Wiener-Hopf factorization and distribution of extrema for ... · Wiener-Hopf factorization...

Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion

Wiener-Hopf factorization and distribution ofextrema for a family of Levy processes

Alexey Kuznetsov

Department of Mathematics and StatisticsYork University

June 20, 2009

Research supported by the Natural Sciences and Engineering Research Council of Canada

W-H factorization and distribution of extrema Alexey Kuznetsov 0/29

1 IntroductionWiener-Hopf factorizationWell-known examples

2 Processes with meromorphic characteristic exponentCompound Poisson processProcess with jumps of infinite variation (analogue of NIGprocess)A β-family of Levy processes

3 Numerical results

4 Conclusion and future work

Wiener-Hopf factorization

Review of Wiener-Hopf factorization

The characteristic exponent Ψ(z) is

E[eizXt

]= exp(−tΨ(z)),

and by the Levy-Khintchine representation

Ψ(z) = −iµz +σ2z2

2−∫R

(eizx − 1− izx

)ν(x)dx

We define extrema processes Mt = sup{Xs : s ≤ t} andNt = inf{Xs : s ≤ t}, introduce an exponential random variableτ = τ(q) with parameter q > 0, which is independent of the processXt, and use the following notation for characteristic functions of Mτ ,Nτ :

φ+q (z) = E

[eizMτ

], φ−q (z) = E

[eizNτ

]W-H factorization and distribution of extrema Alexey Kuznetsov 1/29

Wiener-Hopf factorization

Review of Wiener-Hopf factorization

TheoremRandom variables Mτ and Xτ −Mτ are independent, randomvariables Nτ and Xτ −Mτ have the same distribution, and for z ∈ Rwe have

Ψ(z) + q= E

[eizXτ

[eizMτ

]E[eiz(Xτ−Mτ )

]= φ+

q (z)φ−q (z)

Moreover, random variable Mτ [Nτ ] is infinitely divisible, positive[negative] and has zero drift.

Well-known examples

Outline

3 Numerical results

Well-known examples

WH for Brownian motion with drift

Let Xt = Wt + µt. Then Ψ(z) = z2

2 − iµz and Ψ(z) + q = 0 has twosolutions

z1,2 = i(µ±√µ2 + 2q)

Thus function q(Ψ(z) + q)−1 can be factorized as

Ψ(z) + q=

2 − iµz + q

√µ2 + 2q

iz + µ+√µ2 + 2q

× µ−√µ2 + 2q

iz + µ−√µ2 + 2q

Well-known examples

WH for Brownian motion with drift

The main idea: since the random variable Mτ [Nτ ] is positive[negative], its characteristic function must be analytic and have nozeros in C+ [C−], where

C+ = {z ∈ C : Im(z) > 0}, C− = {z ∈ C : Im(z) < 0}, C± = C± ∪ R.

φ+q (z) =

µ−√µ2 + 2q

iz + µ−√µ2 + 2q

and Mτ is an exponential random variable with parameter√µ2 + 2q − µ.

Well-known examples

Computing density of Mt

Let pt(x) be the density function of Mt and pM (q, x) the densityfunction of Mτ(q). Then

pM (q, x) = E[pτ(q)(x)] = q

∞∫0

e−qtpt(x)dt

and therefore

pt(x) =1

∫q0+iR

pM (q, x)eqtdq

In our example we have

pt(x) =1

∫q0+iR

(√µ2 + 2q − µ

)e−(√µ2+2q−µ)x+qt dq

Well-known examples

Kou model: double exponential jump diffusion model

Xt is a Levy process with jumps defined by

ν(x) = a1e−b1xI{x>0} + a2e

b2xI{x<0}

Then the characteristic exponent is

Ψ(z) =σ2z2

2− iµz − a1

b1 − iz− a2

b2 + iz+a1

Thus equation Ψ(z) + q = 0 is a fourth degree polynomial equation,and we have explicit solutions and exact WH factorization.

Well-known examples

Phase-type distributed jumps

DefinitionThe distribution of the first passage time of the finite state continuoustime Markov chain is called phase-type distribution.

q(x) = p0exLe1 =

N∑k=1

aiebix

where bi are eigenvalues of the Markov generator L. Thus if Xt hasphase-type jumps, its characteristic exponent Ψ(z) is a rationalfunction, and Ψ(z) + q = 0 is reduced to a polynomial equation, andthe Wiener-Hopf factors are given in closed form (in terms of roots ofthis polynomial equation).

