Exact replication under portfolio constraints: a viability approach

Exact replication under portfolio constraints: a viability approach CEREMADE, Université Paris-Dauphine Joint work with Jean-Francois Chassagneux & Idris Kharroubi

Motivation Complete market with no interest rate and one stock : ds t = σ(s t)dw t Price and Hedge of a European option with regular payoff h(s T ) : [ P t = E t [h(s T )] t = E t h (S T ) S ] T S t where S is the tangent process with dynamics d S t = σ (S t) S tdw t. Addition of no short sell regulatory constraints : need t 0 ] h is increasing = t = E t [h (S T ) S T S t 0 = If h is increasing, the super-replication price under no short sell constraints of h(s T ) is the replication price. In general, the super-replication price under no short sell constraints of h(s T ) is the replication price of ĥ(s T ) with ĥ the smallest increasing function above h. For which couple [model,constraints] is this property satisfied?

Agenda 1 Super-replication under portfolio constraints 2 3 4 5

Super-replication price The market model Portfolio process X t,x, s = x + S t = S 0 + s t t 0 uds u = x + σ(s u)dw u, 0 t T. s In addition to classical admissibility conditions, we impose t uσ(s u)dw u, 0 t s T. A K t := { A such that s K P a.s., t s T }, where K is a closed convex set. The super-replication price of h(s T ) at time t under K-constraints defines as { } pt K [h] := inf x R, A K t such that X t,x, T h(s T ) P a.s.

Condition at maturity : Facelift transform The super-replication price of h(s T ) at time t under K-constraints defines as { } pt K [h] := inf x R, A K t such that X t,x, T h(s T ) P a.s. At maturity T, we need T K. = Need to change the terminal condition. = Smallest function above h whose "derivatives" belong to K. Definition of the facelift operator : F K [h](x) := sup y R d h(x + y) δ K (y), x R d, where δ K : y sup z K y, z is the support function of K. F K [h] identifies as the smallest viscosity super-solution of { } min u h, inf δ K (ζ) ζ, xu = 0 ζ =1

Condition at maturity : Facelift transform The super-replication price of h(s T ) at time t under K-constraints defines as { } pt K [h] := inf x R, A K t such that X t,x, T h(s T ) P a.s. At maturity T, we need T K = Need to change the terminal condition. = Smallest function above h whose "derivatives" belong to K. Definition of the facelift operator : F K [h](x) := sup y R d h(x + y) δ K (y), x R d, where δ K : y sup z K y, z is the support function of K. F K [h] identifies as the smallest viscosity super-solution of { } min u h, inf ζ =1 δ K (ζ) ζ, xu = 0

Characterizations of the super-replication price Direct PDE characterization pt K [h] = v K [h](t, S t) where v K [h] is the unique viscosity solution of the PDE { } min L σ u, inf ζ =1 δ K (ζ) ζ, xu = 0 for t < T and u(t, x) = F K [h], with L the Dynkin operator of the diffusion S. Dual representation in terms of pricing measure : [ T ] v K [h](t, x) = sup E Qν t,x h(x t,x T ) δ K (ν s)ds ν s.t. δ K (ν)< t with Q ν the equivalent measure for which W t t νsds is a Brownian motion. 0 BSDE characterization : Minimal solution of the Z-constrained BSDE Y t = F K [h](s T ) T t Z sdw s + T t dl s, with Z t Kσ(S t),

The question of interest super-replicate h(s T ) under K-constraints We always have. super-replicate F K [h](s T ) under K-constraints super-replicate h(s T ) under K-constraints When do we have? replicate F K [h](s T ) without constraints In the Black Scholes model : True for intervals in dimension 1 [Broadie, Cvitanic, Soner] True for any convex set K and money or wealth proportion constraints For general local volatility model : [Our contribution] A necessary and sufficient condition for the previous property to hold for a large class of payoff functions h.

The question of interest super-replicate h(s T ) under K-constraints We always have. super-replicate F K [h](s T ) under K-constraints super-replicate h(s T ) under K-constraints When do we have? replicate F K [h](s T ) without constraints In the Black Scholes model : True for intervals in dimension 1 [Broadie, Cvitanic, Soner] True for any convex set K and money or wealth proportion constraints For general local volatility models : [Our contribution] A necessary and sufficient condition for the previous property to hold for a large class of payoff functions h.

Intervals in dimension 1 Dimension 1 stock : ds t = σ(s t)dw t with σ regular. Interval convex constraint K := [a, b]. Let h be a payoff function such that F K [h] is differentiable. Do we have p K t [h] = p t[f K [h]]? The unconstrained hedging strategy of F K [h] at time t is [ t := E t F K [h](s T ) S ] T, with d S t = σ (S t) S tdw t. S t = S interprets as a probability change and we can find a proba Q s. t. [ ] t := E Q t F K [h](s T ) K, 0 t T, since F K [h] is valued in the convex K. = Revisit and generalize this known result for the Black Scholes model.

