1,721,018 research outputs found
Market consistent bid-ask option pricing under Dempster-Shafer uncertainty
No full textWe refer to the discrete-time market model under ambiguity introduced in [Cinfrignini, A., Petturiti, D. and Vantaggi, B., Dynamic bid–ask pricing under Dempster-Shafer uncertainty. J. Math. Econ., 2023a, 107, 102871], formed by a frictionless risk-free bond and a non-dividend paying stock with bid-ask spread. For a European derivative, we generalize the classical binomial pricing formula by allowing for bid-ask prices and investigate the properties of the ensuing replicating strategies. Next, for an American derivative, we propose a backward bid-ask pricing procedure and prove that the resulting discounted price processes are the bid-ask Choquet-Snell envelopes of the discounted payoff process, respectively. Moreover, for an American call option, we prove a generalization of the well-known Merton's theorem holding for both the bid and the ask price processes. Finally, we introduce a market consistent calibration procedure and show the use of the calibrated model in bid-ask option pricing
Market consistent bid-ask option pricing under Dempster-Shafer uncertainty
Full text linkWe refer to the discrete-time market model under ambiguity introduced in [Cinfrignini, A., Petturiti, D. and Vantaggi, B., Dynamic bid-ask pricing under Dempster-Shafer uncertainty. J. Math. Econ., 2023a, 107, 102871], formed by a frictionless risk-free bond and a non-dividend paying stock with bid-ask spread. For a European derivative, we generalize the classical binomial pricing formula by allowing for bid-ask prices and investigate the properties of the ensuing replicating strategies. Next, for an American derivative, we propose a backward bid-ask pricing procedure and prove that the resulting discounted price processes are the bid-ask Choquet-Snell envelopes of the discounted payoff process, respectively. Moreover, for an American call option, we prove a generalization of the well-known Merton's theorem [Merton, R.C., Theory of rational option pricing. Bell J. Econ. Manage. Sci., 1973, 4, 141-183] holding for both the bid and the ask price processes. Finally, we introduce a market consistent calibration procedure and show the use of the calibrated model in bid-ask option pricing
The extent of partially resolving uncertainty in assessing coherent conditional plausibilities
Full text linkHandling uncertainty and reasoning under partial knowledge are challenging tasks that require to deal with coherent assessments and their extensions. Plausibility theory is shown to rest upon the principle of partially resolving uncertainty due to Jaffray, together with a systematically optimistic behavior. This means that we allow situations in which the agent may only acquire the information that a non-impossible event occurs, without knowing which is the true state of the world. This leads to assume that a target event is plausibly true if it is compatible with the acquired piece of information. The aim of the paper is to provide coherence conditions for a conditional plausibility assessment (namely, Pl-coherence), by referring to a suitable axiomatic definition based on the Dempster's rule of conditioning. We provide different equivalent notions of Pl-coherence in terms of consistency, betting scheme, and penalization that, as a by-product, highlight different interpretations. We then specialize the Pl-coherence conditions to the subclasses of (finitely additive) conditional probabilities and (finitely maxitive) conditional possibilities
Modeling agent's conditional preferences under objective ambiguity in Dempster-Shafer theory
No full textWe manage decisions under “objective” ambiguity by considering generalized Anscombe-Aumann acts, mapping states of the world to generalized lotteries on a set of consequences. A generalized lottery is modeled through a belief function on consequences, interpreted as a partially specified randomizing device. Preference relations on these acts are given by a decision maker focusing on different scenarios (conditioning events). We provide a system of axioms which are necessary and sufficient for the representability of these “conditional preferences” through a conditional functional parametrized by a unique full conditional probability P on the algebra of events and a cardinal utility function u on consequences. The model is able to manage also “unexpected” (i.e., “null”) conditioning events and distinguishes between a systematically pessimistic or optimistic behavior, either referring to “objective” belief functions or their dual plausibility functions. Finally, an elicitation procedure is provided, reducing to a Quadratically Constrained Linear Program (QCLP)
Dempster-Shafer Approximations and Probabilistic Bounds in Statistical Matching
No full textMany economic applications require to integrate information coming from different data sources. In this work we consider a specific integration problem called statistical matching, referring to probabilistic distributions of Y|X, Z|X and X, where X, Y, Z are categorical (possibly multi-dimensional) random variables. Here, we restrict to the case of no logical relations among random variables X, Y, Z. The non-uniqueness of the conditional distribution of (Y, Z)|X suggests to deal with sets of probabilities. For that we consider different strategies to get a conditional belief function for (Y, Z)|X that approximates the initial assessment in a reasonable way. In turn, such conditional belief function, together with the marginal probability distribution of X, gives rise to a joint belief function for the distribution of V= (X, Y, Z)
On the identification of discrete graphical models with hidden nodes
No full textWe focus on the identification of discrete undirected graphical
models with one unobserved binary variable and establish a
necessary and sufficient condition for the rank of the
transformation from the natural parameters to the parameters of
the model to be full. This ensures local identification of this
class of models. These models generalize the latent class model,
by allowing associations between the observed variables
conditionally on the latent one. The practical importance of this
issue is witnessed by several applied papers.
