pricing bermudan options under local levy models with default we consider a defaultable asset whose risk neutral pricing dynamics are described by an exponential levy type martingale this class of models allows for a local volatility local default intensity and a locally dependent levy measure we present a pricing method for bermudan options based on an analytical approximation of the characteristic function combined with the cos method due to a special form of the obtained characteristic function the price can be computed using a fast fourier transform based algorithm resulting in a fast and accurate calculation the greeks can be computed at almost no additional computational cost error bounds for the approximation of the characteristic function as well as for the total option price are given

a comparison of some methods for the evaluation of highly oscillatory integrals abstract in part i the extended clenshaw curtis method for finite fourier integrals is discussed and a number of timed comparisons are made between the various implementations which appear in the literature part ii deals with irregular oscillatory integrals and outlines the various methods which have been proposed for their evaluation their effectiveness is discussed in the light of a set of test examples

stability and size dependency of cauchy born hypothesis in three dimensional applications abstract the cauchy born hypothesis cb provides a hierarchical approach in the molecular theory of crystal elasticity to relate the continuum and atomic deformations this kinematic theory has been extensively used as the constitutive law of continuum regions in multi scale models in these models the fine scale is proposed to describe the real behavior of crystalline structure wherever the continuum description fails the main objective of this article is to investigate the stability and size dependency of cb hypothesis in three dimensional applications by direct comparison of information between atomistic and continuous description of a medium the sutton chen many body potential is used for the gold metal to consider the real metallic behavior in numerical simulations two failure criteria are introduced in the strain and stress domains the validity surfaces are derived for the cauchy born hypothesis and the size effect of specimens is investigated on the convergency of results it is shown that the gold crystal deforms homogeneously inside the validity surface in which the material is elastic and the cb has remained valid it is observed that although the deformation is inhomogeneous and the cb is invalid outside the validity surface the crystalline structure may exhibit elastic or plastic behavior in this region moreover it is numerically shown that the size dependency of validity surface decreases with the increase of the size of specimens these observations are meticulously investigated by loading and unloading several cubic specimens using molecular dynamics simulation

harnack inequalities and discrete continuous error estimates for a chain of atoms with two body interactions we consider deformations in r 3 of an infinite linear chain of atoms where each atom interacts with all others through a two body potential we compute the effect of an external force applied to the chain at equilibrium the positions of the particles satisfy an euler lagrange equation for large classes of potentials we prove that every solution is well approximated by the solution of a continuous model we establish an error estimate between the discrete and the continuous solution based on a harnack lemma of indepen dent interest finally we apply our results to some lennard jones potentials

overall properties of a discrete membrane with randomly distributed defects a prototype for variational percolation problems with surface energies is considered the description of the macroscopic properties of a two dimensional discrete membrane with randomly distributed defects in the spirit of the weak membrane model of blake and zisserman quadratic springs that may break at a critical length of the elongation after introducing energies depending on suitable independent identically distributed random variables this is done by exhibiting an equivalent continuum energy computed using convergence geometric measure theory and percolation arguments we show that below a percolation threshold the effect of the defects is negligible and the continuum description is given by the dirichlet integral while above that threshold an additional griffith fracture term appears in the energy which depends only on the defect probability through the chemical distance on the weak cluster of defects

a fast and accurate fft based method for pricing early exercise options under levy processes a fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented the method is based on a quadrature technique and relies heavily on fourier transformations the main idea is to reformulate the well known risk neutral valuation formula by recognizing that it is a convolution the resulting convolution is dealt with numerically by using the fast fourier transform this novel pricing method which we dub the convolution method is applicable to a wide variety of payoffs and requires only the knowledge of the characteristic function of the model as such the method is applicable within many regular affine models among which is the class of exponential levy models for an m times exercisable bermudan option the overall complexity is o mn log 2 n with n grid points used to discretize the price of the underlying asset american options are priced efficiently by applying richardson extrapolation to the prices of bermudan options

effective cohesive behavior of layers of interatomic planes a simple model of cleavage in brittle crystals consists of a layer of material containing n atomic planes separating in accordance with an interplanar potential under the action of an opening displacement prescribed on the boundary of the layer the problem addressed in this work concerns the characterization of the constrained minima of the energy en of the layer as a function of as n becomes large these minima determine the effective or macroscopic cohesive law of the crystal the main results presented in this communication are i the computation of the limit e0 of en as n ii the characterization of the minimum values of e0 as a function of the macroscopic opening displacement iii a proof of uniform convergence of the minima of en for the case of nearest neighbor interactions and iv a proof of uniform convergence of the derivatives of en or tractions in the same case the scaling on which the present convergence analysis is based has the effect of separating the bulk and surface contributions to the energy it differs crucially from other scalings employed in the past in that it renders both contributions of the same order

on contingent claim valuation in continuous time for volatility models of ornstein uhlenbeck type the paper addresses the valuation of contingent claims in stochastic volatility models of ornstein uhlenbeck type stressing the situation when volatility is driven by purely discontinuous levy processes a reduction series methodology is developed for this purpose which also provides a way for the numerical study of the value functionals the methodology is illustrated in the options case and in models based on gig distributions numerical examples are provided these examples show how the series enable computation accuracies of some three decimal places with just a single digit number of terms

pricing and hedging european style options in levy based stochastic volatility models considering the leverage effect abstract this paper develops a semi analytical exact solution for the discrete time variance optimal hedging strategy based on two dimensional fourier cosine series expansions this approach is novel and is more efficient than those previously suggested in the literature numerical studies for different models also suggest that the algorithm has better accuracy and stability for the pricing and hedging problems a levy based stochastic volatility model considering the leverage effect is also proposed we have employed data on apple stock prices and the corresponding option contracts to test this model to evaluate its ability to perform pricing and hedging in summary our model design is relatively reliable in the context of the known empirical patterns such as the existence of jumps the volatility smile and the negative correlation between asset returns and volatilities

