Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

American-style options, stochastic volatility, and degenerate variational inequalities Paul Feehan Department of Mathematics Rutgers, The State University of New Jersey

Joint with Panagiota Daskalopoulos and Camelia Pop — May 20, 2011 Stochastic Analysis in Finance & Insurance – Ann Arbor, Michigan

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Introduction and motivation from mathematical finance I

We consider continuous processes defined by stochastic differential equations with degenerate/non-Lipschitz coefficients and motivated by option pricing.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Introduction and motivation from mathematical finance I

We consider continuous processes defined by stochastic differential equations with degenerate/non-Lipschitz coefficients and motivated by option pricing.

I

In particular, we stochastic volatility processes, such as the Heston process, and their generalizations (from R2 to Rn ).

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Introduction and motivation from mathematical finance I

We consider continuous processes defined by stochastic differential equations with degenerate/non-Lipschitz coefficients and motivated by option pricing.

I

In particular, we stochastic volatility processes, such as the Heston process, and their generalizations (from R2 to Rn ).

I

We consider their Kolmogorov PDEs and initial/boundary value and obstacle problems arising in option pricing.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Introduction and motivation from mathematical finance I

We consider continuous processes defined by stochastic differential equations with degenerate/non-Lipschitz coefficients and motivated by option pricing.

I

In particular, we stochastic volatility processes, such as the Heston process, and their generalizations (from R2 to Rn ).

I

We consider their Kolmogorov PDEs and initial/boundary value and obstacle problems arising in option pricing.

I

We explore questions of existence, uniqueness, and regularity of solutions to variational inequalities, as well as the regularity and geometric properties of the free boundary.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Formulations of the perpetual American-style put option problem Explicit solution and its regularity properties

A simple example Suppose a financial asset share price process. S(t), is modeled as geometric Brownian motion (Black-Scholes-Merton model), dS(t) = S(t) (r dt + σ dW (t)) ,

t ≥ 0,

where r ≥ 0, σ > 0, X (t) = log S(t) is the log price process, and W (t) is Brownian motion.

Question What is the price, u(x), of an American-style perpetual put option with strike K > 0, as a function of x = log S(0) ∈ R?

Remark An American-style put (call) option gives holder the right but not obligation to sell (buy) one share of asset for a strike price K > 0 at any time t ≤ T (maturity). The option is perpetual if T = ∞. Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Formulations of the perpetual American-style put option problem Explicit solution and its regularity properties

An obstacle or linear complementarity problem The price, u ∈ W 2,∞ (R), may be characterized as the solution to the obstacle (or linear complementarity) problem, min{Lu − f , u − ψ} = 0 on R, ¯ = C 0,α (R), ¯ the Banach space of bounded α-H¨older where f ∈ C α (R) ¯ the Banach space of continuous functions, 0 < α ≤ 1, and ψ ∈ C 0,1 (R), bounded Lipschitz continuous functions, is given by ψ(x) := (K − e x )+ , x ∈ R, and σ2 Lu := − uxx − rux + ru. 2 (We allow f 6= 0 for completeness, but f ≡ 0 in financial applications.)

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Formulations of the perpetual American-style put option problem Explicit solution and its regularity properties

A stationary variational inequality problem The price, u ∈ W 1,2 (R, e −γ|x| dx), may also be characterized as the solution to the stationary variational inequality problem, which is to find u ≥ ψ such that a(u, v − u) ≥ (f , v − u)L2 (R,e −γ|x| dx) , ∀v ∈ W 1,2 (R, e −γ|x| dx), v ≥ ψ, where a(w , v ) := (Lw , v )L2 (R,e −γ|x| dx) ,

∀w , v ∈ W 1,2 (R, e −γ|x| dx),

and, for existence and uniqueness of weak solutions, it suffices to assume f ∈ L2 (R, e −γ|x| dx) and ψ ∈ W 1,2 (R, e −γ|x| dx). Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Formulations of the perpetual American-style put option problem Explicit solution and its regularity properties

A free boundary value problem The price, u ∈ W 2,∞ (R), may be viewed as the solution to the free boundary value problem, Lu = f on C ,

u = ψ on E ,

ux = ψx on ∂C ,

and u ≥ ψ on R,

where C := {x ∈ R : u(x) > ψ(x)} = (`∗ , ∞), E := {x ∈ R : u(x) = ψ(x)} = (−∞, `∗ ], and the free boundary point, `∗ ∈ R, is determined as part of the solution to the problem.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Formulations of the perpetual American-style put option problem Explicit solution and its regularity properties

Explicit solution to obstacle problem

This obstacle problem may be solved explicitly: ( K − ex , −∞ < x ≤ `∗ , u(x) = 2r /σ 2 −2rx/σ 2 (K − L∗ )(L∗ ) e , x ≥ `∗ , where L∗ :=

2r σ2 K = K − K and `∗ := log(L∗ ). 2r + σ 2 2r + σ 2

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Formulations of the perpetual American-style put option problem Explicit solution and its regularity properties

Smooth pasting property for solution to obstacle problem

We see that ( u 0 (x) =

−∞ < x < `∗ , −e x , 2r 2r /σ 2 −2rx/σ 2 −(K − L∗ ) σ2 (L∗ ) e , x > `∗ ,

giving u 0 (−`∗ ) = −L∗ = u 0 (+`∗ ), and thus u 0 (x) is continuous across the free boundary point, `∗ , where ¯ u(`∗ ) = ψ(`∗ ), and so u ∈ C 1 (R).

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Formulations of the perpetual American-style put option problem Explicit solution and its regularity properties

Optimal regularity for solution to obstacle problem Finally, we have ( u 00 (x) =

−e x , −∞ < x < `∗ , 4r 2 2r /σ 2 −2rx/σ 2 , x > `∗ , e (K − L∗ ) σ4 (L∗ )

giving u 00 (`∗ −) = −L∗ and u 00 (`∗ +) = (K − L∗ )

4r 2 . σ4

¯ Hence, u 00 (x) is discontinuous across the free boundary but u ∈ C 1,1 (R).

