Risk arbitrage and hedging to acceptability under transaction costs

Abstract

The classical discrete-time model of proportional transaction costs relies on the assumption that a feasible portfolio process has solvent increments at each step. We extend this setting in two directions, allowing convex transaction costs and assuming that increments of the portfolio process belong to the sum of a solvency set and a family of multivariate acceptable positions, e.g. with respect to a dynamic risk measure. We describe the sets of superhedging prices, formulate several no (risk) arbitrage conditions and explore connections between them. In the special case when multivariate positions are converted into a single fixed asset, our framework turns into the no-good-deals setting. However, in general, the possibilities of assessing the risk with respect to any asset or a basket of assets lead to a decrease of superhedging prices and the no-arbitrage conditions become stronger. The mathematical techniques rely on results for unbounded and possibly non-closed random sets in Euclidean space.

Appendix: Random sets and their selections

Let $\mathbb{R}^{d}$ be the Euclidean space with norm $\vert \cdot \vert$ and the Borel $\sigma$-algebra ${\mathcal{B}}(\mathbb{R}^{d})$. The closure of a set $A\subseteq \mathbb{R}^{d}$ is denoted by ${\mathrm{cl\,}}A$. A set-valued function $\omega \mapsto X(\omega )$ from a complete probability space $(\Omega ,{\mathcal{F}},P)$ to the family of all subsets of $\mathbb{R}^{d}$ is called ℱ-measurable (or graph-measurable) if its graph

$$ \mathrm{Gr\,}X=\{(\omega ,x)\in \Omega \times \mathbb{R}^{d}: x\in X( \omega ) \}\subseteq \Omega \times \mathbb{R}^{d} $$

belongs to the product $\sigma$-algebra ${\mathcal{F}}\otimes {\mathcal{B}}(\mathbb{R}^{d})$. In this case, $X$ is said to be a random set. In the same way, the ℋ-measurability of $X$ with respect to a sub-$\sigma$-algebra ℋ of ℱ is defined. The random set $X$ is said to be closed (convex, open) if $X(\omega )$ is a closed (convex, open) set for almost all $\omega$.

Definition A.1

An ℱ-measurable random element $\xi$ in $\mathbb{R}^{d}$ is said to be an ℱ-measurable selection (selection in short) of $X$ if $\xi (\omega )\in X(\omega )$ for almost all $\omega \in \Omega$. We denote by $L^{0}({\mathcal{F}};X)$ the family of all ℱ-measurable selections of $X$, and $L^{p}({\mathcal{F}};X)$ is the family of $p$-integrable ones.

It is known that an a.s. non-empty random set has at least one selection; see Hess [22, Theorem 4.4]. Let ℋ be a sub-$\sigma$-algebra of ℱ and recall Definition 2.3.

The decomposable subsets of $L^{0}({\mathcal{F}};\mathbb{R}^{d})$ are called stable, and infinitely decomposable ones are called $\sigma$-stable in Cheridito et al. [6]. The following result for ${\mathcal{H}}={\mathcal{F}}$ is well known in case $p=1$ (see Hiai and Umegaki [23] where the decomposability concept was first introduced); see also Molchanov [30, Theorem 2.1.10] for ${\mathcal{H}}={\mathcal{F}}$ and Kabanov and Safarian [26, Proposition 5.4.3] for $p=0$.

Theorem A.2

(See [28, Theorem 2.4] and [30, Theorem 2.1.10])

Let $\varXi$ be a non-empty subset of $L^{p}({\mathcal{F}};\mathbb{R}^{d})$ for $p=0$ or $p\in [1,\infty ]$. Then

$$ \varXi \cap L^{p}({\mathcal{H}};\mathbb{R}^{d})=L^{p}({\mathcal{H}};X) $$

for an ℋ-measurable random closed set $X$ if and only if $\varXi$ is ℋ-decomposable and closed.

For $A_{1},A_{2}\subseteq \mathbb{R}^{d}$, define their elementwise (Minkowski) sum as

$$ A_{1}+A_{2}=\{x_{1}+x_{2}: x_{1}\in A_{1},x_{2}\in A_{2}\}\,. $$

The same definition applies for the sum of subsets of $L^{0}({\mathcal{F}};\mathbb{R}^{d})$. The set of pairwise differences of points from $A_{1}$ and $A_{2}$ is obtained as $A_{1}+(-A_{2})$, or shortly $A_{1}-A_{2}$, where $-A_{2}=\{-x: x\in A_{2}\}$ is the centrally symmetric variant of $A_{2}$. For the sum $A+\{x\}$ of a set and a singleton we write shortly $A+x$. Note that the sum of two closed sets is not necessarily closed unless at least one of the closed summands is compact. The following result differs from [30, Theorem 1.3.25] in considering the possibly non-closed sum of two random closed sets.

Lemma A.3

Let $X$ and $Y$ be two random sets. Then

$$ L^{0}({\mathcal{F}};X)+L^{0}({\mathcal{F}};Y)=L^{0}({\mathcal{F}};X+Y). $$

If both $X$ and $Y$ are random closed sets, then $X+Y$ is measurable.

Proof

It is trivial that $L^{0}({\mathcal{F}};X)+L^{0}({\mathcal{F}};Y)\subseteq L^{0}({ \mathcal{F}};X+Y)$. To prove the converse inclusion, consider $\xi \in L^{0}({\mathcal{F}};X+Y)$. Since $X$ and $Y$ are ℱ-measurable, the measurable selection theorem [26, Theorem 5.4.1] yields that there exist ℱ-measurable selections $\xi '\in L^{0}({\mathcal{F}};X)$ and $\xi ''\in L^{0}({\mathcal{F}};Y)$ such that $\xi =\xi '+\xi ''$.

Assume that $X$ and $Y$ are closed. Consider their Castaing representations (see [30, Definition 1.3.6]) as $X(\omega )={\mathrm{cl\,}}\{\xi '_{i}(\omega ),i\geq 1\}$ and $Y(\omega )={\mathrm{cl\,}}\{\xi ''_{i}(\omega ),i\geq 1\}$. The measurability of $X+Y$ follows from the representation

$$ \mathrm{Gr\,}(X+Y) =\bigcup _{k\geq 1}\bigcap _{m\geq 1} \bigcup _{i,j \geq 1}\bigg\{ (\omega ,x): \vert x-\xi '_{i}(\omega )-\xi ''_{j}( \omega )\vert \le \frac{1}{m}, \vert \xi '_{i}(\omega )\vert \le k \bigg\} . $$

Indeed, if $(\omega ,x)\in \mathrm{Gr\,}(X_{1}+X_{2})$, then $x=a+b$ for $a\in X_{1}(\omega )$ and $b\in X_{2}(\omega )$. Let $k\geq 1$ be such that $\vert a\vert +1\le k$. Since $a\in X_{1}$, there exists a subsequence $(\xi '_{n_{\ell }})_{\ell \geq 1}$ such that $\xi '_{n_{\ell }}(\omega )\to a$. We may assume without loss of generality that $\vert \xi '_{n_{\ell }}(\omega )\vert \le k$. Similarly, $\xi ''_{n_{\ell }}(\omega )\to b$. Therefore $\vert x-\xi '_{i}(\omega )-\xi ''_{j}(\omega )\vert \le \frac{1}{m}$ if $m>0$, and $\vert \xi '_{i}(\omega )\vert \le k$ for some $i,j$. □