forward starting options pricing with double stochastic volatility stochastic interest rates and double jumps abstract we present an extension of double heston stochastic volatility model by introducing cir stochastic interest rate and double exponential jumps in the stock price process we derive the characteristic function and forward characteristic function of the log asset price and thereby forward starting options are well evaluated by the cos method we also provide efficient simulation of the proposed model and monte carlo solutions to forward starting options pricing based on the qe scheme numerical results show that the cos method is fast and efficient for pricing forward starting options

application of levy processes and esscher transformed martingale measures for option pricing in fuzzy framework in this paper we consider the european option valuation problem we assume that the underlying asset follows a geometric levy process the log price is a sum of a brownian motion with drift and a linear combination of poisson processes describing jumps in price in our approach we use martingale method and theory of fuzzy sets to obtain the european call and put option pricing formulas we use the mean correcting and the esscher transformed martingale measures application of the first mentioned measure required deep analysis of transformation of characteristics of levy process we assume that some model parameters cannot be precisely described and therefore we apply fuzzy numbers application of fuzzy arithmetic enables us to consider different sources of uncertainty and introduce experts opinions or imprecise estimates to the model in contradistinction to our previous papers where the european call option price at time zero was analysed we introduce the valuation expressions of the european call and put options for arbitrary time t numerical simulations conducted in the paper are used to analyse and illustrate the theoretical results this numerical approach is based on l r numbers and the exact shape of the fuzzy numbers which give us the possibility of comparing behaviour of option prices for various values of the parameters of the underlying asset

a novel pricing method for european options based on fourier cosine series expansions here we develop an option pricing method for european options based on the fourier cosine series and call it the cos method the key insight is in the close relation of the characteristic function with the series coefficients of the fourier cosine expansion of the density function in most cases the convergence rate of the cos method is exponential and the computational complexity is linear its range of application covers underlying asset processes for which the characteristic function is known and various types of option contracts we will present the method and its applications in two separate parts the first one is this paper where we deal with european options in particular in a follow up paper we will present its application to options with early exercise features

forest fire model with immune trees abstract we present a generalization of the forest fire model of p bak et al by including the immunity g which is the probability that a tree is not ignited although one of its neighbors is burning when g reaches a critical value gc p which depends on the tree growth rate p the fire cannot survive any more i e a continuous phase transition takes place from a steady state with fire to a steady state without fire we present results of computer simulations and explain them by analytic calculations the fire spreading at the phase transition represents a new type of percolation which is called fluctuating site percolation

one dimensional generalized branching annihilating random walker process with stochastic generation of offsprings abstract we introduce a branching annihilating random walker process with two species particles a and b which diffuse creating new particles and annihilating instantaneously a b 0 when they meet each kind of particle branches stochastically having offsprings of the same or different type the model is defined and studied by means of epidemic simulations in a one dimensional discrete lattice the phase diagram of the model exhibits two states the vacuum and the active one separated by a critical line along that line the system undergoes irreversible second order phase transitions monte carlo results show that the transitions belong to the same universality class as directed percolation in the limiting case when the generation of offsprings is forbidden the model is mapped into the standard diffusion limited reaction a b 0 which asymptotically evolves towards the vacuum state the transition between the stationary regime and such vacuum states is also studied

the energy of some microscopic stochastic lattices we introduce a notion of energy for some microscopic stochastic lattices such lattices are broad generalizations of simple periodic lattices for which the question of the definition of an energy was examined in a series of previous works 14 18 note that slightly more general deterministic geometries were also considered in 6 these lattices are involved in the modelling of materials whose microscopic structure is a perturbation in a sense made precise in the article of the periodic structure of a perfect crystal the modelling considered here is either a classical modelling where the sites of the lattice are occupied by ball like atomic systems that interact by pair potentials or a quantum modelling where the sites are occupied by nuclei equipped with an electronic structure spread all over the ambient space the corresponding energies for the infinite stochastic lattices are derived consistently with truncated systems of finite size by application of a thermodynamic limit process subsequent works 7 8 will be devoted to the macroscopic limits of the energies of such microscopic lattices thereby extending to a stochastic context the results of 4 5 such convergences in a stochastic setting in dimension 1 have been studied in 21 22 we will also study in 8 some variants and extensions of the stationary setting presented here

self organized critical forest fire model mean field theory and simulation results in 1 to 6 dimenisons we argue that the critical forest fire model ffm introduced by drossel and schwabl phys rev lett 69 1629 1992 is a critical branching process in the mean field approximation where the number s of trees burned in a forest fire is power law distributed with exponent s 5 2 the mean field model of the ffm in finite dimension is the percolation model and as in percolation the upper critical dimension is 6 simulations show that the ffm becomes increasingly percolationlike with increasing dimension d and is within error bars eully consistent with the percolation results when d 3

scaling laws and simulation results for the self organized critical forest fire model we discuss the properties of a self organized critical forest fire model which has been introduced recently we derive scaling laws and define critical exponents the values of these critical exponents are determined by computer simulations in 1 to 8 dimensions the simulations suggest a critical dimension d c 6 above which the critical exponents assume their mean field values changing the lattice symmetry and allowing trees to be immune against fire we show that the critical exponents are universal

a hybrid finite difference scheme for pricing asian options in this paper we apply a hybrid finite difference scheme to evaluate the prices of asian call options with fixed strike price we use the crank nicolson method to discretize the time variable and a hybrid finite difference scheme to discretize the spatial variable the hybrid difference scheme uses the central difference approximation whenever the mesh points are sufficiently away from the left hand side of the domain to ensure the stability of the scheme otherwise a midpoint upwind difference scheme is used the matrix associated with the discrete operator is an m matrix which ensures that the spatial discretization scheme is maximum norm stable it is proved that the scheme is second order convergent with respect to both time and spatial variables numerical experiments support these theoretical results

