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

Homogenizations of L´ evy operators 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) dX t = X t ( µdt + σdW t ) Riskless bond db ( t ) = rb ( t ) dt Final value of the European option V ( x, T ) = ( x − K ) + B-S equation 2 σ 2 x 2 ∂ 2 V ∂t ( x, t ) = 1 ∂V ∂x 2 ( 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 ): = C BS ( x, t ) = xN ( d 1 ) − Ke − r ( T − t ) N ( d 2 ) −∞ e − y 2 � z 1 2 dy . where N ( z ) = √ 2 π • Statistical study of the stock price indi- cates the jumps. • The constant volatility is not used in the market. • Implied volatility I : C BS ( t, x ; K, T ; ∃ ! I ) = C obs , and I ( K ) shows the smile curve.

2. Stochastic volatility models : Risky asset dX t = X t ( µdt + σ ( t ) dW t ) , where for f > 0 a function, σ ( t ) = f ( Y t ), Volatiliy process dY t = α ( m − Y t ) dt + d ˆ Z t , ˆ Z t : Brownian motion correlated with W t ; α : rate of mean reversion; m : long-run average of Y t . Y t ’s characteristics (cf. J.P. Fouque, G. Pa- panicolaou, K.R. Sircar) • Y t must be ergodic. • Y t can be a jump process. • As α = 1 ε → + ∞ , Y t oscillates very rapidly.

Examples ( f ( y ) = √ y , e y , | y | ...) • Ornstein-Uhlenbeck process � 1 − ρ 2 dZ t ) dY t = α ( m − Y t ) dt + β ( ρdW t + • Log-normal process � 1 − ρ 2 dZ t ) dY t = c 1 Y t dt + c 2 Y t ( ρdW t + • Cox-Ingersoll-Ross (CIR) process � 1 − ρ 2 dZ t ) � dY t = κ ( m − Y t ) dt + ν Y t ( ρdW t + • 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) dX t = rX t dt + f ( Y t ) X t dW t , √ √ dY t = [1 ε ( m − Y t ) − ν √ ε Λ( Y t )] dt + ν 2 2 √ ε d ˆ Z t , � where Λ( y ) = ρ µ − r 1 − ρ 2 , f ( y ) + γ ( y ) γ : market price of volatility risk. ⇓ P ε ( t, X t , Y t ) : Price of the European option : √ 2 f ( y ) 2 x 2 ∂ 2 P √ ε xf ( y ) ∂ 2 P ∂t + 1 2 ∂P ∂x 2 + ρν ∂x∂y + √ ν 2 ∂ 2 P ∂x − P )+[1 2 ∂y 2 + r ( x∂P ε ( m − y ) − ν √ ε Λ( y )] ∂P ∂y = 0 , ε 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. dX ε t = rX ε t dt + f ( Y ε t , ξ ε t ) X ε t dW t , √ √ t = [1 2 2 t ) − ν t )] dt + ν dY ε ε ( m − Y ε √ ε Λ( Y ε t , ξ ε √ ε d ˆ Z t , ξ ε t = ξ t ε . ⇓ P ε ( t, X ε t , Y ε t , ξ ε t ) : Price of the European option 2 f ( y, ξ ) 2 x 2 ∂ 2 P ε ∂P ε ∂x 2 + r ( x∂P ε ∂t + 1 ∂x − P ε ) 2 ρνxf ( y, ξ ) ∂ 2 P ε 2Λ( y, ξ ) ∂P ε √ √ + 1 √ ε [ ∂x∂y − ∂y ] ε [ ν 2 ∂ 2 P ε ∂y 2 + ( m − y ) ∂P ε +1 ∂y + � P ε ( · , ξ + η ) − P ε ( · , ξ ) − � η, ∇ P ε � dq ( η )] = 0 ,

P ( T, x, y ) = h ( y ) . 1 Change of variables : For dq ( η ) = | η | 1+ µ , y = z ξ = η √ ε, 1 µ ε Q ( t, x, z, η ) = P ( t, x, y, ξ ) ⇓ √ ε ) 2 x 2 ∂ 2 Q √ ε ) ∂ 2 Q √ ∂t + 1 ∂Q 2 f ( z 2 xf ( z ∂x 2 + ρν ∂x∂z + ν 2 ∂ 2 Q √ ∂x − Q )+[ 1 ∂z 2 + r ( x∂Q √ ε ( m − z 2Λ( z √ ε )] ∂Q √ ε − ν ∂z � Q ( · , η + η ′ ) − Q ( · , η ) − � η ′ , ∇ Q � dq ( η ′ ) = 0 . + ⇓ Q = Q 0 + √ εQ 1 + εQ 2 + ε 3 2 Q 3 + ... 1 2 3 + ε µ R 1 + ε µ R 2 + ε µ R 3 + ...

(iii) Examples of 1D Levy processes L´ evy operator : � x ∈ R N . − R N [ u ( x + z ) − u ( x ) −� z, ∇ u ( x ) � ] q ( dz ) (1) Compound poisson process with one point L´ evy measure . q ( dz ) = cδ a ( 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 ) = c Σ d j =1 p j δ a j ( dz ) where p j ≥ 0, Σ d j =1 p j = 1, c > 0: frequency of the jump, a i : jump lengths.

