Position of the problem
Spot model
Pricing & hedging
Conclusion
A structural risk-neutral model
for pricing and hedging power derivatives
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities Paris 13 & Paris 7
EDF R&D - FiME Research Centre
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
1 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Contents
1
Position of the problem
Electricity prices modeling
Related works
2
Spot model
Design
Estimation
3
Pricing & hedging
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
4
Conclusion
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
2 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Looking for a power spot price model
Applications
pricing of derivatives on the spot
asset valuation (strip of hourly fuel spread options)
hedging
energy market risk management
Model requirements
realistic
robust
tractable
consistent
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
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hedging
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power
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derivatives
Research
3 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Two types of modeling
Modeling futures prices
pros modeling the real available instruments
cons introduction of many parameters to reconstruct
hourly futures prices
Modeling spot prices
1
Exogeneous
pros tractability
cons correlation
2
Equilibrium
pros correlation
cons complexity
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
4 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Related works
Electricity prices exogeneous dynamics
Deng (00), Benth et al. (03, 07, 09), Burger et al. (04), Kolodnyi
(04), Cartea & Figueroa (05), Geman & Roncoroni (06)
Equilibrium model
Spot
Futures
Pirrong & Jermakyan (00)
×
Barlow (02)
×
Kanamura & Ohashi (07)
×
Cartea & Villaplana (08)
×
×
Coulon & Howison (09)
×
×
Lyle & Elliot (09)
×
×
René Aı̈d, Luciano Campi, Nicolas Langrené
Options
×
×
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
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R&D
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Research
5 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
This talk
Objectives
pricing and hedging power derivatives...
... using an improved version of Aı̈d, C., Nguyen & Touzi
(09) Structural Risk-Neutral model
Spot
Aı̈d, C., Nguyen & Touzi (09)
improved SRN model
René Aı̈d, Luciano Campi, Nicolas Langrené
Futures
×
×
×
×
Options
×
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
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hedging
R&D
power
- FiME
derivatives
Research
6 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN Model
Variables
n
fuels, 1 ≤ i ≤ n
Dt
demand (MW)
Cti
Sti
capacities (en MW)
hi
fuel prices
heat rates (hi Sti en e/MWh, % en i)
Electricity price (e/MWh)
bt =
P
n
X
hi Sti 1{Pi−1 Ctk ≤ Dt ≤ Pi
k=1
k=1
Ctk }
i=1
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
7 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN Model
Variables
n
fuels, 1 ≤ i ≤ n
Dt
demand (MW)
Cti
Sti
capacities (en MW)
hi
fuel prices
heat rates (hi Sti en e/MWh, % en i)
Electricity price (e/MWh)
bt =
P
n
X
hi Sti 1{Pi−1 Ctk ≤ Dt ≤ Pi
k=1
k=1
Ctk }
i=1
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
7 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN Model
Variables
n
fuels, 1 ≤ i ≤ n
Dt
demand (MW)
Cti
Sti
capacities (en MW)
hi
fuel prices
heat rates (hi Sti en e/MWh, % en i)
Electricity price (e/MWh)
bt =
P
n
X
hi Sti 1{Pi−1 Ctk ≤ Dt ≤ Pi
k=1
k=1
Ctk }
i=1
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
7 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN Model
Variables
n
fuels, 1 ≤ i ≤ n
Dt
demand (MW)
Cti
Sti
capacities (en MW)
hi
fuel prices
heat rates (hi Sti en e/MWh, % en i)
Electricity price (e/MWh)
bt =
P
n
X
hi Sti 1{Pi−1 Ctk ≤ Dt ≤ Pi
k=1
k=1
Ctk }
i=1
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
7 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN Model
Variables
n
fuels, 1 ≤ i ≤ n
Dt
demand (MW)
Cti
Sti
capacities (en MW)
hi
fuel prices
heat rates (hi Sti en e/MWh, % en i)
Electricity price (e/MWh)
bt =
P
n
X
hi Sti 1{Pi−1 Ctk ≤ Dt ≤ Pi
k=1
k=1
Ctk }
i=1
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
7 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN Model
Variables
n
fuels, 1 ≤ i ≤ n
Dt
demand (MW)
Cti
Sti
capacities (en MW)
hi
fuel prices
heat rates (hi Sti en e/MWh, % en i)
Electricity price (e/MWh)
bt =
P
n
X
hi Sti 1{Pi−1 Ctk ≤ Dt ≤ Pi
k=1
k=1
Ctk }
i=1
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
7 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN Model
