circulant preconditioners for pricing options abstract we use the normalized preconditioned conjugate gradient method with strang s circulant preconditioner to solve a nonsymmetric toeplitz system a n x b which arises from the discretization of a partial integro differential equation in option pricing by using the definition of family of generating functions introduced in 16 we prove that strang s circulant preconditioner leads to a superlinear convergence rate under certain conditions numerical results exemplify our theoretical analysis

on finite difference schemes for partial integro differential equations of levy type abstract in this article we introduce a finite difference approximation for integro differential operators of levy type we approximate solutions of possibly degenerate integro differential equations by treating the nonlocal operator as a second order operator on the whole unit ball eliminating the need for truncation of the levy measure which is present in the existing literature this yields an approximation scheme with significantly reduced computational cost especially for levy measures corresponding to processes with jumps of infinite variation

financial options pricing with regime switching jump diffusions we present a numerical method to evaluate financial derivatives under regime switching jump diffusion models the prices of european and american options are derived by solving a partial integro differential equation pide and a linear complementarity problem lcp respectively we use the implicit method with three time levels to solve the pide and apply it coupled with the operator splitting method to solve the lcp the proposed method has the advantage not only to avoid any fixed point iteration techniques at each time step but also to evaluate directly the prices of the european and american options at all states of the economy it is proved that the implicit method to solve the pide for the european option problem is stable with the second order accuracy in the discrete 2 norm we perform some numerical simulations to illustrate the analysis of the proposed method under the regime switching jump diffusion models

a penalty method for american options with jump diffusion processes the fair price for an american option where the underlying asset follows a jump diffusion process can be formulated as a partial integral differential linear complementarity problem we develop an implicit discretization method for pricing such american options the jump diffusion correlation integral term is computed using an iterative method coupled with an fft while the american constraint is imposed by using a penalty method we derive sufficient conditions for global convergence of the discrete penalized equations at each timestep finally we present numerical tests which illustrate such convergence

statistical inference procedures for bivariate archimedean copulas abstract a bivariate distribution function h x y with marginals f x and g y is said to be generated by an archimedean copula if it can be expressed in the form h x y 1 f x g y for some convex decreasing function defined on 0 1 in such a way that 1 0 many well known systems of bivariate distributions belong to this class including those of gumbel ali mikhail haq thelot clayton frank and hougaard frailty models also fall under that general prescription this article examines the problem of selecting an archimedean copula providing a suitable representation of the dependence structure between two variates x and y in the light of a random sample x 1 y 1 x n y n the key to the estimation procedure is a one dimensional empirical distribution function that can be constructed whether the uniform representation of x and y is archimedean or not and independently of their marginals this semiparametric estimator based on a decomposition of kendall s tau statistic

difference quadrature schemes for nonlinear degenerate parabolic integro pde we derive and analyze monotone difference quadrature schemes for bellman equations of controlled levy jump diffusion processes these equations are fully nonlinear degenerate parabolic integro pdes interpreted in the sense of viscosity solutions we propose new direct discretizations of the nonlocal part of the equation that give rise to monotone schemes capable of handling singular levy measures furthermore we develop a new general theory for deriving error estimates for approximate solutions of integro pdes which thereafter is applied to the proposed difference quadrature schemes

bivariate survival models induced by frailties abstract when two observed survival times depend via a proportional hazards model on the same unobserved variable called a frailty this common dependence induces an association between the observed times this article considers the class of bivariate survival distributions that can arise in this way extends it to allow negative association and shows that the observable bivariate distribution determines the unobserved frailty distribution up to a scale parameter a cross ratio function easily estimated even from censored data is the key to both the characterization results and inferential procedures and diagnostic plots that are introduced and illustrated by real examples the models considered are the natural counterpart for survival data of the latent variable models that have long been used in factor analysis for continuous data or in latent structure analysis for binary data the main distinguishing feature of survival data is the possibility of censoring which makes the standard methods difficu

semiparametric estimation of random effects using the cox model based on the em algorithm consider a survival experiment where individuals within a certain subset of the population share a common unobservable random frailty such a frailty could be an unobservable genetic or early environmental effect if individuals were in sibling groups or an environmental effect if individuals were grouped by households suppose that if the frailty omega is known the cox proportional hazards model for the observable covariates is valid with the consequence of the random effect being a multiplicative factor on the hazard rate assuming tht the random frailties follow a gamma distribution estimates of the fixed and random effects are obtained by using an em algorithm based on a profile likelihood construction the method developed is applied to the framingham heart study to examine the risks of smoking and cholesterol levels adjusting for potential random effects