Compound Poisson process

Outline

3 Numerical results

Definition

Let Xt be a compound Poisson process defined by

ν(x) =eαx

cosh(x)

The characteristic exponent of Xt is given by

Ψ(z) = −∫R

(eixz − 1

)ν(x)dx =

cos(π2α) − π

cosh(π2 (z − iα)

WH factorization

Theorem

Assume that q > 0. Define

η =2π

arccos

q + π sec(π2α))

p0 =Γ(

14 (1− α)

14 (3− α)

14 (η − α)

14 (4− η − α)

)Then for Im(z) > i(α− η) we have

φ+q (z) = p0

14 (η − α− iz)

14 (4− η − α− iz)

14 (1− α− iz)

14 (3− α− iz)

)We have P(Mτ = 0) = p0 and the density of Mτ is given by explicitformula in terms of 2F1(a, b; c; z) – the Gauss hypergeometricfunction.

Uniquenes of Wiener-Hopf factorization

Assume we have two functions f+(z) and f−(z), such that f±(0) = 1,f±(z) is analytic in C±, continuous and has no roots in C± andz−1 ln(f±(z))→ 0 as z →∞, z ∈ C±. If

Ψ(z) + q= f+(z)f−(z), z ∈ R

then f±(z) = φ±q (z).

Process with jumps of infinite variation

Outline

3 Numerical results

Definition

Let Xt be defined by a triple (µ, σ, ν), where the jump measure ν(x)is given by

ν(x) =eαx[

sinh(x2 )]2

with | α |< 1.

Proposition

The characteristic exponent of Xt is given by

Ψ(z) =σ2z2

2+ iρz + 4π(z − iα) coth (π(z − iα))− 4γ,

γ = πα cot (πα) , ρ = 4π2α+4γ(γ − 1)

α− µ

Theorem on zeros of Ψ(z) + q

TheoremAssume that q > 0.(i) Equation Ψ(iζ) + q = 0 has infinitely many solutions, all of which

are real and simple. They are located as follows:

ζ−0 ∈ (α− 1, 0)ζ+0 ∈ (0, α+ 1)ζn ∈ (n+ α, n+ α+ 1), n ≥ 1ζn ∈ (n+ α− 1, n+ α), n ≤ −1

(ii) If σ 6= 0 we have as n→ ±∞

ζn = (n+ α) +8σ2

(n+ α)−1

− 8σ2

(2ρσ2

)(n+ α)−2 +O(n−3)

Theorem

(iii) If σ = 0 we have as n→ ±∞

ζn = (n+ α+ ω0) + c0(n+ α+ ω0)−1

− c0ρ

(4γ − q − 4π2c0

)(n+ α+ ω0)−2 +O(n−3),

c0 = −4(4γ − q + αρ)16π2 + ρ2

, ω0 =1π

arcctg( ρ

)− 1.

(iv) Function q(Ψ(z) + q)−1 can be factorized as follows

Ψ(z) + q=

1(1 + iz

)(1 + iz

ζ−0

) ∏|n|≥1

1 + izn+α

1 + izζn

4π(ζ − α) cot(π(ζ − α))− (ρ+ µ)ζ − 4γ =σ2ζ2

2− µζ − q

LemmaAssume that α and β are not equal to a negative integer, andbn = O(n−ε1) for some ε1 > 0 as n→∞. Then

∏n≥0

1 + zn+α

1 + zn+β+bn

≈ Czβ−α,

as z →∞, | arg(z) |< π − ε2 < π, where C = Γ(α)Γ(β)

∏n≥0

(1 + bn

Main Results

TheoremFor q > 0

φ−q (z) =1

1 + izζ+0

∏n≥1

1 + izn+α

1 + izζn

, φ+q (z) =

11 + iz

ζ−0

∏n≤−1

1 + izn+α

1 + izζn

Infinite products converge uniformly on compact subsets of C \ iR.The density of Mτ is given by

pM (q, x) =d

dxP(Mτ < x) = −c−0 ζ

−0 e

ζ−0 x −∑k≥1

c−k ζ−keζ−kx

c−0 =∏n≤−1

1− ζ−kn+α

1− ζ−kζn

, c−k =1− ζ−k

−k+α

1− ζ−kζ−0

∏n≤−1, n 6=−k

1− ζ−kn+α

1− ζ−kζn

A β-family of Levy processes

Outline

3 Numerical results

Definition of the β-family

DefinitionWe define a β-family of Levy processes by the generating triple(µ, σ, ν), where the jump density is

ν(x) = c1e−α1β1x

(1− e−β1x)λ1I{x>0} + c2

eα2β2x

(1− eβ2x)λ2I{x<0}

and parameters satisfy αi > 0, βi > 0, ci ≥ 0 and λi ∈ (0, 3).