pricing early exercise and discrete barrier options by fourier cosine series expansions we present a pricing method based on fourier cosine expansions for early exercise and discretely monitored barrier options the method works well for exponential levy asset price models the error convergence is exponential for processes characterized by very smooth rm c infty a b in mathbb r transitional probability density functions the computational complexity is o m 1 n log n with n a small number of terms from the series expansion and m the number of early exercise monitoring dates this paper is the follow up of fang and oosterlee in siam j sci comput 31 2 826 848 2008 in which we presented the impressive performance of the fourier cosine series method for european options

a fourier cosine method for an efficient computation of solutions to bsdes we develop a fourier method to solve backward stochastic differential equations bsdes a general theta discretization of the time integrands leads to an induction scheme with conditional expectations these are approximated by using fourier cosine series expansions relying on the availability of a characteristic function the method is applied to bsdes with jumps numerical experiments demonstrate the applicability of bsdes in financial and economic problems and show fast convergence of our efficient probabilistic numerical method

pricing early exercise and discrete barrier options by shannon wavelet expansions we present a pricing method based on shannon wavelet expansions for early exercise and discretely monitored barrier options under exponential levy asset dynamics shannon wavelets are smooth and thus approximate the densities that occur in finance well resulting in exponential convergence application of the fast fourier transform yields an efficient implementation and since wavelets give local approximations the domain boundary errors can be naturally resolved which is the main improvement over existing methods

interval analysis theory and applications we give an overview on applications of interval arithmetic among others we discuss verification methods for linear systems of equations nonlinear systems the algebraic eigenvalue problem initial value problems for odes and boundary value problems for elliptic pdes of second order we also consider the item software in this field and give some historical remarks

an iterative splitting method for pricing european options under the heston model abstract in this paper we propose an iterative splitting method to solve the partial differential equation pde in option pricing problems we focus on the heston stochastic volatility model and the derived two dimensional pde we take the european option as an example and conduct numerical experiments using different boundary conditions the iterative splitting method transforms the two dimensional equation into two quasi one dimensional equations with the variable on the other dimension fixed which helps to lower the computational cost numerical results show that the iterative splitting method together with an artificial boundary condition abc based on the method by li and huang 2019 gives the most accurate option price and greeks compared to the classic finite difference method with the commonly used boundary conditions in heston 1993

accurate evaluation of european and american options under the cgmy process a finite difference method for integro differential equations arising from le vy driven asset processes in finance is discussed the equations are discretized in space by the collocation method and in time by an explicit backward differentiation formula the discretization is shown to be second order accurate for a relevant parameter range determining the degree of the singularity in the le vy measure the singularity is dealt with by means of an integration by parts technique an application of the fast fourier transform gives the overall amount of work o n t n log n rendering the method fast

solving partial integro differential option pricing problems for a wide class of infinite activity levy processes in this paper numerical analysis of finite difference schemes for partial integro differential models related to european and american option pricing problems under a wide class of levy models is studied apart from computational and accuracy issues qualitative properties such as positivity are treated consistency of the proposed numerical scheme and stability in the von neumann sense are included gauss laguerre quadrature formula is used for the discretization of the integral part numerical examples illustrating the potential advantages of the presented results are included

exact results for the one dimensional self organized critical forest fire model we present the analytic solution of the self organized critical soc forest fire model in one dimension proving soc in systems without conservation laws by analytic means under the condition that the system is in the steady state and very close to the critical point we calculate the probability that a string of ital n neighboring sites is occupied by a given configuration of trees the critical exponent describing the size distribution of forest clusters is exactly tau 2 and does not change under certain changes of the model rules computer simulations confirm the analytic results

dual control monte carlo method for tight bounds of value function under heston stochastic volatility model abstract the aim of this paper is to study the fast computation of the lower and upper bounds on the value function for utility maximization under the heston stochastic volatility model with general utility functions it is well known there is a closed form solution to the hjb equation for power utility due to its homothetic property it is not possible to get closed form solution for general utilities and there is little literature on the numerical scheme to solve the hjb equation for the heston model in this paper we propose an efficient dual control monte carlo method for computing tight lower and upper bounds of the value function we identify a particular form of the dual control which leads to the closed form upper bound for a class of utility functions including power non hara and yaari utilities finally we perform some numerical tests to see the efficiency accuracy and robustness of the method the numerical results support strongly our proposed scheme

efficient pricing of european options on two underlying assets by frame duality abstract the proj method for pricing european options on one underlying asset was proposed by j lars kirkby and then was applied to price bermudan and asian options in this paper we extend the method to higher dimensions especially two dimensions in which some exotic options can be priced our method does not rely on a priori truncation of the integration range and exhibits excellent performance compared with other state of the art methods particularly for fatter tailed short maturity models we also discuss the errors introduced in each approximation and give corresponding error bounds numerical results on implementation of this method to price for popular two assets options under both the geometric brownian motion and variance gamma dynamics demonstrate remarkable accuracy and robustness

radial basis function partition of unity methods for pricing vanilla basket options meshfree methods based on radial basis function rbf approximation are becoming widely used for solving pde problems they are flexible with respect to the problem geometry and highly accurate a disadvantage of these methods is that the linear system to be solved becomes dense for globally supported rbfs a remedy is to introduce localisation techniques such as partition of unity rbf partition of unity methods rbf pum allow for a significant sparsification of the linear system and lower the computational effort in this work we apply a global rbf method as well as rbf pum to problems in option pricing we consider one and two dimensional vanilla options in order to price american options we employ a penalty approach a penalty term suitable for american multi asset call options has been designed rbf pum is shown to be competitive compared with a finite difference method and a global rbf method it is as accurate as the global rbf method but significantly faster the results for rbf pum look promising for extension to higher dimensional problems