an rbf fd method for pricing american options under jump diffusion models abstract american option prices under jump diffusion models are determined as solutions to partial integro differential equations pide in this paper a new combination of a time and spatial discretization applied to a linear complementary formulation lcp of the free boundary pide is proposed first a coordinate stretching transformation is performed to the asset price so that the computation of the prices can be focused on regions of real interest instead of on the whole solution domain an implicit explicit time discretization applied to the reformulated lcp on a uniform temporal grid is followed by a spatial discretization to get a fully discrete system the radial basis function rbf finite difference method is a local method resulting in a sparse linear system in contrast to global rbf methods which lead to ill conditioned dense matrix systems for the corresponding european option we prove consistency stability and second order convergence in a discrete l 2 norm we derive mild conditions for the model parameters under which these results hold numerical experiments are performed with european and american options and a comparison with numerical results available in the literature illustrates the accuracy and efficiency of the proposed algorithm

numerical methods for the fractional laplacian a finite difference quadrature approach the fractional laplacian delta alpha 2 is a nonlocal operator which depends on the parameter alpha and recovers the usual laplacian as alpha to 2 a numerical method for the fractional laplacian is proposed based on the singular integral representation for the operator the method combines finite differences with numerical quadrature to obtain a discrete convolution operator with positive weights the accuracy of the method is shown to be o h 3 alpha convergence of the method is proven the treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method numerical experiments on known exact solutions validate the predicted convergence rates computational examples include exponentially and algebraically decaying solutions with varying regularity the generalization to nonlinear equations involving the operator is discussed the obstacle problem for the fractional laplacian is computed

regression analysis of multivariate incomplete failure time data by modeling marginal distributions abstract many survival studies record the times to two or more distinct failures on each subject the failures may be events of different natures or may be repetitions of the same kind of event in this article we consider the regression analysis of such multivariate failure time observations each marginal distribution of the failure times is formulated by a cox proportional hazards model no specific structure of dependence among the distinct failure times on each subject is imposed the regression parameters in the cox models are estimated by maximizing the failure specific partial likelihoods the resulting estimators are shown to be asymptotically jointly normal with a covariance matrix that can be consistently estimated simultaneous inferential procedures are then proposed extensive monte carlo studies indicate that the normal approximation is adequate for practical use the new methods allow time dependent covariates missing observations and arbitrary patterns of censorship they are illustrat

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

a class of multivanate failure time distributions summary a class of continuous multivariate lifetime distributions is proposed the dependence between individuals in a group is modelled by a group specific quantity which can be interpreted as an unobserved covariate common to the individuals in the group and assumed to follow a positive stable distribution it is possible to include covariates in the model and discuss whether the dependence is still present after specific covariates are taken into account if the conditional hazards are proportional then the hazards in the marginal distributions are also proportional but with different constants of proportionality also the hazard for the minimum in a group is proportional to the marginal hazards if the conditional distributions given the group quantity are weibull then the marginal distributions are also weibull this class can be used to test the hypothesis of independence of litter mates in the proportional hazards model

toward a curse of dimensionality appropriate coda asymptotic theory for semi parametric models we argue that due to the curse of dimensionality there are major difficulties with any pure or smoothed likelihood based method of inference in designed studies with randomly missing data when missingness depends on a high dimensional vector of variables we study in detail a semi parametric superpopulation version of continuously stratified random sampling we show that all estimators of the population mean that are uniformly consistent or that achieve an algebraic rate of convergence no matter how slow require the use of the selection randomization probabilities we argue that in contrast to likelihood methods which ignore these probabilities inverse selection probability weighted estimators continue to perform well achieving uniform n 1 2 rates of convergence we propose a curse of dimensionality appropriate coda asymptotic theory for inference in non and semi parametric models in an attempt to formalize our arguments we discuss whether our results constitute a fatal blow to the likelihood principle and study the attitude toward these that a committed subjective bayesian would adopt finally we apply our coda theory to analyse the effect of the curse of dimensionality in several interesting semi parametric models including a model for a two armed randomized trial with randomization probabilities depending on a vector of continuous pretreatment covariates x we provide substantive settings under which a subjective bayesian would ignore the randomization probabilities in analysing the trial data we then show that any statistician who ignores the randomization probabilities is unable to construct nominal 95 per cent confidence intervals for the true treatment effect that have both i an expected length which goes to zero with increasing sample size and ii a guaranteed expected actual coverage rate of at least 95 per cent over the ensemble of trials analysed by the statistician during his or her lifetime however we derive a new interval estimator depending on the randomization probabilities that satisfies i and ii