Remark ¯ the best possible regularity for the Even if we choose ψ ∈ C 2 (R), ¯ solution, u, one obtains is C 1,1 (R). Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Heston process and degenerate elliptic/parabolic PDEs Weighted Sobolev spaces and energy estimates

Heston’s stochastic volatility process The asset price process proposed by Heston (1993) is defined by S(u) = exp(X (u)), where {(X (u), Y (u))}t≥0 solve the stochastic differential equation (SDE), p dX (u) = (r − q − Y (u)/2)) du + Y (u) dB1 (u), X (t) = x, p dY (u) = κ(θ − Y (u)) du + σ Y (u) dB2 (u), Y (t) = y , p where B1 (u) := W1 (u), B2 (u) := ρW1 (u) + 1 − ρ2 W2 (u), {(W1 (u), W2 (u))}u≥0 is two-dimensional Brownian motion, κ, θ, σ are positive constants, ρ ∈ (−1, 1), q ≥ 0, and r > 0.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Heston process and degenerate elliptic/parabolic PDEs Weighted Sobolev spaces and energy estimates

Generator of the Heston process

Standard arguments show that the generator of the Heston process is −A, where Au := −


y uxx + 2ρσuxy + σ 2 uyy − (r − q − y /2)ux − κ(θ − y )uy + ru, 2

for u ∈ C02 (R × (0, ∞)). The operator A is not uniformly elliptic on R × (0, ∞) since it becomes degenerate along y = 0.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Heston process and degenerate elliptic/parabolic PDEs Weighted Sobolev spaces and energy estimates

Elliptic and parabolic partial differential operators Suppose (t, x) ∈ Q = [0, T ) × O, where O ⊂ Rn is a domain, and Au(t, x) := −

X 1X aij (t, x)uxi xj (t, x) − bi (t, x)uxi + c(t, x)u(t, x). 2 i,j

i

If ha(t, x)ξ, ξi ≥ w (t, x)|ξ|2 ,

∀ξ ∈ Rn , (t, x) ∈ Q,

where w > 0 on Q, then A is I

Elliptic on O (parabolic on Q) if w > 0 on O (on Q), and

I

Uniformly elliptic on O (uniformly parabolic on Q) if w ≥ δ on O (on Q), for some constant δ > 0.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Heston process and degenerate elliptic/parabolic PDEs Weighted Sobolev spaces and energy estimates

Extension from Heston to other degenerate partial differential operators The Heston process and its generator, −A, as important paradigms for degenerate processes and partial differential operators, but our methods and results can be extended to a broad class of degenerate partial differential operators. If O ⊂ Rn−1 × (0, ∞) is a domain, α > 0, and Au(x) := −

X 1X aij (x)uxi xj (x) − bi (x)uxi + c(x)u(x), 2 i,j

aij (x) = xnα ¯aij (x),

i

x ∈ O,

where λ|ξ|2 ≤ h¯a(x)ξ, ξi ≤ Λ|ξ|2 , ∀ξ ∈ Rn , x ∈ O, and 0 < λ < Λ < ∞.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Heston process and degenerate elliptic/parabolic PDEs Weighted Sobolev spaces and energy estimates

Weighted Sobolev spaces Definition (H 1 weighted Sobolev space) We need a weight function when defining our Sobolev spaces, w(x, y ) := y β−1 e −γ|x|−µy ,

β=

2κθ 2κ ,µ = 2, 2 σ σ

for (x, y ) ∈ O and a suitable positive constant, γ. Then H 1 (O, w) := {u ∈ L2 (O, w) : (1 + y )1/2 u ∈ L2 (O, w), and y 1/2 Du ∈ L2 (O, w)}, where kuk2H 1 (O,w) :=

Z


y |Du|2 + 1 + y )u 2 w dxdy .

O Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Heston process and degenerate elliptic/parabolic PDEs Weighted Sobolev spaces and energy estimates

Weighted Sobolev spaces and the domain boundary Definition (Weighted Sobolev spaces and boundaries) I

¯1 = Γ ¯0 ∪ Γ1 , where Γ0 := ∂O ∩ (R × {0}) and Denote ∂O = Γ0 ∪ Γ ¯2 t Γ0 (transverse intersection) Γ1 := ∂O ∩ (R × (0, ∞)) is C 2 and Γ

I

Let H01 (O ∪ Γ0 , w) (respectively, H01 (O, w)) be the closure in H 1 (O, w) of C0∞ (O ∪ Γ0 ) (respectively, C0∞ (O)).

Lemma (Equivalence of Sobolev spaces when β ≥ 1) If β ≥ 1, then H01 (O ∪ Γ0 , w) = H01 (O, w).

Remark A function u ∈ H01 (O ∪ Γ0 , w) obeys u = 0 along Γ1 (trace sense) but can have arbitrary values along Γ0 . Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Heston process and degenerate elliptic/parabolic PDEs Weighted Sobolev spaces and energy estimates

A bilinear form

Definition (Bilinear form associated with the Heston operator) The Heston generator, −A, may be written in divergence form and defines a bilinear map, a : V × V → R, via a(u, v ) := (Au, v )H ,

u, v ∈ C0∞ (O).

where V := H01 (O ∪ Γ0 , w) and H := L2 (O, w).

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Heston process and degenerate elliptic/parabolic PDEs Weighted Sobolev spaces and energy estimates

G˚ arding inequality Proposition (G˚ arding inequality) Let r , σ, κ, θ ∈ R be constants such that β :=

2κθ > 0, σ2

σ 6= 0,

and

− 1 < ρ < 1.

Then, there are positive constants, C1 , C2 , depending at most on the coefficients r , κ, θ, ρ, σ, such that for all u ∈ V , a(u, u) ≥

1 C2 kuk2V − C3 k(1 + y )1/2 uk2H . 2

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Heston process and degenerate elliptic/parabolic PDEs Weighted Sobolev spaces and energy estimates

Continuity estimate

Proposition (Continuity estimate) There is a positive constant, C1 , depending at most on the coefficients r , κ, θ, ρ, σ such that |a(u, v )| ≤ C1 kukV kv kV ,

Paul Feehan

∀(u, v ) ∈ V × V .

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Heston process and degenerate elliptic/parabolic PDEs Weighted Sobolev spaces and energy estimates

Working with weighted Sobolev spaces ...

I

The weighted Sobolev spaces we describe provide a natural framework within which to pose problems of existence and uniqueness of weak solutions to variational equations and inequalities for the Heston and similar degenerate partial differential operators.

I

There is a vast literature discussing use and applications of weighted Sobolev spaces (... the foundations of Sturm-Liouville theory or work of McKean, Kufner, Stredulinsky, and many others provide just a small sample ...) but it tends to be more a collection of special cases then a unified framework for PDEs.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Heston process and degenerate elliptic/parabolic PDEs Weighted Sobolev spaces and energy estimates

... and working around their deficiencies Some examples of complications: I

While the Sobolev embedding theorems are extensively employed in the usual proofs of existence, uniqueness, and regularity of weak solutions for non-degenerate elliptic PDEs on bounded domains, their counterparts for weighted Sobolev spaces are typically false.

I

The Rellich-Kondrachov compact embedding theorem is employed in the usual proof of existence of weak solutions for non-degenerate elliptic PDEs on bounded domains, but its counterpart for either weighted Sobolev spaces or unbounded domains is typically false.

I

Existence of boundary trace operators is often delicate.

