Homogenizations of L evy operators with asymmetric densities - - - PDF document

Homogenizations of L´ evy

• perators with asymmetric

densities

• application to the stochastic

volatility model -

Mariko Arisawa Wilfgang Pauli Institut, University of Wien E-mail: mariko.arisawa@univie.ac.at

• I. Introduction
• 1. Black-Scholes models :

Risky asset (constant volatility) dXt = Xt(µdt + σdWt) Riskless bond db(t) = rb(t)dt Final value of the European option V (x, T) = (x − K)+ B-S equation ∂V ∂t (x, t) = 1 2σ2x2∂2V ∂x2 (x, t)+rx∂V ∂x (x, t)−rV (x, t) x ∈ [0, ∞), t ≥ 0, V (x, T) = (x − K)+ x ∈ [0, ∞).

Observations:

• Analytical solution exists V (x, t):

= CBS(x, t) = xN(d1) − Ke−r(T−t)N(d2) where N(z) =

1 √ 2π

z

−∞ e−y2

2 dy.

• Statistical study of the stock price indi-

cates the jumps.

• The constant volatility is not used in the

market.

• Implied volatility I:

CBS(t, x; K, T; ∃!I) = Cobs, and I(K) shows the smile curve.

• 2. Stochastic volatility models :

Risky asset dXt = Xt(µdt + σ(t)dWt), where for f > 0 a function, σ(t) = f(Yt), Volatiliy process dYt = α(m − Yt)dt + d ˆ Zt, ˆ Zt : Brownian motion correlated with Wt; α : rate of mean reversion; m : long-run average of Yt. Yt’s characteristics (cf. J.P. Fouque, G. Pa- panicolaou, K.R. Sircar)

• Yt must be ergodic.
• Yt can be a jump process.
• As α = 1

ε → +∞, Yt oscillates very rapidly.

Examples (f(y) = √y, ey, |y|...)

• Ornstein-Uhlenbeck process

dYt = α(m − Yt)dt + β(ρdWt +

• 1 − ρ2dZt)
• Log-normal process

dYt = c1Ytdt + c2Yt(ρdWt +

• 1 − ρ2dZt)
• Cox-Ingersoll-Ross (CIR) process

dYt = κ(m−Yt)dt+ν

• Yt(ρdWt+
• 1 − ρ2dZt)
• Well-posedness (P.-L. Lions and M. Musiela,

Ann.I.H.P. 2007) : at least the correlation ρ ∈ [−1, −1

√ 2] is necessary.

3. PIDE for stochastic volatility models. (i) Rescaled stochastic volatility model (PDE case). (Fouque,Papanicolaou, Sircar) dXt = rXtdt + f(Yt)XtdWt, dYt = [1 ε(m − Yt) − ν √ 2 √ε Λ(Yt)]dt + ν √ 2 √ε d ˆ Zt, where Λ(y) = ρ µ−r

f(y) + γ(y)

• 1 − ρ2,

γ : market price of volatility risk. ⇓ P ε(t, Xt, Yt) : Price of the European option : ∂P ∂t + 1 2f(y)2x2∂2P ∂x2 + ρν √ 2 √ε xf(y) ∂2P ∂x∂y+ ν2 ε ∂2P ∂y2 +r(x∂P ∂x −P)+[1 ε(m−y)−ν √ 2 √ε Λ(y)]∂P ∂y = 0, P(T, x, y) = h(y).

1 ε = α: Rate of mean reversion. What is the

limit as α → ∞ ?, and the asymptotics ?

(ii) Rescaled stochastic volatility model with jumps. dXε

t = rXε t dt + f(Y ε t , ξε t )Xε t dWt,

dY ε

t = [1

ε(m − Y ε

t ) − ν

√ 2 √ε Λ(Y ε

t , ξε t )]dt + ν

√ 2 √ε d ˆ Zt, ξε

t = ξt

ε.

