Robust hedging with proportional transaction costs

Abstract

A duality for robust hedging with proportional transaction costs of path-dependent European options is obtained in a discrete-time financial market with one risky asset. The investor’s portfolio consists of a dynamically traded stock and a static position in vanilla options, which can be exercised at maturity. Trading of both options and stock is subject to proportional transaction costs. The main theorem is a duality between hedging and a Monge–Kantorovich-type optimization problem. In this dual transport problem, the optimization is over all probability measures that satisfy an approximate martingale condition related to consistent price systems, in addition to an approximate marginal constraint.

Appendix

In this appendix, we prove Theorem 3.2. We proceed in several lemmas. We first use Lemma 3.1 to reduce the problem to bounded claims. Then, using a compactness argument as in [3], we further reduce it to bounded and continuous claims.

Lemma 5.1

Suppose that (2.8) holds for all bounded upper semicontinuous functions. Then it also holds for all G satisfying Assumption 2.1.

Proof

Suppose that G satisfies Assumption 2.1. Let φ be any smooth function satisfying

$$0 \le\varphi\le1, \quad \varphi(\mathbb {S}) =1 \quad \forall \| \mathbb {S}\| \le1,\qquad \varphi(\mathbb {S}) =0 \quad \forall \| \mathbb {S}\| \ge2. $$

For a constant M>1, set

$$\varphi_M(\mathbb {S}):= \varphi(\mathbb {S}/M),\qquad G_M := G \varphi_M. $$

G _M is bounded and upper semicontinuous. Then, by the hypothesis, inequality (2.8) and the duality formula stated in Theorem 2.6 hold for G _M . In view of Assumption 2.1,

$$\left|G(\mathbb {S}) - G_M(\mathbb {S})\right| \le L(1+\|\mathbb {S}\|^2) \chi_{\{ \| \mathbb {S}\| \ge M\}}. $$

Let α _M be as in Lemma 3.1. Then, for all sufficiently large M,

$$\left|G(\mathbb {S}) - G_M(\mathbb {S})\right| \le2L \alpha_{M}(\mathbb {S}). $$

Since G _M satisfies (2.8),

$$V(G_M) \le\sup_{\mathbb{Q}\in\mathcal{M}_{\kappa,\mathcal {P}} }\mathbb {E}_{\mathbb{Q}} \left[G_M(\mathbb {S})\right] \le\sup_{\mathbb{Q}\in\mathcal{M}_{\kappa,\mathcal {P}} }\mathbb{E}_{\mathbb{Q}} \left[G(\mathbb {S})\right] + 2L \sup_{\mathbb{Q}\in\mathcal{M}_{\kappa,\mathcal {P}} }\mathbb{E}_{\mathbb{Q}} \left[\alpha_M(\mathbb {S})\right] . $$

By the subadditivity of the minimal superreplication cost V,

$$V(G) \le V(G_M) + 2L V (\alpha_{M}). $$

Combining the above inequalities and Lemma 3.1, we arrive at

$$\begin{aligned} V(G) \le& \liminf_{M\to\infty}\big( V(G_M) + 2L V (\alpha_{M} )\big)\\ \le&\sup_{\mathbb{Q}\in\mathcal{M}_{\kappa,\mathcal {P}} }\mathbb{E}_{\mathbb{Q}} \left[G(\mathbb {S})\right] + 2L \liminf_{M\to\infty} \Big( V (\alpha_{M} ) + \sup_{\mathbb{Q}\in\mathcal{M}_{\kappa,\mathcal {P}} }\mathbb{E}_{\mathbb{Q}} \left[\alpha_M(\mathbb {S})\right] \Big) \\ \leq&\sup_{\mathbb{Q}\in\mathcal{M}_{\kappa,\mathcal {P}} }\mathbb{E}_{\mathbb{Q}} \left[G(\mathbb {S})\right] . \end{aligned}$$

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The above proof also yields the following equivalence.

Lemma 5.2

Suppose that (2.8) holds for all nonnegative, bounded, uniformly continuous functions. Then it also holds for all G that are bounded and continuous.

Proof

Let G be a bounded continuous function. By adding to G an appropriate constant, we may assume that it is nonnegative as well. Given an integer N, define G _N as before. Since G _N is compactly supported and continuous, it is also uniformly continuous. We then proceed exactly as in the previous lemma to conclude the proof. □

We need the following elementary result.

Lemma 5.3

Let G be bounded and upper semicontinuous. Then there exists a uniformly bounded sequence of continuous functions $G_{n}:\mathbb {R}_{+}^{d} \to \mathbb {R}$ with G _n ≥G and

$$ \limsup_{n\rightarrow\infty}G_n(x_n)\leq G(x) $$

(5.1)

for every $x\in\mathbb{R}^{d}_{+}$ and every sequence $\{x_{n}\}_{n=1}^{\infty}\subset\mathbb{R}^{d}_{+}$ with lim _n→∞ x _n =x.

