Model-independent bounds for option prices -- a mass transport approach
The paper under review is to investigate model-independent bounds for exotic options written on a risky asset by infinite-dimensional linear programming methods, and to prove that there is no duality gap from the theory of Monge-Kantorovich mass transport. The novelty of this paper is to adapt the theory of optimal transport to obtain robust model-independent bounds on option prices. The problem of robust pricing in a multi-period setting for exotic options seems difficult to provide explicitly the optimal lower/upper bounds. An exotic option of the underlying risky asset \(S_t\) at discrete times \(t_1< \cdots < t_n\) has the payoff \(\Phi (S_1, \dots, S_n)\), where \(S_i(s_1, \dots, s_n) = s_i\) and \(\Phi_{i, K} (S_i) = (S_i-K)^+\) at time \(t_i\) with price \(E^Q[\Phi_{i, K}]\) and \(Q^i =\mathrm{Law}_{S_i}=\mu_i\). The set \(M(\mu_1, \dots, \mu_n)\) consists of all martingale measures \(Q\) on the path space with \(Q^i= \mu_i\) and mean \(s_0\). The primal problem in the optimal transport is \(P = \inf_{Q\in M(\mu_1, \dots, \mu_n)} E^Q[\Phi]\). The dual formulation is a construction of a semi-static subhedging strategy consisting of the sum of a static vanilla portfolio and a delta strategy \(\Psi_{u_i, \Delta_j}(s_1, \dots, s_n)= \sum_{i=1}^n u_i(s_i) = \sum_{j=1}^{n-1}\Delta_j(s_1, \dots, s_j)(s_{j+1}-s_j)\). For any \(Q\in M(\mu_1, \dots, \mu_n)\), \(E^Q[\Phi] \geq E^Q[\Psi_{u_i, \Delta_j}] = \sum_{i=1}^nE^{Q_i}[u_i]\). The dual problem \(D=\sup \{\sum_{i=1}^nE^{Q_i}[u_i]: \Psi_{u_i, \Delta_j} \leq \Phi\}\) shows that \([P \geq D]\). The main result of the paper under review (Theorem 1.1) states that \(P=D\) if \(\Phi\) is a lower-semi-continuous function with \(\Phi(s_1, \dots, s_n) \geq - K(1+|s_1|+\cdots + |s_n|)\), and \(P\) is attained by a martingale measure \(Q\in M(\mu_1, \dots, \mu_n)\). The dual supremum \(D\) is not attained in general with an artificial example provided in Proposition 4.1. Section 2 puts the above primal problem into the primal Monge-Kantorovich problem in optimal transport so that assumptions on the measures \(Q^i\) to be increasing in the convex order and \(M(\mu_1, \dots, \mu_n) \) to be non-empty, are required. Proposition 2.4 shows that \(M(\mu_1, \dots, \mu_n)\) is compact in weak topology. Theorem 1.1 basically follows from the minimax theorem of decision theory and a Monge-Kantorovich duality theorem in Theorem 3.1. The attainable measure \(Q\) for the primal problem follows from the compactness of the set of all martingale transport plans, but the dual set of semi-static subheadgings does not possess the compactness. A complete proof of the main result is given in Section 3. Section 4 provides an example for the non-attainable dual problem. It would be interesting to find proper conditions such that the dual supremum \(D\) is attained by choices of \(\Delta_j\) with \(\Psi_{u_i, \Delta_j} \leq \Phi\). It would be even more interesting to see the proposed multi-period exotic option in this paper tradable in the financial world.
scientific article; zbMATH DE number 6189062
scientific article