When terminal facelift enforces delta constraints

Abstract

This paper deals with the superreplication of non-path-dependent European claims under additional convex constraints on the number of shares held in the portfolio. The corresponding superreplication price of a given claim has been widely studied in the literature, and its terminal value, which dominates the claim of interest, is the so-called facelift transform of the claim. We investigate under which conditions the superreplication price and strategy of a large class of claims coincide with the exact replication price and strategy of the facelift transform of this claim. In one dimension, we observe that this property is satisfied for any local volatility model. In any dimension, we exhibit an analytical necessary and sufficient condition for this property, which combines the dynamics of the stock together with the characteristics of the closed convex set of constraints. To obtain this condition, we introduce the notion of first order viability property for linear parabolic PDEs. We investigate in detail several practical cases of interest: multidimensional Black–Scholes model, non-tradable assets, and short-selling restrictions.

Appendices

Appendix A: Facelift properties

The first lemma collects some useful properties of the facelift transform. Lemma A.2 is an approximation result, and Lemma A.3 is a (minimal) PDE characterization of the facelift.

Lemma A.1

• (i) If h is lower semi-continuous, then so is F _K [h].

• (ii) If h(x)≥g(x) for all x, then F _K [h](x)≥F _K [g](x) for all $x\in\mathbb{R}^{d}$.

• (iii) If 0∈K and h( ⋅ )=c with c a given constant, then F _K [h]=h.

• (iv) F _K [h∨g]=F _K [h]∨F _K [g].

• (v) F _K [h]≥h and F _K [h]=F _K [F _K [h]].

Proof

Property (i) holds true since F _K [h] is the pointwise supremum of lower semi-continuous functions. Properties (ii)–(iv) are obvious consequences of Definition 2.3 of the facelift transform. Property (v) follows from the fact that $0 \in\tilde{K}$ and from the computation, for any $x\in\mathbb{R}^{d}$,

$$\begin{aligned} \mbox {\textsc {F}}_K\big[\mbox {\textsc {F}}_K[h]\big](x) &= \sup_{y_2 \in\mathbb {R}^d} \big(\mbox {\textsc {F}}_K[h](x+y_2) - \delta_K(y_2)\big) \\ &= \sup_{y_1,y_2 \in\mathbb{R}^d} \big(h(x+y_2+y_1)- \delta_K(y_1) - \delta_K(y_2)\big) \le \mbox {\textsc {F}}_K[h](x). \end{aligned}$$

 □

Lemma A.2

Assume that h is lower semi-continuous and bounded from below by −m _h for some m _h ≥0. Then there exists an increasing sequence (h _n ) _n≥1 of bounded Lipschitz functions, uniformly bounded from below by −m _h , converging to h and such that F _K [h _n ]↑F _K [h].

Proof

We define the function sequence (g _n ) by

$$\begin{aligned} g_n(x) = & \inf_{y \in\mathbb{R}^d}\big(h(y) + n|x-y|\big) ,\quad x\in \mathbb{R}^d, n \geq1. \end{aligned}$$

It is clear that the sequence (g _n ) is nondecreasing, that −m _h ≤g _n ≤h and that g _n is n-Lipschitz-continuous for all n≥1. We now prove that (g _n ) converges pointwise to h. Fix some $x\in \mathbb{R}^{d}$. Since h is lower semi-continuous and bounded from below, there exists a sequence (x _n ) in $\mathbb{R}^{d}$ such that

$$\begin{aligned} g_n(x) = & h(x_n) + n |x-x_n|, \qquad n\ge1. \end{aligned}$$

(A.1)

Since h is bounded from below by −m _h , we deduce

$$\begin{aligned} n |x-x_n| \le& h(x) - h(x_n) \le h(x) + m_h, \qquad n\ge1, \end{aligned}$$

so that lim _n→∞ x _n =x. Together with (A.1) and the lower semi-continuity of h, this yields

$$\begin{aligned} \lim_{n\rightarrow\infty} g_n(x) \ge& \liminf_{n \rightarrow \infty} h(x_n) \ge h(x). \end{aligned}$$

Thus, g _n (x)↑h(x) as n↑∞, for all $x \in \mathbb{R}^{d}$.