Levy processes similar to β-family

The generalized tempered stable family

ν(x) = c+

e−α+x

xλ+I{x>0} + c−

eα−x

|x|λ−I{x<0}.

can be obtained as the limit as β → 0+ if we let

c1 = c+βλ+ , c2 = c−β

λ− , α1 = α+β−1, α2 = α−β

−1, β1 = β2 = β

Particular cases:λ1 = λ2 −→ tempered stable, or KoBoL processesc1 = c2, λ1 = λ2 and β1 = β2 −→ CGMY processesci = 4, βi = 1/2, λi = 2, α1 = 1− α and α2 = 1 + α −→ theprocess discussed in previous section

Computing characteristic exponent

Proposition

If λi ∈ (0, 3) \ {1, 2} then

Ψ(z) =σ2z2

2+ iρz

− c1β1

B(α1 −

β1; 1− λ1

)− c2β2

B(α2 +

β2; 1− λ2

where constants γ and ρ are expressed in terms of beta and digammafunctions.

TheoremAssume that q > 0.(i) Equation Ψ(iζ) + q = 0 has infinitely many solutions, all of which

are real and simple. They are located as follows:

ζ−0 ∈ (−β1α1, 0)ζ+0 ∈ (0, β2α2)ζn ∈ (β2(α2 + n− 1), β2(α2 + n)), n ≥ 1ζn ∈ (β1(−α1 + n), β1(−α1 + n+ 1)), n ≤ −1

(ii) If σ 6= 0 we have as n→ +∞

ζn = β2(n+ α2) +2c2

σ2β22Γ(λ2)

(n+ α2)λ2−3 +O(nλ2−3−ε)

Theorem

(iii) If σ = 0 we have as n→ +∞

ζn = β2(n+ α2 + ω0) +A(n+ α2 + ω0)λ +O(nλ−ε)

(iv) Function q(Ψ(z) + q)−1 can be factorized as follows

Ψ(z) + q=

1(1 + iz

)(1 + iz

ζ−0

∏n≥1

1 + izβ2(n+α2)

1 + izζn

∏n≤−1

1 + izβ1(n−α1)

1 + izζn

The proof of (ii) and (iii) is based on the following two asymptoticformulas as ζ → +∞

B(α+ ζ; γ) = Γ(γ)ζ−γ[1− γ(2α+ γ − 1)

2ζ+O(ζ−2)

]B(α− ζ; γ) = Γ(γ)

sin(π(ζ − α− γ))sin(π(ζ − α))

ζ−γ[1 +

γ(2α+ γ − 1)2ζ

+O(ζ−2)]

If σ 6= 0 and ζ → +∞ we use the above formulas and rewrite equationΨ(iζ) + q = 0 as

sin(π(ζβ2− α2 + λ2

))sin(π(ζβ2− α2

)) =σ2βλ2

2c2Γ(1− λ2)ζ3−λ2 +O(ζ1−λ2) +O(ζλ1−λ2)

Main results

TheoremFor q > 0

φ−q (z) =1

1 + izζ+0

∏n≥1

1 + izβ2(n+α2)

1 + izζn

, φ+q (z) =

11 + iz

ζ−0

∏n≤−1

1 + izβ1(n−α1)

1 + izζn

Infinite products converge uniformly on compact subsets of C \ iR.The density of Mτ is given by

pM (q, x) =d

dxP(Mτ < x) = −c−0 ζ

−0 e

ζ−0 x −∑k≥1

c−k ζ−keζ−kx

c−0 =∏n≤−1

1− ζ−kβ1(n−α1)

1− ζ−kζn

, c−k =1− ζ−k

β1(−k−α1)

1− ζ−kζ−0

∏n≤−1, n 6=−k

1− ζ−kβ1(n−α1)

1− ζ−kζn

Outline

3 Numerical results

Computing density of Mτ

(i) compute ζ±0 , ζn - Newton’s method for large n. For small n –localization, bisection and then Newton’s method.

(ii) compute coefficients c−k – use asymptotic expansion of ζn toaccelerate convergence of products.