Variables
n
fuels, 1 ≤ i ≤ n
Dt
demand (MW)
Cti
Sti
capacities (en MW)
hi
fuel prices
heat rates (hi Sti en e/MWh, % en i)
Electricity price (e/MWh)
bt =
P
n
X
hi Sti 1{Pi−1 Ctk ≤ Dt ≤ Pi
k=1
k=1
Ctk }
i=1
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
7 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN model
Pros
Consistency between electricity prices and fuel prices
Consistency between electricity prices and demand
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
8 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN model
Pros
Consistency between electricity prices and fuel prices
Consistency between electricity prices and demand
Cons
Marginal fuel cost is not the spot price
1
2
3
Non-convex technical constraints
Strategic behaviour (Hortaçsu & Puller, RAND J. of
Economics 2008)
Fixed cost recovery problem for peak-load generation plants
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
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hedging
R&D
power
- FiME
derivatives
Research
8 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN Model - illustration
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
9 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN Model - illustration
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
10 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN Model - illustration
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
11 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model
bt := Pn hi S i 1 Pi−1 k
P
Marginal fuel cost P
t {
i=1
C ≤ Dt ≤ i
René Aı̈d, Luciano Campi, Nicolas Langrené
k=1
t
k=1
Ctk }
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
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hedging
R&D
power
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Research
12 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model
bt := Pn hi S i 1 Pi−1 k
P
Marginal fuel cost P
t {
i=1
C ≤ Dt ≤ i
k=1
Available capacity C t :=
René Aı̈d, Luciano Campi, Nicolas Langrené
t
k=1
Ctk }
Pn
k
k=1 Ct
Universities
A structural risk-neutral
Paris 13model
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Research
12 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model
bt := Pn hi S i 1 Pi−1 k
P
Marginal fuel cost P
t {
i=1
C ≤ Dt ≤ i
k=1
Available capacity C t :=
t
k=1
Ctk }
Pn
k
k=1 Ct
Price spikes occur when the electric system is under stress, i.e.
C t − Dt is small
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
12 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model
bt := Pn hi S i 1 Pi−1 k
P
Marginal fuel cost P
t {
i=1
C ≤ Dt ≤ i
k=1
Available capacity C t :=
t
k=1
Ctk }
Pn
k
k=1 Ct
Price spikes occur when the electric system is under stress, i.e.
C t − Dt is small
yt :=
Pt
bt
P
as a (nonlinear) function of xt := C t − Dt
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
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hedging
R&D
power
- FiME
derivatives
Research
12 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Estimation
Observation
Decreasing relation
Difficult estimation
Idea
Quantiles
Figure: PowerNext - 19th hours
Nov, 13th 06 to April 30th 10
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
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hedging
R&D
power
- FiME
derivatives
Research
13 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Estimation
Observation
Decreasing relation
Difficult estimation
Idea
Quantiles
Figure: PowerNext - 19th hours
Nov, 13th 06 to April 30th 10
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
13 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Estimation
Observation
Decreasing relation
Difficult estimation
Idea
Quantiles
Figure: PowerNext - 19th hours
Nov, 13th 06 to April 30th 10
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
13 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Estimation
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
14 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Estimation
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
15 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Estimation
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
16 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Estimation
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
17 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Estimation
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
18 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Estimation
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
19 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Estimation
Estimated relation : yt =
γ
xtν
Improved SRN model
Pt = g
n
X
!
