European option pricing under cumulative prospect theory with constant relative sensitivity probability weighting functions

Computational Management Science - In this contribution, we evaluate European financial options under continuous cumulative prospect theory. In prospect theory, risk attitude and loss aversion are...     

Special functions and characterizations of probability distributions by zero regression properties

Characterizations of the binomial, negative binomial, gamma, Poisson, and normal distributions are obtained by the property of zero regression of certain polynomial statistics of arbitrary degree, on the mean. In each case, the equations which express zero regression are derived from the recurrence relations of a set of special functions. The differential recurrence relations of these special functions are used in the proofs of the characterization theorems.     

Some new approaches to multivariate probability distributions

We extend and generalize to the multivariate set-up our earlier investigations related to expected remaining life functions and general hazard measures including representations and stability theorems for arbitrary probability distributions in terms of these concepts. (The univariate case is discussed in detail in Kotz and Shanbhag, Advan. Appl. Probab. 12 (1980), 903–921.)     

Fitting combinations of exponentials to probability distributions

Two techniques are described for approximating distributions on the positive half‐line by combinations of exponentials. One is based on Jacobi polynomial expansions, and the other on the logbeta distribution. The techniques are applied to some well‐known distributions (degenerate, uniform, Pareto, lognormal and others). In theory, the techniques yield sequences of combination of exponentials that always converge to the true distribution, but their numerical performance depends on the particular distribution being approximated. An error bound is given in the case the logbeta approximations. Copyright © 2006 John Wiley & Sons, Ltd.     

Generalization of the theorem of Marcinkiewicz on entire characteristic functions of probability distributions

The relation is studied between the distribution of the zeros and the order of growth of entire analytic functions for which ¦p(z)¦ (i Imz) for Imz 0, in particular, of entire characteristic functions of probability distributions. The main result is the following: if ₁ is the exponent of convergence of the sequence of zeros of such a function of order which lie in a half plane Imz d > 0, then the inequality ₁ < implies the inequality p 3. This estimate is precise.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 85, pp. 94–103, 1979.The author is grateful to I. V. Ostrovskii for posing the problem and for his constant assistance with the work.     

Extending pricing rules with general risk functions

The paper addresses pricing issues in imperfect and/or incomplete markets if the risk level of the hedging strategy is measured by a general risk function. Convex Optimization Theory is used in order to extend pricing rules for a wide family of risk functions, including Deviation Measures, Expectation Bounded Risk Measures and Coherent Measures of Risk. Necessary and sufficient optimality conditions are provided in a very general setting. For imperfect markets the extended pricing rules reduce the bid-ask spread. The findings are particularized so as to study with more detail some concrete examples, including the Conditional Value at Risk and some properties of the Standard Deviation. Applications dealing with the valuation of volatility linked derivatives are discussed.     

Parameter estimation by Hellinger type distance for multivariate distributions based upon probability generating functions

Maximum likelihood (ML) estimation is a popular method for parameter estimation when modeling discrete or count observations but unfortunately it may be sensitive to outliers. Alternative robust methods like minimum Hellinger distance (MHD) have been proposed for estimation. However, in the multivariate case, the MHD method leads to computer intensive estimation especially when the joint probability density function is complicated. In this paper, a Hellinger type distance measure based on the probability generating function is proposed as a tool for quick and robust parameter estimation. The proposed method yields consistent estimators, performs well for simulated and real data, and can be computationally much faster than ML or MHD estimation.     

Contingent claim pricing using probability distortion operators: methods from insurance risk pricing and their relationship to financial theory

This paper considers the pricing of contingent claims using an approach developed and used in insurance pricing. The approach is of interest and significance because of the increased integration of insurance and financial markets and also because insurance-related risks are trading in financial markets as a result of securitization and new contracts on futures exchanges. This approach uses probability distortion functions as the dual of the utility functions used in financial theory. The pricing formula is the same as the Black-Scholes formula for contingent claims when the underlying asset price is log-normal. The paper compares the probability distortion function approach with that based on financial theory. The theory underlying the approaches is set out and limitations on the use of the insurance-based approach are illustrated. The probability distortion approach is extended to the pricing of contingent claims for more general assumptions than those used for Black-Scholes option pricing.     

