Model-independent bounds for option prices—a mass transport approach

Abstract

In this paper we investigate model-independent bounds for exotic options written on a risky asset using infinite-dimensional linear programming methods. Based on arguments from the theory of Monge–Kantorovich mass transport, we establish a dual version of the problem that has a natural financial interpretation in terms of semi-static hedging. In particular we prove that there is no duality gap.

Appendix

As a special case of [28, Theorem 2.14], we have the duality equation

for every lower semi-continuous cost function $\varPhi:\mathbb{R}^{n}\to [0,\infty]$. The main task in the subsequent proof of Proposition 2.1 is to show that the duality equation is obtained if one restricts to functions in the class $\mathcal{S}$ in the dual problem.

Proof of Proposition 2.1

As in the proof of Theorem 1.1, it is sufficient to prove the duality equation in the case Φ≥0.

Given a bounded continuous function f and ε>0, there is for every i=1,…,n some $u\in\mathcal{S}$ such that f≥u and ∫(f−u)dμ _i <ε. Therefore we may change the class of admissible functions from $\mathcal{S}$ to $C_{b}(\mathbb{R})$, i.e., it suffices to prove

(A.1)

We first show this under the additional assumption that $\varPhi \in C_{c}(\mathbb{R}^{n})$. By [28, Theorem 2.14], there exist for each η>0 μ _i -integrable functions u _i , i=1,…,n, such that

$$P_\mathrm{MK}(\varPhi)-\sum_{i=1}^n \int u_i\, d\mu_i\leq\eta $$

and u ₁⊕⋯⊕u _n ≤Φ. Note that the latter inequality implies that u ₁,…,u _n are uniformly bounded since Φ is uniformly bounded from above.

To replace u ₁ by a function in C _b , we consider H=Φ−(u ₁⊕⋯⊕u _n ) and define

$$ \tilde{u}_1 (x_1):= \inf_{x_2,\ldots, x_n\in\mathbb{R}} H(x_1, \ldots, x_n) $$

for $x_{1}\in\mathbb{R}$. We claim that $\tilde{u}_{1}$ is (uniformly) continuous. Indeed, as Φ is uniformly continuous, for every ε>0 there exists δ>0 such that whenever $x,x'\in \mathbb{R}$, |x−x′|<δ, then

Thus we obtain

$$| \tilde{u}_1 (x)-\tilde{u}_1 (x')|=\Big|\inf_{x_2,\ldots, x_n\in \mathbb{R}} H(x,x_2, \ldots, x_n) - \inf_{x_2,\ldots, x_n\in\mathbb{R}} H(x',x_2, \ldots, x_n)\Big|\leq\varepsilon $$

whenever |x−x′|<δ. By definition $\tilde{u}_{1}$ is also bounded from below and satisfies $\tilde{u}_{1}\geq u_{1}$ as well as

$$\tilde{u}_1 \oplus u_2\oplus\cdots\oplus u_n\leq\varPhi. $$

Iteratively replacing the functions u ₂,…,u _n in the same fashion, we obtain (A.1) in the case $\varPhi\in C_{c}(\mathbb{R}^{n})$.

Finally, the same argument as in the proof of Theorem 1.1 gives the duality relation in the case of a general, lower semi-continuous function $\varPhi:\mathbb{R}^{n}\to[0,\infty ]$. □