Ctk − Dt
×
n
X
!
hi Sti 1{Pi−1 Ctk ≤ Dt ≤ Pi
k=1
k=1
Ctk }
i=1
k=1
with scarcity function
g (x) := min
René Aı̈d, Luciano Campi, Nicolas Langrené
γ
,
M
1{x>0} + M1{x60}
xν
Universities
A structural risk-neutral
Paris 13model
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20 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Back-testing
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
21 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Back-testing
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
22 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Backtesting
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
23 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Pricing & hedging
Pricing
incomplete market
need for a hedging criterion
super-replication, utility indifference or mean-variance
our choice : Local Risk Minimization
Local Risk Minimization (Pham (00), Schweizer (01))
b
valuation : expected discounted payoff under Q
allows to decompose contingent claim into hedgeable part
(fuels) and non-hedgeable part (demand, capacities)
allows explicit formulas
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
24 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Pricing & hedging
Pricing
incomplete market
need for a hedging criterion
super-replication, utility indifference or mean-variance
our choice : Local Risk Minimization
Local Risk Minimization (Pham (00), Schweizer (01))
b
valuation : expected discounted payoff under Q
allows to decompose contingent claim into hedgeable part
(fuels) and non-hedgeable part (demand, capacities)
allows explicit formulas
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
24 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Dynamics of fuels
Assume r constant, convenience yields and storage costs zero for
simplicity.
Fuels
n ≥ 1 fuels (as coal, gas, oil ...) whose cost hi S i to produce 1
MWh of electricity follows
dSti = Sti (µit dt + σti dWtS,i )
where W S,i are correlated BMs and coeff’s are chosen so that
h1 S 1 < . . . < hn S n (model spreads Y i = hi+1 S i+1 − hi S i as
independent geometric BMs).
NA and completeness assumption
There exists a unique risk-neutral probability Q ∼ P for S.
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
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7 and
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hedging
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Research
25 / 45
Ce
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Position of the problem
Spot model
Pricing & hedging
Conclusion
Local risk minimization I
Roughly speaking, let X be a multidimensional (discounted) price
process
Introduced by Föllmer-Schweizer (1991)
Under regularity condition, any payoff H with maturity T
Z
H = H0 +
T
ξtH dXt + LH
T
0
where H0 ∈ R and LH martingale orthogonal to X .
RT
H
H
0 ξdX is the hedgeable part, LT the residual risk, H0 + LT
the cost of the strategy
How to compute H0 , ξ H , LH ? Easy when X is a martingale :
use Kunita-Watanabe decomposition !
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
26 / 45
Ce
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Position of the problem
Spot model
Pricing & hedging
Conclusion
Local risk minimization I
Roughly speaking, let X be a multidimensional (discounted) price
process
Introduced by Föllmer-Schweizer (1991)
Under regularity condition, any payoff H with maturity T
Z
H = H0 +
T
ξtH dXt + LH
T
0
where H0 ∈ R and LH martingale orthogonal to X .
RT
H
H
0 ξdX is the hedgeable part, LT the residual risk, H0 + LT
the cost of the strategy
How to compute H0 , ξ H , LH ? Easy when X is a martingale :
use Kunita-Watanabe decomposition !
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
26 / 45
Ce
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Position of the problem
Spot model
Pricing & hedging
Conclusion
Local risk minimization I
Roughly speaking, let X be a multidimensional (discounted) price
process
Introduced by Föllmer-Schweizer (1991)
Under regularity condition, any payoff H with maturity T
Z
H = H0 +
T
ξtH dXt + LH
T
0
where H0 ∈ R and LH martingale orthogonal to X .
RT
H
H
0 ξdX is the hedgeable part, LT the residual risk, H0 + LT
the cost of the strategy
How to compute H0 , ξ H , LH ? Easy when X is a martingale :
use Kunita-Watanabe decomposition !
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
EDF
hedging
R&D
power
- FiME
derivatives
Research
26 / 45
Ce
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Position of the problem
Spot model
Pricing & hedging
Conclusion
Local risk minimization I
Roughly speaking, let X be a multidimensional (discounted) price
process
Introduced by Föllmer-Schweizer (1991)
Under regularity condition, any payoff H with maturity T
Z
H = H0 +
T
ξtH dXt + LH
T
0
where H0 ∈ R and LH martingale orthogonal to X .