⇓ P ε(t, Xε

t , Y ε t , ξε t ) : Price of the European option

∂P ε ∂t + 1 2f(y, ξ)2x2∂2P ε ∂x2 + r(x∂P ε ∂x − P ε) + 1 √ε[ √ 2ρνxf(y, ξ)∂2P ε ∂x∂y − √ 2Λ(y, ξ)∂P ε ∂y ] +1 ε[ν2∂2P ε ∂y2 + (m − y)∂P ε ∂y +

• P ε(·, ξ + η) − P ε(·, ξ) − η, ∇P εdq(η)] = 0,

P(T, x, y) = h(y). Change of variables : For dq(η) =

1 |η|1+µ,

y = z √ε, ξ = η ε

1 µ

Q(t, x, z, η) = P(t, x, y, ξ) ⇓ ∂Q ∂t + 1 2f( z √ε)2x2∂2Q ∂x2 + ρν √ 2xf( z √ε) ∂2Q ∂x∂z +ν2∂2Q ∂z2 +r(x∂Q ∂x −Q)+[ 1 √ε(m− z √ε−ν √ 2Λ( z √ε)]∂Q ∂z +

• Q(·, η + η′) − Q(·, η) − η′, ∇Qdq(η′) = 0.

⇓ Q = Q0 + √εQ1 + εQ2 + ε

3 2Q3 + ...

+ε

1 µR1 + ε 2 µR2 + ε 3 µR3 + ...

(iii) Examples of 1D Levy processes L´ evy operator : −

• RN[u(x+z)−u(x)−z, ∇u(x)]q(dz)

x ∈ RN. (1) Compound poisson process with one point L´ evy measure . q(dz) = cδa(dz), a ∈ R, where c > 0: frequency of the jump, a: the jump length. (2) Compound poisson process with discrete L´ evy measure. q(dz) = cΣd

j=1pjδaj(dz)

where pj ≥ 0, Σd

j=1pj = 1, c > 0: frequency of

the jump, ai: jump lengths.

(3) Compound poisson process with Gaussian distribution . q(dz) = c 1 √ 2πv exp(−|z − m|2 2v )dz where c > 0: frequency of the jump, jump dis- tribution is the normal distribution. (4) Variance gamma process (σ2 = 0, b, dq(z)) q(dz) = c(Iz<0e−c1|z| + Iz>0e−c2|z|) 1 |z|dz, where c, c1, c2 > 0 are constants. (5) Stable process (σ2 = 0, b, dq(z)) q(dz) = c1 1 |z|1+α(dz) z < 0, = c2 1 |z|1+α(dz) z > 0 where c1, c2 ≥ 0 at least one ci = 0, 0 < α < 2.

(6) CGMY model (σ2 = 0, b, q(dz)) q(dz) = c(Iz<0e−G|z| + Iz>0e−M|z|)|z|−(1+Y )dz where c > 0, G ≥ 0, M ≥ 0, Y < 2, such that the following are assumed: (1) Y <0 => G > 0, M > 0 (2) Y = 0 => VG process (3) G = M = 0, 0 < Y < 2 => symmetric stable process

• II. PIDE Analysis

Purposes :

• Determination of the ergodic jump process.
• Homogenization technique to the asypm-

totic analysis of α → ∞ (mean reverting rate). Methods :

• Viscosity solutions.
• Regularity of solutions.
• Ergodoc (cell) problem.
• Homogenizations.

• 1. Viscosity solutions

(a) Let u(x) ∈ LSC(Ω). Sub-differential of u at x ∈ Ω : (p, X) ∈ J2,+

Ω

u(x) s.t. ∀δ > 0, ∃ε > 0, u(x+z)<u(x)+p, z+ 1 2Xz, z+δ|z|2 ∀|z|<ε. (b) Let u(x) ∈ USC(Ω). Super-differential of u at x ∈ Ω : (p, X) ∈ J2,−

Ω

u(x) s.t. ∀δ > 0, ∃ε > 0, u(x+z) ≥ u(x)+p, z+1 2Xz, z+δ|z|2 ∀|z|<ε PDE problem F(x, u, ∇u, ∇2u) = 0 in Ω. (∗)

• F(x, u, ∇u, ∇2u) = −Tr(∇2u) = −∆u.
• F(x, u, ∇u, ∇2u) = supα∈A{− N

i,j=1 aij ∂2u ∂xi∂xj−

N

i=1 bi ∂u ∂xi − f(x, α)}.

Definition. (i) u(x) ∈ LSC(Ω) is a viscosity subsolution of (*) iff F(x, u, p, X)<0 ∀(p, X) ∈ J2,+

Ω

u(x). (ii) u(x) ∈ USC(Ω) is a viscosity supersolution

• f (*) iff

F(x, u, p, X) ≥ 0 ∀(p, X) ∈ J2,−

Ω

u(x). Definition. (i) u(x) ∈ LSC(Ω) is a viscosity subsolution of (*) iff for any φ ∈ C2(Ω) s.t. u − φ takes a loc.

