Degenerate processes and degenerate parabolic PDEs Elliptic variational inequalities for the Heston operator Parabolic variational inequalities for the Heston operator American-style options, stochastic volatility, and degenerate parabolic variational inequalities Panagiota Daskalopoulos 1 Paul Feehan 2 1 Department of Mathematics Columbia University 2 Department of Mathematics Rutgers University July 12, 2010 – Vienna University Analysis, Stochastics, and Applications Conference in Honour of Walter Schachermayer Daskalopoulos and Feehan Stochastic volatility and degenerate variational inequalities
Degenerate processes and degenerate parabolic PDEs Overview Elliptic variational inequalities for the Heston operator Heston process and degenerate elliptic and parabolic equations Parabolic variational inequalities for the Heston operator Introduction ◮ Degenerate Markov processes and their associated parabolic PDEs are pervasive in finance. ◮ Degenerate parabolic PDEs give rise to challenging terminal/boundary value problems (European-style options) and terminal/boundary value obstacle problems (American-style options). ◮ What boundary conditions are appropriate or necessary? Daskalopoulos and Feehan Stochastic volatility and degenerate variational inequalities
Degenerate processes and degenerate parabolic PDEs Overview Elliptic variational inequalities for the Heston operator Heston process and degenerate elliptic and parabolic equations Parabolic variational inequalities for the Heston operator Degenerate elliptic and degenerate parabolic partial differential equations ◮ Research goes back to Kohn and Nirenberg (1965). ◮ A highly selective list includes Daskalopoulos and her collaborators, Feller, Freidlin, Koch, Kufner, Levendorskii, Opic, Pinsky, Stredulinsky, ... ◮ Although previous research on degenerate elliptic/parabolic PDEs is extensive, more often than not, results often exclude even simple examples of interest in finance (CIR, Heston, etc). ◮ Recent research due to Ekstrom and Tysk for CIR PDEs and Laurence and Salsa for solutions of American-style, multi-asset BSM option pricing problems. Daskalopoulos and Feehan Stochastic volatility and degenerate variational inequalities
Degenerate processes and degenerate parabolic PDEs Overview Elliptic variational inequalities for the Heston operator Heston process and degenerate elliptic and parabolic equations Parabolic variational inequalities for the Heston operator Heston’s Stochastic Volatility Process Heston’s asset price process, S ( u ) = exp( X ( u )), is defined by � dX ( u ) = ( r − q − Y ( u ) / 2)) du + Y ( u ) dW 1 ( u ) , X ( t ) = x , � dY ( u ) = κ ( θ − Y ( u )) du + σ Y ( u ) dW 2 ( u ) , Y ( t ) = y , where ( W 1 ( u ) , dW 3 ( u )) is two-dimensional Brownian motion, 1 − ρ 2 W 3 ( u ), κ, θ, σ are positive constants, � W 2 ( u ) := ρ W 1 ( u ) + ρ ∈ ( − 1 , 1), r ≥ 0, q ≥ 0, and Y ( u ) is the variance process. Daskalopoulos and Feehan Stochastic volatility and degenerate variational inequalities
Degenerate processes and degenerate parabolic PDEs Overview Elliptic variational inequalities for the Heston operator Heston process and degenerate elliptic and parabolic equations Parabolic variational inequalities for the Heston operator Degenerate parabolic PDEs and variational inequalities Option pricing problems for the Heston process lead to ◮ Degenerate parabolic differential equations, ◮ Degenerate parabolic variational inequalities, for European and American-style option pricing problems, respectively. Daskalopoulos and Feehan Stochastic volatility and degenerate variational inequalities
Degenerate processes and degenerate parabolic PDEs Overview Elliptic variational inequalities for the Heston operator Heston process and degenerate elliptic and parabolic equations Parabolic variational inequalities for the Heston operator Heston parabolic differential equation If −∞ ≤ x 0 < x 1 < ∞ , let O := ( x 0 , x 1 ) × (0 , ∞ ) and Q := [0 , T ) × O . If ψ : Q → R is a suitable function, for example, ψ ( t , x , y ) = ( K − e x ) + or ( e x − K ) + , and r ≥ 0, define u ( t , x , y ) := e − r ( T − t ) E t , x , y [ ψ ( T , X ( T ) , Y ( T ))] , Q then we expect − u ′ + Au = 0 on Q , u ( T , · ) = ψ ( T , · ) on O , where − Au := y u xx + 2 ρσ u xy + σ 2 u yy � � +( r − q − y / 2) u x + κ ( θ − y ) u y − ru . 2 Daskalopoulos and Feehan Stochastic volatility and degenerate variational inequalities
