Model-independent bounds for option prices -- a mass transport approach
type: Item
instance of: scholarly article
title: Model-independent bounds for option prices -- a mass transport approach
zbMATH Open document ID: 1277.91162
author: Mathias Beiglböck
author: Pierre Henry-Labordère
author: F. Penkner
published in: Finance and Stochastics
publication date: 2013-07-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/1106.5929
review text: 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 M…
Mathematics Subject Classification ID: 91G20
Mathematics Subject Classification ID: 91G80
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 90C08
zbMATH DE Number: 6189062
zbMATH Keywords: model-independent pricing
zbMATH Keywords: Monge-Kantorovich transport problem
zbMATH Keywords: option arbitrage
zbMATH Keywords: semi-static subhedging
zbMATH Keywords: minimax
zbMATH Keywords: martingale transport plan
zbMATH Keywords: lower semi-continuouity
zbMATH Keywords: duality
MaRDI profile type: Publication
OpenAlex ID: W3122203059
arXiv ID: 1106.5929
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DOI: 10.1007/S00780-013-0205-8
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reviewed by: Weiping Li
description: scientific article; zbMATH DE number 6189062