Quantization of probability distributions under norm-based distortion measures II: Self-similar distributions

For a probability measure P on Rd and n∈N consider where the infimum is taken over all subsets α of Rd with card(α)?n and V is a nondecreasing function. Under certain conditions on V, we derive the precise n-asymptotics of en for self-similar distributions P and we find the asymptotic performance of optimal quantizers using weighted empirical measures.     

The application of the principle of minimum cross-entropy to the characterization of the exponential-type probability distributions

Summary Systematic and simple characterizations are presented for several familiar distributions in exponential family by means of the principle of minimum cross-entropy (minimum discrimination information). The suitable prior distributions and the appropriate constraints on expected values are given for the underlying distributions.     

Some probability inequalities connected with Schur functions

Bounds for several integrals (tail probabilities, for example) are established by showing that each integral is a Schur function.     

Integral transform and fractional derivative formulas involving the extended generalized hypergeometric functions and probability distributions

In the present paper, our aim is to establish several formulas involving integral transforms, fractional derivatives, and a certain family of extended generalized hypergeometric functions. As corollaries and consequences, many interesting results are shown to follow from our main results. A probability density function involving the extended generalized hypergeometric function is introduced, and its properties are studied. The corresponding properties of some of the classical probability distributions and their associated probability density functions are easily derivable as special cases of our general results. Copyright © 2016 John Wiley & Sons, Ltd.     

Asymptotic equivalence of probability distributions with applications to some problems of asymptotic independence

Summary The problem of asymptotic approximation is formulated in a generalized form, and the basic concepts of asymptotic equivalence of probability distributions, which are fundamental to the study of asymptotic approximation problems, are established precisely. For the purpose of practical applications, some useful criterions for the asymptotic equivalence are given.     

Iterated probability distributions and extremes with random sample size

In this paper the possible nondegenerated limit distributions for the n-fold mapping of a given probability distribution are considered. If the mapping used for the iteration procedure is a probability generating function of a positive integer-valued random variable then the results can be applied to the max-stability of distributions of random variables with random sample size.     

Abelian theorems for a class of probability distributions inR d and their application

A class of multidimensional absolutely continuous distributions is considered. Each of them has a moment-generating function that is finite in a bounded set S and, therefore, generates a family of so-called conjugate or associated distributions. At the focus of our attention are the limiting distributions for this family that appear as the conjugating parameter tends to the boundary of S. As in the one-dimensional case, each such limiting distribution can be obtained as a consequence of an Abelian theorem. Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part II.     

Stochastic optimization on social networks with application to service pricing

In this paper we develop a combined simulation and optimization approach for solving difficult decision problems on complex dynamic networks. For a specific reference problem we consider a telecommunication service provider who offers a telecommunication service to a market with network effects. More particularly, the service consumption of an individual user depends on both idiosyncratic characteristics and the popularity of this service among the customer’s immediate neighborhood. Both the social network and the individual user preferences are largely heterogeneous and changing over time. In addition the service provider’s decisions are made in absence of perfect knowledge about user preferences. The service provider pursues the strategy of stimulating the demand by offering differentiated prices to the customers. For finding the optimal pricing we apply a stochastic quasi-gradient algorithm that is integrated with a simulation model that drives the evolution of the network and user preferences over time. We show that exploiting the social network structure and implementing differentiated pricing can substantially increase the revenues of a service provider operating on a social network. More generally, we show that stochastic gradient methods represent a powerful methodology for the optimization of decisions in social networks.     

A Bayesian approach to pricing longevity risk based on risk-neutral predictive distributions

We present a Bayesian approach to pricing longevity risk under the framework of the Lee-Carter methodology. Specifically, we propose a Bayesian method for pricing the survivor bond and the related survivor swap designed by Denuit et al. (2007). Our method is based on the risk neutralization of the predictive distribution of future survival rates using the entropy maximization principle discussed by Stutzer (1996). The method is illustrated by applying it to Japanese mortality rates.