I

Approximation of functions in weighted Sobolev spaces by smooth functions with compact support is better understood. Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness H 2 regularity

Variational equality problem

Problem (Weak solutions to the Heston equation) Given f ∈ L2 (O, w), we call u ∈ H01 (O ∪ Γ0 , w) a weak solution to Au = f

on O

and u = 0

on Γ1 ,

if a(u, v ) = (f , v )L2 (O,w) ,

Paul Feehan

∀v ∈ H01 (O ∪ Γ0 , w).

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness H 2 regularity

Interpretation of weak solutions Lemma (Interpretation of weak solutions) Let f ∈ L2 (O, w). If u ∈ H01 (O ∪ Γ0 , w) obeys ∀v ∈ H01 (O ∪ Γ0 , w),

a(u, v ) = (f , v )H ,

(1)

and in addition u ∈ H 2 (O, w), then Au = f and

a.e. on O,

u = 0 on Γ1 (trace sense),

β

y (ρux + σuy ) = 0 on Γ0 (trace sense).

(2) (3)

Conversely, if u ∈ H 2 (O, w) obeys (2), then u ∈ H01 (O ∪ Γ0 , w) and is a solution to (1).

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness H 2 regularity

Existence and uniqueness of weak solutions Theorem (Existence and uniqueness) Assume r > 0. Suppose there are M, m ∈ H 2 (O, w) such that Am ≤ AM

a.e. on O

and

m≤0≤M

on Γ1 .

If f ∈ L2 (O, w) obeys Am ≤ f ≤ AM

a.e. on O,

then there exists a unique solution u ∈ H 1 (O ∪ Γ0 , w) to the stationary variational equation for the Heston operator. Moreover, m≤u≤M

a.e. on O,


kukV 1 (O,w) ≤ C kf kL2 (O,w) + k(1 + y )ukL2 (O,w) . Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness H 2 regularity

Role of growth bounds in the proof of existence

Remark (Role of growth bounds) I

I

Growth bounds m, M are required, in part, because of the failure of a(u, v ) to be coercive. Growth bounds m, M compensate for the failure of the continuous Sobolev and compact Rellich-Kondrachov embedding theorems.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness H 2 regularity

Examples of upper and lower bounds Lemma (Examples of upper and lower bounds) Suppose f ∈ L2 (O, w) obeys n(x, y ) ≤ f (x, y ) ≤ N(x, y ) a.e. (x, y ) ∈ O, where n(x, y ) := c0 + c1 x + c2 y + c3 (1 + y )e `x + c4 (1 + y )e ky , N(x, y ) := C0 + C1 x + C2 y + C3 (1 + y )e Lx + C4 (1 + y )e Ky , for constants c0 , . . . , c4 , C0 , . . . , C4 ∈ R and positive constants k, K , `, L obeying 2k < µ, 2K < µ, 2` < γ, 2L < γ. Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness H 2 regularity

Examples of upper and lower bounds Lemma (Continued) Then there are constants d0 , . . . , d4 ∈ R, depending only on c0 , . . . , c4 , k, ` and the coefficients of A and constants D0 , . . . , D4 ∈ R, depending only on C0 , . . . , C4 , K , L and the coefficients of A, such that if m(x, y ) := d0 + d1 x + d2 y + d3 e `x + d4 e ky , M(x, y ) := D0 + D1 x + D2 y + D3 e

Lx

+ D4 e

Ky

(x, y ) ∈ R × R++ , ,

(x, y ) ∈ R × R++ ,

then m, M ∈ H 2 (O, w) and Am(x, y ) ≤ f (x, y ) ≤ AM(x, y ) a.e. (x, y ) ∈ O, with Am(x, y ) ≤ n(x, y ) and Paul Feehan

N(x, y ) ≤ AM(x, y ), Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness H 2 regularity

Examples of upper and lower bounds Lemma (Continued) ... provided we also require I

If c0 6= 0 or c1 6= 0, then r > 0;

I

If c2 6= 0, then min{κ, r } > 0;

I

If c3 6= 0, then r > `(r − q)+ and 0 < ` < 1;

I

If c4 6= 0, then 0 < k < min{2κ, r /κθ};

and similarly for C0 , . . . , C4 , K , L.

Remark ¯1 , then the coefficients In addition, if O ⊂ R × (0, ∞) with ∂O = Γ0 ∪ Γ ci , Ci , i = 0, . . . , 4 may be chosen so that m ≤ 0 ≤ M on Γ1 since, a fortiori, m ≤ 0 ≤ M on R × (0, ∞). Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness H 2 regularity

A second-order weighted Sobolev space Definition (Weighted H 2 Sobolev space) Let H 2 (O, w) := {u ∈ L2 (O, w) : (1 + y )1/2 u, (1 + y )Du, yD 2 u ∈ L2 (O, w)}, where kuk2H 2 (O,w)

Z :=


y 2 |D 2 u|2 + (1 + y )2 |Du|2 + (1 + y )u 2 w dxdy .

O

2 Let Hloc (O, w) := {u ∈ L2 (O, w) : u ∈ H 2 (O 0 , w), ∀O 0 b O}.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness H 2 regularity

H 2 regularity and solution to the strong problem Theorem (H 2 regularity) Assume the hypotheses required for existence and uniqueness of weak solutions. If u ∈ H01 (O ∪ Γ0 , w) is a weak solution to the stationary variational equation for the Heston operator, then u ∈ H 2 (O, w) and Au = f

a.e. on O

and

u = 0 on Γ1 .

Moreover,   kukH 2 (O,w) ≤ C k(1 + y )1/2 f kL2 (O,w) + k(1 + y )ukL2 (O,w) .

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness H 2 regularity

H¨older continuity and Sobolev embeddings Remark (H¨older continuity) I

I

I

¯ for α ∈ (0, 1) If v ∈ H 2 (O, w), we cannot conclude that v ∈ C α (O) since the Sobolev embedding theorem does not hold in general for weighted Sobolev spaces. 2 However, if v ∈ Hloc (O), then v ∈ C α (O¯0 ) for any domain O 0 ⊂ O with O¯0 ⊂ R2 compact and such that O 0 obeys the uniform interior cone condition. The condition v ∈ H 2 (O, w), together with the standard Sobolev embeddings H 2 (O 0 ) → C α (O¯0 ), merely implies that v ∈ C α (O ∪ Γ1 ).

