Stochastic Modelling of Electricity and Related Markets: Chapter 3

Transcription

1 Sochasic Modelling of Elecriciy and Relaed Markes: Chaper 3 Fred Espen Benh, Jūraė Šalyė Benh and Seen Koekebakker Presener: Tony Ware Universiy of Calgary Ocober 14, 2009

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4 The Schwarz model S() = S(0) exp X(), wih dx() = κ ( α X() ) d + σdw(). The saring poin for he models in his chaper (and indeed for much of commodiy spo price modelling) is he Schwarz one-facor model from his 1997 paper The sochasic behavior of commodiy prices: implicaions for valuaion and hedging (Journal of Finance, Vol. 52(3), ).

5 Spo price modelling wih OU processes I() is an II process wih a Lévy-Kinchine represenaion ψ(, s; θ) = iθ ( γ(s) γ() ) 1 2 θ2( C(s) C() ) where γ is of finie variaion. + R {e izθ 1 izθ1 z <1 } l(dz, du), An RCLL process X(s) ( s T) is an OU process if i is he unique srong soluion o dx(s) = ( µ(s) α(s)x(s) ) ds + σ(s)di(s), X() = x. The unique soluion can be wrien X(s) = x e R s α(v)dv + µ(u)e R s u α(v)dv du + σ(u)e R s u α(v)dv di(u).

6 Spo price modelling wih OU processes I() is an II process wih a Lévy-Kinchine represenaion ψ(, s; θ) = iθ ( γ(s) γ() ) 1 2 θ2( C(s) C() ) where γ is of finie variaion. + R {e izθ 1 izθ1 z <1 } l(dz, du), An RCLL process X(s) ( s T) is an OU process if i is he unique srong soluion o dx(s) = ( µ(s) α(s)x(s) ) ds + σ(s)di(s), X() = x. The unique soluion can be wrien X(s) = x e R s α(v)dv + µ(u)e R s u α(v)dv du + σ(u)e R s u α(v)dv di(u).

7 Spo price modelling wih OU processes I() is an II process wih a Lévy-Kinchine represenaion ψ(, s; θ) = iθ ( γ(s) γ() ) 1 2 θ2( C(s) C() ) where γ is of finie variaion. + R {e izθ 1 izθ1 z <1 } l(dz, du), An RCLL process X(s) ( s T) is an OU process if i is he unique srong soluion o dx(s) = ( µ(s) α(s)x(s) ) ds + σ(s)di(s), X() = x. The unique soluion can be wrien X(s) = x e R s α(v)dv + µ(u)e R s u α(v)dv du + σ(u)e R s u α(v)dv di(u).

8 Spo price modelling wih OU processes The characerisic funcion of an OU process [ ] E e iθx(s) X() = x = e χ(,s;θ), wih ( χ(, s; θ) = iθ x e R s α(v)dv + where ψ (, s; g( ) ) is defined by ψ (, s; g( ) ) = i g(u)dγ(u) R µ(u)e R ) s u α(v)dv du + ψ (, s; θσ( )e R s α(v)dv), g 2 (u)dc(u) {e izg(u) 1 izg(u)1 z <1 } l(dz, du).

9 Spo price modelling wih OU processes The expeced value of an OU process The above resul can be used o show ha, if Brownian moion E [ X(s) X() = x ] = x e R s α(v)dv z 1 σ(u)e R s u α(v)dv dγu z 1 z l(dz, du) <, µ(u)e R s u α(v)dv du zσ(u)e R s u α(v)dv l(dz, du). If I() = B(), hen X(s) (condiioned on X() = x) is normal wih mean and variance x e R s α(v)dv + µ(u)e R s u α(v)dv du and σ 2 (u)e 2 R s u α(v)dv du.

10 Spo price modelling wih OU processes The expeced value of an OU process The above resul can be used o show ha, if Brownian moion E [ X(s) X() = x ] = x e R s α(v)dv z 1 σ(u)e R s u α(v)dv dγu z 1 z l(dz, du) <, µ(u)e R s u α(v)dv du zσ(u)e R s u α(v)dv l(dz, du). If I() = B(), hen X(s) (condiioned on X() = x) is normal wih mean and variance x e R s α(v)dv + µ(u)e R s u α(v)dv du and σ 2 (u)e 2 R s u α(v)dv du.