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

(6) CGMY model ( σ 2 = 0, b , q ( dz )) q ( dz ) = c ( I z< 0 e − G | z | + I z> 0 e − 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 ( p, X ) ∈ J 2 , + at x ∈ Ω : u ( x ) s.t. ∀ δ > 0, Ω ∃ ε > 0, u ( x + z ) <u ( x )+ � p, z � + 1 2 � Xz, z � + δ | z | 2 ∀| z | <ε. (b) Let u ( x ) ∈ USC (Ω). Super-differential of ( p, X ) ∈ J 2 , − u at x ∈ Ω : u ( x ) s.t. ∀ δ > 0, Ω ∃ ε > 0, u ( x + z ) ≥ u ( x )+ � p, z � +1 2 � Xz, z � + δ | z | 2 ∀| z | <ε PDE problem F ( x, u, ∇ u, ∇ 2 u ) = 0 in Ω . ( ∗ ) • F ( x, u, ∇ u, ∇ 2 u ) = − Tr ( ∇ 2 u ) = − ∆ u. ∂ 2 u • F ( x, u, ∇ u, ∇ 2 u ) = sup α ∈A {− � N i,j =1 a ij ∂x i ∂x j − i =1 b i ∂u � N ∂x i − f ( x, α ) } .

Definition. (i) u ( x ) ∈ LSC (Ω) is a viscosity subsolution of (*) iff ∀ ( p, X ) ∈ J 2 , + F ( x, u, p, X ) < 0 u ( x ) . Ω (ii) u ( x ) ∈ USC (Ω) is a viscosity supersolution of (*) iff ∀ ( p, X ) ∈ J 2 , − F ( x, u, p, X ) ≥ 0 u ( x ) . Ω Definition. (i) u ( x ) ∈ LSC (Ω) is a viscosity subsolution of (*) iff for any φ ∈ C 2 (Ω) s.t. u − φ takes a loc. max. at x 0 ∈ Ω and u ( x 0 ) = φ ( x 0 ), F ( x 0 , u ( x 0 ) , ∇ φ ( x 0 ) , ∇ 2 φ ( x 0 )) < 0 . (ii) u ( x ) ∈ USC (Ω) is a viscosity supersolution of (*) iff for any φ ∈ C 2 (Ω) s.t. u − φ takes a loc. min. at x 0 ∈ Ω and u ( x 0 ) = φ ( x 0 ), F ( x 0 , u ( x 0 ) , ∇ φ ( x 0 ) , ∇ 2 φ ( x 0 )) ≥ 0 . Remark Two definitions are equivalent.

PIDE problem � F ( x, u, ∇ u, ∇ 2 u ) − R N [ u ( x + z ) − u ( x ) ( ∗∗ ) R N . −� z, ∇ u � ] dq ( z ) = 0 in Definition A. (i) u ( x ) ∈ LSC (Ω) : visc. sub- solution of (**) < = > ∀ x ∈ R N , ∀ ( p, X ) ∈ J 2 , + u ( x ), ∀ ( ε, δ ) (satisfying (a)): Ω 1 � F ( x, u ( x ) , p, X ) − 2 � ( X + 2 δI ) z, z � dq ( z ) | z | <ε � − | z | >ε [ u ( x + z ) − u ( x ) − � z, p � ] dq ( z ) < 0 (ii) u ( x ) ∈ USC (Ω) : visc. supersolution of (**) < = > ∀ x ∈ R N , ∀ ( p, X ) ∈ J 2 , − u ( x ), ∀ ( ε, δ ) Ω (satisfying (b)): 1 � F ( x, u ( x ) , p, X ) − 2 � ( X − 2 δI ) z, z � dq ( z ) | z | <ε � − | z | >ε [ u ( x + z ) − u ( x ) − � z, p � ] dq ( z ) ≥ 0

Definition B. (i) u ( x ) ∈ LSC (Ω) : visc. subsolution of (**) < = > ∀ φ ∈ C 2 ( R N ), ∀ x ∈ R N s.t. global maximum of u − φ : � F ( x, u ( x ) , φ ( x ) , φ 2 ( x )) − R N [ φ ( x + z ) − φ ( x ) −� z, ∇ φ ( x ) � ] dq ( z ) < 0 . (ii) u ( x ) ∈ USC (Ω) : visc. supersolution of (**) < = > ∀ φ ∈ C 2 ( R N ), ∀ x ∈ R N s.t. global minimum of u − φ : � F ( x, u ( x ) , φ ( x ) , φ 2 ( x )) − R N [ 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, ∇ 2 u ) − R N [ u ( x + z ) − u ( x ) − −� z, ∇ u � ] dq ( z ) = 0 in Ω , x ∈ Ω c , u ( x ) = Ψ( x ) Neumann problem � F ( x, u, ∇ u, ∇ 2 u ) − 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 R N : 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, ∇ 2 u ) − R N [ u ( x + β ( x, z )) − u ( x ) − −� β ( x, z ) , ∇ u � ] q ( z ) dz = 0 in Ω , x ∈ Ω c , u ( x ) = Ψ( x ) where 1 q ( z ) = γ ∈ (1 , 2) , | z | N + γ which corresponds to the case of the jump size β ( x, z ) depends on x ∈ Ω.