Proof

For $n \in \mathbb {N}$, consider the grid $O_{n}= \{ (\frac{k_{1}}{n},\dots,\frac{k_{d}}{n} ), k_{1},\dots,k_{d}\in\mathbb{Z}_{+} \}$. Define the function $G_{n}:O_{n}\rightarrow\mathbb{R}_{+}$ by

$$G_n(x)=\sup_{ u\in\mathbb{R}^d_{+}, \|u-x\|\leq\frac{2}{n} } G(u), \quad x\in O_n. $$

Next, we extend G _n to the domain $\mathbb{R}^{d}_{+}$.

For any $k_{1},\dots,k_{d}\in\mathbb{Z}_{+}$ and a permutation σ:{1,…,d}→{1,…,d}, consider the d-simplex

$$U^\sigma_{k_1,\dots,k_d}=\Big\{ (x_1,\dots,x_d)\Big| \frac{k_i}{n}\leq x_i\leq\frac{k_i+1}{n} \ {\mbox{and}}\ x_{\sigma(i)}\leq x_{\sigma(j)}, 1\le i<j\le d\Big\} . $$

Fix a simplex $U^{\sigma}_{k_{1},\dots,k_{d}}$. Any $u\in U^{\sigma}_{k_{1},\dots,k_{d}}$ can be represented uniquely as a convex combination of the simplex vertices u ₁,…,u _d+1 (which belong to O _n ). Thus, define the continuous function $G^{n,\sigma}_{k_{1},\dots,k_{d}}:U^{\sigma}_{k_{1},\dots,k_{d}}\rightarrow\mathbb{R}$ by $G^{n,\sigma}_{k_{1},\dots,k_{d}}(u)=\sum_{i=1}^{d+1}\lambda_{i} G_{n}(u_{i})$, where λ ₁,…,λ _d+1∈[0,1] with $\sum _{i=1}^{d+1}\lambda_{i}=1$, and $\sum_{i=1}^{d+1}\lambda_{i} u_{i}=u$ are uniquely determined.

Any element $u\in\mathbb{R}^{d}_{+}$ belongs to at least one simplex of the above form. Moreover, observe that if u belongs to two simplexes, say $U^{\sigma}_{k_{1},\dots,k_{d}}$ and $U^{\sigma'}_{k'_{1},\dots,k'_{d}}$, then $G^{n,\sigma}_{k_{1},\dots,k_{d}}(u)=G^{n,\sigma'}_{k'_{1},\dots,k'_{d}}(u)$. Thus, we can extend the function $G_{n}:O_{n}\rightarrow\mathbb{R}$ to a function $G_{n}:\mathbb{R}^{d}_{+}\rightarrow\mathbb{R}$ by setting $G_{n}(u)=G^{n,\sigma}_{k_{1},\dots,k_{d}}(u)$ for $u\in U^{\sigma}_{k_{1},\dots,k_{d}}$, where $k_{1},\dots,k_{d}\in\mathbb{Z}_{+}$, and σ:{1,…,d}→{1,…,d} is a permutation.

This sequence has the desired properties. □

The following result completes the proof of Theorem 3.2.

Lemma 5.4

Suppose that (2.8) holds for all bounded continuous functions. Then it also holds for all bounded upper semicontinuous G.

Proof

Let G be bounded and upper semicontinuous. Let G _n be the sequence of bounded continuous functions constructed in Lemma 5.3. Hence, (2.8) and Theorem 2.6 hold for G _n .

Using Theorem 2.6, we choose a sequence of probability measures $\mathbb{Q}_{n}\in\mathcal{M}_{\kappa,\mathcal {P}}$ satisfying

$$ \mathbb{E}^{(n)} [G_n(\mathbb{S})]>V(G_n)-\frac{1}{n}. $$

(5.2)

Using similar compactness arguments as in Lemma 3.6, we construct a subsequence $\mathbb{Q}_{n_{\ell}}$, $\ell\in\mathbb{N}$, that converges weakly to a probability measure $\tilde{\mathbb{Q}}\in\mathcal{M}_{\kappa,\mathcal {P}}$. Recall that the G _n are uniformly bounded. Thus, by (5.1) and the Skorokhod representation theorem,

$$\limsup_{\ell\rightarrow\infty} \mathbb{E}^{(n_\ell)} [G_{n_\ell} (\mathbb{S})] \leq\tilde{\mathbb{E}} [G(\mathbb{S})]. $$

This together with (5.2) yields that

$$V(G)\leq\liminf_{n\rightarrow\infty}V(G_n)\leq \liminf_{n\rightarrow\infty}\mathbb{E}^{(n)} [G_n(\mathbb{S})] \leq \tilde{\mathbb{E}}[ G(\mathbb{S})]\leq \sup_{\mathbb{Q}\in\mathcal{M}_{\kappa,\mathcal {P}} }\mathbb{E}_{\mathbb{Q}} \left[G(\mathbb {S})\right]. $$

This completes the proof. □