RT
H
H
0 ξdX is the hedgeable part, LT the residual risk, H0 + LT
the cost of the strategy
How to compute H0 , ξ H , LH ? Easy when X is a martingale :
use Kunita-Watanabe decomposition !
René Aı̈d, Luciano Campi, Nicolas Langrené
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Local risk minimization II
When X is not a martingale but not far from being so ...
b for X
Föllmer-Schweizer (1991) : there exists a risk-neutral Q
s.t.
Z T
b
H = E[H]
+
LH
ξbH dXt + b
t
T
0
b
H0 = E[H],
ξ H = ξbH , LH = b
LH
b is called minimal equivalent martingale measure
Q
Hobson (2005) : in diffusion models with non-tradable assets,
b
E[H]
is an upper bound for U-indifference bid price.
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Local risk minimization II
When X is not a martingale but not far from being so ...
b for X
Föllmer-Schweizer (1991) : there exists a risk-neutral Q
s.t.
Z T
b
H = E[H]
+
LH
ξbH dXt + b
t
T
0
b
H0 = E[H],
ξ H = ξbH , LH = b
LH
b is called minimal equivalent martingale measure
Q
Hobson (2005) : in diffusion models with non-tradable assets,
b
E[H]
is an upper bound for U-indifference bid price.
René Aı̈d, Luciano Campi, Nicolas Langrené
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Local risk minimization II
When X is not a martingale but not far from being so ...
b for X
Föllmer-Schweizer (1991) : there exists a risk-neutral Q
s.t.
Z T
b
H = E[H]
+
LH
ξbH dXt + b
t
T
0
b
H0 = E[H],
ξ H = ξbH , LH = b
LH
b is called minimal equivalent martingale measure
Q
Hobson (2005) : in diffusion models with non-tradable assets,
b
E[H]
is an upper bound for U-indifference bid price.
René Aı̈d, Luciano Campi, Nicolas Langrené
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Local risk minimization II
When X is not a martingale but not far from being so ...
b for X
Föllmer-Schweizer (1991) : there exists a risk-neutral Q
s.t.
Z T
b
H = E[H]
+
LH
ξbH dXt + b
t
T
0
b
H0 = E[H],
ξ H = ξbH , LH = b
LH
b is called minimal equivalent martingale measure
Q
Hobson (2005) : in diffusion models with non-tradable assets,
b
E[H]
is an upper bound for U-indifference bid price.
René Aı̈d, Luciano Campi, Nicolas Langrené
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Research
27 / 45
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Local risk minimization II
When X is not a martingale but not far from being so ...
b for X
Föllmer-Schweizer (1991) : there exists a risk-neutral Q
s.t.
Z T
b
H = E[H]
+
LH
ξbH dXt + b
t
T
0
b
H0 = E[H],
ξ H = ξbH , LH = b
LH
b is called minimal equivalent martingale measure
Q
Hobson (2005) : in diffusion models with non-tradable assets,
b
E[H]
is an upper bound for U-indifference bid price.
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
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Research
27 / 45
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Dynamics of demand and capacities
Demand & capacities
dDt = a (t, Dt ) dt + b (t, Dt ) dWtD
dCti = αi t, Cti dt + βi t, Cti dWtC ,i
where W D , W C and W S are independent BMs.
Minimal martingale measure
b is given by
The minimal martingale measure Q
b F = Q| S ⊗ P| C ,D
Q|
t
Ft
F
t
where Ft is the filtration generated by W S , W D , W C .
René Aı̈d, Luciano Campi, Nicolas Langrené
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Futures
Under our assumptions, we can prove the following
b t [PT ]
Futures prices Fte (T ) = E
Fte
(T ) =
n
X
hi GiT(t, Ct , Dt ) Fti (T )
i=1
with :
"
GiT(t,Ct ,Dt ) = Et g
René Aı̈d, Luciano Campi, Nicolas Langrené
n
X
#
!