the β meixner model we propose to approximate the meixner model by a member of the b family introduced by kuznetsov 2010 in 2 the advantage of the approximation is the semi explicit formulae for the running extrema under the b family processes which enables us to produce more efficient algorithms for pricing path dependent options through the wiener hopf factors we will explore the performance of the approximation both in an equity framework and in the credit risk setting where we use the approximation to calibrate a surface of credit default swaps the paper follows the approach of the study made by schoutens and damme 2010 in 1 where the aim was to approximate the variance gamma we will contextualize the results by schoutens and damme 2010 in 1 and the ones here with respect to the approach taken by jeannin and pistorius 2010 in 15 an asymptotic expression for the rate of convergence of the approximation is derived

analytical pricing of geometric asian power options on an underlying driven by a mixed fractional brownian motion abstract in this paper we study the pricing problem of the continuously monitored fixed and floating strike geometric asian power options in a mixed fractional brownian motion environment first we derive both closed form solutions and mixed fractional partial differential equations for fixed and floating strike geometric asian power options based on delta hedging strategy and partial differential equation method second we present the lower and upper bounds of the prices of fixed and floating strike geometric asian power options under the assumption that both risk free interest rate and volatility are interval numbers finally numerical studies are performed to illustrate the performance of our proposed pricing model

spreading analysis and finite size scaling study of the critical behavior of a forest fire model with immune trees abstract in a forest fire model with immune trees ffmit recently proposed by drossel and schwabl physica a 199 1993 183 the sites of the lattice are either empty a tree or a burning tree and at each time step the system is updated in parallel according to the following rules i a burning tree becomes an empty site ii trees grow with probability p from empty sites and iii a green tree becomes a burning tree with probability 1 g if at least one next neighbor is burning fixing an arbitrary grow probability p and starting with a small immunity g increments of g causes the fire density of the steady state to decrease until the fire becomes irreversibly extinguished at a certain critical point of coordinates pc gc the set of critical points defines a critical curve gc p which in the l limit divides the p g plane in two regions a steady state with fire fronts for g

model risk and discretisation of locally risk minimising strategies we consider two models for the price process a time continuous jump diffusion and a time discretisation of it then we study the robustness of the related locally risk minimising strategy to this model choice where we focus mainly on hedging asian and spread options using the discretisation scheme and the convergence results on backward stochastic differential equations as studied in khedher and vanmaele 2016 we show that the discrete time locally risk minimising strategies converge to the corresponding continuous time strategies in an l 2 sense we present different numerical examples to illustrate our results

peaks and jumps reconstruction with b splines scaling functions we consider a methodology based on b splines scaling functions to numerically invert fourier or laplace transforms of functions in the space l 2 r the original function is approximated by a finite combination of jth order b splines basis functions and we provide analytical expressions for the recovered coefficients the methodology is particularly well suited when the original function or its derivatives present peaks or jumps due to discontinuities in the domain we will show in the numerical experiments the robustness and accuracy of the method

the evaluation of american options in a stochastic volatility model with jumps an efficient finite element approach we consider the problem of pricing american options in the framework of a well known stochastic volatility model with jumps the bates model according to this model the asset price is described by a jump diffusion stochastic differential equation in which the jump term consists of a levy process of compound poisson type while the volatility is modeled as a cir type process correlated with the asset price pricing american options under the bates model requires us to solve a partial integro differential equation with the final condition and boundary conditions prescribed on a free boundary in this paper a numerical method for solving such a problem is proposed in particular first of all using a richardson extrapolation technique the problem is reduced to a problem with fixed boundary then the problem obtained is solved using an ad hoc finite element method which efficiently combines an implicit explicit time stepping an operator splitting technique and a non uniform mesh of right angled triangles numerical experiments are presented showing that the option pricing algorithm developed in this paper is extremely accurate and fast in particular it is significantly more efficient than other numerical methods that have recently been proposed for pricing american options under the bates model

convergence analysis of a quasi continuum approximation for a two dimensional material without defects in many applications materials are modeled by a large number of particles or atoms where any particle can interact with any other the computational cost is very high since the number of atoms is huge recently much attention has been paid to a so called quasi continuum qc method which is a mixed atomistic continuum model the qc method uses an adaptive finite element framework to effectively integrate the majority of the atomistic degrees of freedom in regions where there is no serious defect however numerical analysis of this method is still in its infancy in this paper we will conduct a convergence analysis of the qc method in the case when there is no defect we will also remark on the case when the defect region is small the difference between our analysis and conventional analysis is that our exact atomistic solution is not a solution of a continuous partial differential equation but a discrete lattice scale solution which is not approximately related to any conventional partial differential equation

recombined multinomial tree based on saddle point approximation and its application to levy models options pricing abstract this paper studies the constructing methods of a recombined multinomial tree based on saddle point approximation and its application to levy models options pricing firstly the levy process and the european option pricing are introduced then we used the characteristic function of levy process to generate density function value at discrete point in a certain range based on the saddle point approximation method then we provide the method to construct recombined multinomial tree and give the pricing formula of european option and path dependent option pricing based on the backward iteration finally we used the cgmy process to demonstrate its application to european option american option and american barrier option pricing it proves that saddle point approximation turns the inverse fourier integral transform problem to several function value calculation comparing to ifft the speed of calculation using this method is faster and it avoid the negative probability density function value based on ifft because of the linear growth of the number of nodes we can extend the saddle point approximation method to the calculation of path dependent option

american option valuation under time changed tempered stable levy processes abstract given that the underlying assets in financial markets exhibit stylized facts such as leptokurtosis asymmetry clustering properties and heteroskedasticity effect this paper presents a novel model for pricing american option under the assumptions that the stock price processes are governed by time changed tempered stable levy process as this model is constructed by introducing random time changes into tempered stable ts processes which specially refer to normal tempered stable nts distribution as well as classical tempered stable cts distribution it permits infinite jumps as well as capturing random varying time in stochastic volatility consequently taking into account the empirical facts such as leptokurtosis skewness and volatility clustering behaviors we employ the fourier cosine technique to calculate american option and propose the improved particle swarm optimization ipso intelligent algorithm for model calibration to demonstrate the advantage of the constructed model we carry out empirical research on american index option in financial markets across wide ranges of models with the time changing normal tempered stable distribution model yielding a superior performance than others