Define now the function sequence (h _n ) by

$$\begin{aligned} h_n (x) := & g_n(x) \wedge n,\quad x\in\mathbb{R}^d, n\ge1. \end{aligned}$$

Since g _n is Lipschitz-continuous and bounded from below, h _n is bounded and Lipschitz-continuous, for all n≥1. Moreover, since (g _n ) is nondecreasing and converges pointwise to h, we also get that (h _n ) is nondecreasing and converges pointwise to h. It remains to prove the convergence of F _K [h _n ] to F _K [h]. For any $x\in\mathbb{R}^{d}$, we simply observe that

$$\begin{aligned} \mbox {\textsc {F}}_K[h](x) = & \sup_{u \in\tilde{K}} \big(h(x+u)-\delta_K(u) \big) \\ = & \sup_{n\ge1, u \in\tilde{K}} \big(h_n(x+u)-\delta_K(u) \big) = \uparrow \mbox{-} \lim_{n\rightarrow\infty} \mbox {\textsc {F}}_K[h_n](x). \end{aligned}$$

 □

Lemma A.3

Let h be a lower semi-continuous function from $\mathbb{R}^{d}$ to $\mathbb{R}$.

• (i) If F _K [h] is locally bounded, then F _K [h] is a viscosity supersolution of

  $$\begin{aligned} \min\{\mathcal{C}_K(\partial_x u), u-h \} = & 0 \qquad\textit {on } \mathbb{R}^d. \end{aligned}$$

  (A.2)

• (ii) If v is a differentiable supersolution of (A.2), then

  $$\begin{aligned} v(x) \ge& F_K[v](x) \ge F_K[h](x) , \qquad x\in\mathbb{R}^d. \end{aligned}$$

  In particular, if h is differentiable and ∇h∈K, then F _K [h]=h.

Proof

(i) We first recall that since h is lower semi-continuous, so is F _K [h] by Lemma A.1. Let $\bar{x} \in\mathbb{R}^{d}$ and $\phi \in C^{1}(\mathbb{R}^{d};\mathbb{R})$ be a test function such that

$$\begin{aligned} 0 = & \mbox {\textsc {F}}_K[h](\bar{x}) - \phi(\bar{x}) = \mbox{(strict)}\min_{x \in\mathbb{R}^d} (\mbox {\textsc {F}}_K[h]-\phi)(x) . \end{aligned}$$

(A.3)

Observe that Lemma A.1(v) implies F _K [F _K [h]]=F _K [h], so that

$$\begin{aligned} \mbox {\textsc {F}}_K[h](\bar{x}) \ge& \mbox {\textsc {F}}_K[h](\bar{x} + y) - \delta _K(y) , \qquad y\in\mathbb{R}^d. \end{aligned}$$

Using (A.3), we deduce that

$$\begin{aligned} \phi(\bar{x}) \ge& \phi(\bar{x} + y) - \delta_K(y) , \qquad y\in \mathbb{R}^d. \end{aligned}$$

In particular, for y=εζ where ε>0 and $\zeta\in \tilde{K}$ with |ζ|=1, we obtain

$$\begin{aligned} \frac{\phi(\bar{x}) - \phi(\bar{x} + \epsilon\zeta) }{\epsilon} \le& - \delta_K(\zeta) , \qquad y\in\tilde{K}. \end{aligned}$$

Letting ε goes to 0 yields $\delta_{K} (\zeta) - \partial_{x} \phi(\bar{x} ) \zeta\ge0$. Since ζ is arbitrarily chosen in $\tilde{K}$, this yields $\mathcal{C}_{K}(\partial_{x}\phi)(\bar{x})\geq0$.