estimation of a bivariate distribution function under random censorship summary estimators are proposed of the underlying distribution function of independent bivariate random vectors which are subject to random right censorship in each coordinate the estimators satisfy the monotonicity requirements of a distribution function are uniformly strongly consistent at a rate equal to that of the empirical distribution function and are fairly simple to compute

on profile likelihood abstract we show that semiparametric profile likelihoods where the nuisance parameter has been profiled out behave like ordinary likelihoods in that they have a quadratic expansion in this expansion the score function and the fisher information are replaced by the efficient score function and efficient fisher information the expansion may be used among others to prove the asymptotic normality of the maximum likelihood estimator to derive the asymptotic chi squared distribution of the log likelihood ratio statistic and to prove the consistency of the observed information as an estimator of the inverse of the asymptotic variance

locally efficient estimation of a multivariate survival function in longitudinal studies we consider estimation of the joint distribution of multivariate survival times t t1 tk which are subject to right censoring by a common censoring variablec two estimators are proposed an initial inverse probability of censoring weighted ipcw estimator and a one step estimator both estimators incorporate information on available time independent and time dependent prognostic factor covariate data the ipcw estimator is consistent and asymptotically normal can under coarsening at random car and a correct specification of a model for the hazard of censoring given the past covariate and failure data the one step estimator is a locally efficient doubly robust estimator that is i it is can under the assumption of car and either but not necessarily both correct specification of a model for the hazard of censoring given the past or correct specification of a model for the conditional distribution of t given past failure and covariate information and ii it is efficient when both these

a high order finite element scheme for pricing options under regime switching jump diffusion processes this paper considers the numerical pricing of european american and butterfly options whose asset price dynamics follow the regime switching jump diffusion process in an incomplete market structure and using the no arbitrage pricing principle the option pricing problem under the jump modulated regime switching process is formulated as a set of coupled partial integro differential equations describing different states of a markov chain we develop efficient numerical algorithms to approximate the spatial terms of the option pricing equations using linear and quadratic basis polynomial approximations and solve the resulting initial value problem using exponential time integration various numerical examples are given to demonstrate the superiority of our computational scheme with higher level of accuracy and faster convergence compared to existing methods for pricing options under the regime switching model

a novel and accurate finite difference method for the fractional laplacian and the fractional poisson problem abstract in this paper we develop a novel finite difference method to discretize the fractional laplacian 2 in hypersingular integral form by introducing a splitting parameter we formulate the fractional laplacian as the weighted integral of a weak singular function which is then approximated by the weighted trapezoidal rule compared to other existing methods our method is more accurate and simpler to implement and moreover it closely resembles the central difference scheme for the classical laplace operator we prove that for u c 3 2 r our method has an accuracy of o h 2 uniformly for any 0 2 while for u c 1 2 r the accuracy is o h 1 2 the convergence behavior of our method is consistent with that of the central difference approximation of the classical laplace operator additionally we apply our method to solve the fractional poisson equation and study the convergence of its numerical solutions the extensive numerical examples that accompany our analysis verify our results as well as give additional insights into the convergence behavior of our method

new perturbation results on pseudo inverses of linear operators in banach spaces abstract let x and y be banach spaces let t x y be a bounded linear operator with closed range and let s x y be another bounded linear operator we study conditions on s t that guarantee the closeness of the range of s and obtain some new bounds on the pseudo inverse of s