rbf pu method for pricing options under the jump diffusion model with local volatility abstract meshfree methods based on radial basis functions rbfs are of general interest for solving partial differential equations pdes because they can provide high order or spectral convergence for smooth solutions in complex geometries for global rbf methods one of the major disadvantages is the computational cost associated with the dense linear systems that arise therefore this paper is currently directed toward localized rbf approximations known as the rbf partition of unity rbf pu method for partial integro differential equation pide arisen in option pricing problems in jump diffusion model rbf pu method produces algebraic systems with sparse matrices which have small condition number also for comparison some stable time discretization schemes are combined with the operator splitting method to get a fully discrete problem numerical examples are presented to illustrate the convergence and stability of the proposed algorithms for pricing european and american options with merton and kou models

atomic to continuum passage for nanotubes a discrete saint venant principle and error estimates we consider general infinite nanotubes of atoms in mathbb r 3 where each atom interacts with all the others through a two body potential at the equilibrium the positions of the atoms satisfy a euler lagrange equation when there are no exterior forces and for a suitable geometry a particular family of nanotubes is the set of perfect nanotubes at the equilibrium when exterior forces are applied on the nanotube we compare the nanotube to nanotubes of the previous family in part i of the paper this quantitative comparison is formulated in our first main result as a discrete saint venant principle as a corollary we also give a liouville classification result our saint venant principle can be derived for a large class of potentials including the lennard jones potential when the perfect nanotubes at the equilibrium are stable the approach is designed to be applicable to nanotubes that can have general shapes like for instance carbon nanotubes or dna under the oversimplified assumption that all the atoms are identical in part ii of the paper we derive from our saint venant principle a macroscopic mechanical model for general nanotubes we prove that every solution is well approximated by the solution of a continuum model involving stretching and twisting but no bending we establish error estimates between the discrete and the continuous solution more precisely we give two error estimates one at the microscopic level and one at the macroscopic level

extinction survival and dynamical phase transition of branching annihilating random walk we analyze statistical properties of random walkers which disappear when they meet and make offsprings by a controllable rate numerical results for one two and three dimensions and for the sierpinski gasket are assessed in view of the mean field theory predictions universality classes are found to depend on the number of offsprings in space dimension less than 3

on a forest fire model with supposed self organized criticality we study a stochastic forest fire model introduced by p baket al as a model showing self organized criticality this model involves a growth parameterp and the criticality is supposed to show up in the limitp 0 by simulating the model on much larger lattices and with much smaller values ofp we find that the correlations with longest range do not show a nontrivial critical phenomenon in this limit though we cannot rule out percolation like critical behavior on a smaller but still divergent length scale in contrast the model shows nontrivialdeterministic evolution over time scales 1 p in the limitp 0

fourier cosine method for ruin probabilities in theory ruin probabilities in classical insurance risk models can be expressed in terms of an infinite sum of convolutions but its inherent complexity makes efficient computation almost impossible in contrast fourier transforms of convolutions could be evaluated in a far simpler manner this feature aligns with the heuristic of the recently popular work by fang and oosterlee in particular they developed a numerical method based on fourier transform for option pricing we here promote their philosophy to ruin theory in this paper we not only introduce the fourier cosine method to ruin theory which has o n computational complexity but we also enhance the error bound for our case that are not immediate from fang and oosterlee 2009 we also suggest a robust method on estimation of ruin probabilities with respect to perturbation of the moments of both claim size and claim arrival distributions rearrangement inequality will also be adopted to amplify the fourier cosine method resulting in an effective global estimation

a highly efficient shannon wavelet inverse fourier technique for pricing european options in the search for robust accurate and highly efficient financial option valuation techniques we here present the swift method shannon wavelets inverse fourier technique based on shannon wavelets swift comes with control over approximation errors made by means of sharp quantitative error bounds the nature of the local shannon wavelets basis enables us to adaptively determine the proper size of the computational interval numerical experiments on european style options show exponential convergence and confirm the bounds robustness and efficiency

on the data driven cos method in this paper we present the data driven cos method ddcos it is a fourier based financial option valuation method which assumes the availability of asset data samples a characteristic function of the underlying asset probability density function is not required as such the presented technique represents a generalization of the well known cos method 1 the convergence of the proposed method is o 1 n in line with monte carlo methods for pricing financial derivatives the ddcos method is then particularly interesting for density recovery and also for the efficient computation of the options sensitivities delta and gamma these are often used in risk management and can be obtained at a higher accuracy with ddcos than with plain monte carlo methods

two dimensional fourier cosine series expansion method for pricing financial options the cos method for pricing european and bermudan options with one underlying asset was developed in f fang and c w oosterlee siam j sci comput 31 2008 pp 826 848 and f fang and c w oosterlee numer math 114 2009 pp 27 62 in this paper we extend the method to higher dimensions with a multidimensional asset price process the algorithm can be applied to for example pricing two color rainbow options but also to pricing under the popular heston stochastic volatility model for smooth density functions the resulting method converges exponentially in the number of terms in the fourier cosine series summations otherwise we achieve algebraic convergence the use of an fft algorithm for asset prices modeled by levy processes makes the algorithm highly efficient we perform extensive numerical experiments

fast quadrature methods for options with discrete dividends abstract discontinuities in the stock price at ex dividend dates make it hard to derive mathematically elegant solutions for european style options with discrete dividends under the piecewise lognormal model numerical schemes such as non recombining trees are either computationally intensive lack accuracy when computations are fast or are unable of handling discrete barriers quadrature techniques circumvent the latter two problems but due to the occurrence of time consuming multifold integrals for multi dividend options their implementations have been limited to three dividend european options only this research proposes a faster computational method where the integrand is approximated by a chebyshev polynomial and the fast fejer quadrature is used to evaluate integral representations of european american bermudan continuous and discrete barrier options the method has the capability of accurately handling an arbitrary number of discrete dividend payments in an efficient manner the computational superiority and higher accuracy of the proposed approach is demonstrated via an extensive set of numerical results