CTk − DT
1{Pi−1 C k ≤ DT ≤ Pi
k=1 T
k
k=1 CT
}
k=1
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Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures prices - hedging
Futures price dynamics
P
dFte (T ) = ni=1 hi GiT (t, Ct , Dt )dFti (T ) + Fti (T )dGiT (t, Ct , Dt )
dGiT (t, Ct , Dt )
=
+
n
X
∂G T
i
k=1
∂GiT
∂z
∂ck
(t, Ct , Dt )βk (t, Ctk )dWtC ,k
(t, Ct , Dt )b(t, Dt )dWtD
so that
dFte (T ) = θtS dWt + θtC dWtC + θtD dWtD
for adapted suitable processes θS , θC , θD , which are explicitly
computable.
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Futures prices - hedging
To go further, need to choose more specific dynamics for
demand and capacities
deterministic part for seasonality + Ornstein-Uhlenbeck
GiT explicite as function of extended incomplete
Goodwin-Staton integral :
Z ∞
1
−z 2
e
dz
G (x, y ; ν) =
νe
(y + z)
x
... for which efficient numerical algorithms are provided in Aı̈d,
C. & Langrené (10).
René Aı̈d, Luciano Campi, Nicolas Langrené
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Futures prices - hedging : spot simulations
René Aı̈d, Luciano Campi, Nicolas Langrené
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Research
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Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Futures prices - hedging : spot simulations
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
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Research
33 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Futures prices - hedging : spot simulations
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
Paris 13model
& Paris
for pricing
7 and
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hedging
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derivatives
Research
34 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Futures prices - hedging : spot simulations
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
A structural risk-neutral
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hedging
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Research
35 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Futures prices - hedging : spot simulations
René Aı̈d, Luciano Campi, Nicolas Langrené
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Research
36 / 45
Ce
Position of the problem
Spot model
Pricing & hedging
Conclusion
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Futures prices - hedging
Numerical test
Hedging a 3-months
electricity futures with a
delivery period of 1 hour
with a daily rebalanced
basket of futures
contracts on fuels
René Aı̈d, Luciano Campi, Nicolas Langrené
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Research
37 / 45
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Futures prices - hedging
Numerical test
Hedging a 3-months
electricity futures with a
delivery period of 1 hour
with a daily rebalanced
basket of futures
contracts on fuels
René Aı̈d, Luciano Campi, Nicolas Langrené
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37 / 45
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Futures prices - hedging
Remarks
Positive values are
losses
Far from maturity :
perfect hedge ;
electricity futures is
equivalent to a
basket of fuels
Close to maturity :
inefficient hedge
René Aı̈d, Luciano Campi, Nicolas Langrené
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Spread options
Spread option with a 2 fuel model
The price π0 at time t = 0 of a call spread option with pay-off
H = (PT − h1 ST1 − K )+ is given by :
Z
π0 =
fC 1 −DT (z)fC 2 (c) φ1 (c, z)1{z>0} + φ2 (c, z)1{z≤0} dcdz,
R2
T
T
φ1 = (g − 1)BS0 (σ1 , K )1{g >1}
Z ∞
K + (1 − g )y fˆY 1 (y )BS0 σ2 ,
φ2 = g
1{g ≤1} + 1{g >1} 1{y < K } dy
T
g −1
g
0




+ gY02 N 
r−
σ12
2
T − ln
√
σ1 T
K
(g −1)Y01


 + (g − 1) BS0

K

σ1 ,
 1{g >1}
g −1
with g := g (c + z).
René Aı̈d, Luciano Campi, Nicolas Langrené
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Spread options
semi-explicit formula : numerical integration
partial hedging with futures on fuels and electricity,
semi-explicit formulae for partial hedging strategy (not only
for spread options)
applied on European dark spread (i.e. energy - gas) call option
with a period of delivery of 1 hour
René Aı̈d, Luciano Campi, Nicolas Langrené
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Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Position of the problem
Spot model
Pricing & hedging
Conclusion
Hedging with futures on electricity
Consider H = ϕ(FTe (T ∗ ), FT (T ∗ ), CT , DT ) with T ∗ > T .
b
By Markov, its Q-price
in t is φ(t, Ft , Ct , Dt ) with φ(t, x, c, z)
regular
H’s decomposition into hedgeable part/residual risk is
Z T
Z T
b
b
H = E[H] +
ξt dFt +
ξbte dFte + b
LT
0
0
where
ξbte
ξbti
=
1
C
||(θt , θtD )||2
!