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness H 2 regularity

H¨older continuity and classical solutions Theorem (H¨older continuity up to the boundary – with C. Pop) Assume the hypotheses required for existence and uniqueness of weak ¯ w) for q > 2 + β and solutions. If in addition f ∈ Lqloc (O, 1 u ∈ H0 (O ∪ Γ0 , w) is a weak solution to the stationary variational ¯ for some α ∈ (0, 1). equation for the Heston operator, then u ∈ C α (O)

Theorem (Classical solutions) Assume the hypotheses required for H¨ older continuity of solutions up to the boundary. If in addition f ∈ C (O) and u ∈ H01 (O ∪ Γ0 , w) is a weak solution to the stationary variational equation for the Heston operator, ¯ ∩ C 2 (O) ∩ H 2 (O, w) for some α ∈ (0, 1) and then u ∈ C α (O) Au = f

on O Paul Feehan

and

u = 0 on Γ1 .

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness Outline of proof of existence of solutions H 2 regularity

Variational inequality problem Problem Let f ∈ L2 (O, w) and g , ψ ∈ H 1 (O, w) such that ψ ≤ g on Γ1 . We call u a weak solution to min{Au − f , u − ψ} = 0 u=g

a.e. on O, on Γ1 ,

if u ∈ H 1 (O, w) obeys u ≥ ψ a.e. on O and a(u, v − u) ≥ (f , v − u)L2 (O,w) ,

u − g ∈ H01 (O ∪ Γ0 , w),

∀v ∈ H 1 (O, w) with v − g ∈ H01 (O ∪ Γ0 , w),

Paul Feehan

v ≥ ψ a.e. on O.

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness Outline of proof of existence of solutions H 2 regularity

Variational inequality boundary and obstacle conditions

Remark (Dirichlet boundary and obstacle conditions) Note that I

I

I

v , g ∈ H 1 (O, w) with v = g on Γ1 (trace sense) if and only if v − g ∈ H01 (O ∪ Γ0 , w). g , ψ ∈ H 1 (O, w) with ψ ≤ g on Γ1 (trace sense) if and only if (ψ − g )+ ∈ H01 (O ∪ Γ0 , w). By a standard reduction, we assume g ≡ 0 without loss of generality.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness Outline of proof of existence of solutions H 2 regularity

Existence and uniqueness of solutions Theorem (Existence and uniqueness of solutions) Assume r > 0. Suppose there are M, m, ϕ ∈ H 2 (O, w) such that (1 + y )1/2 M, (1 + y )1/2 m, (1 + y )1/2 ϕ ∈ Lq (O, w), for some q > 2 and Am ≤ AM,

Aϕ ≥ 0, ϕ≥0

and

and

A(m + ϕ) > 0

m≤0≤M

a.e. on O,

on Γ1

Given f ∈ L2 (O, w) such that Am ≤ f ≤ AM

a.e. on O,

and ψ ∈ H 1 (O, w) such that ψ ≤ 0 on Γ1 (trace sense), then ... Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness Outline of proof of existence of solutions H 2 regularity

Existence and uniqueness of solutions Theorem (Existence and uniqueness of solutions - continued) ... there exists a unique solution u ∈ H 1 (O ∪ Γ0 , w) to the stationary variational inequality for the Heston operator and u obeys max{m, ψ} ≤ u ≤ M

a.e. on O,


kukH 1 (O,w) ≤ C kf kL2 (O,w) + k(1 + y )ukL2 (O,w) + kψ + kH 1 (O,w) .

Remark (Method of proof) The result is proved using penalization and adapting arguments of Bensoussan and Lions (1982) from the case of uniformly elliptic differential operators with bounded coefficients on a bounded domain to the case of a non-coercive, degenerate elliptic differential operator with unbounded coefficients on an unbounded domain. Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness Outline of proof of existence of solutions H 2 regularity

Role of growth bounds in the proof

Remark (Role of growth bounds) 1. Uniqueness. I I I

When O is bounded, m, M, ϕ could be replaced by constants. ϕ is used to reduce to the case of positive solutions. One can then prove uniqueness by a contradiction argument.

2. Existence. Growth bounds m, M compensate for the failure of the continuous Sobolev and compact Rellich-Kondrachov embedding theorems.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness Outline of proof of existence of solutions H 2 regularity

Outline of proof of existence. I 1. Consider the regularized bilinear form, aλ (u, v ) := a(u, v ) + λ((1 + y )u, v )H ,

∀u, v ∈ V ,

where λ is (greater than or equal to) the constant appearing in G˚ arding’s inequality, so aλ is coercive: there is an α > 0 so that aλ (v , v ) ≥ αkv k2V ,

∀v ∈ V .

2. Use coercivity of aλ to show there is at most one solution u ∈ K := {v ∈ H01 (O ∪ Γ0 ) : v ≥ ψ a.e. on O} to the variational inequality defined by aλ : aλ (u, v − u) ≥ (f , v − u)H , Paul Feehan

∀v ∈ K.

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness Outline of proof of existence of solutions H 2 regularity

Outline of proof of existence. II 3. For ε > 0, define a penalization function 1 βε (v ) := − (ψ − v )+ ∈ V 0 , ε

v ∈ V,

and the penalized equation for the regularized bilinear form, aλ (u, v ) + (βε (u), v )H = (f , v )H ,

∀v ∈ V .

4. Observe that the penalization function is monotone: (βε (v ) − βε (v 0 ), v − v 0 )H ≥ 0,

∀v , v 0 ∈ V .

5. Using coercivity of aλ and monotonicity of βε , show that there is at most one solution, uε , to the penalized equation defined by aλ : aλ (u, v ) + (βε (u), v )H = (f , v )H , Paul Feehan

∀v ∈ V .

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness Outline of proof of existence of solutions H 2 regularity

Outline of proof of existence. III 6. Using a Galerkin approximation method, show there is at least one solution, uε , to the penalized equation defined by aλ , aλ (u, v ) + (βε (u), v )H = (f , v )H ,

∀v ∈ V ,

and that an error estimate holds: √ |(ψ − uε )+ |H ≤ C ε,

ε > 0.

7. Letting ε → 0 and passing to weakly convergent subsequences, show that uε ∈ V converges to a solution u ∈ V to the variational inequality defined by the regularized bilinear form: aλ (u, v − u) ≥ (f , v − u)H , Paul Feehan

∀v ∈ K.

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness Outline of proof of existence of solutions H 2 regularity

Outline of proof of existence. IV 8. Prove existence of solutions to the variational inequality, a(u, v − u) ≥ (f , v − u)H ,

∀v ∈ K .

using a priori estimates, an iterative method relying on existence and uniqueness of solutions to the variational equality and the variational inequality defined by the regularized bilinear form, and passage to a weakly convergent subsequence.

Remark I

The proof of existence solutions to the variational inequality leverages existence and uniqueness for the solutions to the variational equation.