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

a semi lagrangian method for the weather options of mean reverting brownian motion with jump diffusion abstract the weather options were created to enable companies to hedge against climate risks however the valuation of weather options is complex since the underlying temperature process has no negotiable price this paper presents a new weather option pricing model which is governed by a stochastic underlying temperature following a mean reverting browning motion with jump diffusion under the assumption of mean self financing consequently a two dimensional partial integro differential equation pide is derived to value the weather based options the numerical techniques applied in this paper are based on a semi lagrangian method slm in which at each time step a set of one dimensional partial integro differential equations are solved and the solution of each pide is updated using semi lagrangian time stepping in addition monotonicity stability and the convergence results of the discrete schemes are also derived in this paper lastly this paper values a series of european hdd based weather put options using slm and highlights the key details of the numerical implementation

error estimates for approximate solutions to bellman equations associated with controlled jump diffusions we derive error estimates for approximate viscosity solutions of bellman equations associated to controlled jump diffusion processes which are fully nonlinear integro partial differential equations two main results are obtained i error bounds for a class of monotone approximation schemes which under some assumptions includes finite difference schemes and ii bounds on the error induced when the original levy measure is replaced by a finite measure with compact support an approximation process that is commonly used when designing numerical schemes for integro partial differential equations our proofs use and extend techniques introduced by krylov and barles jakobsen

retrieving levy processes from option prices regularization of an ill posed inverse problem we propose a stable nonparametric method for constructing an option pricing model of exponential levy type consistent with a given data set of option prices after demonstrating the ill posedness of the usual and least squares version of this inverse problem we suggest to regularize the calibration problem by reformulating it as the problem of finding an exponential levy model that minimizes the sum of the pricing error and the relative entropy with respect to a prior exponential levy model we prove the existence of solutions for the regularized problem and show that it yields solutions which are continuous with respect to the data stable with respect to the choice of prior and which converge to the minimum entropy least squares solution of the initial problem when the noise level in the data vanishes

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

fast numerical solution of parabolic integrodifferential equations with applications in finance we numerically solve parabolic problems u t ca u 0 in 0 t times omega t infty where omega subset mathbb r is a bounded interval and ca is a strongly elliptic integrodifferential operator of order rho in 0 2 a discontinuous galerkin dg discretization in time and a wavelet discretization in space are used the densely populated matrices in the corresponding linear systems of equations are replaced by sparse ones using appropriate wavelet compression techniques the linear systems in each time step are solved by an incomplete gmres iteration under these conditions we show that the complexity of our algorithm is linear up to some logarithmic terms in the number of spatial degrees of freedom and present error estimates applications to purely discontinuous levy processes arising in finance are given

nonparametric estimation of the gap time distributions for serial events with censored data summary in many follow up studies each subject can potentially experience a series of events which may be repetitions of essentially the same event or may be events of entirely diverent natures this paper provides a simple nonparametric estimator for the multivariate distribution function of the gap times between successive events when the follow up time is subject to right censoring the estimator is consistent and upon proper normalisation converges weakly to a zero mean gaussian process with an easily estimated covariance function numerical studies demonstrate that both the distribution function estimator and its covariance function estimator perform well for practical sample sizes an application to a colon cancer study is presented

coarsening at random in general sample spaces and random censoring in continuous time heitjan and rubin proposed a concept coarsening at random generalizing rubin s theory of missing at random their analysis was done in discrete sample spaces we propose a generalization to general sample spaces among heitjan and rubin s applications was right censoring in survival analysis we discuss the application of the generalized theory to various censoring patterns in continuous time and connect to the modern theory of random censoring

estimation of the failure time distribution in the presence of informative censoring we present a method for estimating the survival curve of a continuous failure time random variable from right censored data our method allows adjustment for informative censoring due to measured prognostic factors for time to event and censoring while simultaneously quantifying the sensitivity of the inference to residual dependence between failure and censoring due to unmeasured factors we present the results of a simulation study and illustrate our approach using data from the aids clinical trial group 175 study

