Levy processes in the energy markets

Technical presentation about Levy processes applied to the energy market.

1. Levy Processes In The Energy Markets
EDFTrading
——
O. Senhadji El Rhazi
August 06, 2005

2. EDF Trading Derivatives Desk
Principal Models
O. Senhadji El Rhazi 1 August 06, 2005

3. EDF Trading Derivatives Desk2-Factor Model
dF(t,T)
F(t,T)
=σce
−a(T−t)
dWc(t)+σ
ldW
l(t)
whereF(t,T)is the price at time t for a 1-hour forward delivered at time T
•Continuous and log-normal process (no spike in the spot process)
•Pricing : European, Asian, Swing, at the money
O. Senhadji El Rhazi 2 August 06, 2005

4. EDF Trading Derivatives DeskExplanatory variables : Exode
lnSi(t) =
n
X
j=1
α
j
i
VEj(t)+ui(t)
whereSi(t)represents the spot price at the daytand the houri,(VEj)1≤1≤nexplanatory
variables, based on the temperature,ui(t)anAR(1)non gaussian process, whose variance
depends on(VEj)1≤1≤n
•Only a spot model and non log-normal process (spike)
•Pricing : European out the money and specific products (pay-off depending on the
temperature)
O. Senhadji El Rhazi 3 August 06, 2005

5. EDF Trading Derivatives DeskLevy Process
F(t,T) = Λ(T,t)e
X
T
t
dX
t
t=−aX
t
tdt+σdLt,
whereΛ(t,T)represents the seasonality andLtis a Levy process
•Non continuous and non log-normal process (spike in the spot process)
•Pricing : European, Swing, at and out the money
O. Senhadji El Rhazi 4 August 06, 2005

6. EDF Trading Derivatives Desk
Levy Process
O. Senhadji El Rhazi 5 August 06, 2005

7. EDF Trading Derivatives DeskDefinitions and Basic Assumptions
Ltis a Levy process ifL0= 0and has independent and stationary increments, and it’s
continuous in probability
∀t≥0,∀ε >0, lims→t(|Lt−Ls|>0) = 0
The Fourier transform ofLtfollows Levy-Khintchine formula :
[e
zLt
] =e
tψ(z)
∀z∈, ψ(z) =imz−
σ
2
2
z
2
+
Z
(e
izu
−1−iuz)ν(du)
•The measureν(dx)is called the Levy mesur ofLt
•Levy process consists of three independent parts
O. Senhadji El Rhazi 6 August 06, 2005

8. EDF Trading Derivatives DeskEsscher Transforms
WecallEsschertransformanychangeof toalocallyequivalentmeasure withadensity
process
Zt=
d
d
, Zt=e
ωLt−ϕ(ω)t
ϕ(z) =mz+
σ
2
2
z
2
+
Z
(e
zu
−1−uz)ν(du)
•The Girsanov transform is an especial case of Esscher transform
•The technique in commodity is to single outωfrom forward or spot curves
O. Senhadji El Rhazi 7 August 06, 2005

9. EDF Trading Derivatives DeskAsset Price Model 1
The generalization of Schwartz’ model must incorporates this conditions :
•Mean-reversion on energy prices
•Seasonal variations of Forward curves
•Capture leptokurtic behaviour of log-spot
O. Senhadji El Rhazi 8 August 06, 2005

10. EDF Trading Derivatives DeskAsset Price Model 2
AssumeF(t,T)the forward price at timetwith delivery at timeT, which we model as
stochastic process
F(t,T) = Λ(t,T)e
X
T
t
Using the condition of arbitrage-free price of forward, the market price of riskω, we
postulate the model (under ) :
Λ(t,T) =F(0,T)exp(
Z
t
0
[ϕ(ω+σe
−a(T−s)
)−ϕ(ω)]ds)
X
T
t=
Z
t
0
σe
−a(T−s)
dLs
O. Senhadji El Rhazi 9 August 06, 2005