a geometrically regularized gradient damage model with energetic equivalence abstract aiming to bridge the gap between damage and fracture mechanics for modeling localized failure in solids this paper addresses a novel geometrically regularized gradient damage model with energetic equivalence for cracking evolution with the free energy potential defined as usual in terms of the local strain and damage fields the constitutive relations are derived consistently from the standard framework of thermodynamics upon the sharp crack topology geometrically regularized by the damage localization band with a length scale the ensuing energetic equivalence naturally yields the damage evolution law of gradient type and the associated boundary condition of neumann type compared to other gradient damage models no extra assumptions like the nonlocal energy residual and insulation condition are introduced moreover the damage gradient physically accounting for microscopic nonlocal interactions is fully dissipative as expected in line with the unified phase field theory recently proposed by the author wu 2017 during failure processes the material behavior is uniquely characterized by two constitutive functions i e the degradation function defining the free energy potential of the bulk and the geometric function regularizing the sharp crack topology in particular optimal constitutive functions defining an equivalent cohesive zone model of general softening laws are postulated with the involved parameters calibrated from standard material properties the proposed model is numerically implemented into the multi field finite element method and applied to several benchmark tests of concrete under mode i and mixed mode failure it is found that the incorporated length scale can be regarded either a numerical parameter or a material property for the former considered in this work the length scale has negligible if not no effects on the global responses so long as the sharp crack topology and the damage field of high gradients within the localization band are well resolved furthermore the localization bandwidth does not exhibit spurious widening but rather it approaches to a finite value proportional to the length scale comparison between the numerical and experimental results regarding the curve of load versus displacement and crack path confirms validity of the proposed gradient damage model for characterizing localized failure in quasi brittle solids

variational formulation of softening phenomena in fracture mechanics the one dimensional case we show that discrete models of atoms subject to nearest neighbour non linear interactions approximate continua allowing for softening and fracture a detailed study of local minima and stationary points is carried out scale effects are discussed

the threshold of coexistence and critical behaviour of a predator prey cellular automaton we study a probabilistic cellular automaton to describe two population biology problems the threshold of species coexistence in a predator prey system and the spreading of an epidemic in a population by carrying out mean field approximations and numerical simulations we obtain the phase boundaries thresholds related to the transition between an active state where prey and predators present a stable coexistence and a prey absorbing state the numerical estimates for the critical exponents show that the transition belongs to the directed percolation universality class in the limit where the cellular automaton maps into a model for the spreading of an epidemic with immunization we observe a crossover from directed percolation class to the dynamic percolation class patterns of growing clusters related to species coexistence and spreading of epidemic are shown and discussed

robust pricing of european options with wavelets and the characteristic function we present a novel method for pricing european options based on the wavelet approximation method and the characteristic function we focus on the discounted expected payoff pricing formula and compute it by means of wavelets we approximate the density function associated to the underlying asset price process by a finite combination of jth order b splines and recover the coefficients of the approximation from the characteristic function two variants for wavelet approximation will be presented where the second variant adaptively determines the range of integration the compact support of a b splines basis enables us to price options in a robust way even in cases where fourier based pricing methods may show weaknesses the method appears to be particularly robust for pricing long maturity options fat tailed distributions as well as staircase like density functions encountered in portfolio loss computations

efficient pricing of discrete arithmetic asian options under mean reversion and jumps based on fourier cosine expansions we propose an efficient pricing method for arithmetic asian options based on fourier cosine expansions in particular we allow for mean reversion and jumps in the underlying price dynamics there is an extensive body of empirical evidence in the current literature that points to the existence and prominence of such anomalies in the prices of certain asset classes such as commodities our efficient pricing method is derived for the discretely monitored versions of the european style arithmetic asian options the analytical solutions obtained from our fourier cosine expansions are compared to the benchmark fast fourier transform based pricing for the examination of its accuracy and computational efficiency

efficient numerical fourier methods for coupled forward backward sdes we develop three numerical methods to solve coupled forward backward stochastic differential equations we propose three different discretization techniques for the forward stochastic differential equation a theta discretization of the time integrands is used to arrive at schemes with conditional expectations these conditional expectations are approximated by using the cos method which relies on the availability of the conditional characteristic function of the discrete forward process the numerical methods are applied to different problems including a financial problem richardson extrapolation is used to obtain more accurate results resulting in the observation of second order convergence in the number of time steps advantages and disadvantages of each method are compared against each other

a high order finite difference method for option valuation in this paper we propose the use of an efficient high order finite difference algorithm to price options under several pricing models including the blackscholes model the mertons jumpdiffusion model the hestons stochastic volatility model and the nonlinear transaction costs or illiquidity models we apply a local mesh refinement strategy at the points of singularity usually found in the payoff of most financial derivatives to improve the accuracy and restore the rate of convergence of a non uniform high order five point stencil finite difference scheme for linear models the time stepping is dealt with by using an exponential time integration scheme with carathodoryfejr approximations to efficiently evaluate the product of a matrix exponential with a vector of option prices nonlinear blackscholes equations are solved using an efficient iterative scheme coupled with a richardson extrapolation our numerical experiments clearly demonstrate the high order accuracy of the proposed finite difference method making the latter a method of choice for solving both linear and nonlinear partial differential equations in financial engineering problems

self organized critical forest fire model a forest fire model is introduced which contains a lightning probability f this leads to a self organized critical state in the limit f ensuremath rightarrow 0 provided that the time scales of tree growth and burning down of forest clusters are separated we derive scaling laws and calculate all critical exponents the values of the critical exponents are confirmed by computer simulations for a two dimensional system we show that the forest density in the critical state assumes its minimum possible value i e that energy dissipation is maximum