11 Spo price modelling wih OU processes The saionary disribuion of an OU process Suppose ha all coefficiens are consan. In he Brownian moion case, we find ha lim s X(s) = X, where ( ) µ X N α, σ2. 2α If I() = L() (a Lévy process), and z 2 ln z l(dz) <, hen he cumulan funcion of X is iθ µ α + ψ(θe αs )ds, where φ(θ) is he cumulan funcion of L(1). 0

12 Geomeric models Inroduce n independen pure jump semimaringale II processes I j (), given by I j () = γ j () + z Ñ j (dz, du) + z N j (dz, du), 0 z <1 and p independen Brownian moions B j (). Define S() by where and ln S() = ln Λ() + 0 m X i () + i=1 z 1 n Y j (), j=1 dy j () = ( δ j () β j ()Y j () ) d + η j ()di j (), dx i () = ( µ i () α i ()X i () ) d + wih Λ() modelling he seasonal price level. p σ ik ()db k (), k=1

13 Geomeric models The dynamics of S() are given by ds() S( ) = Λ () Λ() + 1 σ ij ()σ jk () 2 i,j,k ( µi () α i ()X i () ) + j ( δj () β j ()Y j () ) d + i + j + j z <1 z <1 { } e ηj()z 1 η j ()z l j (dz, d) + i,k } Ñj (dz, d) + { e ηj()z 1 j z 1 σ ik db k () { } e ηj()z 1 N j (dz, d). Inegrabiliy condiions apply if we are o use his for opion pricing. In he one-facor Schwarz case his reduces o { ds() Λ S() = () Λ() + α() ln Λ() σ2 () + ( µ() α() ln S() ) } d + σ()db().

14 Geomeric models The dynamics of S() are given by ds() S( ) = Λ () Λ() + 1 σ ij ()σ jk () 2 i,j,k ( µi () α i ()X i () ) + j ( δj () β j ()Y j () ) d + i + j + j z <1 z <1 { } e ηj()z 1 η j ()z l j (dz, d) + i,k } Ñj (dz, d) + { e ηj()z 1 j z 1 σ ik db k () { } e ηj()z 1 N j (dz, d). Inegrabiliy condiions apply if we are o use his for opion pricing. In he one-facor Schwarz case his reduces o { ds() Λ S() = () Λ() + α() ln Λ() σ2 () + ( µ() α() ln S() ) } d + σ()db().

15 Geomeric models Some more special cases dx() = α()x()d + σ()db() dy() = α()y()d + di(). d ln S() = d ln Λ() α() ( ln S() ln Λ() ) d + σ()db() + di(). Benh and Šalyė Benh (2004) Pure jump NIG Lucia and Schwarz (2002) dx 1 () = α 1 X 1 ()d + σ 1 db 1 () dx 2 () = µ 2 d + σ 2 (ρdb 1 () + ) 1 ρ 2 db 2 () Villaplana (2002) replaces µ 2 by ( µ 2 α 2 X 2 () ), and adds dy() = α 1 Y()d + di(), wih I() a ime-inhomogenous compound Poisson process.

16 Geomeric models Some more special cases dx() = α()x()d + σ()db() dy() = α()y()d + di(). d ln S() = d ln Λ() α() ( ln S() ln Λ() ) d + σ()db() + di(). Eberlein Sahl (2003) No mean-reversion (in Y()), I() hyperbolic Lévy Lucia and Schwarz (2002) dx 1 () = α 1 X 1 ()d + σ 1 db 1 () dx 2 () = µ 2 d + σ 2 (ρdb 1 () + ) 1 ρ 2 db 2 () Villaplana (2002) replaces µ 2 by ( µ 2 α 2 X 2 () ), and adds dy() = α 1 Y()d + di(), wih I() a ime-inhomogenous compound Poisson process.

17 Geomeric models Some more special cases dx() = α()x()d + σ()db() dy() = α()y()d + di(). d ln S() = d ln Λ() α() ( ln S() ln Λ() ) d + σ()db() + di(). Carea and Figueroa (2005) I() compound Poisson Lucia and Schwarz (2002) dx 1 () = α 1 X 1 ()d + σ 1 db 1 () dx 2 () = µ 2 d + σ 2 (ρdb 1 () + ) 1 ρ 2 db 2 () Villaplana (2002) replaces µ 2 by ( µ 2 α 2 X 2 () ), and adds dy() = α 1 Y()d + di(), wih I() a ime-inhomogenous compound Poisson process.

18 Geomeric models Some more special cases dx() = α()x()d + σ()db() dy() = α()y()d + di(). d ln S() = d ln Λ() α() ( ln S() ln Λ() ) d + σ()db() + di(). Geman and Roncoroni (2006) di() = h(s())dj(), J() ime-inhomogenous compound Poisson. Lucia and Schwarz (2002) dx 1 () = α 1 X 1 ()d + σ 1 db 1 () dx 2 () = µ 2 d + σ 2 (ρdb 1 () + ) 1 ρ 2 db 2 () Villaplana (2002) replaces µ 2 by ( µ 2 α 2 X 2 () ), and adds dy() = α 1 Y()d + di(), wih I() a ime-inhomogenous compound Poisson process.

