Technical presentation about Levy processes applied to the energy market.

1. Levy Processes In The Energy Markets
EDF Trading
——
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 Desk
2-Factor Model
dF(t, T)
F(t, T)
= σce−a(T−t)dWc(t) + σldWl(t)
where F(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 Desk
Explanatory variables : Exode
lnSi(t) =
n
j=1
α
j
i V Ej(t) + ui(t)
where Si(t) represents the spot price at the day t and the hour i, (V Ej)1≤1≤n explanatory
variables, based on the temperature, ui(t) an AR(1) non gaussian process, whose variance
depends on (V Ej)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 Desk
Levy Process
F(t, T) = Λ(T, t)eXT
t
dXt
t = −aXt
tdt + σdLt,
where Λ(t, T) represents the seasonality and Lt is 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 Desk
Definitions and Basic Assumptions
Lt is a Levy process if L0 = 0 and has independent and stationary increments, and it’s
continuous in probability
∀t ≥ 0, ∀ε > 0, lims→t (|Lt − Ls| > 0) = 0
The Fourier transform of Lt follows Levy-Khintchine formula :
[ezLt] = etψ(z)
∀z ∈ , ψ(z) = imz −
σ2
2
z2 + (eizu − 1 − iuz)ν(du)
• The measure ν(dx) is called the Levy mesur of Lt
• Levy process consists of three independent parts
O. Senhadji El Rhazi 6 August 06, 2005

8. EDF Trading Derivatives Desk
Esscher Transforms
We call Esscher transform any change of to a locally equivalent measure with a density
process
Zt =
d
d
, Zt = eωLt−ϕ(ω)t
ϕ(z) = mz +
σ2
2
z2 + (ezu − 1 − uz)ν(du)
• The Girsanov transform is an especial case of Esscher transform
• The technique in commodity is to single out ω from forward or spot curves
O. Senhadji El Rhazi 7 August 06, 2005

9. EDF Trading Derivatives Desk
Asset 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 Desk
Asset Price Model 2
Assume F(t, T) the forward price at time t with delivery at time T, which we model as
stochastic process
F(t, T) = Λ(t, T)eXT
t
Using the condition of arbitrage-free price of forward, the market price of risk ω, we
postulate the model (under ) :
Λ(t, T) = F(0, T)exp( t
0 [ϕ(ω + σe−a(T−s)) − ϕ(ω)]ds)
XT
t = t
0 σe−a(T−s)dLs
O. Senhadji El Rhazi 9 August 06, 2005

11. EDF Trading Derivatives Desk
Asset Price Model 3
F(t, T) = Λ(T, t)eXT
t
• Mean-reversion :
dXt
t = −aXt
tdt + σdLt
• Seasonal variations :
Λ(t, T) = F(0, T)exp( t
0 [ϕ(ω + σe−a(T−s)) − ϕ(ω)]ds)
• Leptokurtic log-spot :
Lt and not Wt
O. Senhadji El Rhazi 10 August 06, 2005

12. EDF Trading Derivatives Desk
Asset Price Model 4
We denote Yt = 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)] − t
0 [ϕ(ω + σe−as) − ϕ(ω)]ds
We consider discretisation of an interval [0, T], with step h = T
n. We denote Yih = Yi
which follows the following schema :
Yi − φ1Yi−1 − φi
0 = εi
with,
εi = ih
(i−1)h σe−a(ih−s)dLs, φ1 = e−ah
O. Senhadji El Rhazi 11 August 06, 2005

13. EDF Trading Derivatives Desk
and,
φi
0 = − ih
0 [ϕ(ω + σe−as) − ϕ(ω)]ds + φ1
(i−1)h
0 [ϕ(ω + σe−as) − ϕ(ω)]ds
εi are i.i.d if ah ≪ 1, we can make this approximation :
εi ∼ σLh
• With conjugate gradient or maximum likelihood methods, we can estimate the pa-
rameters of the model Lt.
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 Desk
Distributions
Generalized hyperbolic distribution were introduced by Bandorff-Nielsen (1977). Their
Lebesgue densities are given by :
fHG
(x; λ, α, δ, δ, µ) = a(λ, α, β, δ) δ2
+ (x − µ)2λ−1
2 Kλ−1
2
(α δ2
+ (x − µ)2
)exp(β(x − µ))
a(λ, α, β, δ) =
γλ
√
2παλ−1
2 δλKλ(δγ)
, γ = α2
− β2
• Kλ denotes the modified Bessel function of the third kind with index λ
• α determines the shape, β the skewness, µ the location, δ the scaling parameter, and
λ characterizes certain sub-classes
O. Senhadji El Rhazi 14 August 06, 2005