a second order parareal algorithm for fractional pdes we are concerned with using the parareal parallel in time algorithm for large scale odes system u t a u t d a u t f t arising frequently in semi discretizing time dependent pdes with spatial fractional operators where d 0 is a constant 0 1 and a is a spare and symmetric positive definite spd matrix the parareal algorithm is iterative and is characterized by two propagators f and g which are respectively associated with small temporal mesh size t and large temporal mesh size t the two mesh sizes satisfy t j t with j 2 being an integer which is called mesh ratio let t unit f and t unit g be respectively the computational cost of the two propagators for moving forward one time step then it is well understood that the speedup of the parareal algorithm namely e satisfies e o c log 1 where c t unit f t unit g and is the convergence factor a larger e corresponds a more efficient parareal solver for g backward euler and some choices of f previous studies show that can be a satisfactory quantity particularly for f 2nd order dirk diagonally implicit runge kutta it holds 1 3 for any choice of the mesh ratio j in this paper we continue to consider f 2nd order dirk but with a new choice for g the imex implicit explicit euler method where the implicit and explicit computation is respectively associated with a and d a compared to the widely used backward euler method this choice on the one hand increases c this point is apparent and interestingly on the other hand it can also make the convergence factor smaller 1 5 numerical results are provided to support our conclusions

ignorability and coarse data we present a general statistical model for data coarsening which includes as special cases rounded heaped censored partially categorized and missing data formally with coarse data observations are made not in the sample space of the random variable of interest but rather in its power set grouping is a special case in which the degree of coarsening is known and nonstochastic we establish simple conditions under which the possible stochastic nature of the coarsening mechanism can be ignored when drawing bayesian and likelihood inferences and thus the data can be validly treated as grouped data the conditions are that the data be coarsened at random a generalization of the condition missing at random and that the parameters of the data and the coarsening process be distinct applications of the general model and the ignorability condition are illustrated in a numerical example and described briefly in a variety of special cases

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

an imex scheme for pricing options under stochastic volatility models with jumps partial integro differential equation pide formulations are often preferable for pricing options under models with stochastic volatility and jumps especially for american style option contracts we consider the pricing of options under such models namely the bates model and the so called stochastic volatility with contemporaneous jumps svcj model the nonlocality of the jump terms in these models leads to matrices with full matrix blocks standard discretization methods are not viable directly since they would require the inversion of such a matrix instead we adopt a two step implicit explicit imex time discretization scheme the imex cnab scheme where the jump term is treated explicitly using the second order adams bashforth ab method while the rest is treated implicitly using the crank nicolson cn method the resulting linear systems can then be solved directly by employing lu decomposition alternatively the systems can be iterated under a scalable algebraic multigrid amg method f

solving complex pide systems for pricing american option under multi state regime switching jump diffusion model abstract based on exponential time differencing approach an efficient second order method is developed for solving systems of partial integral differential equations the method is implemented to solve american options under multi state regime switching with jumps the method is seen to be strongly stable l stable and avoids any spurious oscillations caused by non smooth initial data the predictor corrector nature of the method makes it highly efficient in solving nonlinear pides in each regime with different volatilities and interest rates penalty method approach is applied to handle the free boundary constraint of american options numerical results are presented to illustrate the performance of the method for american options under merton s jump diffusion models pade approximation of matrix exponential functions and partial fraction splitting technique are applied to construct computationally efficient version of the method efficiency accuracy and reliability of the method are compared with those of the existing methods available in the literature

tri diagonal preconditioner for pricing options the value of a contingent claim under a jump diffusion process satisfies a partial integro differential equation pide we localize and discretize this pide in space by the central difference formula and in time by the second order backward differentiation formula the resulting system t nx b in general is a nonsymmetric toeplitz system we then solve this system by the normalized preconditioned conjugate gradient method a tri diagonal preconditioner l n is considered we prove that under certain conditions all the eigenvalues of the normalized preconditioned matrix l n 1t n l n 1t n are clustered around one which implies a superlinear convergence rate numerical results exemplify our theoretical analysis

a second order finite difference method for option pricing under jump diffusion models we develop a finite difference method to solve partial integro differential equations which describe the behavior of option prices under jump diffusion models with localization to a bounded domain of the spatial variable these equations are discretized on uniform grid points over a finite domain of time and spatial variables the proposed method is based on three time levels and leads to linear systems with tridiagonal matrices in this paper the stability of the proposed method and the second order convergence rate with respect to a discrete ell 2 norm are proved numerical results obtained with european put options under the merton and kou models show the behaviors of the stability and the second order convergence rate

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

on the rate of convergence of finite difference approximations for bellmans equations with variable coefficients the estimates presented here for parabolic bellman s equations with variable coefficients extend the ones earlier obtained for constant coefficients

