babi
lity
-5 0 5
x 104
0
1
2
3
4
5
6
7x 10
-5
profit: y = (r-p)q + x(p)
Pro
babi
lity
= 0 = 0.3 = 0.5 = 0.7 = 0.8
rate) retail(flat /120$
60)(,300][
/56$)(,/70$][
),30,300,7.0,4(~),(log 22,
MWhr
MWhqMWhqE
MWhpMWhpE
NqpQP
Bivariate lognormal-normal distributionDist’n of p
Dist’n of profit
r
Optimal exotic payoff
CARA Mean-Var
Oum, Oren and Deng, March 24, 2006
15
-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
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
Mean-Var
Oum, Oren and Deng, March 24, 2006
16
-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
-1 -0.5 0 0.5 1 1.281.5 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 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
Oum, Oren and Deng, March 24, 2006
17
0 50 100 150 200 250-6
-4
-2
0
2
4
6
8
10
12x 10
6
Spot price p
x*(p
)
m2 = m1-0.1m2 = m1-0.05m2 = m1m2 = m1+0.05m2 = m1+0.1
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) With CARA utility
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
Oum, Oren and Deng, March 24, 2006
18
0 50 100 150 200 250-1
-0.5
0
0.5
1
1.5x 10
7
Spot price p
x*(p
)
a =0.1a =0.5a =1a =1.5a =2
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) with CARA utility
Optimal payoffs by risk aversion
aYea
YU 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.
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
dKKpKxdKpKKxFpFxFxpxF
F
))((''))((''))(('1)()(
0
Forward payoff
Put option payoff
Call option payoff
Bond payoff
Strike < F
Strike > F
Payoff
p
Forward price
F
Payoff
K
Payoff
Strike price
K
Strike price
dKKpKxdKpKKxpsxssxsxpxs
s
))((''))(('')('])(')([)(
0
Oum, Oren and Deng, March 24, 2006
20
Replication of Exotic Payoffs
dKKpKxdKpKKxFpFxFxpxF
F
))((''))((''))(('1)()(
0
Forward payoff
Put option payoff
Call option payoff
Bond payoff
Strike < F Strike > 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?
Oum, Oren and Deng, March 24, 2006
21
For put options with available strikes K1<K2<…<Kn
Thus, use size of long put position for strike price Ki
Similarly, for call options with available strikes k1<k1<…<kn,
Thus, use size of long call position for strike price ki
Discretization
n
ii
iiF
pKpKxpKx
dKpKKx1
11
0)(
2
)),(max()),(max())((
Payoff of put option struck at Ki.0where 110 FKKKK nn
m
ii
ii
FKp
pKxpKxdKKpKx
1
11 )(2
)),(min()),(min())((
.where 110 mm kkkkF
2
)()( 11 ii KxKx
2
)()( 11 ii kxkx
Oum, Oren and Deng, March 24, 2006
22
0 50 100 150 200 250-6
-4
-2
0
2
4
6
8
10
12x 10
6
Spot price p
payo
ff
x*(p)Replicated when K = $1Replicated when K = $5Replicated when K = $10
0 50 100 150 200 250-8
-6
-4
-2
0
2
4
6
8x 10
5
Spot price p
x*'(p
)
=0 =0.3 =0.5 =0.7 =0.8
0 50 100 150 200 2500
1000
2000
3000
4000
5000
6000
7000
8000
Spot price p
x*''(
p)
=0 =0.3 =0.5 =0.7 =0.8
Replicating Portfolio (CARA/lognormal-normal))( px )( px
puts calls
Payoffs from discretized portfolio
0 50 100 150 200 250-2
-1
0
1
2
3
4x 10
7
Spot price p
x*(p
)
=0 =0.3 =0.5 =0.7 =0.8
F
)( px
Oum, Oren and Deng, March 24, 2006
23
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 50 100 150 200 250-400
-300
-200
-100
0
100
200
300
400
500
Spot price p
x*'(p
) =0 =0.3 =0.5 =0.7 =0.8
0 50 100 150 200 2500
2
4
6
8
10
Spot price p
x*''(
p)
=0 =0.3 =0.5 =0.7 =0.8
Replicating Portfolio (MV/lognormal-normal))( px )( px )( px
puts calls
F
Payoffs from discretized portfolio
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
Oum, Oren and Deng, March 24, 2006
24
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 (MV/lognormal-lognormal))( px )( px )( px
puts calls
F
Payoffs from discretized portfolio
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 constraints05.0})(),({..
)](),([)(
VaRpxqpPts
pxqpEMaxpx