I

The proof of uniqueness uses a reduction to the case of positive solutions and an argument by contradiction. Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness Outline of proof of existence of solutions H 2 regularity

H 2 regularity and strong solutions Theorem (H 2 regularity) Assume the hypotheses required for existence and uniqueness of weak solutions and, in addition, that ψ ∈ H 2 (O, w). If u ∈ H01 (O ∪ Γ0 , w) is a weak solution to the stationary variational inequality for the Heston operator, then u ∈ H 2 (O, w) and min{Au − f , u − ψ} = 0 a.e. on O, u = 0 on Γ1 . There is a constant C depending only on the coefficients of A so that  kukH 2 (O,w) ≤ C k(1 + y )5/2 ukL2 (O,w) + k(1 + y )3/2 f kL2 (O,w)
+ k(1 + y )ψkH 2 (O,w) . Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness Outline of proof of existence of solutions H 2 regularity

Remarks on the proof and extensions Remark I

The preceding result is proved by adapting the arguments of Bensoussan and Lions (1982), again taking account of the I I I I

I

Degeneracy of the Heston operator, Non-coercivity of the bilinear form, a(u, v ), Unboundedness of the coefficients, Unboundedness of the domain.

We expect by work in progress that I I

u ∈ W 2,p (O, w), for 2 < p < ∞ and ψ ∈ W 2,p (O, w), and ¯ when ψ(x, y ) = (K − e x )+ . u ∈ C 1,1 (O)

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness Outline of proof of existence of solutions H 2 regularity

H¨older continuity

Theorem (H¨older continuity up to the boundary) Assume the hypotheses required for existence and uniqueness of weak ¯ w) for q > 2 + β and solutions. If in addition f ∈ Lqloc (O, u ∈ H01 (O ∪ Γ0 , w) is a weak solution to the stationary variational ¯ for some α ∈ (0, 1). inequality for the Heston operator, then u ∈ C α (O)

Proof. Combine the corresponding result for the case of the weak solution to the variational equality with the method of penalization.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness Outline of proof of existence of solutions H 2 regularity

Classical solutions

Theorem (Classical solutions) Assume the hypotheses required for existence and uniqueness of weak ¯ w) ∩ C (O) for solutions and, in addition, that ψ ∈ H 2 (O, w), f ∈ Lqloc (O, 1 q > 2 + β. If u ∈ H0 (O ∪ Γ0 , w) is a weak solution to the stationary variational inequality for the Heston operator, then ¯ ∩ H 2 (O, w) for some α ∈ (0, 1), and u ∈ C 2 (O) ∩ C α (O) min{Au − f , u − ψ} = 0 on O, u = 0 on Γ1 .

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Existence and uniqueness Outline of proof of existence of solutions H 2 regularity

Local H 2 regularity In finance, ψ is usually only Lipschitz and ψ ∈ / H 2 (O, w).

Theorem (Local H 2 regularity) Assume the hypotheses required for existence and uniqueness of weak solutions and, in addition, that (1 + y )3/2 f ∈ L2 (O, w) and (1 + y )5/2 m, (1 + y )5/2 M ∈ L2 (O, w), and let U ⊆ O be an open subset such that (1 + y )ψ ∈ H 2 (U , w). If u ∈ H01 (O ∪ Γ0 ) is the unique solution to the variational inequality for the Heston operator, then u ∈ H 2 (U 0 , w) for every U 0 ⊂ U¯0 ⊂ U and  kukH 2 (U 0 ,w) ≤ C k(1 + y )5/2 ukL2 (O,w) + k(1 + y )3/2 f kL2 (O,w)
+ k(1 + y )ψkH 2 (U ,w) , where C depends only on the coefficients of A and dist(U 0 , U ). Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Analytical tools Existence and uniqueness for the strong problem H 2 regularity

A failure of coercivity in the parabolic equation

I

Simple attempts to adapt the argument Bensoussan and Lions (1982) in their proof existence and uniqueness of solutions to the “strong” evolutionary variational inequality to the Heston generator, −A, fail because the bilinear form defined by A is non-coercive.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Analytical tools Existence and uniqueness for the strong problem H 2 regularity

A change of dependent variable

I

To circumvent the lack of coerciveness, we employ the change of dependent variable u˜(t, x, y ) = e −λ(1+y )(T −t) u(t, x, y ),

u ∈ V , (t, x, y ) ∈ Q,

by analogy with the familiar exponential shift change of dependent variable u˜ = e −λ(T −t) u.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Analytical tools Existence and uniqueness for the strong problem H 2 regularity

Transformation of the equations

I

One finds that the non-coercive parabolic problem, −u 0 + Au = f on Q,

u(T ) = h on O,

u = g on Σ,

is transformed, for t ∈ [T − δ, T ] and sufficiently small δ, into an equivalent coercive parabolic problem, ˜ u = f˜ on Q, −˜ u 0 + A˜ I

u˜(T ) = h on O,

u˜ = g˜ on Σ,

An obstacle condition u ≥ ψ is transformed into an equivalent ˜ obstacle condition u˜ ≥ ψ.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Analytical tools Existence and uniqueness for the strong problem H 2 regularity

Transformation of the bilinear form

The bilinear form on V × V (defined by the weight w) associated to the ˜ operator A(t) (with suitable boundary conditions) is ˜ u (t), v )L2 (O,w) . ˜a(t; u˜(t), v ) := (A(t)˜

(4)

We then obtain the key continuity estimate and G˚ arding inequality for ˜a(t).

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Analytical tools Existence and uniqueness for the strong problem H 2 regularity

Continuity estimate and G˚ arding inequality for the transformed Heston operator Proposition For a sufficiently large positive constant λ, depending only the coefficients of A, and a sufficiently small positive constant δ < T , depending only on λ and the coefficients of A, the bilinear map ˜a(t) : V × V → R obeys |˜a(t; u, v )| ≤ C kukV kv kV , α ˜a(t; v , v ) ≥ kv k2V , 2 for all u, v ∈ V and t ∈ [T − δ, T ].

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Analytical tools Existence and uniqueness for the strong problem H 2 regularity

Change of Sobolev weight and transformation back to original problem The weight in our previous definition of weighted Sobolev spaces, w(x, y ) := y β−1 e −γ|x|−µy ,

(x, y ) ∈ O,

is replaced, when transforming back from a solution u˜ to a solution u to the original problem, by ˜ w(x, y ) := e −2λM(1+y ) w(x, y ) = y β−1 e −γ|x|−µy −2δλ(1+y ) ,

(x, y ) ∈ O,

where M > T is a constant.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Analytical tools Existence and uniqueness for the strong problem H 2 regularity

Setup for the evolutionary problem Definition I

Recall that V = H01 (O ∪ Γ0 , w) and H = L2 (O, w). Denote V := L2 (0, T ; V ), V 0 := L2 (0, T ; V 0 ), H := L2 (0, T ; H), and K := {v ∈ V : v ≥ ψ}, given ψ ∈ V .