• max. at x0 ∈ Ω and u(x0) = φ(x0),

F(x0, u(x0), ∇φ(x0), ∇2φ(x0))<0. (ii) u(x) ∈ USC(Ω) is a viscosity supersolution

• f (*) iff for any φ ∈ C2(Ω) s.t. u − φ takes a
• loc. min. at x0 ∈ Ω and u(x0) = φ(x0),

F(x0, u(x0), ∇φ(x0), ∇2φ(x0)) ≥ 0. Remark Two definitions are equivalent.

PIDE problem F(x, u, ∇u, ∇2u) −

• RN[u(x + z) − u(x)

(∗∗) −z, ∇u]dq(z) = 0 in

RN.

Definition A. (i) u(x) ∈ LSC(Ω) : visc. sub- solution of (**) <=> ∀x ∈ RN, ∀(p, X) ∈ J2,+

Ω

u(x), ∀(ε, δ) (satisfying (a)): F(x, u(x), p, X) −

• |z|<ε

1 2(X + 2δI)z, zdq(z) −

• |z|>ε[u(x + z) − u(x) − z, p]dq(z)<0

(ii) u(x) ∈ USC(Ω) : visc. supersolution of (**) <=> ∀x ∈ RN, ∀(p, X) ∈ J2,−

Ω

u(x), ∀(ε, δ) (satisfying (b)): F(x, u(x), p, X) −

• |z|<ε

1 2(X − 2δI)z, zdq(z) −

• |z|>ε[u(x + z) − u(x) − z, p]dq(z) ≥ 0

Definition B. (i) u(x) ∈ LSC(Ω) : visc. subsolution of (**) <=> ∀φ ∈ C2(RN), ∀x ∈ RN s.t. global maximum of u − φ : F(x, u(x), φ(x), φ2(x)) −

• RN[φ(x + z) − φ(x)

−z, ∇φ(x)]dq(z)<0. (ii) u(x) ∈ USC(Ω) : visc. supersolution of (**) <=> ∀φ ∈ C2(RN), ∀x ∈ RN s.t. global minimum of u − φ : F(x, u(x), φ(x), φ2(x)) −

• RN[φ(x + z) − φ(x)

−z, ∇φ(x)]dq(z) ≥ 0. Theorem. (M.A. (2008)) Definitions A and B are equivalent.

• Remark. Gneral comparison and existence re-

sults are studied. For example, Dirichlet problem F(x, u, ∇u, ∇2u) −

• RN[u(x + z) − u(x)−

−z, ∇u]dq(z) = 0 in Ω, u(x) = Ψ(x) x ∈ Ωc, Neumann problem F(x, u, ∇u, ∇2u) −

• x+z∈Ω[u(x + z) − u(x)−

−z, ∇u]dq(z) = 0 in Ω, ∇u(x), n(x) = 0 x ∈ ∂Ω,

• References. Problems in RN : O. Alvarez and
• A. Tourin; G. Barles, R. Buckdahn, E. Par-

doux; E. Jacobsen and K. Karlsen, etc. Prob- lems in Ω : M. Arisawa; G. Barles and C. Im- bert, etc. Neumann problem : J. Menardi, and

• M. Garroni; M. Arisawa, etc.

There are many open problems, for example the comparison of solutions of F(x, u, ∇u, ∇2u) −

• RN[u(x + β(x, z)) − u(x)−

−β(x, z), ∇u]q(z)dz = 0 in Ω, u(x) = Ψ(x) x ∈ Ωc, where q(z) = 1 |z|N+γ γ ∈ (1, 2), which corresponds to the case of the jump size β(x, z) depends on x ∈ Ω.

• 2. Ergodic problem for PIDE

(P) Find a unique number dg such that the following has a viscosity solution u : dg + F(x, ∇u, ∇2u) −

• RN[u(x + z)

−u(x) − z, ∇u]q(z)dz} − g(x) = 0 in

TN.

Meaning : Finite horizon problem ∂u ∂t (x, t) + F(x, ∇u, ∇2u) −

• RN[u(x + z, t)

−u(x, t) − z, ∇u]q(z)dz − g(x) = 0 in

TN

Infinite horizon problem λuλ(x) + F(x, ∇uλ, ∇2uλ) −

• RN[uλ(x + z)

−uλ(x) − z, ∇uλ]q(z)dz − g(x) = 0 in

TN

⇓ dg = lim

T→∞

u(x, T) T = lim

λ→0 λuλ(x)

∀x ∈ TN.

