� Elliptic and parabolic obstacle problems with thin and Lipschitz obstacles Arshak Petrosyan Math Finance and PDEs 2011 Rutgers University November 4, 2011 Arshak Petrosyan (Purdue) Obstacle problems with Lip obstacles Rutgers Math Fin PDEs 2011 1 / 30
Multi-asset American options Let S , S , ... S n denote the prices of n risky dividend paying assets that 1 satisfy the stochastic differential equations dS i ( t ) = ( µ i − δ i ) S i ( t ) dt + σ i S i ( t ) dW i , where dW i ( t ) are standard Brownian motions such that E ( dW i ) = , Var ( dW i ) = dt , Cov ( dW i , dW j ) = ρ i j dt . Arshak Petrosyan (Purdue) Obstacle problems with Lip obstacles Rutgers Math Fin PDEs 2011 2 / 30
Multi-asset American options Let S , S , ... S n denote the prices of n risky dividend paying assets that 1 satisfy the stochastic differential equations dS i ( t ) = ( µ i − δ i ) S i ( t ) dt + σ i S i ( t ) dW i , where dW i ( t ) are standard Brownian motions such that E ( dW i ) = , Var ( dW i ) = dt , Cov ( dW i , dW j ) = ρ i j dt . If V ( S , . . . , S n , t ) is the price of the European style option derived from 2 these assets, with payoff function Φ ( S , . . . , S n ) at time T , then V must satisfy the Black-Scholes equation n ∂ V n � V ∶= ∂ ∂t V + ∑ + ∑ ( r − δ i ) S i ∂V − rV = ( t < T ) α i j S i S j ∂S i ∂S j ∂S i i , j = i = V ( S , . . . , S n , T ) = Φ ( S , . . . , S n ) . Arshak Petrosyan (Purdue) Obstacle problems with Lip obstacles Rutgers Math Fin PDEs 2011 2 / 30
Multi-asset American options If V ( S , . . . , S n , t ) is the price of an American type option with payoff 1 function Φ ( S , . . . , S n ) , then V satisfies the variational inequality on ( � + ) n × (−∞ , T ) � V ≤ , V ≥ Φ, � V ( V − Φ ) = V ( S , T ) = Φ ( S ) . Arshak Petrosyan (Purdue) Obstacle problems with Lip obstacles Rutgers Math Fin PDEs 2011 3 / 30
Multi-asset American options If V ( S , . . . , S n , t ) is the price of an American type option with payoff 1 function Φ ( S , . . . , S n ) , then V satisfies the variational inequality on ( � + ) n × (−∞ , T ) � V ≤ , V ≥ Φ, � V ( V − Φ ) = V ( S , T ) = Φ ( S ) . Of special interest is the exercise region 2 � = {( S , t ) ∶ V ( S , t ) = Φ ( S ) , t ≤ T } . Arshak Petrosyan (Purdue) Obstacle problems with Lip obstacles Rutgers Math Fin PDEs 2011 3 / 30
Multi-asset American options If V ( S , . . . , S n , t ) is the price of an American type option with payoff 1 function Φ ( S , . . . , S n ) , then V satisfies the variational inequality on ( � + ) n × (−∞ , T ) � V ≤ , V ≥ Φ, � V ( V − Φ ) = V ( S , T ) = Φ ( S ) . Of special interest is the exercise region 2 � = {( S , t ) ∶ V ( S , t ) = Φ ( S ) , t ≤ T } . Typically Φ ( S ) is only Lipschitz continuous 3 ▸ n = : Φ ( S ) = ( S − K ) + American call option ▸ n = : Φ ( S ) = ( K − S ) + American put option ▸ n = : Φ ( S ) = ( max { S , S } − K ) + American call max-options ▸ n = : Φ ( S ) = ( min { S , S } − K ) + American call min-options Not that these Φ’s are also piecewise smooth (important!) Arshak Petrosyan (Purdue) Obstacle problems with Lip obstacles Rutgers Math Fin PDEs 2011 3 / 30
Multi-asset American options Φ ( S ) = ( S − K ) + Φ ( S ) = ( K − S ) + Φ ( S , S ) = ( max { S , S } − K ) + Φ ( S , S ) = ( min { S , S } − K ) + Arshak Petrosyan (Purdue) Obstacle problems with Lip obstacles Rutgers Math Fin PDEs 2011 4 / 30
Parabolic obstacle problem With an appropriate transformation of variables (including x i = log S i ), 1 this can be rewritten as a variational inequality for the heat operator for a function v = v ( x , t ) in � n × ( , ∞) ( ∆ − ∂ t ) v ≤ , v − φ ≥ , ( ∆ − ∂ t ) v ( v − φ ) = v ( x , ) = φ ( x , ) . Tis is nothing but a parabolic obstacle problem with obstacle φ . Arshak Petrosyan (Purdue) Obstacle problems with Lip obstacles Rutgers Math Fin PDEs 2011 5 / 30
Parabolic obstacle problem With an appropriate transformation of variables (including x i = log S i ), 1 this can be rewritten as a variational inequality for the heat operator for a function v = v ( x , t ) in � n × ( , ∞) ( ∆ − ∂ t ) v ≤ , v − φ ≥ , ( ∆ − ∂ t ) v ( v − φ ) = v ( x , ) = φ ( x , ) . Tis is nothing but a parabolic obstacle problem with obstacle φ . Te exercise region � transforms to the coincidence set 2 Λ = {( x , t ) ∶ v ( x , t ) = φ ( x , t )} . Arshak Petrosyan (Purdue) Obstacle problems with Lip obstacles Rutgers Math Fin PDEs 2011 5 / 30