11. EDF Trading Derivatives DeskAsset Price Model 3
F(t,T) = Λ(T,t)e
X
T
t
•Mean-reversion :
dX
t
t=−aX
t
tdt+σdLt
•Seasonal variations :
Λ(t,T) =F(0,T)exp(
Z
t
0
[ϕ(ω+σe
−a(T−s)
)−ϕ(ω)]ds)
•Leptokurtic log-spot :
Ltand notWt
O. Senhadji El Rhazi 10 August 06, 2005

12. EDF Trading Derivatives DeskAsset Price Model 4
We denoteYt=ln(
St
F(0,t)
), this process is a solution of the SDE given by
dYt=a(mt−Yt)dt+σdLt
mt=−
1
a
[ϕ(ω+σe
−at
)−ϕ(ω)]−
Z
t
0
[ϕ(ω+σe
−as
)−ϕ(ω)]ds
We consider discretisation of an interval[0,T], with steph=
T
n
. We denoteY
ih=Yi
which follows the following schema :
Yi−φ1Yi−1−φ
i
0=εi
with,
εi=
Z
ih
(i−1)h
σe
−a(ih−s)
dLs, φ1=e
−ah
O. Senhadji El Rhazi 11 August 06, 2005

13. EDF Trading Derivatives Deskand,
φ
i
0=−
Z
ih
0
[ϕ(ω+σe
−as
)−ϕ(ω)]ds+φ1
Z
(i−1)h
0
[ϕ(ω+σe
−as
)−ϕ(ω)]ds
εiare i.i.d ifah≪1, we can make this approximation :
εi∼σL
h
•With conjugate gradient or maximum likelihood methods, we can estimate the pa-
rameters of the modelLt.
O. Senhadji El Rhazi 12 August 06, 2005

14. EDF Trading Derivatives Desk
Generalized Hyperbolic Process
O. Senhadji El Rhazi 13 August 06, 2005

15. EDF Trading Derivatives DeskDistributions
Generalized hyperbolic distribution were introduced by Bandorff-Nielsen (1977). Their
Lebesgue densities are given by :
f
HG
(x;λ,α,δ,δ,μ) =a(λ,α,β,δ)
r
δ
2
+ (x−μ)
2λ−
1
2K
λ−
1
2
(α
r
δ
2
+ (x−μ)
2
)exp(β(x−μ))
a(λ,α,β,δ) =
γ
λ
√
2πα
λ−
1
2δ
λ
Kλ(δγ)
, γ=
r
α
2
−β
2
•K
λdenotes the modified Bessel function of the third kind with indexλ
•αdetermines the shape,βthe skewness,μthe location,δthe scaling parameter, and
λcharacterizes certain sub-classes
O. Senhadji El Rhazi 14 August 06, 2005

16. EDF Trading Derivatives DeskProperties
•The normal distribution is obtained as a limit case if
δ→ ∞and δ/α→σ
2
•4 degree of freedom (Mean, Variance, Skewwness and Kurtosis)
•Heavy-tailed distribution (heavier than the normal)
•λandαrepresent the number of spikes and volatility andβrepresent the sign of
those spikes and their intensity.
O. Senhadji El Rhazi 15 August 06, 2005

17. EDF Trading Derivatives DeskNormal Inverse GaussianNIG
λ=−1/2, f
N IG
(x;α,β,δ,μ) =
αδ
π
e
(δ
√
α
2
−β
2
+β(x−μ))
K1(α
r
δ
2
+ (x−μ)
2
)
r
δ
2
+ (x−μ)
2
Xi∼ NIG(α,β,δi,μi), X1+X2∼ NIG(α,β,δ1+δ2,μ1+μ2)
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Densities: Normal(left),NIG(0.3,0.3,0.8,0)(centre),NIG(0.3,0,0.8,2)(right) andNIG(0.3,0,0.8,0)in stipple.O. Senhadji El Rhazi 16 August 06, 2005