∃µ : ∀g ∈ C(TN) → dg =

• g(x)dµ.

”Long time average”=”Average by an invari- ant measure µ”. Assumptions : Suppose that either one of

• F is uniformly elliptic.
• F = H(x, ∇u), ∃K ∈ B(r) ⊂ RN s.t.

q(z) ≥ 1 |z|N+α α ∈ (1, 2) ∀z ∈ K.

• Example. N = 1, α ∈ (1, 2),

q(z) = 0 z < 0; = 1 |z|1+α z > 0.

Theorem. (M.A.(2007, World Scientics) ) Assume the above. There exists a unique num- ber dg such that for any e > 0 there exists a periodic viscosity solution u of dg + F(x, ∇u, ∇2u) −

• Rn u(x + z)

−u(x) − ∇u(x), zdq(z) − g ≥ −ε x ∈ TN, dg + F(x, ∇u, ∇2u) −

• Rn u(x + z)

−u(x) − ∇u(x), zdq(z) − g<ε x ∈ TN.

• Remarks. 1. The above ε > 0 can be 0.

2. The above L´ evy process is ergodic, and can be the mean-reverting stochastic volatility model.

• 3. We use this (even if ε > 0, called approxi-

mated cell problem) to solve the homogeniza- tion problems of the PIDE.

• 3. H¨
• lder continuity of u :

H(x, ∇u) −

• Rn[u(x + z) − u(x)

−∇u(x), z] 1 |z|N+γ − g = 0 x ∈ TN. Assupmtions

• RN min(1, |z|2)q(z)dz<C
• |g(x) − g(y)|<M|x − y|θ0

∀x, y ∈ RN

• |H(x, p) − H(y, p)|<w(|x − y|)|p|q
• supx∈RN |u(x)|<M

• Theorem. (M.A.)

Assume the above, where where there exist constants L > 0, ρ1 > 0 such that lim

s↓0 w(s)s−ρ1<L,

ρ1 + γ > q. Then, if γ > 1, for any θ ∈ (0, min{1, θ0 + γ}), there exists a constant Cθ > 0 such that |u(x) − u(y)|<Cθ|x − y|θ ∀x, y ∈ RN, where Cθ depends only on M > 0 and Ci (i = 1, 2). Theorem 3.2. Proof Idea (N = 1) :

• Utilize Definition A to localize the singular

integral, i.e. by (ε, δ) explicite, s.t. u(x+z)<u(x)+p, z+1 2Xz, z+δ|z|2 ∀|z|<ε,

extract the effect of the singularity in −

• |z|<ε(X + 2δ)|z|2

1 |z|1+γdz.

• A comparison argument used by Lions-Ishii

for 2nd order elliptic case. Fix r > 0, and put Cθ so that Cθrθ = 2M. Then, if ∃ˆ x, ˆ y |u(ˆ x) − u(ˆ y)| > Cθ|ˆ x − ˆ y|θ, then |ˆ x − ˆ y| < r. Put Φ(x, y) = u(x) − u(y) − Cθ|x − y|θ, and let (ˆ x, ˆ y) be its maximum. Re- mark that ˆ x = ˆ

• y. For φ(x, y) = Cθ|x − y|θ, put

p = ∇xφ(ˆ x, ˆ y) = Cθθ|ˆ x − ˆ y|θ−2(ˆ x − ˆ y), Q = ∇xxφ(ˆ x, ˆ y) = Cθθ(θ − 1)|ˆ x − ˆ y|θ−2. ⇓ u(ˆ x+z)−u(ˆ x)<pz+1 2(Q+2δ)|z|2 for |z| < ε,

u(ˆ y+z)−u(ˆ y) ≥ pz−1 2(Q+2δ)|z|2 for |z| < ε, where δ = Cθθ(1 − θ) 4 |ˆ x − ˆ y|θ−2, ε = c|ˆ x − ˆ y|. From the definition of viscosity solutions, H(ˆ x, p) −

• |z|<ε

1 2(Q + 2δ)z2q(z)dz −

• |z|≥ε[u(ˆ

x + z) − u(ˆ x) − pz]q(z)dz<g(ˆ x), H(ˆ y, p) −

• |z|<ε

1 2 − (Q + 2δ)z2q(z)dz −

• |z|≥ε[u(ˆ

y + z) − u(ˆ y) − pz]q(z)dz<g(ˆ y). ⇓ −

• |z|<ε(Q + 2δ)z2q(z)dz

< − H(ˆ x, p) + H(ˆ y, p) + g(ˆ x) − g(ˆ y).

