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![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](https://image.slidesharecdn.com/levyprocessesintheenergymarkets-170629135106/85/Levy-processes-in-the-energy-markets-7-320.jpg)
![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](https://image.slidesharecdn.com/levyprocessesintheenergymarkets-170629135106/85/Levy-processes-in-the-energy-markets-10-320.jpg)
![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](https://image.slidesharecdn.com/levyprocessesintheenergymarkets-170629135106/85/Levy-processes-in-the-energy-markets-11-320.jpg)
![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](https://image.slidesharecdn.com/levyprocessesintheenergymarkets-170629135106/85/Levy-processes-in-the-energy-markets-12-320.jpg)
![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](https://image.slidesharecdn.com/levyprocessesintheenergymarkets-170629135106/85/Levy-processes-in-the-energy-markets-13-320.jpg)
![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](https://image.slidesharecdn.com/levyprocessesintheenergymarkets-170629135106/85/Levy-processes-in-the-energy-markets-19-320.jpg)
![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](https://image.slidesharecdn.com/levyprocessesintheenergymarkets-170629135106/85/Levy-processes-in-the-energy-markets-22-320.jpg)
Uploaded byOtmane Senhadji El Rhazi
Levy processes in the energy markets
1) The document discusses several models for pricing derivatives in energy markets, including a 2-factor model, an Exode model, and models based on Levy processes. 2) It proposes an asset price model that incorporates mean-reversion, seasonal variations in forward curves, and the leptokurtic behavior of log-spot prices using a Levy process. 3) The model discretizes the Levy process using a Euler scheme and estimates the model parameters using maximum likelihood methods.
![11.[104 111]analytical solution for telegraph equation by modified of sumudu ...](https://cdn.slidesharecdn.com/ss_thumbnails/11-104-111analyticalsolutionfortelegraphequationbymodifiedofsumudutransformelzakitransform-120513000219-phpapp01-thumbnail.jpg?width=640&height=640&fit=bounds)
Levy processes in the energy markets
- 1. Levy Processes InThe Energy Markets EDF Trading —— O. Senhadji El Rhazi August 06, 2005
- 2. EDF Trading DerivativesDesk Principal Models O. Senhadji El Rhazi 1 August 06, 2005
- 3. EDF Trading DerivativesDesk 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 DerivativesDesk 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 DerivativesDesk 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 DerivativesDesk Levy Process O. Senhadji El Rhazi 5 August 06, 2005
- 7. EDF Trading DerivativesDesk 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 DerivativesDesk 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 DerivativesDesk 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 DerivativesDesk 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 DerivativesDesk 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 DerivativesDesk 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 DerivativesDesk 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 DerivativesDesk Generalized Hyperbolic Process O. Senhadji El Rhazi 13 August 06, 2005
- 15. EDF Trading DerivativesDesk 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 DerivativesDesk 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 DerivativesDesk 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 ××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× × × × × × × × × × × × × ××× × × × × × × × × × × × × ××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× −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 ××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× × × × × × × × × × × × × ××× × × × × × × × × × × × × ××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× −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 ××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× × × × × × × × × × × × × ××× × × × × × × × × × × × × ××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× 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 DerivativesDesk 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 ××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× × × × × × × × × × × × × × × × ××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× −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 ××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× × × × × × × × × × × × × × × × ××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× −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 ××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× × × × × × × × × × × × × ×××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× 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 DerivativesDesk 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 DerivativesDesk 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 DerivativesDesk 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 DerivativesDesk 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