(ii) Let v be a differentiable supersolution of (A.2). We then have

$$\begin{aligned} \delta_{K}(y) - \partial_xv(x+ty)y \ge& 0 , \qquad(t,x,y)\in [0,1]\times\mathbb{R}^d\times\tilde{K}. \end{aligned}$$

Fix now $\bar{x} \in\mathbb{R}^{d}$. We get from the previous inequality that

$$\begin{aligned} \int_0^1\Big(\delta_{K}(y) - \frac{d}{dt}v(\bar{x}+ty) \Big) \,d t \ge& 0 , \qquad y\in\mathbb{R}^d. \end{aligned}$$

Therefore, we compute

$$\begin{aligned} v(\bar{x}) \ge& v(\bar{x}+y) -\delta_{K}(y) , \qquad y\in\mathbb{R}^d. \end{aligned}$$

Taking the supremum over y, we obtain $v (\bar{x})\ge \mbox {\textsc {F}}_{K}[h] (\bar{x})$.

Suppose now that h is differentiable and [∂ _x h]^⊤∈K. Since we already know that F _K [h]≥h, we conclude that F _K [h]=h. □

Appendix B: Proof of Corollary 2.7

For the sake of clarity, let us define $\tilde{v}^{h}_{K}$ by

$$\begin{aligned} \begin{array}{rcl@{\quad}l} \tilde{v}^h_K(t,x) &:= & v^h_K(t,x) &\text{ for } (t,x)\in [0,T)\times\mathbb{R}^d, \\ \tilde{v}^h_K(T,x) &:= & v^h_K(T-,x) &\text{ for } x \in\mathbb{R}^d. \end{array} \end{aligned}$$

Recall the definitions of $u^{\mbox {\scriptsize \textsc {F}}_{K}[h]}$ and $u^{\mbox {\scriptsize \textsc {F}}_{K}[h_{n}]}$ in the proof of Theorem 3.1, Sect. 5.2, Step 1 and Substep 2(a) and the fact that

$$\begin{aligned} \mbox {\textsc {F}}_K[h](x) = \uparrow \mbox{-} \lim_{n\infty} \mbox {\textsc {F}}_K[h_n](x) \mbox{ and } u^{\mbox {\scriptsize \textsc {F}}_K[h]}(t,x) = \uparrow \mbox{-} \lim _{n\infty} u^{\mbox {\scriptsize \textsc {F}}_K[h_n]}(t,x) \end{aligned}$$

for $(t,x) \in[0,T]\times\mathbb{R}^{d}$. From the left equality of the above statement and Proposition 2.6, we deduce that $\tilde{v}^{h}_{K}$ is a viscosity supersolution of

$$\begin{aligned} \left\{ \begin{array}{rcl@{\quad}l} - \mathcal{L}_{\sigma} u(t,x) &=& 0 & \mbox{ for } (t, x) \in[0,T)\times\mathbb{R}^d, \\ u(T,x) &=& \mbox {\textsc {F}}_K[h_n](x) & \mbox{ for } x \in\mathbb{R}^d. \end{array} \right. \end{aligned}$$

(B.1)

Since F _K [h _n ] is Lipschitz-continuous, it is also well known (see, e.g. [19]) that $u^{\mbox {\scriptsize \textsc {F}}_{K}[h_{n}]}$ is a viscosity solution of (B.1) for any n≥1. The PDE (B.1) satisfies the assumptions of Theorem 4.4.5 in [20], which provides a strong comparison theorem for viscosity solutions with polynomial growth. Since the functions $u^{\mbox {\scriptsize \textsc {F}}_{K}[h_{n}]}$ and $\tilde{v}^{h}_{K}$ have linear growth, this yields

$$\begin{aligned} \tilde{v}^h_K(t,x) \ge& u^{\mbox {\scriptsize \textsc {F}}_K[h_n]}(t,x) , \qquad(t,x) \in [0,T]\times\mathbb{R}^d, \end{aligned}$$

for any n≥1. The proof is concluded by letting n go to infinity.  □

Appendix C: Proof of Lemma 4.12

Fix $(t,x)\in[0,T]\times\mathbb{R}^{d}$. We first notice that since γ∈S _d , the terminal condition $\gamma(X^{t,x}_{T}-x)$ can be written as $\gamma(X^{t,x}_{T}-x)= \partial_{x} \bar{h}(X^{t,x}_{T})$ with $\bar{h}$ defined by

$$\begin{aligned} \bar{h}(x') = & {1\over2} (x'-x)^\top\gamma(x'-x) ,\qquad x'\in \mathbb{R}^d. \end{aligned}$$