asymptotic behaviour of a pile up of infinite walls of edge dislocations we consider a system of parallel straight edge dislocations and we analyse its asymptotic behaviour in the limit of many dislocations the dislocations are represented by points in a plane and they are arranged in vertical walls each wall is free to move in the horizontal direction the system is described by a discrete energy depending on the one dimensional horizontal positions x i 0 of the n walls the energy contains contributions from repulsive pairwise interactions between all walls a global shear stress forcing the walls to the left and a pinned wall at x 0 that prevents the walls from leaving through the left boundary we study the behaviour of the energy as the number of walls n tends to infinity and characterise this behaviour in terms of convergence there are five different cases depending on the asymptotic behaviour of the single dimensionless parameter n corresponding to beta n ll 1 n 1 n ll beta n ll 1 and beta n gg 1 and the two critical regimes n 1 n and n 1 as a consequence we obtain characterisations of the limiting behaviour of stationary states in each of these five regimes the results shed new light on the open problem of upscaling large numbers of dislocations we show how various existing upscaled models arise as special cases of the theorems of this paper the wide variety of behaviour suggests that upscaled models should incorporate more information than just dislocation densities this additional information is encoded in the limit of the dimensionless parameter n

a numerical method for pricing discrete double barrier option by legendre multiwavelet in this article a fast numerical algorithm for pricing discrete double barrier option is presented according to blackscholes model the price of option in each monitoring date can be evaluated by a recursive formula upon the heat equation solution these recursive solutions are approximated by using legendre multiwavelets as orthonormal basis functions and expressed in operational matrix form the most important feature of this method is that its cpu time is nearly invariant when monitoring dates increase besides the rate of convergence of presented algorithm was obtained the numerical results verify the validity and efficiency of the numerical method

the double exponential transformation in numerical analysis the double exponential transformation was first proposed by takahasi and mori in 1974 for the efficient evaluation of integrals of an analytic function with end point singularity afterwards this transformation was improved for the evaluation of oscillatory functions like fourier integrals recently it turned out that the double exponential transformation is useful not only for numerical integration but also for various kinds of sinc numerical methods the purpose of the present paper is to review the double exponential transformation in numerical integration and in a variety of sinc numerical methods

a fast numerical method to price american options under the bates model we consider the problem of pricing american options in the framework of a well known stochastic volatility model with jumps the bates model according to this model the price of an american option can be obtained as the solution of a linear complementarity problem governed by a partial integro differential equation in this paper a numerical method for solving such a problem is proposed in particular first of all using a bermudan approximation and a richardson extrapolation technique the linear complementarity problem is reduced to a set of standard linear partial differential problems see for example ballestra and sgarra 2010 chang et al 2007 2012 then these problems are solved using an ad hoc pseudospectral method which efficiently combines the chebyshev polynomial approximation an implicit explicit time stepping and an operator splitting technique numerical experiments are presented showing that the novel algorithm is very accurate and fast and significantly outperforms other methods that have recently been proposed for pricing american options under the bates model

option pricing with legendre polynomials here we develop an option pricing method based on legendre series expansion of the density function the key insight relying on the close relation of the characteristic function with the series coefficients allows to recover the density function rapidly and accurately based on this representation for the density function approximations formulas for pricing european type options are derived to obtain highly accurate result for european call option the implementation involves integrating high degree legendre polynomials against exponential function some numerical instabilities arise because of serious subtractive cancellations in its formulation 96 in proposition a 1 to overcome this difficulty we rewrite this quantity as solution of a second order linear difference equation and solve it using a robust and stable algorithm from olver derivation of the pricing method has been accompanied by an error analysis errors bounds have been derived and the study relies more on smoothness properties which are not provided by the payoff functions but rather by the density function of the underlying stochastic models this is particularly relevant for options pricing where the payoffs of the contract are generally not smooth functions the numerical experiments on a class of models widely used in quantitative finance show exponential convergence

analysis of three dimensional crack initiation and propagation using the extended finite element method an erratum has been published for this article in international journal for numerical methods in engineering 2005 63 8 1228 we present a new formulation and a numerical procedure for the quasi static analysis of three dimensional crack propagation in brittle and quasi brittle solids the extended finite element method xfem is combined with linear tetrahedral elements a viscosity regularized continuum damage constitutive model is used and coupled with the xfem formulation resulting in a regularized crack band version of xfem the evolving discontinuity surface is discretized through a c0 surface formed by the union of the triangles and quadrilaterals that separate each cracked element in two the element s properties allow a closed form integration and a particularly efficient implementation allowing large scale 3d problems to be studied several examples of crack propagation are shown illustrating the good results that can be achieved copyright 2005 john wiley sons ltd

the kolmogorov forward fractional partial differential equation for the cgmy process with applications in option pricing abstract in this paper we derive the kolmogorov forward fractional partial differential equation fpde for the cgmy process the resulting fpde is solved numerically with a second order method in space and backward differentiation formula of order two in time we price options by integrating the resulting probability density function multiplied by the pay off function of the option hence we only have to solve one fpde to price several options this is useful in practical applications where it is common to price many options simultaneously for the same underlying diffusion model the traditional way to price options requires the solution of one fpde per option since the fpdes are of similar type and complexity the advantage of our suggested method is obvious in the case of pricing multiple european type options on the same underlying asset when only the pay off function differs the numerical experiments verify that the extra cost of pricing several options compared to only one option is very small with our suggested method

cauchy born rule and the stability of crystalline solids static problems we study the connection between atomistic and continuum models for the elastic deformation of crystalline solids at zero temperature we prove under certain sharp stability conditions that the correct nonlinear elasticity model is given by the classical cauchy born rule in the sense that elastically deformed states of the atomistic model are closely approximated by solutions of the continuum model with stored energy functionals obtained from the cauchy born rule the analysis is carried out for both simple and complex lattices and for this purpose we develop the necessary tools for performing asymptotic analysis on such lattices our results are sharp and they also suggest criteria for the onset of instabilities of crystalline solids