X
θtC ,i βi ∂ci φ + θtD b∂z φ
i
hi GiT ∗
= ∂yi φ +
||(θtC , θtD )||2
!
X
θtC ,i βi ∂ci φ
+
θtD b∂z φ
i
b
LT can be computed explicitly as well
René Aı̈d, Luciano Campi, Nicolas Langrené
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Dynamics of fuels etc
Local risk minimization
Futures
Options
Hedging with futures on electricity
Position of the problem
Spot model
Pricing & hedging
Conclusion
Hedging with futures on electricity
Consider H = ϕ(FTe (T ∗ ), FT (T ∗ ), CT , DT ) with T ∗ > T .
b
By Markov, its Q-price
in t is φ(t, Ft , Ct , Dt ) with φ(t, x, c, z)
regular
H’s decomposition into hedgeable part/residual risk is
Z T
Z T
b
b
H = E[H] +
ξt dFt +
ξbte dFte + b
LT
0
0
where
ξbte
ξbti
=
1
C
||(θt , θtD )||2
!
X
θtC ,i βi ∂ci φ + θtD b∂z φ
i
hi GiT ∗
= ∂yi φ +
||(θtC , θtD )||2
!
X
θtC ,i βi ∂ci φ
+
θtD b∂z φ
i
b
LT can be computed explicitly as well
René Aı̈d, Luciano Campi, Nicolas Langrené
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Conclusion
Conclusions
SRN electricity spot price model with a scarcity function
allows futures and derivatives pricing and hedging
nevertheless, only fuels dependent part can be hedged ...
... unless we use energy future for partially hedging demand
and capacities risk (see the paper Aı̈d-C.-Langrené)
Perspectives
comparison with ”real” quoted futures, calibration dynamics
utility-based pricing of futures, options ...
optimal investment/production problem, optimal switching
René Aı̈d, Luciano Campi, Nicolas Langrené
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Conclusion
Conclusions
SRN electricity spot price model with a scarcity function
allows futures and derivatives pricing and hedging
nevertheless, only fuels dependent part can be hedged ...
... unless we use energy future for partially hedging demand
and capacities risk (see the paper Aı̈d-C.-Langrené)
Perspectives
comparison with ”real” quoted futures, calibration dynamics
utility-based pricing of futures, options ...
optimal investment/production problem, optimal switching
René Aı̈d, Luciano Campi, Nicolas Langrené
Universities
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Paris 13model
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Position of the problem
Spot model
Pricing & hedging
Conclusion
References
Aı̈d, C., Nguyen Huu & Touzi, Int. J. Theoretical & Applied
Finance, 2010
Aı̈d, C., Langrené, 2010, available on HAL
Barlow, Math. Finance, 2002
Benth & Koekebakker, J. of Energy Economics, 2007
Benth & Vos, Tech. Rep., Math. Dept. Oslo, 2009
Benth, Ekeland, Hauge & Nielsen, Applied Math. Finance,
2003
Burger, Klar, Müller & Schlindlmayr, Quantitative Finance,
2004
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Position of the problem
Spot model
Pricing & hedging
Conclusion
References
Cartea & Figueroa, Applied Math. Finance, 2005
Cartea & Villaplana, J. of Banking & Finance, 2008
Coulon & Howison, J. of Energy Markets, 2009
Deng, Tech. Rept.,California Energy Institute, 2000
Geman & Roncoroni, J. of Business, 2006
Kanamura & Ohashi, Energy Economics, 2007
Kolodnyi, J. of Engineering Mathematics, 2004
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Position of the problem
Spot model
Pricing & hedging
Conclusion
References
Lyle & Elliott, Energy Economics, 2009
Pham, Math. Meth. of Operations Research, 2000
Pirrong & Jermakyan, J. of Banking & Finance, 2008 1
Schweizer, Handbook Math. Finance, Cambridge Univ. Press,
2001
1. Olin Business School Tech. Rep. 2000
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René Aı̈d, Luciano Campi, Nicolas Langrené