I

The transformed Heston generator, −A(t), defines a linear map A (t) ∈ L (V , V 0 ) and a bilinear map a(t) : V × V → R by a(t; u, v ) := A (t)u(v ),

u, v ∈ V ,

and A (t)u(v ) := (A(t)u, v )H ,

Paul Feehan

u, v ∈ C (0, T ; C0∞ (O)).

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Analytical tools Existence and uniqueness for the strong problem H 2 regularity

Boundary conditions

Remark (Dirichlet boundary conditions) I

If u, g ∈ L2 (0, T ; H 1 (O, w)), then u = g on Γ1 × [0, T ) (trace sense) means that u − g ∈ L2 (0, T ; H01 (O ∪ Γ0 , w)).

I

We assume that g ≡ 0 and so the condition ψ ≤ g on [0, T ) × Γ1 (trace sense), that is (ψ − g )+ ∈ L2 (0, T ; H01 (O ∪ Γ0 , w)), is replaced by ψ ≤ 0 on [0, T ) × Γ1 (trace sense), that is ψ + ∈ L2 (0, T ; H01 (O ∪ Γ0 , w)).

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Analytical tools Existence and uniqueness for the strong problem H 2 regularity

Evolutionary variational inequality problem

Problem Suppose f , ψ ∈ H and h ∈ V with h ≥ ψ(T , ·) on O and ψ ≤ 0 on [0, T ) × Γ1 . Find u ∈ K , with u 0 ∈ H , such that −(u 0 (t), v − u(t))H + a(t; u(t), v − u(t)) ≥ (f (t), v − u(t))H , ∀v ∈ V with v ≤ ψ(t, ·),

Paul Feehan

t ∈ [0, T ].

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Analytical tools Existence and uniqueness for the strong problem H 2 regularity

Existence and uniqueness of solutions Theorem There exists a unique solution to the evolutionary variational inequality for the Heston generator.

Remark As with the stationary variational inequality, the result is proved by the penalization method and adapting the arguments of Bensoussan and Lions (1982), extending their methods from the case of uniformly parabolic differential operators with bounded coefficients on a bounded domain to the case of the Heston operator, a degenerate parabolic differential operator with unbounded coefficients on an unbounded domain.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Analytical tools Existence and uniqueness for the strong problem H 2 regularity

Regularity for solutions to the strong problem for the parabolic Heston variational inequality

Using our weighted Sobolev spaces and estimates, we adapt the Bensoussan-Lions regularity theory to establish

Theorem If ψ ∈ L2 (0, T ; H 2 (O, w)) and u is the solution to the evolutionary variational inequality for the Heston generator, then u ∈ L2 (0, T ; H 2 (O, w)).

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Analytical tools Existence and uniqueness for the strong problem H 2 regularity

Solution to the strong form of the obstacle problem Given H 2 regularity, a solution to the strong problem for the parabolic Heston variational inequality is a solution to the more familiar “complementarity” or strong form of the obstacle problem.

Theorem Given f ∈ L2 (0, T ; L2 (O, w)), g ∈ L2 (0, T ; H 1 (O, w)), and h ∈ H 1 (O, w) obeying g ≥ ψ on Γ1 × [0, T ), h ≥ ψ on O, there is a unique u ∈ L2 (0, T ; H 2 (O, w)) solving min{−u 0 + Au − f , u − ψ} = 0 a.e. on Q, u = g on Γ1 × [0, T ), u(T ) = h on O.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Analytical tools Existence and uniqueness for the strong problem H 2 regularity

Improved regularity

Remark We expect (in progress) u ∈ L2 (0, T ; W 2,p (O, w)).

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Solutions to stationary variational equalities Solutions to stationary variational inequalities Solutions to evolutionary variational equalities Solutions to evolutionary variational inequalities Consequences of stochastic representations

Stochastic representations and their consequences

I

Given a stochastic representation of a solution, certain regularity and geometric properties follow readily.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Solutions to stationary variational equalities Solutions to stationary variational inequalities Solutions to evolutionary variational equalities Solutions to evolutionary variational inequalities Consequences of stochastic representations

Stochastic representations and their consequences

I

Given a stochastic representation of a solution, certain regularity and geometric properties follow readily.

I

Paradigms for such technques may be found in the work of Broadie & Detemple (1997), Detemple (2006), Jaillet, Lamberton, & Lapeyre (1990), Villeneuve (1999), and Laurence & Salsa (2009).

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Solutions to stationary variational equalities Solutions to stationary variational inequalities Solutions to evolutionary variational equalities Solutions to evolutionary variational inequalities Consequences of stochastic representations

Stochastic representations and their consequences

I

Given a stochastic representation of a solution, certain regularity and geometric properties follow readily.

I

Paradigms for such technques may be found in the work of Broadie & Detemple (1997), Detemple (2006), Jaillet, Lamberton, & Lapeyre (1990), Villeneuve (1999), and Laurence & Salsa (2009). The following stochastic representations may be derived by adapting methods of Bensoussan & Lions (1982), Friedman (1976), and Øksendal (2003).

I

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Solutions to stationary variational equalities Solutions to stationary variational inequalities Solutions to evolutionary variational equalities Solutions to evolutionary variational inequalities Consequences of stochastic representations

Probabilistic solutions to stationary variational equalities I Problem (Stationary variational equality) Let O ⊂ R × (0, ∞) be a domain with C 2 boundary, ∂O, let f ∈ C (O) obey |f (x, y )| ≤ C1 (1 + y ), (x, y ) ∈ O, ¯ such that and let g ∈ Cb (∂O). Find u ∈ C 2 (O) ∩ C (O) Au = f u=g

on O, on ∂β O.

¯0 ∪ Γ1 when 0 < β < 1 and ∂β O = Γ1 when β ≥ 1. Here, ∂β O := Γ

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Solutions to stationary variational equalities Solutions to stationary variational inequalities Solutions to evolutionary variational equalities Solutions to evolutionary variational inequalities Consequences of stochastic representations

Probabilistic solutions to stationary variational equalities II

Theorem (Uniqueness of solutions to Problem 7.1) ¯ with Let u be a solution to Problem 7.1. Then u = u ∗ on O,   u ∗ (x, y ) := EQ e −r τ g (X (τ ), Y (τ ))1{τ <∞} Z τ  −rs + EQ e f (X (s), Y (s)) ds , (x, y ) ∈ O,

(5)

0

where r > 0 and τ is the exit time from O of the process (X (s), Y (s))s≥0 starting at (x, y ) ∈ O.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Solutions to stationary variational equalities Solutions to stationary variational inequalities Solutions to evolutionary variational equalities Solutions to evolutionary variational inequalities Consequences of stochastic representations

Probabilistic solutions to stationary var’l inequalities I Problem (Stationary variational inequality) ¯ obey ψ ≤ g on ∂β O, Let O, f , g be as in Problem 7.1 and let ψ ∈ C (O) and |f (x, y )| ≤ C1 (1 + y ) and |ψ(x, y )| ≤ C2 (1 + e C3 x ),

(x, y ) ∈ O.

