A fuzzy approach to option pricing in a Levy process setting

Nowak, Piotr; Romaniuk, Maciej

doi:10.2478/amcs-2013-0046

Cited by 29 publications

(14 citation statements)

References 33 publications

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“…Similar approach was also applied by us in Nowak and Romaniuk (2010a, 2013c, 2014, where the stochastic analysis, including the Jacod-Grigelionis characteristics (see, e.g., Shiryaev 1999;Nowak 2002), and the fuzzy sets theory were employed to find the European option pricing formulas.…”

Section: Introductionmentioning

confidence: 99%

Valuing catastrophe bonds involving correlation and CIR interest rate model

Nowak

Romaniuk

2016

Comp. Appl. Math.

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Natural catastrophes lead to problems of insurance and reinsurance industry. Classic insurance mechanisms are often inadequate for dealing with consequences of catastrophic events. Therefore, new financial instruments, including catastrophe bonds (cat bonds), were developed. In this paper we price the catastrophe bonds with a generalized payoff structure, assuming that the bondholder's payoff depends on an underlying asset driven by a stochastic jump-diffusion process. Simultaneously, the risk-free spot interest rate has also a stochastic form and is described by the multi-factor Cox-Ingersoll-Ross model. We assume the possibility of correlation between the Brownian part of the underlying asset and the components of the interest rate model. Using stochastic methods, we prove the valuation formula, which can be applied to the cat bonds with various payoff functions. We use adaptive Monte Carlo simulations to analyze the numerical properties of the obtained pricing formula for various settings, including some similar to the practical cases.

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Section: Introductionmentioning

confidence: 99%

Valuing catastrophe bonds involving correlation and CIR interest rate model

Nowak

Romaniuk

2016

Comp. Appl. Math.

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“…We consider a pricing oligopoly model (Karpowicz, 2012;Nowak and Romaniuk, 2013) with a leader and two followers. For instance, let us consider AA, one of the most profitable airlines globally, which has always had the reputation of a leader and industry challenger.…”

Section: Using This Inequality In the Identitymentioning

confidence: 99%

Computing the Stackelberg/Nash equilibria using the extraproximal method: Convergence analysis and implementation details for Markov chains games

Trejo¹,

Clempner

Poznyak³

2015

International Journal of Applied Mathematics and Computer Science

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In this paper we present the extraproximal method for computing the Stackelberg/Nash equilibria in a class of ergodic controlled finite Markov chains games. We exemplify the original game formulation in terms of coupled nonlinear programming problems implementing the Lagrange principle. In addition, Tikhonov's regularization method is employed to ensure the convergence of the cost-functions to a Stackelberg/Nash equilibrium point. Then, we transform the problem into a system of equations in the proximal format. We present a two-step iterated procedure for solving the extraproximal method: (a) the first step (the extra-proximal step) consists of a "prediction" which calculates the preliminary position approximation to the equilibrium point, and (b) the second step is designed to find a "basic adjustment" of the previous prediction. The procedure is called the "extraproximal method" because of the use of an extrapolation. Each equation in this system is an optimization problem for which the necessary and efficient condition for a minimum is solved using a quadratic programming method. This solution approach provides a drastically quicker rate of convergence to the equilibrium point. We present the analysis of the convergence as well the rate of convergence of the method, which is one of the main results of this paper. Additionally, the extraproximal method is developed in terms of Markov chains for Stackelberg games. Our goal is to analyze completely a three-player Stackelberg game consisting of a leader and two followers. We provide all the details needed to implement the extraproximal method in an efficient and numerically stable way. For instance, a numerical technique is presented for computing the first step parameter (λ) of the extraproximal method. The usefulness of the approach is successfully demonstrated by a numerical example related to a pricing oligopoly model for airlines companies.