Parabolic obstacle problem With an appropriate transformation of variables (including x i = log S i ), 1 this can be rewritten as a variational inequality for the heat operator for a function v = v ( x , t ) in � n × ( , ∞) ( ∆ − ∂ t ) v ≤ , v − φ ≥ , ( ∆ − ∂ t ) v ( v − φ ) = v ( x , ) = φ ( x , ) . Tis is nothing but a parabolic obstacle problem with obstacle φ . Te exercise region � transforms to the coincidence set 2 Λ = {( x , t ) ∶ v ( x , t ) = φ ( x , t )} . Te solutions of the obstacle problem are well understood when φ is 3 smooth. However, the general theory of free boundaries with nonsmooth (say Lipschitz) obstacles φ is still lacking. We will discuss what complication arise when φ is piecewise-smooth. Arshak Petrosyan (Purdue) Obstacle problems with Lip obstacles Rutgers Math Fin PDEs 2011 5 / 30
Classical obstacle problem Given ▸ Ω domain in � n Ω Arshak Petrosyan (Purdue) Obstacle problems with Lip obstacles Rutgers Math Fin PDEs 2011 6 / 30
Classical obstacle problem Given ▸ Ω domain in � n ▸ φ ∶ Ω → � ( obstacle ) ∶ ∂ Ω → � ( boundary values ), > φ on ∂ Ω Ω φ Arshak Petrosyan (Purdue) Obstacle problems with Lip obstacles Rutgers Math Fin PDEs 2011 6 / 30
Classical obstacle problem Given ▸ Ω domain in � n ▸ φ ∶ Ω → � ( obstacle ) ∶ ∂ Ω → � ( boundary values ), > φ on ∂ Ω φ Ω Ω φ Arshak Petrosyan (Purdue) Obstacle problems with Lip obstacles Rutgers Math Fin PDEs 2011 6 / 30
Classical obstacle problem Given ▸ Ω domain in � n ▸ φ ∶ Ω → � ( obstacle ) ∶ ∂ Ω → � ( boundary values ), > φ on ∂ Ω φ Minimize the Dirichlet integral D Ω ( u ) = ∫ Ω ∣ ∇ u ∣ dx Ω on the closed convex set K = { u ∈ W , ( Ω ) ∣ u = on ∂ Ω, u ≥ φ on Ω } . Arshak Petrosyan (Purdue) Obstacle problems with Lip obstacles Rutgers Math Fin PDEs 2011 6 / 30
Classical obstacle problem Given ▸ Ω domain in � n ▸ φ ∶ Ω → � ( obstacle ) ∶ ∂ Ω → � ( boundary values ), > φ on ∂ Ω u Minimize the Dirichlet integral D Ω ( u ) = ∫ Ω ∣ ∇ u ∣ dx u = φ on the closed convex set K = { u ∈ W , ( Ω ) ∣ u = on ∂ Ω, u ≥ φ on Ω } . Te minimizer u solves the variational inequality ∆ u ≤ , u ≥ φ , ( ∆ u )( u − φ ) = in Ω Arshak Petrosyan (Purdue) Obstacle problems with Lip obstacles Rutgers Math Fin PDEs 2011 6 / 30
Classical obstacle problem It is know that u is as regular as φ , up to C , [ Caffar elli 1998]. u u = φ Arshak Petrosyan (Purdue) Obstacle problems with Lip obstacles Rutgers Math Fin PDEs 2011 7 / 30
Classical obstacle problem It is know that u is as regular as φ , up to C , [ Caffar elli 1998]. If φ ∈ C , then u is also C , and satisfies ∆ u = ∆ φχ { u = φ } in Ω. u u = φ Arshak Petrosyan (Purdue) Obstacle problems with Lip obstacles Rutgers Math Fin PDEs 2011 7 / 30
Classical obstacle problem It is know that u is as regular as φ , up to C , [ Caffar elli 1998]. If φ ∈ C , then u is also C , and satisfies ∆ u = ∆ φχ { u = φ } in Ω. u Te set Λ Λ = Λ ( u ) ∶= { x ∈ Ω ∣ u = φ } is known as the coincidence set . Arshak Petrosyan (Purdue) Obstacle problems with Lip obstacles Rutgers Math Fin PDEs 2011 7 / 30
Classical obstacle problem It is know that u is as regular as φ , up to C , [ Caffar elli 1998]. If φ ∈ C , then u is also C , and satisfies ∆ u = ∆ φχ { u = φ } in Ω. u Te set Λ Λ = Λ ( u ) ∶= { x ∈ Ω ∣ u = φ } Γ is known as the coincidence set . One of the main objects of study is the free boundary Γ ( u ) ∶= ∂ Λ ( u ) . Arshak Petrosyan (Purdue) Obstacle problems with Lip obstacles Rutgers Math Fin PDEs 2011 7 / 30
Classical obstacle problem It is know that u is as regular as φ , up to C , [ Caffar elli 1998]. If φ ∈ C , then u is also C , and satisfies ∆ u = ∆ φχ { u = φ } in Ω. u Te set Λ Λ = Λ ( u ) ∶= { x ∈ Ω ∣ u = φ } Γ is known as the coincidence set . One of the main objects of study is the free boundary Γ ( u ) ∶= ∂ Λ ( u ) . Te regularity properties of u and Γ are fairly well studied when φ ∈ C , and ∆ φ < . Arshak Petrosyan (Purdue) Obstacle problems with Lip obstacles Rutgers Math Fin PDEs 2011 7 / 30
Piecewise smooth Lipschitz obstacles Arshak Petrosyan (Purdue) Obstacle problems with Lip obstacles Rutgers Math Fin PDEs 2011 8 / 30
Recommend
More recommend
