Probability Distributions of Assets Inferred from Option Prices via the Principle of Maximum Entropy

Borwein, Jonathan M.; Choksi, Rustum; Maréchal, Pierre

doi:10.1137/s1052623401400324

Cited by 36 publications

(51 citation statements)

References 12 publications

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“…The Buchen-Kelly density has the greatest entropy among all densities over [0, ∞[ matching call prices (1), and its implied digital prices verify (4). In other words, the Buchen-Kelly is the element of G with the greatest entropy.…”

Section: The Med Obtained From Call and Digital Pricesmentioning

confidence: 98%

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A Family of Maximum Entropy Densities Matching Call Option Prices

Neri

Schneider

2013

Applied Mathematical Finance

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We investigate the position of the Buchen-Kelly density [10] in the family of entropy maximising densities from [21] which all match European call option prices for a given maturity observed in the market. Using the Legendre transform which links the entropy function and the cumulant generating function, we show that it is both the unique continuous density in this family and the one with the greatest entropy. We present a fast root-finding algorithm that can be used to calculate the Buchen-Kelly density, and give upper boundaries for three different discrepancies that can be used as convergence criteria. Given the call prices, arbitrage-free digital prices at the same strikes can only move within upper and lower boundaries given by left and right call spreads. As the number of call prices increases, these bounds become tighter, and we give two examples where the densities converge to the Buchen-Kelly density in the sense of relative entropy when we use centered call spreads as proxies for digital prices. As pointed out by Breeden and Litzenberger [6], in the limit a continuous set of call prices completely determines the density.

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Section: The Med Obtained From Call and Digital Pricesmentioning

confidence: 98%

“…Making hard to justify ad-hoc assumptions becomes unnecessary, and it is therefore no wonder that this approach is becoming more and more popular in the financial literature (see [3], [4], [7], [8], [9], [10], [12], [15], [16], [17]). …”

Section: Introductionmentioning

confidence: 99%

A Family of Maximum Entropy Densities Matching Call Option Prices

Neri

Schneider

2013

Applied Mathematical Finance

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“…Applications range from the estimation of the power spectrum of a signal [3], to the more recent inference of the probability distribution for the price of an asset from option prices [14], [12].…”

Section: (27)mentioning

confidence: 99%

“…Such objects naturally fit within the context of spectral estimation, where the objective function is entropy-like. For entropy optimization problems, which are pervasive in engineering fields [26] and have recently found application in finance [14,12], the goal is to describe the properties of a stochastic process based on the knowledge of its moments. In an effort to rigorously formalize the arguments used in the published literature on this subject to derive optimal solutions, Borwein and Lewis [8, 10, 11] developed a general mathematical framework based on duality under which to study these kind of problem.…”

Section: Introductionmentioning

confidence: 99%

Strategic dynamic vehicle routing with spatio-temporal dependent demands

Feijer

Savla

2012

2012 American Control Conference (ACC)

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Dynamic vehicle routing problems address the issue of determining optimal routes for a set of vehicles, to serve a given set of demands that arrive sequentially in time. Traditionally, demands are assumed to be generated over time by an exogenous stochastic process. This thesis is concerned with the study of dynamic vehicle routing problems where demands are strategically placed in the space by an agent with selfish interests and physical constraints. In particular, we focus on the following problem: a team of vehicles seek to device dynamic routing policies that minimize the average waiting time of a typical demand, from the moment it is placed in the space until its location is visited; while an adversarial agent operating from a central depot with limited capacity aims at the opposite, strategically choosing the spatio-temporal point process according to which place demands.We model the above problem and its inherent pure conflict of interests as a zero-sum game, and characterize equilibria under heavy load regime. For the analysis we discriminate between two cases: bounded and unbounded domains. In both cases we show that a routing policy based on performing successive TSP tours through outstanding demands and a power-law spatial distribution of demands are optimal, saddle point of the utility function of the game. The latter emerges as the unique solution of maximizing a non-convex nowhere differentiable functional over the infinite-dimensional space of probability densities; the non-convexity is the result of the spatio-temporal dependence induced by the physical constraints imposed on the behavior of the agent, and the non-differentiability is due to the emptiness of the interior of the positive cone of integrable functions. We solve this problem applying Fenchel conjugate duality for partially finite programming in the case of bounded domains; and a direct duality approach exploiting the structure of a concave integral functional part of the objective and the linear equality constraints, for unbounded domains. Remarkably, all the results obtained hold for any domain with a sufficiently smooth boundary, clossedness or connectedness is not needed. We provide numerical simulations to validate the theory.

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“…Besides the works mentioned above, the maximum entropy method could be used to estimate the implied correlations between different currency pairs [20], to retrieve the neutral density of future stock risks or other asset risks [21], and to infer the implied probability density and distribution from option prices [22,23]. Stutzer and Hawkins [24,25] even used the MEP to price derivative securities such as futures and swaps.…”

Section: Introductionmentioning

confidence: 99%

Applications of Entropy in Finance: A Review

Zhou

Cai

Tong

2013

Entropy

219

129

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Although the concept of entropy is originated from thermodynamics, its concepts and relevant principles, especially the principles of maximum entropy and minimum cross-entropy, have been extensively applied in finance. In this paper, we review the concepts and principles of entropy, as well as their applications in the field of finance, especially in portfolio selection and asset pricing. Furthermore, we review the effects of the applications of entropy and compare them with other traditional and new methods.

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Probability Distributions of Assets Inferred from Option Prices via the Principle of Maximum Entropy

Cited by 36 publications

References 12 publications

A Family of Maximum Entropy Densities Matching Call Option Prices

A Family of Maximum Entropy Densities Matching Call Option Prices

Strategic dynamic vehicle routing with spatio-temporal dependent demands

Applications of Entropy in Finance: A Review

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