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

pricing options in jump diffusion models an extrapolation approach we propose a new computational method for the valuation of options in jump diffusion models the option value function for european and barrier options satisfies a partial integrodifferential equation pide this pide is commonly integrated in time by implicit explicit imex time discretization schemes where the differential diffusion term is treated implicitly while the integral jump term is treated explicitly in particular the popular imex euler scheme is first order accurate in time second order accuracy in time can be achieved by using the imex midpoint scheme in contrast to the above approaches we propose a new high order time discretization scheme for the pide based on the extrapolation approach to the solution of odes that also treats the diffusion term implicitly and the jump term explicitly the scheme is simple to implement can be added to any pide solver based on the imex euler scheme and is remarkably fast and accurate we demonstrate our approach on the examples of merton s and kou s jump diffusion models the diffusion extended variance gamma model as well as the two dimensional duffie pan singleton model with correlated and contemporaneous jumps in the stock price and its volatility by way of example pricing a one year double barrier option in kou s jump diffusion model our scheme attains accuracy of 10 5 in 72 time steps in 0 05 seconds in contrast it takes the first order imex euler scheme more than 1 3 million time steps in 873 seconds and the second order imex midpoint scheme 768 time steps in 0 49 seconds to attain the same accuracy our scheme is also well suited for bermudan options combining simplicity of implementation and remarkable gains in computational efficiency we expect this method to be very attractive to financial engineering modelers

water wave propagation in unbounded domains part ii numerical methods for fractional pdes this paper concerns numerical solutions for a fractional partial differential equation arising as a nonreflecting boundary condition in water wave propagation the fractional derivative operator is written as divergence of a singular integrable convolution which allows the equation to be viewed as a conservation law with a linear nonlocal flux a semi discrete finite volume scheme is presented using conservative piecewise polynomial reconstruction of the solution the convolution with the singular kernel is then integrated exactly time integration uses runge kutta schemes of matching order stability is discussed convergence is established and numerical examples are presented

european rainbow option values under the two asset merton jump diffusion model abstract in this paper we present a semi closed analytical formula for the values of european call and put options on the minimum or maximum of two assets under the two asset merton jump diffusion model in addition useful formulas for several first and second order greeks of these options are derived

locally efficient estimation with current status data and time dependent covariates abstract in biostatistical applications interest often focuses on the estimation of the distribution of a failure time variable t if one observes only whether or not t exceeds an observed monitoring time c then the data structure is called current status data also known as interval censored data case i we extend the data structure by allowing the presence of a possibly time dependent covariate process which is observed up till the monitoring time c we follow the approach of robins and rotnitzky by modeling the hazard of c conditional on the failure time variable and the covariate process i e the missingness or censoring process under the restriction that the missingness monitoring process satisfies coarsening at random because of the curse of dimensionality no globally efficient nonparametric estimators with a good practical performance at moderate sample sizes exist we introduce an inverse probability of censoring weighted estimator of the distribution and of smooth functionals of this di

kaplan meier estimate on the plane estimation of the bivariate survival function from censored data is considered the product integral representation of univariate survival functions is generalized to the bivariate case and used to determine identifiability of the survival function of the partially observed data a bivariate analogue of the kaplan meier estimate is introduced and its almost sure consistency is studied extensions to the general multivariate case are sketched

fast and efficient numerical methods for an extended black scholes model we study fast and efficient numerical approximations of a partial integro differential equation pide that arises in option pricing theory financial problems as well as in various scientific modeling and deal with two different topics as a first part of this article we rigorously study different preconditioned iterative schemes for the full discrete model equation focusing in particular on the choice of a fast and efficient solver we develop preconditioners using wavelet basis transformations and fourier sine transformations to improve the convergence criteria of iterative solvers for the full discrete model we implement a multigrid mg iterative method for the discrete model problem as well in fact we approximate the problem using a finite difference scheme then implement a few preconditioned krylov subspace methods as well as a mg method to speed up the computation a series of numerical experiments demonstrates that the number of iterations required for convergence is small and independent of the system size in this sense the methods are optimal then in the second part in this study we analyze the stability and the accuracy of two different one step schemes to approximate the model