Degenerate processes and degenerate parabolic PDEs Overview Elliptic variational inequalities for the Heston operator Heston process and degenerate elliptic and parabolic equations Parabolic variational inequalities for the Heston operator Degenerate elliptic or parabolic PDEs Suppose ( t , x ) ∈ Q = [0 , T ) × O and O ⊂ R n , and a ij ( t , x ) ∂ 2 u − Au ( t , x ) := 1 � ( t , x ) 2 ∂ x i ∂ x j i , j b i ( t , x ) ∂ u � ( t , x ) − c ( t , x ) u ( t , x ) . + ∂ x i i If ξ T A ( t , x ) ξ ≥ µ ( t , x ) | ξ | 2 , ξ ∈ R n , where µ ( x ) > 0, then A is elliptic (parabolic) on Q if µ > 0 on Q , and A is uniformly elliptic (parabolic) on Q if µ ≥ δ on Q , for some constant δ > 0. This condition fails for the Heston operator, as µ = 0 along { y = 0 } component of ¯ O and the operator is “degenerate”. Daskalopoulos and Feehan Stochastic volatility and degenerate variational inequalities
Degenerate processes and degenerate parabolic PDEs Weighted Sobolev spaces and energy estimates Elliptic variational inequalities for the Heston operator Existence and uniqueness for the elliptic variational inequality Parabolic variational inequalities for the Heston operator Weighted Sobolev spaces Definition We need a weight function when defining our Sobolev spaces, w ( x , y ) := 2 β = 2 κθ σ 2 , µ = 2 κ σ 2 y β − 1 e − γ | x |− µ y , σ 2 , for ( x , y ) ∈ O and a suitable positive constant, γ . Then H 1 ( O , w ) := { u ∈ L 2 ( O , w ) : (1 + y ) 1 / 2 u ∈ L 2 ( O , w ) , and y 1 / 2 Du ∈ L 2 ( O , w ) } , where � � � u � 2 u 2 x + u 2 (1 + y ) u 2 w dxdy . � � H 1 ( O , w ) := y w dxdy + y O O Daskalopoulos and Feehan Stochastic volatility and degenerate variational inequalities
Degenerate processes and degenerate parabolic PDEs Weighted Sobolev spaces and energy estimates Elliptic variational inequalities for the Heston operator Existence and uniqueness for the elliptic variational inequality Parabolic variational inequalities for the Heston operator Weighted Sobolev spaces (continued) Let H 1 0 ( O , w ) be the closure in H 1 ( O , w ) of C 1 c ( O ) ∩ H 1 ( O , w ). For i = 0 , 1, let H 1 0 ( O ∪ Γ i , w ) be the closure in H 1 ( O , w ) of C 1 c ( O ∪ Γ i ) ∩ H 1 ( O , w ), where Γ 0 = ( x 0 , x 1 ) × { 0 } and Γ 1 = { x 0 , x 1 } × (0 , ∞ ) , and Γ 1 = { x 0 } × (0 , ∞ ) if x 1 = + ∞ , Γ 1 = { x 1 } × (0 , ∞ ) if x 0 = −∞ , and Γ 1 = ∅ if x 0 = −∞ and x 1 = + ∞ . Daskalopoulos and Feehan Stochastic volatility and degenerate variational inequalities
Degenerate processes and degenerate parabolic PDEs Weighted Sobolev spaces and energy estimates Elliptic variational inequalities for the Heston operator Existence and uniqueness for the elliptic variational inequality Parabolic variational inequalities for the Heston operator G˚ arding inequality Proposition Let q , r , σ, κ, θ ∈ R be constants such that β := 2 κθ σ 2 > 0 , σ � = 0 , and − 1 < ρ < 1 . Then for all u ∈ V such that u = 0 on Γ 1 , where V = H 1 ( O , w ) , a ( u , u ) ≥ 1 2 C 2 � u � 2 V − C 3 � (1 + y ) 1 / 2 u � 2 L 2 ( O , w ) . Daskalopoulos and Feehan Stochastic volatility and degenerate variational inequalities
Degenerate processes and degenerate parabolic PDEs Weighted Sobolev spaces and energy estimates Elliptic variational inequalities for the Heston operator Existence and uniqueness for the elliptic variational inequality Parabolic variational inequalities for the Heston operator Continuity estimates Proposition Choose ◮ β < 1 : V = H 1 ( O , w ) and W = H 1 0 ( O ∪ Γ 1 , w ) ; ◮ β > 1 : V = W = H 1 ( O , w ) . Then | a ( u , v ) | ≤ C 1 � u � V � v � W , ∀ ( u , v ) ∈ V × W , where C 1 is a positive constant depending at most on the coefficients r , q , κ, θ, ρ, σ . Daskalopoulos and Feehan Stochastic volatility and degenerate variational inequalities
Degenerate processes and degenerate parabolic PDEs Weighted Sobolev spaces and energy estimates Elliptic variational inequalities for the Heston operator Existence and uniqueness for the elliptic variational inequality Parabolic variational inequalities for the Heston operator Elliptic variational inequality with (nonhomogeneous) Dirichlet boundary conditions Let f ∈ L 2 ( O , w ) and g , ψ ∈ H 1 ( O , w ) such that ψ ≤ g on O . For β > 1, find u ∈ H 1 ( O , w ) such that a ( u , v − u ) ≥ ( f , v − u ) L 2 ( O , w ) , with u ≥ ψ on O and u = g on Γ 1 , ∀ v ∈ H 1 ( O , w ) with v ≥ ψ on O and v = g on Γ 1 , that is, u − g , v − g ∈ H 1 0 ( O ∪ Γ 0 , w ). For β < 1, the statement is identical, except that the Dirichlet conditions are u = g and v = g on Γ, that is, u − g , v − g ∈ H 1 0 ( O , w ). Daskalopoulos and Feehan Stochastic volatility and degenerate variational inequalities
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