Hypercubes for d stocks with separate dynamics Dimension d stock with separate dynamics : ds i t = σ i (S i t)dw t, 1 i d. Hypercube constraints K := Π d i=1[a i, b i ]. Let h be a payoff function such that F K [h] is differentiable. Do we have p K t [h] = p t[f K [h]]? The unconstrained hedging strategy of F K [h] at time t is [ ] i t := E t ( F K [h](s T )) i ST i with d S i St i t = σ i (St) i StdW i t. = Since F K [h] is valued in the hypercube K, K because [ ] [ ] S i a i = a i E T t i S i St i t b i E T t = b St i i, 0 t T, Does it generalize to any convex set or any model?

General convex set K and model dynamics σ Consider A model dynamics : Portfolio constraints : σ Lipschitz, differentiable and invertible K closed convex set with non empty interior Problem of interest : Is there a structural condition on the coupe [K, σ] under which for any payoff h in a given class, we have p K [h] = p[f K [h]]? First, simplified version : Is there a structural condition on the couple [K, σ] under which For any payoff h CK 1, we have p K [h] = p[h]? where CK 1 denotes the class of C 1 functions with derivatives valued in K. (i.e. regular and stable under F K )

General convex set K and model dynamics σ Consider A model dynamics : Portfolio constraints : σ Lipschitz, differentiable and invertible K closed convex set with non empty interior Problem of interest : Is there a structural condition on the coupe [K, σ] under which for any payoff h in a given class, we have p K [h] = p[f K [h]]? First, simplified version : Is there a structural condition on the couple [K, σ] under which For any payoff h CK 1, we have p K [h] = p[h]? where CK 1 denotes the class of Cb 1 functions with derivatives valued in K. (i.e. regular and stable under F K )

BSDE representation for the For any payoff h CK 1, the unconstrained price (p(t, S t)) 0 t T of h(s T ) is solution of the BSDE Y t = h(s T ) T t Z r dw r, 0 t T. The corresponding hedging strategy h t identifies to xp(t, S t) = Y t( X t) 1. Hence satisfies the (linear) BSDE : T h t = h(s T ) + t We know that h(s T ) K. d [ xσ j (S r )] Γ h r σ(s r )dr j=1 = End up on a viability problem : T t Γ h r σ(s r )dw r, For any h valued in K, does the solution h of the BSDE remains in K?

Viability property for BSDE [Buckdahn, Quincampoix, Rascanu] provide a Necessary and Sufficient condition for viability property on BSDEs (or PDEs) : For any terminal condition ξ K, the solution of the BSDE Y t = ξ + T t F (Y s, Z s)ds T t Z sdw s satisfies Y t K P a.s., for 0 t T. There exists C > 0 such that 2 y π K (y), F (y, z) 1 2 2 xx[d 2 K (y)]z, z + Cd 2 K (y), (y, z) R d M d where π K and d K are the projection and distance operators on K. = This provides a sufficient condition for our problem. Is it necessary?

Revisiting the condition of [BQR] for "regular" convex set K There exists C > 0 such that 2 y π K (y), F (y, z) 1 2 2 xx[dk 2 (y)]z, z + CdK 2 (y), (y, z) R d M d Restriction to Polyhedral convex Denoting by n the unit normal vector to K, there exists C > 0 s.t. 2 y π K (y), F (y, z) 1 2 n(y)z, n(y)z + Cd K 2 (y), y / Int(K), z M d Restriction to Polyhedral convex There exists C > 0 s.t. y / Int(K), z M d satisfying n(y) z = 0, y π K (y), F (y, z) CdK 2 (y) Restriction to Polyhedral convex n(y), F (y, z) 0, (y, z) K M d s.t. n(y) z = 0

Adapting the condition to our framework 2 n(y), F (y, z) 0, (y, z) K M d s.t. n(y) z = 0 rewrites d 2 n(y), [ xσ j (x)] γσ(x) = 0, (y, γ) K M d s.t. n(y) γ = 0 j=1 Condition too strong in our context. But γ is symmetric and we shall work under the condition : d n(y), [ xσ j (x)] γσ(x) = 0, (x, y, γ) R d K S d s.t. n(y) γ = 0 j=1 Technical point : What about points with multiple normal vectors? = Need to restrict to border points K with unique normal vector

The main result For a closed convex set K s.t. Int K and an elliptic volatility σ, we have : For any payoff h CK 1, the hedging strategy of h(s t) belongs to K, i.e. p K (h) = p(h) d n(y), [ xσ j (x)] γσ(x) = 0, (x, y, γ) R d K S d s.t. n(y) γ = 0 j=1 This provides a structural condition on the couple [K,σ] under which portfolio restrictions have no effect on payoff functions whose derivatives satisfy the constraint.