numerical valuation of european and american options under kou s jump diffusion model numerical methods are developed for pricing european and american options under kou s jump diffusion model which assumes the price of the underlying asset to behave like a geometrical brownian motion with a drift and jumps whose size is log double exponentially distributed the price of a european option is given by a partial integro differential equation pide while american options lead to a linear complementarity problem lcp with the same operator spatial differential operators are discretized using finite differences on nonuniform grids and time stepping is performed using the implicit rannacher scheme for the evaluation of the integral term easy to implement recursion formulas are derived which have optimal computational cost when pricing european options the resulting dense linear systems are solved using a stationary iteration for american options two ways to solve the lcps are described an operator slitting method and a penalty method numerical experiments confirm that the developed methods are very efficient as fairly accurate option prices can be computed in a few milliseconds on a pc

a finite difference scheme for option pricing in jump diffusion and exponential levy models we present a finite difference method for solving parabolic partial integro differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a levy process or more generally a time inhomogeneous jump diffusion process we discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the levy measure we propose an explicit implicit finite difference scheme which can be used to price european and barrier options in such models we study stability and convergence of the scheme proposed and under additional conditions provide estimates on the rate of convergence numerical tests are performed with smooth and nonsmooth initial conditions

numerical schemes for pricing asian options under state dependent regime switching jump diffusion models we study the pricing problem of asian options when the underlying asset price follows a very general state dependent regime switching jump diffusion process via a partial differential equation approach under this model the price of the option can be obtained by solving a highly complex system of coupled two dimensional parabolic partial integro differential equations pides we prove existence of the solution to this system of pides by the method of upper and lower solutions via constructing a monotonic sequence of approximating solutions whose limit is a strong solution of the pide system we then propose several numerical schemes for solving the system of pides one of the proposed schemes is built upon the constructive proof hence its results are provably convergent to the solution of the system of pides we illustrate the accuracy of the proposed methods by several numerical examples

an approximation of small time probability density functions in a general jump diffusion model we propose a method for approximating probability density functions related to multidimensional jump diffusion processes for small time horizons a closed form approximation of the characteristic function is derived based on the ito taylor expansion the probability density function is then approximated numerically by inverting the characteristic function using fast fourier transform as application we consider a general stochastic volatility model which involves time state dependent drift and diffusion functions as well as jump components we test our approach under the heston model and the bates model and show that our method provides accurate approximations

hedging with a correlated asset solution of a nonlinear pricing pde hedging a contingent claim with an asset which is not perfectly correlated with the underlying asset results in unhedgeable residual risk even if the residual risk is considered diversifiable the option writer is faced with the problem of uncertainty in the estimation of the drift rates of the underlying and the hedging instrument if the residual risk is not considered diversifiable then this risk can be priced using an actuarial standard deviation principle in infinitesimal time in both cases these models result in the same nonlinear partial differential equation pde a fully implicit monotone discretization method is developed for solution of this pricing pde this method is shown to converge to the viscosity solution certain grid conditions are required to guarantee monotonicity an algorithm is derived which given an initial grid inserts a finite number of nodes in the grid to ensure that the monotonicity condition is satisfied at each timestep the nonlinear discretized algebraic equations are solved using an iterative algorithm which is shown to be globally convergent monte carlo hedging examples are given to illustrate the profit and loss distribution at the expiry of the option

a convergent difference scheme for a class of partial integro differential equations modeling pricing under uncertainty in this paper we present a finite difference scheme to approximate viscosity solutions of a class of partial integro differential equations describing pricing under model uncertainty we establish

an efficient parareal algorithm for a class of time dependent problems with fractional laplacian time dependent diffusion equations with fractional laplacian have received considerable attention in recent years for which numerical methods play an important role because a simple and analytic solution is often unavailable we analyze in this paper a parareal algorithm for this kind of problem which realizes parallel in time computation the algorithm is iterative and uses the 3rd order sdirk singly diagonally implicit runge kutta method with a small step size t as the f propagator and the implicit explicit euler method with a large step size t as the g propagator the two step sizes satisfy t t j with j 2 being an integer using the implicit explicit euler method as the g propagator potentially improves the parallel efficiency but complicates the convergence analysis by employing some technical analysis we provide a sharp estimate of the convergence rate which is independent of the mesh ratio j and the distribution of the eigenvalues of the coefficient matrix an extension of the results to problems with time periodic conditions is also given several numerical experiments are carried out to verify the theoretical results