¯ such that Find u ∈ C 2 (O) ∩ C (O) min{Au − f , u − ψ} = 0 on O, u = g on ∂β O.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Solutions to stationary variational equalities Solutions to stationary variational inequalities Solutions to evolutionary variational equalities Solutions to evolutionary variational inequalities Consequences of stochastic representations

Probabilistic solutions to stationary var’l inequalities II Theorem (Uniqueness of solutions to Problem 7.3) ¯ with Let u be a solution to Problem 7.3. Then u = u ∗ on O, ( "Z # τ ∧θ

u ∗ (x, y ) := sup θ∈T

e −rs f (X (s), Y (s)) ds

EQ 0

  + EQ e −r θ ψ(X (θ), Y (θ))1{θ<τ }   + EQ e −r τ g (X (τ ), Y (τ ))1{τ ≤θ} ,

¯ (x, y ) ∈ O,

where r > 0, τ is the exit time from O of the process (X (s), Y (s))s≥0 starting at (x, y ) ∈ O, and T is the set of F-stopping times with values in [0, ∞).

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Solutions to stationary variational equalities Solutions to stationary variational inequalities Solutions to evolutionary variational equalities Solutions to evolutionary variational inequalities Consequences of stochastic representations

Probabilistic solutions to evolutionary var’l equalities I Problem (Evolutionary variational equality) Let O be as in Problem 7.1 and let 0 < T < ∞ and Q := [0, T ) × O. Let f ∈ C (Q) obey |f (t, x, y )| ≤ C1 (1 + y ),

(t, x, y ) ∈ Q,

let g ∈ Cb ([0, T ) × ∂O), and let h ∈ C (O) obey |h(x, y )| ≤ C2 (1 + e C3 x ),

(x, y ) ∈ O,

¯ such that Find u ∈ C 1,2 (Q) ∩ C (Q) −u 0 + Au = f on Q, u = g on [0, T ) × ∂β O, u(T , ·) = h Paul Feehan

on O.

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Solutions to stationary variational equalities Solutions to stationary variational inequalities Solutions to evolutionary variational equalities Solutions to evolutionary variational inequalities Consequences of stochastic representations

Probabilistic solutions to evolutionary var’l equalities II Theorem (Uniqueness of solutions to Problem 7.5) ¯ with Let u be a solution to Problem 7.5. Then u = u ∗ on Q, Z τ  e −rs f (X (s), Y (s)) ds u ∗ (t, x, y ) := EQ t h i + EQ e −r (τ −t) h(X (T ), Y (T ))1{τ =T } h i + EQ e −r (τ −t) g (τ, X (τ ), Y (τ ))1{τ 0, τ is the exit time from O of (X (s), Y (s))s≥t starting at (t, x, y ) ∈ Q, if such a time exists and τ = T otherwise.

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Solutions to stationary variational equalities Solutions to stationary variational inequalities Solutions to evolutionary variational equalities Solutions to evolutionary variational inequalities Consequences of stochastic representations

Probabilistic solutions to evolutionary var’l inequalities I Problem (Evolutionary variational inequality) Let O, T , Q, f , g , h be as in Problem 7.5, and let ψ obey |ψ(t, x, y )| ≤ C4 (1 + e C5 x ), ψ ≤ g on [0, T ) × ∂β O

(t, x, y ) ∈ Q,

and ψ(T , ·) ≤ h on O.

¯ such that Find u ∈ C 1,2 (Q) ∩ C (Q) min{−u 0 + Au − f , u − ψ} = 0 u=g u(T , ·) = h

Paul Feehan

on Q, on [0, T ) × ∂β O, on O.

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Solutions to stationary variational equalities Solutions to stationary variational inequalities Solutions to evolutionary variational equalities Solutions to evolutionary variational inequalities Consequences of stochastic representations

Probabilistic solutions to evolutionary var’l inequalities II Theorem (Uniqueness of solutions to Problem 7.7) ¯ with Let u be a solution to Problem 7.7. Then u = u ∗ on Q, # ( "Z τ ∧θ

u ∗ (t, x, y ) := sup

θ∈Tt,T

e −rs f (s, X (s), Y (s)) ds

EQ t

h i + EQ e −r (θ−t) ψ(θ, X (θ), Y (θ))1{θ<τ ∧T } h i + EQ e −r (T −t) h(X (T ), Y (T ))1{T =τ ∧θ} h io + EQ e −r (τ −t) g (τ, X (τ ), Y (τ ))1{τ ≤θ,τ
Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Solutions to stationary variational equalities Solutions to stationary variational inequalities Solutions to evolutionary variational equalities Solutions to evolutionary variational inequalities Consequences of stochastic representations

Applications to finance I

In applications to option pricing, we need only consider solutions to the evolutionary variational inequality with f = 0, while ψ(x, y ) = (K − e x )+ or (e x − K )+ and h(x, y ) = ψ(x, y ).

I

We denote U(t, S, y ) = u(t, x, y ) and Ψ(t, S, y ) = ψ(t, x, y ), where S = ex .

The results on the next few slides provide a small sample of what may be proved by adapting arguments of Broadie & Detemple (1997), Jaillet, Lamberton, and Lapeyre (1990), Laurence and Salsa (2009), Touzi (1999), and Villeneuve (1999).

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Solutions to stationary variational equalities Solutions to stationary variational inequalities Solutions to evolutionary variational equalities Solutions to evolutionary variational inequalities Consequences of stochastic representations

Properties of the solution

Lemma Let U(t, S, y ) be as above. Then 1. U(t, S, y ) is a non-increasing function of t ∈ [0, T ]. 2. If Ψ(S) is a convex function of S ∈ (0, ∞), then U(t, S, y ) is a convex function of S ∈ (0, ∞), ∀(t, y ) ∈ [0, T ] × (0, ∞). 3. If Ψ(S) is a non-increasing (non-decreasing) function of S ∈ (0, ∞), then U(t, S, y ) is a non-increasing (non-decreasing) function of S ∈ (0, ∞), ∀(t, y ) ∈ [0, T ] × (0, ∞).