19 Geomeric models Some more special cases dx() = α()x()d + σ()db() dy() = α()y()d + di(). d ln S() = d ln Λ() α() ( ln S() ln Λ() ) d + σ()db() + di(). Geman and Roncoroni (2006) di() = h(s())dj(), J() ime-inhomogenous compound Poisson. Lucia and Schwarz (2002) dx 1 () = α 1 X 1 ()d + σ 1 db 1 () dx 2 () = µ 2 d + σ 2 (ρdb 1 () + ) 1 ρ 2 db 2 () Villaplana (2002) replaces µ 2 by ( µ 2 α 2 X 2 () ), and adds dy() = α 1 Y()d + di(), wih I() a ime-inhomogenous compound Poisson process.

20 Arihmeic models Here S() = Λ() + m X i () + i=1 n Y j (), where X i () and Y j () are as before. Again, inegrabiliy condiions will apply. Negaive values If Y j () = 0, j = 1,..., n, hen S() is a Gaussian OU process (mixure) and can become negaive. In fac, ( P[S() < 0] = Φ m() ), Σ() where m() = Λ() + X i (0)e R 0 α i(s)ds i j=1 Σ 2 () = k 0 σ 2 ik(s)e 2 R s α i(u)du ds.

21 Arihmeic models The nonnegaive model of Benh, Kallsen and Meyer-Brandis (2007) m = 0. The pure jump processes I j () are increasing. The mean-revering levels δ j () = 0. Λ() is now inerpreed as a seasonal floor for S(). The mean level of spo prices is given by Λ m () = Λ() + Y 1 (0)e R 0 β 1(v)dv + j + j where γ j () = γ j () zl j(dz, du). 0 0 zη j (u)e R u β 1(v)dv l j (dz, du), 0 η j (u)e R u β 1(v)dv d γ j (u)

22 Arihmeic models The nonnegaive model of Benh, Kallsen and Meyer-Brandis (2007) m = 0. The pure jump processes I j () are increasing. The mean-revering levels δ j () = 0. Λ() is now inerpreed as a seasonal floor for S(). The mean level of spo prices is given by Λ m () = Λ() + Y 1 (0)e R 0 β 1(v)dv + j + j where γ j () = γ j () zl j(dz, du). 0 0 zη j (u)e R u β 1(v)dv l j (dz, du), 0 η j (u)e R u β 1(v)dv d γ j (u)

23 The auocorrelaion funcion of muli-facor OU processes Le Z() be a deseasonal addiive OU process: Z() = X i () + Y j () wih consan coefficiens. i j The auocorrelaion funcion a ime wih lag τ is given by ρ(, τ) = Corr[Z(), Z( + τ)]. ρ(, τ) = i ˆω i (, τ)e α iτ + j ω j (, τ)e β jτ, where ˆω i (, τ) = P i k σ ikσ i k α i +α i ( 1 e (α i +α i ) ) Var[Z( + τ)]var[z()] ω j (, τ) = Var[Y j ()] Var[Z( + τ)]var[z()].

24 The auocorrelaion funcion of muli-facor OU processes Le Z() be a deseasonal addiive OU process: Z() = X i () + Y j () wih consan coefficiens. i j The auocorrelaion funcion a ime wih lag τ is given by ρ(, τ) = Corr[Z(), Z( + τ)]. ρ(, τ) = i ˆω i (, τ)e α iτ + j ω j (, τ)e β jτ, where ˆω i (, τ) = P i k σ ikσ i k α i +α i ( 1 e (α i +α i ) ) Var[Z( + τ)]var[z()] ω j (, τ) = Var[Y j ()] Var[Z( + τ)]var[z()].

25 Simulaion of saionary OU processes: a case sudy of wih he arihmeic spo model where Z() = S() = Λ() + Y 1 () + Y 2 (). Λ() = sin 2π 365. Y 1 ( + ) = e β ( Y 1 () + Z() ), 0 N(1) e β1u dl(u) = µ J ln(c 1 i )e β 1 u i, wih u i independen samples from U([0, 1]), c i arrival imes of a Poisson process wih inensiy νβ 1, N(1) he number of jumps up o ime 1, wih = 1, ν = 8.06, β 1 = 0.085, µ J = 7.7. Y 2 () is an inhomogeneous compound Poisson process, wih exponenial jump 0.14 sizes wih mean 180, and inensiy λ() = π( 90) sin 365

26 Simulaion of saionary OU processes: a case sudy of wih he arihmeic spo model

27 Simulaion of saionary OU processes: a case sudy of wih he arihmeic spo model

28 Simulaion of saionary OU processes: a case sudy of wih he arihmeic spo model