16. EDF Trading Derivatives Desk
Properties
• The normal distribution is obtained as a limit case if
δ → ∞ and δ/α → σ2
• 4 degree of freedom (Mean, Variance, Skewwness and Kurtosis)
• Heavy-tailed distribution (heavier than the normal)
• λ and α represent the number of spikes and volatility and β represent the sign of
those spikes and their intensity.
O. Senhadji El Rhazi 15 August 06, 2005

17. EDF Trading Derivatives Desk
Normal Inverse Gaussian NIG
λ = −1/2 , fNIG
(x; α, β, δ, µ) =
αδ
π
e(δ
√
α2
−β2
+β(x−µ))K1(α δ2
+ (x − µ)2
)
δ2
+ (x − µ)2
Xi ∼ NIG(α, β, δi, µi), X1 + X2 ∼ NIG(α, β, δ1 + δ2, µ1 + µ2)
−10 −8 −6 −4 −2 0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
−10 −8 −6 −4 −2 0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
×××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××
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−10 −8 −6 −4 −2 0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
−10 −8 −6 −4 −2 0 2 4 6 8 10
0
0.1
0.2
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0.5
×××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××
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−10 −8 −6 −4 −2 0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
−10 −8 −6 −4 −2 0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
×××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××
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Densities: Normal(left), NIG(0.3, 0.3, 0.8, 0)(centre), NIG(0.3, 0, 0.8, 2)(right) and NIG(0.3, 0, 0.8, 0) in stipple.
O. Senhadji El Rhazi 16 August 06, 2005

18. EDF Trading Derivatives Desk
Variance Gamma VG
δ = 0 , fV G
(x; λ, α, β, µ) =
γ2λ
√
πΓ(λ)(2α)λ−1
2
|x − µ|λ−1
2 Kλ−1
2
(α|x − µ|)eβ(x−µ)
Xi ∼ VG(λi, α, β, µi), X1 + X2 ∼ VG(λ1 + λ2, α, β, µ1 + µ2)
−10 −8 −6 −4 −2 0 2 4 6 8 10
0
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
−10 −8 −6 −4 −2 0 2 4 6 8 10
0
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
×××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××
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−10 −8 −6 −4 −2 0 2 4 6 8 10
0
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
−10 −8 −6 −4 −2 0 2 4 6 8 10
0
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
×××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××
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−10 −8 −6 −4 −2 0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
−10 −8 −6 −4 −2 0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
×××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××
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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) and VG(0.6, 0.1, 0, 0.2) in stipple.
O. Senhadji El Rhazi 17 August 06, 2005

19. EDF Trading Derivatives Desk
Subordination
The class of HG distribution can be obtained by subordination of the Brownian time, if
we define
Lt = µt + βτt + Wτt
where Wt is a standard Brownian motion and τt (business time) is generated by a GIG(λ, δ, γ),
which has the following distribution
fGIG
(x) = (
γ
δ
)λ 1
2Kλ(δγ)
xλ−1
exp(−
1
2
(
δ2
x
+ γ2
x))
✒ Period of agitation τt+dt − τt > dt, ar[Wτt+dt
|τt+dt] > ar[Wτt
|τt] + ar[Wτdt
]
✒ Period of calm τt+dt − τt ≤ dt, ar[Wτt+dt
|τt+dt] ≤ ar[Wτt
|τt] + ar[Wτdt
]
The process Lt can be seen as a stochastic volatility model.
O. Senhadji El Rhazi 18 August 06, 2005

20. EDF Trading Derivatives Desk
Residue NIG
Mean and Variance : = (µ +
δβ
γ
)t, ar =
δα2
γ3
0 100 200 300 400 500
−9
−7
−5
−3
−1
1
3
5
7
9
11
0 100 200 300 400 500
−10
0
10
20
30
40
50
60
0 100 200 300 400 500
−70
−60
−50
−40
−30
−20
−10
0
10
20
Simulation of NIG(0.3, 0, 1, 0)(left), NIG(0.3, 0.3, 1, 0)(centre) and NIG(0.03, 0, 1, 0)(right).
O. Senhadji El Rhazi 19 August 06, 2005

21. EDF Trading Derivatives Desk
Residue VG
Mean and Variance : = (µ +
2λβ
γ2 )t, ar =
2λ
γ2(1 + 2(
β
γ
)2)t
0 100 200 300 400 500
−16
−12
−8
−4
0
4
8
12
16
0 100 200 300 400 500
−10
30
70
110
150
190
230
0 100 200 300 400 500
−9
−7
−5
−3
−1
1
3
5
7
9
Simulation of VG(0.7, 0.3, 0, 0)(left), VG(0.7, 0.3, 0.3, 0)(centre) and VG(0.07, 0.3, 0, 0)(right).
O. Senhadji El Rhazi 20 August 06, 2005

22. EDF Trading Derivatives Desk
Bibliographie
[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