Some properties of American option prices in exponential Lévy models Damien Lamberton Mohammed Mikou Universit´ e Paris-Est Workshop on Optimization and Optimal Control Linz, October 2008 Some properties of American option prices in exponential L´ evy models – p. 1/30
Outline Some properties of American option prices in exponential L´ evy models – p. 2/30
Outline 1 Optimal stopping of Lévy processes Some properties of American option prices in exponential L´ evy models – p. 2/30
Outline 1 Optimal stopping of Lévy processes 2 The American put price in an exponential Lévy model Some properties of American option prices in exponential L´ evy models – p. 2/30
Outline 1 Optimal stopping of Lévy processes 2 The American put price in an exponential Lévy model 3 The exercise boundary Some properties of American option prices in exponential L´ evy models – p. 2/30
Outline 1 Optimal stopping of Lévy processes 2 The American put price in an exponential Lévy model 3 The exercise boundary 4 The smooth fit property Some properties of American option prices in exponential L´ evy models – p. 2/30
Optimal stopping of Lévy processes Consider a d -dimensional Lévy process X = ( X t ) t ≥ 0 , with characteristic exponent ψ and generating triplet ( A, ν, γ ) , which means � e iz.X t � z ∈ R d , = exp[ tψ ( z )] , E where ψ ( z ) = − 1 � � e iz.x − 1 − iz.x 1 {| x |≤ 1 } � 2 z.Az + iγ.z + ν ( dx ) , the matrix A = ( A ij ) is the covariance matrix of the Brownian part, the measure ν on R d \ { 0 } is the Lévy ( | x | 2 ∧ 1) ν ( dx ) < ∞ , and γ is � measure of X , which satisfies a vector in R d . Some properties of American option prices in exponential L´ evy models – p. 3/30
Given a bounded and continuous function f on R d , we introduce ( t, x ) ∈ [0 , + ∞ ) × R d , u f ( t, x ) = sup E ( f ( x + X τ )) , τ ∈T 0 ,t where T 0 ,t is the set of all stopping times with values in [0 , t ] . We want to characterize u f as the unique solution of a variational inequality. Some properties of American option prices in exponential L´ evy models – p. 4/30
Given a bounded and continuous function f on R d , we introduce ( t, x ) ∈ [0 , + ∞ ) × R d , u f ( t, x ) = sup E ( f ( x + X τ )) , τ ∈T 0 ,t where T 0 ,t is the set of all stopping times with values in [0 , t ] . We want to characterize u f as the unique solution of a variational inequality. Denote by L the infinitesimal generator of X . The operator L can be written as a sum L = A + B , where A is the local (differential) part and B is the non-local (integral) part. Some properties of American option prices in exponential L´ evy models – p. 4/30
For g ∈ C 2 b ( R d ) , we have d d ∂ 2 g A g ( x ) = 1 ∂g � � A i,j ( x ) + γ i ( x ) , 2 ∂x i ∂x j ∂x i i,j =1 i =1 and � � � B g ( x ) = ν ( dy ) g ( x + y ) − g ( x ) − y. ∇ g ( x ) 1 {| y |≤ 1 } , where ∇ g denotes the gradient of g . Some properties of American option prices in exponential L´ evy models – p. 5/30
For g ∈ C 2 b ( R d ) , we have d d ∂ 2 g A g ( x ) = 1 ∂g � � A i,j ( x ) + γ i ( x ) , 2 ∂x i ∂x j ∂x i i,j =1 i =1 and � � � B g ( x ) = ν ( dy ) g ( x + y ) − g ( x ) − y. ∇ g ( x ) 1 {| y |≤ 1 } , where ∇ g denotes the gradient of g . The local part A g can be defined in the sense of distributions if g is a locally integrable function. Some properties of American option prices in exponential L´ evy models – p. 5/30
We will show that B g can be defined in the sense of distributions if g is bounded and Borel measurable. Some properties of American option prices in exponential L´ evy models – p. 6/30
We will show that B g can be defined in the sense of distributions if g is bounded and Borel measurable. If O is an open subset of R d , we denote by D ( O ) the set of all C ∞ functions with compact support in O and by D ′ ( O ) the space of distributions on O . If u ∈ D ′ ( O ) and ϕ ∈ D ( O ) , � u, ϕ � denotes the evaluation on the test function ϕ of the distribution u . Note that if u is a locally integrable function on O , � � u, ϕ � = u ( x ) ϕ ( x ) dx. O Some properties of American option prices in exponential L´ evy models – p. 6/30