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Solutions to stationary variational equalities Solutions to stationary variational inequalities Solutions to evolutionary variational equalities Solutions to evolutionary variational inequalities Consequences of stochastic representations

Properties of the derivative

Lemma Suppose Ψ(S), S ∈ (0, ∞), obeys m(S2 − S1 ) ≤ Ψ(S2 ) − Ψ1 (S1 ) ≤ M(S2 − S1 ),

0 < S1 < S2 < ∞,

for given −∞ < m ≤ M < ∞. Then, for each (t, y ) ∈ [0, T ) × (0, ∞), U(t, S, y ) is a differentiable function of S ∈ (0, ∞) and m≤

∂U ≤ M, ∂S

∀(t, S, y ) ∈ [0, T ) × (0, ∞) × (0, ∞).

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Solutions to stationary variational equalities Solutions to stationary variational inequalities Solutions to evolutionary variational equalities Solutions to evolutionary variational inequalities Consequences of stochastic representations

Continuation and exericise regions

Definition Given a solution U(t, S, y ) to the evolutionary variational inequality for an obstacle function Ψ(t, S, y ), the continuation and exericise regions are defined by C (U) := {(t, S, y ) ∈ Q : U(t, S, y ) > Ψ(t, S, y )}, E (U) := {(t, S, y ) ∈ Q : U(t, S, y ) = Ψ(t, S, y )}, and similarly for C (u) and E (u), given u(t, x, y ) and ψ(t, x, y ).

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Solutions to stationary variational equalities Solutions to stationary variational inequalities Solutions to evolutionary variational equalities Solutions to evolutionary variational inequalities Consequences of stochastic representations

Characterization of the free boundary The results of Touzi (1999) may be adapted to show

Proposition If Ψ(t, S, y ) = (K − S)+ , there is a S ∗ : [0, T ) × (0, ∞) → [0, K ] such that C (U) = {(t, S, y ) ∈ [0, T ) × (0, ∞) × (0, ∞) : S > S ∗ (t, y )}.

Lemma If Ψ(t, S, y ) = (K − S)+ , then S ∗ : [0, T ) × (0, ∞) → [0, K ] is decreasing with respect to y ∈ (0, ∞).

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Solutions to stationary variational equalities Solutions to stationary variational inequalities Solutions to evolutionary variational equalities Solutions to evolutionary variational inequalities Consequences of stochastic representations

Properties of the free boundary We expect, by work in progress, that I

S ∗ (t, y ) is a continuous function of t ∈ [0, T ), ∀y ∈ (0, ∞).

I

If Ψ(S) = (K − S)+ (respectively, (S − K )+ ), then S ∗ (t, y ) is a non-decreasing (respectively, non-increasing) function of t ∈ [0, T ), ∀y ∈ (0, ∞).

I

If s ∗ (t, y ) = log S ∗ (t, y ), then s ∗ (t, ·) is Lipschitz, uniformly with respect to t ∈ [0, T ).

I

S ∗ : [0, T ) × (0, ∞) → [0, K ] is differentiable with respect to y ∈ (0, ∞).

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Selected references I

E. Bayraktar, A proof of the smoothness of the finite time horizon American put options for jump diffusions, SIAM J. Control and Optimization (2009), 48, 51–572.

I

E. Bayraktar, C. Kardaras and H. Xing, Strict local martingale deflators and pricing American call-type options, Finance and Stochastics, to appear.

I

E. Bayraktar and H. Xing, On The Uniqueness Of Classical Solutions Of Cauchy Problems, Proc. A.M.S. (2010), 138, 2061–2064, arxiv.org/abs/0908.1086v3.

I

E. Bayraktar and H. Xing, Analysis of the optmal exercise boundary of American options for jump diffusions, SIAM J. Control and Optimization (2009), 41, 825–860. Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Selected references (continued) I

I

V. Barbu and C. Marinelli, Variational inequalities in Hilbert spaces with measures and optimal stopping problems, Appl. Math. Optim. 57 (2008), 237–262. A. Bensoussan and J. L. Lions, Applications of variational inequalities in stochastic control, 1982.

I

J. J. Kohn and L. Nirenberg, Degenerate elliptic-parabolic equations of second order, Comm. Pure Appl. Math. 20 (1967), 797–872.

I

P. Daskalopoulos and R. Hamilton, Regularity of the boundary for the porous medium equation, J. American Mathematical Society 11, 1998, pp. 899–965.

I

P. Daskalopoulos and P. Feehan, American-style options, stochastic volatility, and existence and uniqueness of solutions to stationary degenerate variational inequalities, in preparation. Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Selected references (continued) I

P. Daskalopoulos and P. Feehan, American-style options, stochastic volatility, and existence and uniqueness of solutions to evolutionary degenerate variational inequalities, in preparation.

I

P. Daskalopoulos, P. Feehan, and C. Pop, Schauder estimates and regularity of solutions to degenerate elliptic obstacle problems arising in perpetual American-style option pricing problems, in preparation.

I

P. Feehan and C. Pop, Stochastic representation of solutions to degenerate variational equalities and inequalities in the Heston model, in preparation.

I

C. Pop, Ph.D. dissertation, Rutgers University, in preparation.

I

A. Friedman, Variational principles and free boundary problems, Wiley, 1982, New York. Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Selected references (continued) I

P. Jaillet, D. Lamberton, and B. Lapeyre, Variational inequalities and the pricing of American options, Acta Appl. Math. 21 (1990), pp. 263–289.

I

I. Karatzas and S. E. Shreve, Methods of mathematical finance, Springer, New York, 1998.

I

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, 1980.

I

P. Laurence and S. Salsa, Regularity of the free boundary of an American option on several assets, Comm. Pure Appl. Math. 62, 2009, pp. 969–994.

I

L. Mastroeni and M. Matzeu, Parabolic variational inequalities with degenerate elliptic part, Riv. Mat. Univ. Parma 5 (1996), 223–234. Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Selected references (continued) I

E. Ekstr¨ om and J. Tysk, Boundary conditions for the single-factor term structure equation, 2010.

I

E. Ekstr¨ om and J. Tysk, The Black-Scholes equation in stochastic volatility models, 2010.

I

M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate elliptic operators, Ann. Mat. Pura Appl. 80 (1968), 1–122.

I

R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, 1996.

I

N. Touzi, American options exercise boundary when the volatility changes randomly, Applied Mathematics and Optimization 39 (1999), 411–422. Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

Mathematical Finance and Partial Differential Equations 2010 Rutgers University, New Brunswick, NJ November 4, 2011 Web: finmath.rutgers.edu Email: [email protected] 10 invited speakers and up to 15 contributed talks

Paul Feehan

Stochastic volatility and degenerate variational inequalities

Introduction Price of the perpetual American-style put option Heston elliptic/parabolic operator and weighted Sobolev spaces Stationary variational equations for the Heston generator Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

THANK YOU!

Paul Feehan

Stochastic volatility and degenerate variational inequalities