Sketch of proof Half-space decomposition of K K = y K H y with H y half-space containing K and tangent to K at y Due to the linearity of the driver, we observe K is viable any half-space H y is viable = need to verify that each half-space H y with normal vector n(y) is viable iff d n(y), [ xσ j (x)] γσ(x) = 0, (x, γ) R d S d s.t. n(y) γ = 0 j=1 Focus on the dynamics of n(y), t For solution of the BSDE with T H y,ito s formula gives T d n(y), t 0 + n(y), [ xσ j (X r )] Γ r σ(x r ) dr T t j=1 t n(y), Γ r σ(x r )dw r Probability change = the condition is sufficient Terminal condition T = γ(x T x) = the condition is necessary

The constrained super replication problem under constraints What happens if the payoff needs to be facelifted? For any payoff h H, the hedging strategy of F K [h](s t) belongs to K, i.e. p K (h) = p(f K [h]) d n(y), [ xσ j (x)] γσ(x) = 0, (x, y, γ) R d K S d s.t. n(y) γ = 0 j=1 where H it the class of lower semi continuous, bounded from below payoffs s.t. E F K [h](s t,x T ) 2 <, (t, x) [0, T ] R d. When K is bounded, we can restrict to lower semi continuous functions.

The Necessary and sufficient condition d n(y), [ xσ j (x)] γσ(x) = 0, (x, y, γ) R d K S d s.t. n(y) γ = 0 j=1 For a fixed y, let introduce (n(y), n 2(y),..., n d (y)) an orthonormal basis of R d. The family (e kl ) 2 k l d of n(n 1)/2 elements given by e kl = n l (y) n k (y) + n k (y) n l (y), 2 k l d. is an orthonormal basis of { γ S d, s.t. n(y) γ = 0 }. The Necessary and Sufficient condition rewrites d n(y), x n k (y), σ.j (x) n l (y), σ.j (x) j=1 = 0, y K, 2 k, l d.

No short Sell on Asset 1 In dimension 2 No short sell on Asset 1 : n(y) = (1, 0), hence n = (0, 1) and the condition rewrites 1 [ σ 21 2 + σ 22 2] = 0 The quadratic variation of asset 2 does not depend on asset 1. In dimension d No short sell on Asset 1 : n(y) = (1, 0,..., 0), hence n j = (1 {i=j} ) i and the condition rewrites 1 [σ l1 σ k1 +... + σ ld σ kd] = 0, 2 l k d. The quadratic covariation between other assets does not depend on asset 1.

Asset 1 non tradable In dimension 2 Asset 1 not tradable : n(y) = (1, 0), hence n = (0, 1) and the condition rewrites 1 [ σ 21 2 + σ 22 2] = 0 The quadratic variation of asset 2 does not depend on asset 1. Same conditions as for the no short sell case since only the border of the convex set K matters.

Bound on the number of allowed positions Bound of the form 1 + 2 C. The convex set is a losange and we have two type of normal vectors. First n(y) = (1, 1) so that n(y) = ( 1, 1) and the condition rewrites [ 1 σ 11 σ 21 2 + σ 12 σ 22 2] [ + 2 σ 11 σ 21 2 + σ 12 σ 22 2] = 0 Second n(y) = ( 1, 1) so that n(y) = (1, 1) and the condition rewrites 1 [ σ 11 + σ 21 2 + σ 12 + σ 22 2] 2 [ σ 11 + σ 21 2 + σ 12 + σ 22 2] = 0 Conditions on quadratic variations in normal directions

Other applications in dimension 2 Which convex sets work for the Black Scholes model? Only the hypercube ones. Which model dynamics works for any convex set? For assets with separate dynamics, the condition is equivalent to 1σ 11 = 2σ 21 and 1σ 12 = 2σ 22. Hence, the only possible models are of the form dst 1 = σ 11 (St 1 )dbt 1 + σ 12 (St 1 )dbt 2, dst 2 = [σ 11 (St 2 ) + λ 1]dBt 1 + [σ 12 (St 2 ) + λ 2]dBt 2,

Conclusion Necessary and sufficient condition ensuring that in order to super-replicate under constraints, the facelifting procedure of the payoff is sufficient. We can adapt the form of the model to anticipated portfolio constraints. US options. Portfolio constraints in terms of money amount or wealth proportion? How can we compute numerically the solution whenever the condition is not satisfied?