⇓ CCθ|ˆ x − ˆ y|θ−γ<|ˆ x − ˆ y|ρ1−q + M|ˆ x − ˆ y|θ0. Contradiction for r > 0 small enough (|ˆ x − ˆ y| < r).

• 4. Homogenization of PIDE

Problem ∂uε ∂t + H(x, ∇uε) − c(x ε)

• RN[uε(x + z)

−uε(x) − 1|z|<1 ∇uε(x), z]q(z)dz = 0 x ∈ Ω, uε(x, t) = φ(x) x ∈ Ωc t > 0, uε(x, 0) = u0 x ∈ Ω.

• Is there a unique limit :

∃limε↓0uε(x) = u(x) ?

• Characterize u by an effective PIDE:

∂u ∂t + I(x, u, ∇u, I[u]) = 0 t > 0.

Formal asymptotic expansion

• 1. Symmetric α-stable process

(1 < α < 2) ∂uε ∂t + |∇uε| − c(x ε)

• RN[uε(x + z) − uε(x)

−1|z|<1 ∇uε(x), z] 1 |z|N+αdz − g(x ε) = 0 ⇓ uε(x, t) = u(x, t) + εαv(x ε, t) + o(εα) ∇uε(x, t) = ∇xu(x, t) + εα−1∇yv(x ε, t). ⇓ ∂u ∂t + |∇u| − c(x ε)

• RN[u(x + z) − u(x)−

1|z|<1 ∇xu(x), z]

dz |z|N+α − εαc(x ε)

• RN[v(x + z

ε )

−v(x ε) − 1|z|<1

• ∇yv(x

ε), z ε

• ]

dz |z|N+α − g(x ε) = 0 ⇓ ∂u ∂t + |∇u| − c(y)

• RN[u(x + z) − u(x)

−1|z|<1 ∇xu(x), z] 1 |z|N+αdz−c(y)

• RN[v(y+z′)

−v(y)−1|z′|<1

ε

• ∇yv(y), z′

] 1 |z′|N+αdz −g(y) = 0 ⇓ Ergodic problem (Averaging principle) −∃I(x, u, ∇u, I) − c(y)I − c(y)

• RN[v(y + z′)

−v(y)−1|z′|<1

ε

• ∇yv(y), z′

] 1 |z′|N+αdz−g(y) = 0, Effective integro-differential operators

• Uniform sub-ellipticity:

I(x, r, p, I + I′)<I(x, r, p, I) − ∃θI′ ∀I′ > 0

• I(x, r, p, I + I′) ∈ C(Ω × R × RN × R).

⇓ Effective integro-differential equations u is the unique solution of ∂u ∂t + I(x, u, ∇u, I) = 0 t > 0.

• Remark. The formal argument is justified by

the purturbed test fc. method.

• Remark. Asymmetric α-Stable process

q(z)dz = 1z<0 c1 |z|1+αdz + 1z>0 c2 |z|1+αdz can be treated similarly.

• 2. Asymmetric singularity

(N = 1, 0 < α1 < α2 < 2) q(z)dz = 1z<0 c1 |z|1+α1dz + 1z<0 c2 |z|1+α2dz, i.e. ∂uε ∂t + |∇uε| − c(x ε)

−∞[uε(x + z) − uε(x)

−1|z|<1 ∇uε(x), z] c1 |z|1+α1dz −c(x ε)

∞

0 [uε(x + z) − uε(x)

−1|z|<1 ∇uε(x), z] c2 |z|1+α2dz − g(x ε) = 0 ⇓ Stronger singularity dominates: uε(x, t) = u(x, t) + εα2v(x ε, t) + o(εα2) ⇓ Ergodic cell problem with the singular mea- sure supported on a cornea. Summary

• At the mean-reverting rate α → ∞, the

stochastic volatility model converges to the constant volatility BS equation. For the

jump-diffusion stochastic volatility, the dif- fusion term determines the corrector. Thus, the pure jump model had to be considered, to see the effect of the jump.

• Numerical analysis of pure L´

evy operators for the homogenization ? ∂u ∂t −

• RN[u(x + z) − u(x)

−1|z|<1 ∇xu(x), z] 1 |z|N+γdz = 0.

• The comparaison, existence, regularities,

modelization, etc many problems are open in the PIDE.