partial differential equation pricing of contingent claims under stochastic correlation in this paper we study a partial differential equation pde framework for option pricing where the underlying factors exhibit stochastic correlation with an emphasis on computation we derive a multidimensional time dependent pde for the corresponding pricing problem and present a numerical pde solution we prove a stability result and study numerical issues regarding the boundary conditions used moreover we develop and analyze an asymptotic analytical approximation to the solution leading to a novel computational asymptotic approach based on quadrature with a perturbed transition density numerical results are presented to verify second order convergence of the numerical pde solution and to demonstrate its agreement with the asymptotic approximation and monte carlo simulations the effect of certain problem parameters on the pde solution as well as on the asymptotic approximation solution is also studied

tridiagonal implicit method to evaluate european and american options under infinite activity levy models we propose an efficient implicit method to evaluate european and american options when the underlying asset follows an infinite activity levy model since the levy measure of the infinite activity model has the singularity at the origin we approximate infinitely many small jumps by samples of a diffusion the proposed methods to solve partial integro differential equations for european options and linear complementarity problems for american options via an operator splitting method involve solving linear systems with tridiagonal matrices and so can significantly reduce the computations associated with the discrete integral operators the numerical experiments verify that the proposed method has the second order convergence rate under an infinite activity levy model

a finite element like scheme for integro partial differential hamilton jacobi bellman equations we construct a finite element like scheme for fully nonlinear integro partial differential equations arising in optimal control of jump processes special cases of these equations include optimal portfolio and option pricing equations in finance the schemes are monotone and robust we prove that they converge in very general situations including degenerate equations multiple dimensions relatively low regularity of the data and for most if not all types of jump models used in finance in all cases we provide probably optimal error bounds these bounds apply when grids are unstructured and integral terms are very singular two features that are new or highly unusual in this setting

nonparametric estimation of a bivariate survival function in the presence of censoring a new family of estimators of a bivariate survival function based on censored vectors is obtained from a decomposition of the bivariate survival function these estimators are uniformly consistent under bivariate censoring and are self consistent under univariate censoring

discontinuous galerkin method for fractional convection diffusion equations we propose a discontinuous galerkin method for fractional convection diffusion equations with a superdiffusion operator of order alpha 1 alpha 2 defined through the fractional laplacian the fractional operator of order alpha is expressed as a composite of first order derivatives and a fractional integral of order 2 alpha the fractional convection diffusion problem is expressed as a system of low order differential integral equations and a local discontinuous galerkin method scheme is proposed for the equations we prove stability and optimal order of convergence cal o h k 1 for the fractional diffusion problem and an order of convergence of cal o h k frac 1 2 is established for the general fractional convection diffusion problem the analysis is confirmed by numerical examples

jump diffusion processes in random environments abstract in this paper we investigate jump diffusion processes in random environments which are given as the weak solutions of sdes we formulate conditions ensuring existence and uniqueness in law of solutions we investigate the markov property to prove uniqueness we solve a general martingale problem for cadlag processes this result is of independent interest application of our results to generalized exponential levy model are present in the last section

fast numerical valuation of options with jump under merton s model in this paper we consider discontinuous galerkin dg finite element together with finite difference fd scheme for solving merton s jump diffusion model which is given by a partial integro differential equations pides spatial differential operators are discretized using fd on a uniform grid and time stepping is performed using the dg finite element method the treatment of the integral term associated with jumps in models is more challenging the discretization of this integral term will lead to full matrices for the non locality of the integral operator to fast solve this model multigrid method is used for solving such linear algebraical system numerical comparison of multigrid method and gmres method shows that multigrid method is superior to and more effective than gmres method in solving the dense algebraic systems resulting from the fd approximations of the pides

estimating the mean of an increasing stochastic process at a censored stopping time abstract data often come in the form of a longitudinal series say n whose observation has been stopped either by a terminal event u e g death or censoring event c e g lost to follow up when n is not deterministic it is generally not possible to use standard survival analysis tools e g the kaplan meier estimator to consistently estimate functionals of the distribution of n u because of induced informative censoring various seemingly unrelated estimators recently have been proposed to handle this difficulty within the context of specific problems ranging from the analysis of quality adjusted survival data to recurrent event data to lifetime medical cost data in this article i consider a general framework containing each of these specific examples as a special case propose a class of estimators for the mean of the stopped longitudinal process and establish the asymptotic properties of this class of estimators connections between the aforementioned estimators are established