18. EDF Trading Derivatives DeskVariance GammaVG
δ= 0, f
V G
(x;λ,α,β,μ) =
γ
2λ
√
πΓ(λ)(2α)
λ−
1
2
|x−μ|
λ−
1
2K
λ−
1
2
(α|x−μ|)e
β(x−μ)
Xi∼ VG(λi,α,β,μi), X1+X2∼ VG(λ1+λ2,α,β,μ1+μ2)
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0
0.040.080.120.160.200.240.280.320.360.40
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0
2
4
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8
10
0
0.040.080.120.160.200.240.280.320.360.40
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0
2
4
6
8
10
0
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-4
-2
0
2
4
6
8
10
0
0.040.080.120.160.200.240.280.320.360.40
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Densities: Normal(left),VG(0.6,0.1,0,0.01)(centre),VG(0.6,0.3,0,2.2)(right) andVG(0.6,0.1,0,0.2)in stipple.O. Senhadji El Rhazi 17 August 06, 2005

19. EDF Trading Derivatives DeskSubordination
The class ofHGdistribution can be obtained by subordination of the Browniantime, if
we define
Lt=μt+βτt+Wτt
whereWtisastandardBrownianmotionandτt(businesstime)isgeneratedbyaGIG(λ,δ,γ),
which has the following distribution
f
GIG
(x) = (
γ
δ
)
λ
12Kλ(δγ)
x
λ−1
exp(−
1
2
(
δ
2
x
+γ
2
x))
✒Period of agitationτt+dt−τt> dt, ar[Wτ
t+dt
|τt+dt]> ar[Wτ
t
|τt] +ar[Wτ
dt
]
✒Period of calmτt+dt−τt≤dt, ar[Wτ
t+dt
|τt+dt]≤ar[Wτ
t
|τt] +ar[Wτ
dt
]
The processLtcan be seen as a stochastic volatility model.
O. Senhadji El Rhazi 18 August 06, 2005

20. EDF Trading Derivatives DeskResidueNIG
Mean and Variance := (μ+
δβ
γ
)t, ar=
δα
2
γ
3
0
100
200
300
400
500
-9-7-5-3-1
13579
11
0
100
200
300
400
500
-10
0
102030405060
0
100
200
300
400
500
-70-60-50-40-30-20-10
0
1020
Simulation ofNIG(0.3,0,1,0)(left),NIG(0.3,0.3,1,0)(centre) andNIG(0.03,0,1,0)(right).O. Senhadji El Rhazi 19 August 06, 2005

21. EDF Trading Derivatives DeskResidueVG
Mean and Variance := (μ+
2λβ
γ
2
)t, ar=
2λ
γ
2
(1+2(
β
γ
)
2
)t
0
100
200
300
400
500
-16-12
-8-4
048
1216
0
100
200
300
400
500
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3070
110150190230
0
100
200
300
400
500
-9-7-5-3-1
13579
Simulation ofVG(0.7,0.3,0,0)(left),VG(0.7,0.3,0.3,0)(centre) andVG(0.07,0.3,0,0)(right).O. Senhadji El Rhazi 20 August 06, 2005

22. EDF Trading Derivatives DeskBibliographie
[BEHN99] F.E.Benth, L.Ekeland, R.Hauge, B.F.Nielsen ,On arbitrage-free pricing of forward contracts in
energy markets, Preprint, pp. 1-7, (2001).
[CM99] P.Carr and D.B.Madan,Option Valuation Using the Fast Fourier Transform, J.Comp Finance, pp.
61-73, (1999).
[CS00] L.Clewlow and C.Strickland,Energy Derivatives. Pricing and Risk Management, Lacima Publications,
(2000).
[ML02] V.Mignon et S.Lardic,Econométrie des séries temporelles macroéconomiques et financières, Eco-
nomica, pp. 45, 274, 25-52, (2002).
[Rai00] S.Raible,Levy Processes in Finance : Theory, Numerics, and Empirical Facts, PhD thesis, Institut
für Mathematische Stochastik, Universität Freiburg im Breisgau, (2000).
[Sch03] W.Schoutens,Levy Processes in Finance : Pricing Financial Derivatives, Wiley Publications, (2003).
O. Senhadji El Rhazi 21 August 06, 2005