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“…The above method is based on the extension principle introduced by Zadeh (see [31]) and a similar approach using α-level cuts is also applied in financial mathematics for pricing of the derivatives (see, e.g., [23,24]), in optimization of queuing systems (see, e.g., [2]) etc.…”

Section: Simulations In the Fuzzified Environmentmentioning

confidence: 99%

“…Therefore, if f(λ) is an increasing function, then for the given α, the left end point f ̃L [α] is approximated using the crisp value λ ̃L [α] as the intensity of the exponential random variables generated in Monte Carlo simulations. In the same way λ ̃U [α] is used to approximate f ̃U [α] (see also, e.g., [2,23,24]). …”

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confidence: 99%

“…left=0 is applied). Then the obtained intervals are putting on one another to form the approximated fuzzy number.The above method is based on the extension principle introduced by Zadeh (see [31]) and a similar approach using α-level cuts is also applied in financial mathematics for pricing of the derivatives (see, e.g., [23,24]), in optimization of queuing systems (see, e.g., [2]) etc.Because the considered function for the fuzzy result (e.g. PV c  ( ) is not given by an exact, analytical formulae, the simulations are required for its evaluation.…”

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On simulation of maintenance costs for water distribution system with fuzzy parameters

Romaniuk

2016

EiN

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Article citation info: (*) Tekst artykułu w polskiej wersji językowej dostępny w elektronicznym wydaniu kwartalnika na stronie www.ein.org.pl IntroductionThe main aim of each water distribution system (WDS, water pipes network) is to deliver water of desirable quality and assumed quantity for customers. In order to meet these requirements, maintenance services, like repairs and replacements of broken or malfunctioned parts, are required.The literature devoted to the problems of simulation of the WDS behaviour and to analysis of its reliability is very rich (see, e.g, [3, 4, 7, 10, 11, 15, 17, 19, 25, 29] for a description of various approaches, and [16,28] for a more detailed review of methods and literature). Because of a necessary long-time horizon planning (e.g. 20 or even 50 years), an influence of a value of the money in the future on maintenance costs should be taken into account. Also such a problem is addressed in many papers.For example, in [20], the main aim is to estimate and validate various cost functions for different types of assets of a WDS, if their hydraulic (e.g., flow, pump head, pump power) and physical characteristics (e.g., volume, material, nominal diameter) are stored in a specially prepared data base. A method of a linear regression is used to model a dependency between various types of the costs (like an equipment cost) and the mentioned characteristics of a pipe (like a nominal diameter) based on data from several Portuguese urban water utilities. However, a method for a calculation of a present value of the total costs is not developed there in a more detailed way.Contrary to the previous paper, in [12], let's say, a "macro-management" approach for a WDS rehabilitation problem with a real long-time horizon is widely discussed. In this paper, both an economics and hydraulic capacity of a WDS is analysed, if a deterioration of a pipe follows the Hazen-Williams equation. The total costs are related RomAniuk m. on simulation of maintenance costs for water distribution system with fuzzy parameters. Eksploatacja i niezawodnoscmaintenance and Reliability 2016; 18 (4): 514-527, http://dx.doi.org/10.17531/ein.2016.4.6. to a rehabilitation of a pipe and some additional maintenance costs (mainly breakage repair costs). To model the breakage rate, the exponential dependency on an age of a pipe is assumed. The present value of the total costs is then calculated using a constant discount rate. The main aim of the whole procedure is to minimize the total costs associated with a rehabilitation, if some hydraulic constraints, necessary for a quality and a quantity of the supplied water, are preserved. As mentioned in this paper, an analysis period for such a procedure is typically equal to 30-60 years.There are also more specialized approaches to the problem of the costs estimation. For example, in [25], an optimisation procedure for a problem of minimising the costs is also described. In this case, the authors express the total cost of a renewal, risk and an unavailability of a WDS as one func...

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A fuzzy approach to option pricing in a Levy process setting

Cited by 29 publications

References 33 publications

Valuing catastrophe bonds involving correlation and CIR interest rate model

Valuing catastrophe bonds involving correlation and CIR interest rate model

Computing the Stackelberg/Nash equilibria using the extraproximal method: Convergence analysis and implementation details for Markov chains games

On simulation of maintenance costs for water distribution system with fuzzy parameters

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