(iii) series for density of Mτ is exponentially converging

Computing density of Mt

As before, pt(x) can be recovered as the inverse Laplace transform ofpM (q, x)/q. Thus

pt(x) =1

∫q0+iR

eqtpM (q, x)dq

q=eq0t

∞∫0

Re[pM (q0 + iu, x)

q0 + iu

]cos(tu)du

Question: How do we solve Ψ(z) + q = 0 for q ∈ C ?

Solving Ψ(z) + q = 0 for q ∈ C: ODE method

We need to compute the solutions of equation Ψ(iζ) + q = 0 for allq ∈ [q0, q0 + iu0]. First we compute the initial values: the roots ζ±0 , ζnfor real value of q = q0 using the method discussed above. Next weconsider each root as an implicit function of u: ζn(u) is defined as

Ψ(iζn(u)) + (q0 + iu) = 0, ζn(0) = ζn

Using implicit differentiation we obtain a first order differentialequation

dζn(u)du

= − 1Ψ′(iζn(u))

Solving ODE: Runge-Kutta & Newton’s method

Results: surface plot of pM(q, x) and pt(x)

Outline

3 Numerical results

Conclusion and future work

Wiener-Hopf factors can be identified explicitly for certain Levyprocesses, if Ψ(z) is meromorphic and if we know its asymptoticsas z →∞

We have a lot of information about solutions of Ψ(z) + q = 0(localization, asymptotics)Infinite products can be computed efficiently using convergenceacceleration techniquesTo find solutions of Ψ(z) + q = 0 for complex values of q we canuse ODE/Newton’s methodsβ-family of Levy processes is very general, many existing modelscan be obtained as a limitFuture work: applying these processes to price variousderivatives in Mathematical Finance

Wiener-Hopf factors can be identified explicitly for certain Levyprocesses, if Ψ(z) is meromorphic and if we know its asymptoticsas z →∞We have a lot of information about solutions of Ψ(z) + q = 0(localization, asymptotics)

Infinite products can be computed efficiently using convergenceacceleration techniquesTo find solutions of Ψ(z) + q = 0 for complex values of q we canuse ODE/Newton’s methodsβ-family of Levy processes is very general, many existing modelscan be obtained as a limitFuture work: applying these processes to price variousderivatives in Mathematical Finance

Wiener-Hopf factors can be identified explicitly for certain Levyprocesses, if Ψ(z) is meromorphic and if we know its asymptoticsas z →∞We have a lot of information about solutions of Ψ(z) + q = 0(localization, asymptotics)Infinite products can be computed efficiently using convergenceacceleration techniques

To find solutions of Ψ(z) + q = 0 for complex values of q we canuse ODE/Newton’s methodsβ-family of Levy processes is very general, many existing modelscan be obtained as a limitFuture work: applying these processes to price variousderivatives in Mathematical Finance

Wiener-Hopf factors can be identified explicitly for certain Levyprocesses, if Ψ(z) is meromorphic and if we know its asymptoticsas z →∞We have a lot of information about solutions of Ψ(z) + q = 0(localization, asymptotics)Infinite products can be computed efficiently using convergenceacceleration techniquesTo find solutions of Ψ(z) + q = 0 for complex values of q we canuse ODE/Newton’s methods

β-family of Levy processes is very general, many existing modelscan be obtained as a limitFuture work: applying these processes to price variousderivatives in Mathematical Finance

Wiener-Hopf factors can be identified explicitly for certain Levyprocesses, if Ψ(z) is meromorphic and if we know its asymptoticsas z →∞We have a lot of information about solutions of Ψ(z) + q = 0(localization, asymptotics)Infinite products can be computed efficiently using convergenceacceleration techniquesTo find solutions of Ψ(z) + q = 0 for complex values of q we canuse ODE/Newton’s methodsβ-family of Levy processes is very general, many existing modelscan be obtained as a limit

Future work: applying these processes to price variousderivatives in Mathematical Finance

Wiener-Hopf factors can be identified explicitly for certain Levyprocesses, if Ψ(z) is meromorphic and if we know its asymptoticsas z →∞We have a lot of information about solutions of Ψ(z) + q = 0(localization, asymptotics)Infinite products can be computed efficiently using convergenceacceleration techniquesTo find solutions of Ψ(z) + q = 0 for complex values of q we canuse ODE/Newton’s methodsβ-family of Levy processes is very general, many existing modelscan be obtained as a limitFuture work: applying these processes to price variousderivatives in Mathematical Finance

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