Robust option pricing

Bandi, Chaithanya; Bertsimas, Dimitris

doi:10.1016/j.ejor.2014.06.002

Cited by 45 publications

(23 citation statements)

References 17 publications

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“…This is mainly to avoid being overly conservative in each period, instead the uncertainty set U S allows the price to change by a potentially large amount in a single period but forces the change to average out in a longer period of time. As observed in [4], this choice of modeling the cumulative returns leads to linear optimization formulations for the option pricing problem for many kinds of options.…”

Section: The Underlying Primitive For Price Dynamicsmentioning

confidence: 98%

“…In particular, Bandi et al [4] shows that the size of the linear optimization problem scales linearly with T .…”

Section: Propositionmentioning

confidence: 99%

“…In this section, we summarize the major results of Bandi et al [4] and propose to model the underlying price dynamics with uncertainty sets, and then apply robust optimization as opposed to dynamic programming to solve the -arbitrage problem. We use the 1 -norm to measure the error in replication which when combined with polyhedral uncertainty sets results in linear optimization problems.…”

Section: Pricing Multi-dimensional Optionsmentioning

confidence: 99%

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Tractable stochastic analysis in high dimensions via robust optimization

2012

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Modern probability theory, whose foundation is based on the axioms set forth by Kolmogorov, is currently the major tool for performance analysis in stochastic systems. While it offers insights in understanding such systems, probability theory, in contrast to optimization, has not been developed with computational tractability as an objective when the dimension increases. Correspondingly, some of its major areas of application remain unsolved when the underlying systems become multidimensional: Queueing networks, auction design in multi-item, multi-bidder auctions, network information theory, pricing multi-dimensional options, among others. We propose a new approach to analyze stochastic systems based on robust optimization. The key idea is to replace the Kolmogorov axioms and the concept of random variables as primitives of probability theory, with uncertainty sets that are derived from some of the asymptotic implications of probability theory like the central limit theorem. In addition, we observe that several desired system properties such as incentive compatibility and individual rationality in auction design are naturally expressed in the language of robust optimization. In this way, the performance analysis questions become highly structured optimization problems (linear, semidefinite, mixed integer) for which there exist efficient, practical algorithms that are capable of solving problems in high dimensions. We demonstrate that the proposed approach achieves computationally tractable methods for (a) analyzing queueing networks, (b) designing multi-item, multi-bidder auctions with budget constraints, and (c) pricing multi-dimensional options.

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Section: The Underlying Primitive For Price Dynamicsmentioning

confidence: 98%

“…In particular, Bandi et al [4] shows that the size of the linear optimization problem scales linearly with T .…”

Section: Propositionmentioning

confidence: 99%

Section: Pricing Multi-dimensional Optionsmentioning

confidence: 99%

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Tractable stochastic analysis in high dimensions via robust optimization

2012

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“…Here, f is a performance/risk measure that depends on a random element X and a decision variable b that can be chosen from an action space B. The solution to the above problem minimizes worstcase risk over a family of ambiguous probability measures P. Such an ambiguity-averse optimal choice is also referred as a distributionally robust choice, because the performance of the chosen decision variable is guaranteed to be better than OPT irrespective of the model picked from the family P. (2015) for KL-divergence and other likelihood based uncertainty sets, Pflug and Wozabal (2007); Wozabal (2012); Esfahani and Kuhn (2015); Zhao and Guan (2015); Gao and Kleywegt (2016) for Wasserstein distance based neighborhoods, Erdogan and Iyengar (2006) for neighborhoods based on Prokhorov metric, Bandi et al (2015); Bandi and Bertsimas (2014) for uncertainty sets based on statistical tests and Ben-Tal et al (2009); Bertsimas and Sim (2004) for a general overview. As most of the works mentioned above assume the random element X to be R d -valued, it is of our interest in the following example to demonstrate the usefulness of our framework in formulating and solving distributionally robust optimization problems that involve stochastic processes taking values in general Polish spaces as well.…”

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confidence: 99%

Quantifying Distributional Model Risk via Optimal Transport

2019

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This paper deals with the problem of quantifying the impact of model misspecification when computing general expected values of interest. The methodology that we propose is applicable in great generality, in particular, we provide examples involving path-dependent expectations of stochastic processes. Our approach consists in computing bounds for the expectation of interest regardless of the probability measure used, as long as the measure lies within a prescribed tolerance measured in terms of a flexible class of distances from a suitable baseline model. These distances, based on optimal transportation between probability measures, include Wasserstein's distances as particular cases. The proposed methodology is well-suited for risk analysis, as we demonstrate with a number of applications. We also discuss how to estimate the tolerance region non-parametrically using Skorokhod-type embeddings in some of these applications.

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“…The first example of such a work is Bertsimas et al (2011), which analyzes queuing networks with a robust uncertainty set motivated by the probabilistic law of the iterated logarithm. Next, Bandi and Bertsimas (2012) provide an in-depth study of the use of CLT-style uncertainty sets; these sets have been applied to information theory by Bandi and Bertsimas (2011), option pricing by Bandi and Bertsimas (2014b), auction design by Bandi and Bertsimas (2014a), and queueing theory by Bandi et al (2015Bandi et al ( , 2016. More relevant to our paper is Mamani et al (2016), who studied robust inventory management and whose design of uncertainty sets was also motivated by the CLT.…”

Section: Literature Reviewmentioning

confidence: 99%

Robust Inventory Management: An Optimal Control Approach

Wagner

2018

Operations Research

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We formulate and solve static and dynamic models of inventory management that lie at the intersection of robust optimization and optimal control theory. Our objective is to minimize cumulative ordering, holding, and shortage costs over a horizon [0, T], where the variable is a nonnegative ordering rate function q(t) 2 L 2 [0, T]. The demand rate function d(t) is unknown and is only assumed to belong to an uncertainty set ⌦ ⇤by the strong law of large numbers for stochastic processes limWe analyze a static model, where the ordering rate function must be fully specified at time zero, and three dynamic variants, where re-optimizations are allowed during the planning horizon [0, T] at prespecified review epochs. In the dynamic models, at review epoch ⌧ 2 [0, T], the past demand on [0, ⌧) is observable. In the first dynamic model, we ignore this information, and define a variant of ⌦ that is well formed for the remaining planning horizon [⌧, T]. In the second model, we define a variant of ⌦ for [⌧, T] that utilizes the past demand information, though we make a simplifying technical assumption about the consistency of the demand on [0, ⌧) and ⌦. In the third dynamic model, we remove this assumption, and we remedy the arising complications using the Hilbert Projection Theorem. In all cases we derive optimal closed-form ordering rate functions that equal either the bounds a or b, or weighted averages of these bounds (sb + ha)/(s + h) or (sa + hb)/(s + h), where s and h are the shortage and holding costs, respectively. The strategies differ by when these four ordering rates are applied, which is determined by an uncertainty-set-dependent partition of the remaining planning horizon. Computational experiments, focused on studying the dynamic variants, supplement the analytical results, and demonstrate that (1) the three variants exhibit comparable performance under well-behaved stochastic demand and (2) the third variant has a significant advantage when demand is seasonal, especially when the review frequency is appropriately selected. Finally, computational comparisons with the omniscient strategy q(t) ⇤ d(t), for all t 2 [0, T], are encouraging.

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Robust option pricing

Cited by 45 publications

References 17 publications

Tractable stochastic analysis in high dimensions via robust optimization

Tractable stochastic analysis in high dimensions via robust optimization

Quantifying Distributional Model Risk via Optimal Transport

Robust Inventory Management: An Optimal Control Approach

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