For non-full rank models, the obtained characterization allows us to
find the expression of the (sub)space where the identifiability
breaks down. Geometrically, this corresponds to the singularities in
the parameter space. This in turn allows us (a) to derive a reparametrization that leads to an identified model and (b) to compute the correct dimension of the model. The condition is based on the topology of the undirected graph associated with the model and relies on the faithfulness assumption
Conditional decisions under objective and subjective ambiguity in Dempster-Shafer theory
Full text linkThis paper deals with conditional decisions on generalized Anscombe-Aumann acts mapping states of the world to finitely additive probabilities on the set of menus of consequences, the latter conveying a form of “objective” ambiguity. If the decision maker has a systematic pessimistic/optimistic attitude towards “objective” ambiguity, acts reduce to functions mapping states of the world to belief/plausibility functions on consequences. We provide a system of axioms assuring the representability of a family of conditional preference relations on such acts by a conditional functional in which “subjective” uncertainty is modeled through a conditional belief/plausibility function on the states of the world, obeying to a suitable axiomatic definition
Probability envelopes and their Dempster-Shafer approximations in statistical matching
No full textMany economic applications require to integrate information coming from different data sources. In this work we consider a specific integration problem, called statistical matching, referring to integration of data sets where some variables are separately observed and some others are observed in all the data sets. This problem leads to the issue of non-uniqueness for the compatible (conditional) distributions and so it suggests to deal with sets of probabilities. For that we consider different strategies to get a (conditional) belief function that approximates the lower envelope of the class of compatible (conditional) probabilities. We first analyze the case without logical constraints among the variables and then generalize the obtained results by allowing for logical constraints. We finally show an application to real data
Conditional submodular Choquet expected values and conditional coherent risk measures
No full textWe provide an axiomatic definition of conditional submodular capacity that allows conditioning on “null” events and is the basis for the notions of consistency and of consistent extension of a partial assessment. The same definition gives rise to an axiomatic definition of conditional submodular Choquet expected value, which is a conditional functional defined on conditional gambles, that can be expressed as the Choquet integral with respect to its restriction on conditional indicators. Finally, the notion of conditional submodular Choquet expected value is used to provide a definition of conditional submodular coherent risk measure that, locally on every conditioning event, has an upper expected loss interpretation
Dutch book rationality conditions for conditional preferences under ambiguity
No full textWe study preference relations on conditional gambles of a decision maker acting under ambiguity. Dutch book rationality conditions are provided under a linear utility scale, encoding either an optimistic or a pessimistic attitude towards uncertainty. These conditions characterize possibly incomplete preferences representable by totally alternating or monotone conditional functionals. In general, the uniqueness of the representation is not guaranteed, but it can be obtained by adding the hypothesis of existence of a conditional fair price for every conditional gamble. The given rationality conditions have a betting scheme interpretation relying on “penalty fees” for betting on strict preference comparisons
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