We will show that B g can be defined in the sense of distributions if g is bounded and Borel measurable. If O is an open subset of R d , we denote by D ( O ) the set of all C ∞ functions with compact support in O and by D ′ ( O ) the space of distributions on O . If u ∈ D ′ ( O ) and ϕ ∈ D ( O ) , � u, ϕ � denotes the evaluation on the test function ϕ of the distribution u . Note that if u is a locally integrable function on O , � � u, ϕ � = u ( x ) ϕ ( x ) dx. O And the partial derivatives of u are defined by � ∂u � u ( x ) ∂ϕ , ϕ � = − ( x ) dx. ∂x j ∂x j O Some properties of American option prices in exponential L´ evy models – p. 6/30
Introduce the adjoint operator B ∗ of B . For ϕ ∈ C 2 b ( R d ) , let � � B ∗ ϕ ( x ) = x ∈ R d . � ϕ ( x − y ) − ϕ ( x ) + y. ∇ ϕ ( x ) 1 {| y |≤ 1 } ν ( dy ) , Some properties of American option prices in exponential L´ evy models – p. 7/30
Introduce the adjoint operator B ∗ of B . For ϕ ∈ C 2 b ( R d ) , let � � B ∗ ϕ ( x ) = x ∈ R d . � ϕ ( x − y ) − ϕ ( x ) + y. ∇ ϕ ( x ) 1 {| y |≤ 1 } ν ( dy ) , For the next Proposition, we will use the following notations. � � d d ∂ 2 ϕ � � || D 2 ϕ || ∞ = sup � � � � x ∈ R d sup y i y j ( x ) , � � ∂x i ∂x j � � | y |≤ 1 i =1 j =1 � � � y ∈ R d | | y | ≤ 1 � B 1 = . Some properties of American option prices in exponential L´ evy models – p. 7/30
Proposition 1 If ϕ ∈ D ( R d ) , the function B ∗ ϕ is continuous and integrable on R d , and we have ||B ∗ ϕ || L 1 ≤ 1 � 2 || D 2 ϕ || ∞ λ d ( K + B 1 ) | y | 2 ν ( dy ) + 2 || ϕ || L 1 ν ( B c 1 ) , B 1 where K = supp ϕ and λ d is the Lebesgue measure. Moreover, if g ∈ C 2 b ( R d ) , we have � R d g ( x ) B ∗ ϕ ( x ) dx. �B g, ϕ � = For g ∈ L ∞ ( R d ) , the distribution B g can be defined by setting � R d g ( x ) B ∗ ϕ ( x ) dx, ϕ ∈ D ( R d ) . �B g, ϕ � = Some properties of American option prices in exponential L´ evy models – p. 8/30
We can now characterize the value function u f of an optimal stopping problem with reward function f as the solution of a variational inequality. Note that in the following statement ∂ t v + L v is to be understood as a distribution. Theorem 2 Fix T > 0 and let f be a continuous and bounded function on R d . The function v defined by v ( t, x ) = u f ( T − t, x ) is the only continuous and bounded function on [0 , T ] × R d satisfying the following conditions: 1. v ( T, . ) = f , 2. v ≥ f , 3. On (0 , T ) × R d , ∂ t v + L v ≤ 0 , 4. On the open set { ( t, x ) ∈ (0 , T ) × R d | v ( t, x ) > f ( x ) } , ∂ t v + L v = 0 . Some properties of American option prices in exponential L´ evy models – p. 9/30
Proof Continuity of ( t, x ) �→ u f ( t, x ) = sup τ ∈T 0 ,t E ( f ( x + X τ )) . The process ( U t = v ( t, x + X t )) 0 ≤ t ≤ T is the Snell envelope of the process ( Z t = f ( x + X t )) 0 ≤ t ≤ T . Some properties of American option prices in exponential L´ evy models – p. 10/30
Proof Continuity of ( t, x ) �→ u f ( t, x ) = sup τ ∈T 0 ,t E ( f ( x + X τ )) . The process ( U t = v ( t, x + X t )) 0 ≤ t ≤ T is the Snell envelope of the process ( Z t = f ( x + X t )) 0 ≤ t ≤ T . Therefore, ( U t ) 0 ≤ t ≤ T is a supermartingale, and, if τ ∗ = inf { t ∈ [0 , T ] | U t = Z t } , the stopped process ( U t ∧ τ ∗ ) 0 ≤ t ≤ T is a martingale. Some properties of American option prices in exponential L´ evy models – p. 10/30
Proof Continuity of ( t, x ) �→ u f ( t, x ) = sup τ ∈T 0 ,t E ( f ( x + X τ )) . The process ( U t = v ( t, x + X t )) 0 ≤ t ≤ T is the Snell envelope of the process ( Z t = f ( x + X t )) 0 ≤ t ≤ T . Therefore, ( U t ) 0 ≤ t ≤ T is a supermartingale, and, if τ ∗ = inf { t ∈ [0 , T ] | U t = Z t } , the stopped process ( U t ∧ τ ∗ ) 0 ≤ t ≤ T is a martingale. Note that τ ∗ is the exit time from the open set { v > f } for the process ( t, x + X t ) 0 ≤ t ≤ T . Some properties of American option prices in exponential L´ evy models – p. 10/30
Given x ∈ R d and an open subset U of R d , define τ x U = inf { t ≥ 0 | x + X t / ∈ U } . Some properties of American option prices in exponential L´ evy models – p. 11/30
Given x ∈ R d and an open subset U of R d , define τ x U = inf { t ≥ 0 | x + X t / ∈ U } . If g is a bounded continuous function on R d , the following conditions are equivalent 1- For every x ∈ R d , the process ( g ( x + X t ∧ τ x U )) t ≥ 0 is a supermartingale. 2- The distribution L g is a nonpositive measure on U . Some properties of American option prices in exponential L´ evy models – p. 11/30
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