an efficient algorithm based on eigenfunction expansions for some optimal timing problems in finance this paper considers the optimal switching problem and the optimal multiple stopping problem for one dimensional markov processes in a finite horizon discrete time framework we develop a dynamic programming procedure to solve these problems and provide easy to verify conditions to characterize connectedness of switching and exercise regions when the transition or feynman kac semigroup of the markov process has discrete spectrum we develop an efficient algorithm based on eigenfunction expansions that explicitly solves the dynamic programming problem we also prove that the algorithm converges exponentially in the series truncation level our method is applicable to a rich family of markov processes which are widely used in financial applications including many diffusions as well as jump diffusions and pure jump processes that are constructed from diffusion through time change in particular many of these processes are often used to model mean reversion we illustrate the versatility of our method by considering three applications valuation of combination shipping carriers interest rate chooser flexible caps and commodity swing options numerical examples show that our method is highly efficient and has significant computational advantages over standard numerical pde methods that are typically used to solve such problems

a double exponential fast gauss transform algorithm for pricing discrete path dependent options this paper develops algorithms for the pricing of discretely sampled barrier lookback and hindsight options and discretely exercisable american options under the black scholes framework the pricing of these options can be reduced to evaluation of a series of convolutions of the gaussian distribution and a known function we compute these convolutions efficiently using the double exponential integration formula and the fast gauss transform the resulting algorithms have computational complexity of o nn where the number of monitoring exercise dates is n and the number of sample points at each date is n and our results show the error decreases exponentially with n we also extend the approach and provide results for merton s lognormal jump diffusion model

the evaluation of barrier option prices under stochastic volatility this paper considers the problem of numerically evaluating barrier option prices when the dynamics of the underlying are driven by stochastic volatility following the square root process of heston 1993 7 we develop a method of lines approach to evaluate the price as well as the delta and gamma of the option the method is able to efficiently handle both continuously monitored and discretely monitored barrier options and can also handle barrier options with early exercise features in the latter case we can calculate the early exercise boundary of an american barrier option in both the continuously and discretely monitored cases

directed percolation in 2 1 dimensions the author presents results of monte carlo simulations for directed percolation in 2 1 dimensions very close to the percolation threshold his values for pc for bond and site percolation on the bcc and simple cubic lattices are more precise than previous estimates by one to two orders of magnitude the improvement on the critical exponents describing the behaviour at pc is less impressive but they are also substantially more precise than previous ones

on the calibration of the 3 2 model abstract we consider the problem of calibrating the 3 2 stochastic volatility model to option data in comparison to the characteristic function of the heston model the characteristic function of the 3 2 model can be up to 50 times slower to evaluate this makes the standard least squares calibration with finite difference gradients unreasonably slow to address this problem we derive the analytic gradient of the characteristic function in compact form we then propose a computational method for the analytic gradient formula which caches intermediate results across the partial derivatives in addition to the strike dimension and the maturity dimension compared to the fastest method of calibrating the 3 2 model which we could find in the literature the method proposed in this paper is more than 10 times faster we also discuss the issue of apparent non convexity in the least squares calibration of the 3 2 model for market data to tackle it we propose a regularized calibration where the regularization point is obtained using risk neutral mcmc estimation of the model we find that this approach is particularly well suited for the calibration problem as it generates naturally a consistent damping matrix for the parameter estimates in addition to being very fast

a reliable numerical method to price arithmetic asian options abstract we design and analyze two numerical methods for pricing asian options the first one is an explicit finite difference method and therefore as usual only conditionally stable the second method is an implicit finite difference method and unconditionally stable to explore the basic ideas of analysis we discuss the explicit method in detail and then highlight the crucial steps in the analysis of the implicit method numerical results obtained by these two methods are compared with those obtained by an improved monte carlo method the proposed implicit method is very robust as is evident from the comparative numerical results we also provide additional numerical results that confirm theoretical investigations done by many other researchers

efficient and high accuracy pricing of barrier options under the cev diffusion binomial and trinomial lattices are popular techniques for pricing financial options these methods work well for european and american options but for barrier options the need to place a tree node very close to a barrier brings difficulties in their implementation and a large number of time steps are usually required when the barrier is close to the current asset price a finite difference implementation is simpler and we propose a fourth order numerical scheme for continuously and discretely monitored barriers we demonstrate the superior performance of our technique over existing procedures for the black scholes model and we then price barriers under constant elasticity of variance cev diffusion continuously monitored barriers under cev admit an analytical solution but evaluation via this formula is not straightforward furthermore discretely monitored barriers have to be priced numerically our main contribution is therefore a highly accurate and efficient numerical scheme for barrier options under cev and we provide several numerical examples to illustrate the merit of the new technique

self organized criticality in forest fire models we review properties of the self organized critical soc forest fire model ffm self organized critical systems drive themselves into a critical state without fine tuning of parameters after an introduction the rules of the model and the conditions for spiral shaped and soc large scale structures are given for the soc state critical exponents and scaling relations are introduced the existence of an upper critical dimension and the universal behavior of the model are discussed the relations and differences between ffm and percolation systems are outlined considering an extension of the ffm into the regions beyond the critical point phase transitions and the various structures found in these regions are illustrated

application of the fast gauss transform to option pricing in many of the numerical methods for pricing american options based on the dynamic programming approach the most computationally intensive part can be formulated as the summation of gaussians though this operation usually requireso nn work when there aren summations to compute and the number of terms appearing in each summation isn we can reduce the amount of work too n n by using a technique called the fast gauss transform in this paper we apply this technique to the multinomial method and the stochastic mesh method and show by numerical experiments how it can speed up these methods dramatically both for the black scholes model and merton s lognormal jump diffusion model we also propose extensions of the fast gauss transform method to models with non gaussian densities