However, we cannot directly conclude from the H-first order viability of $\mathcal{L}_{\sigma}u=0$ that Δ^t,x belongs to H, since the terminal payoff function h does not belong to $C_{b}^{1}(\mathbb {R}^{d};\mathbb{R})$. We therefore construct a sequence (h _p ) valued in $C^{1}_{b}(\mathbb {R}^{d};\mathbb{R})$ approximating h. Since γn=0, we can write γ=γ ⁺+γ ⁻ with two elements γ ⁺ and γ ⁻ of S _d which are respectively nonnegative and nonpositive and satisfy γ ⁺ n=γ ⁻ n=0.

For all p≥1, define the function h _p by

$$\begin{aligned} h_p(x') = & h_p^+(x')+h_p^-(x'),\quad x'\in\mathbb{R}^d, \end{aligned}$$

where the functions $h^{+}_{p}$, $h^{-}_{p}$ are defined by

$$\begin{aligned} h_p^+(x') = & \left\{ \begin{array}{l@{\quad}l} p( |\sqrt{\gamma^+}(x'-x)|-{p\over2} ) & \mbox{ if } |\sqrt {\gamma^+}(x'-x)| \geq p, \\ {1\over2} (x'-x)^\top\gamma^+ (x'-x) & \mbox{ if } |\sqrt {\gamma^+}(x'-x)| \leq p , \end{array} \right. \\ h_p^-(x') = & \left\{ \begin{array}{l@{\quad}l} -p( |\sqrt{-\gamma^-}(x'-x)|-{p\over2} ) & \mbox{ if } |\sqrt{-\gamma^-}(x'-x)| \geq p, \\ {1\over2} (x'-x)^\top\gamma^- (x'-x) & \mbox{ if } |\sqrt {-\gamma^-}(x'-x)| \leq p , \end{array} \right. \end{aligned}$$

for all $x'\in\mathbb{R}^{d}$. We then easily check that

$$ \partial_x h_p^\pm(x')^\top= \left\{ \begin{array}{l@{\quad}l} p{\gamma^\pm(x'-x) \over|\sqrt{\pm\gamma^\pm} (x'-x)| }& \mbox{ if } |\sqrt{\pm\gamma^\pm}(x'-x)| \geq p, \\ \gamma^\pm(x'-x) & \mbox{ if } |\sqrt{\pm\gamma^\pm }(x'-x)| \leq p , \end{array} \right. $$

for all $x'\in\mathbb{R}^{d}$. Therefore we get from the dominated convergence theorem that

$$\begin{aligned} \mathbb{E}\big[\big|\partial_x h_p(X^{t,x}_T)-\partial_x \bar{h}(X^{t,x}_T)\big|^2\big] \longrightarrow& 0\quad\mbox{ as }p\rightarrow\infty. \end{aligned}$$

(C.1)

Observe that $h_{p}\in C^{1}_{b}(\mathbb{R}^{d};\mathbb{R})$ and ∂ _x h _p is valued in H, for all p≥1. Since the PDE $\mathcal{L}_{\sigma}u=0$ is first order viable for H, we deduce from Proposition 4.3 that

$$\begin{aligned} \Delta_s^{p} \in& H,\qquad t\le s \le T, p\ge1, \end{aligned}$$

(C.2)

where (Δ^p,Λ ^p) is the solution on [t,T] of the BSDE (4.2) associated to the terminal condition $\partial_{x} h_{p}(X^{t,x}_{T})^{\top}$. We get from (C.1) and classical estimates on BSDEs that

$$\begin{aligned} \mathbb{E}\Big[\sup_{t\leq s \leq T}| \Delta^{t,x}_s- \Delta _s^p|^2\Big] \longrightarrow& 0\quad\mbox{ as }p\rightarrow\infty. \end{aligned}$$

Since H is closed, (C.2) together with the previous convergence implies that Δ^t,x is valued in H.  □