Model-independent superhedging under portfolio constraints

Abstract

In a discrete-time market, we study model-independent superhedging where the semi-static superhedging portfolio consists of three parts: static positions in liquidly traded vanilla calls, static positions in other tradable, yet possibly less liquid, exotic options, and a dynamic trading strategy in risky assets under certain constraints. By considering the limit order book of each tradable exotic option and employing the Monge–Kantorovich theory of optimal transport we establish a general superhedging duality, which admits a natural connection to convex risk measures. With the aid of this duality, we derive a model-independent version of the fundamental theorem of asset pricing. The notion “finite optimal arbitrage profit”, weaker than no-arbitrage, is also introduced. It is worth noting that our method covers a large class of delta and gamma constraints.

Appendix: An example related to Definition 2.7(iii)

In this appendix, we provide an example showing that if Definition 2.7(iii) is not satisfied, then the duality in Proposition 3.10 may fail. Let $d=1$, $T=2$, and $x_{0}=1$. Assume that $\mu_{1}(dx)=\frac{1}{2}\delta_{1}(dx)+\frac{1}{2}\delta_{2}(dx)$ and $\mu_{2}(dx)=\delta_{2}(dx)$, where $\delta_{x}$ is the Dirac measure at $x\in \mathbb {R}$. Thus, $\varPi=\{\mathbb {Q}\}$ with ${\mathbb {Q}[S_{1}=1,S_{2}=2]}=\mathbb {Q}[S_{1}=2,S_{2}=2]=\frac{1}{2}$. Consider the collection of trading strategies

$$\mathcal {S}=\{\varDelta=(\varDelta_{0},\varDelta_{1}): \varDelta_{0}\equiv0,\ \varDelta_{1}(x)=\alpha 1_{\{x=1\}}(x)\ \hbox{for some}\ \alpha\in[0,1]\}. $$

While $\mathcal {S}$ trivially satisfies (i) and (ii) in Definition 2.7, (iii) does not hold. To see this, note that $\mathcal {S}_{c}^{\infty}=\{ (0,0)\}$, and thus for any $\varDelta\in \mathcal {S}$ with $\alpha>0$, we have $\mathbb {Q}[\varDelta\neq(0,0)]= 1/2$. In order to superhedge the claim $\varPhi(x_{1},x_{2})\equiv0$, we need to find $n, m\in \mathbb {N}$, $a, b_{i}, c_{j}\in \mathbb {R}$, $K_{i}^{1}, K_{j}^{2}\ge0$, and $\varDelta\in \mathcal {S}$ such that for all $(x_{1},x_{2})\in \mathbb {R}^{2}_{+}$,

$$ 0\le a+\sum_{i=1}^{n}b_{i}(x_{1}-K_{i}^{1})^{+}+\sum_{j=1}^{m}c_{j}(x_{2}-K_{j}^{2})^{+}+\varDelta _{0}(x_{1}-x_{0})+\varDelta_{1}(x_{1})(x_{2}-x_{1}). $$

Since $\varDelta_{0}\equiv0$ and $\varDelta(x_{1})=\alpha1_{\{x_{1}=1\}}$, this inequality reduces to

$$\begin{aligned} f_{\alpha}(x_{1},x_{2}) &:= -\alpha1_{\{x_{1}=1\}}(x_{1})(x_{2}-x_{1}) \\ &\phantom{:}\le a+\sum_{i=1}^{n}b_{i}(x_{1}-K_{i}^{1})^{+}+\sum_{j=1}^{m}c_{j}(x_{2}-K_{j}^{2})^{+} \end{aligned}$$

(A.1)

for all $(x_{1},x_{2})\in \mathbb {R}^{2}_{+}$. Let $f^{*}_{\alpha}$ denote the upper semicontinuous envelope of $f_{\alpha}$. We observe that (A.1) holds for $f_{\alpha}$ if and only if it also holds for $f^{*}_{\alpha}$. It follows that

$$ \begin{aligned} D_{\emptyset}(0)&=\inf_{0\le\alpha\le1}D_{\emptyset}(f_{\alpha})=\inf_{0\le \alpha\le1}D_{\emptyset}(f^{*}_{\alpha})=\inf_{0\le\alpha\le1} P_{\emptyset}(f^{*}_{\alpha})\\ & = \inf_{0\le\alpha\le1}\alpha \mathbb {E}^{\mathbb {Q}}\big[1_{\{S_{1}=1\} }(S_{1})(S_{2}-S_{1})^{-}\big]=0, \end{aligned} $$

where the third equality follows from Proposition 3.10, and the fourth is due to $f^{*}_{\alpha}= \alpha1_{\{x_{1}=1\}}(x_{1})(x_{2}-x_{1})^{-}$. On the other hand, since

$$A^{\mathbb {Q}}_{2}=\sup_{\alpha\in[0,1]}\alpha \mathbb {Q}[S_{1}=1]=\frac{1}{2}, $$

we have $P_{\emptyset}(0)=-\mathbb {E}^{\mathbb {Q}}[A^{\mathbb {Q}}_{2}]=-\frac{1}{2}$, which indicates a duality gap.