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Finance and Stochastics
Erschienen in: Finance and Stochastics 3/2013

01.07.2013

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

verfasst von: Mathias Beiglböck, Pierre Henry-Labordère, Friedrich Penkner

Erschienen in: Finance and Stochastics | Ausgabe 3/2013

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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.
Vorheriger Artikel Duality and convergence for binomial markets with friction
Nächster Artikel Quantitative error estimates for a least-squares Monte Carlo algorithm for American option pricing

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Fußnoten
1
For the sake of simplicity, we assume zero interest rate and no cash/yield dividends. This assumption can be relaxed by considering the process (f t ) introduced in [17] (see Eq. (14)) which has the property to be a local martingale.
 
2
The cumulative distribution function of μ i can be read off from the call prices through \(F_{i}(K)= 1 - \lim_{\varepsilon\downarrow0} 1/\varepsilon[ \mathcal{C}(t_{i},K)-\mathcal{C}(t_{i},K+\varepsilon) ]\) for i=1,…,n. Concerning the mathematical finance application it would be sufficient to consider strikes K≥0 and marginals which are concentrated on the positive half-line. We prefer to go with the more general case since the proofs are not more complicated. A technical difference is that call prices satisfy only \(\lim_{K\to-\infty}\mathcal{C}(t_{i},K) - K = s_{0}\) rather than the simpler \(\mathcal{C}(t_{i},0) = s_{0}\) in the case where S is assumed to be nonnegative.
 
3
Similar strategies are considered in [10, 14] where they are used to subreplicate a European option based on finitely many given call options.
 
4
It might be expected that the delta strategy in (1.2) should also include a constant Δ 0 multiplier of (s 1−s 0) corresponding to an initial forward position. However, this term is not necessary as it can be subsumed into the term u 1.
 
5
In more financial terms, this means that \(\mathcal{C}(t,K)\) is increasing in t for each fixed \(K\in\mathbb{R}\).
 
6
See [36, 37] for an extensive account on the theory of optimal transportation.
 
7
Most of the basic results are equally true for Polish probability spaces (X 1,μ 1),…,(X n ,μ n ), but we do not need this generality here.
 
8
We should like to emphasize that the lower/upper bounds corresponding to different strikes K are attained by different martingale measures. This is not the case if we do not include the martingality constraint, as in this case the upper/lower bounds are attained by the co-monotone resp. anti-monotone coupling for each strike K (see for instance [36, Sect. 2.2.2]).
 
9
In probabilistic terms, the measure \(\mathbb{Q}_{s_{1}}\) is the conditional distribution of S 2 under \(\mathbb{Q}\) given that S 1=s 1.
 
10
We emphasize that while this simple guess works in the present setting, the situation is more subtle for general distributions.
 
11
Formally Hobson and Neuberger are interested to maximize the price of the payoff |S 2−S 1| while we want to minimize the price of−|S 2−S 1|. Mathematically, the two problems are of course the same. We haven chosen the latter formulation to be consistent with the notation in our main result in Theorem 1.1.
 
12
Some progress in this direction is made in [4, Appendix A]. (Note added in revision.)
 
13
I.e., \(g^{**}:\mathbb{R}\to\mathbb{R}\) is the largest convex function smaller than or equal to g.
 
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Metadaten
Titel
Model-independent bounds for option prices—a mass transport approach
verfasst von
Mathias Beiglböck
Pierre Henry-Labordère
Friedrich Penkner
Publikationsdatum
01.07.2013
Verlag
Springer-Verlag
Erschienen in
Finance and Stochastics / Ausgabe 3/2013
Print ISSN: 0949-2984
Elektronische ISSN: 1432-1122
DOI
https://doi.org/10.1007/s00780-013-0205-8

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