constructing robust good lattice rules for computational finance the valuation of many financial derivatives leads to high dimensional integrals the constructions of robust or universal good lattice rules for financial applications are both important and challenging an important common feature of the integrands in computational finance is that they can often be well approximated by a sum of low dimensional functions i e functions that depend on only a small number of variables usually just two variables for numerical integration of such functions the quality of the low order i e low dimensional projections of the node set is crucial in this paper we propose methods to construct good lattice points with optimal low order projections the quality of a point set is measured by a new measure called elementary order ell discrepancy which measures the quality of all order ell projections and is more informative than usual measures two constructions namely the korobov and the component by component constructions are studied such that the low order projections are optimized numerical experiments demonstrate that even in high dimensions it is possible to construct new good lattice points with order 2 projections that are better than those of the sobol points and random points and with higher order projections that are no worse while the sobol points lost the advantage over random points in order 2 projections on the average the new lattice rules have the potential to improve upon the accuracy for favorable functions while doing no harm for unfavorable ones their applications for pricing path dependent options and american options based on the least square monte carlo method are studied and their high efficiency is demonstrated a nice surprise revealed is the robustness property of such lattice rules the good projection property and the suitability for a large range of problems the potential possibility and limitations of good lattice points in achieving good quality of moderate and high order projections is investigated the reason why classical lattice rules may not be efficient for high dimensional finance problems is also discussed

fft based option pricing under a mean reverting process with stochastic volatility and jumps numerous studies present strong empirical evidence that certain financial assets may exhibit mean reversion stochastic volatility or jumps this paper explores the valuation of european options when the underlying asset follows a mean reverting log normal process with stochastic volatility and jumps a closed form representation of the characteristic function of the process is derived for the computation of european option prices via the fast fourier transform

infinite dimensional diffusion processes as gibbs measures on c 0 1 z d an infinite lattice system of interacting diffusion processes is characterized as a gibbs distribution on c 0 1 z d with continuous local conditional probabilities using estimates for the vasserstein metric onc 0 1 dobrushin s contraction technique is applied in order to obtain information about macroscopic properties of the entire diffusion process

unbiased estimation with square root convergence for sde models in many settings in which monte carlo methods are applied there may be no known algorithm for exactly generating the random object for which an expectation is to be computed frequently however one can generate arbitrarily close approximations to the random object we introduce a simple randomization idea for creating unbiased estimators in such a setting based on a sequence of approximations applying this idea to computing expectations of path functionals associated with stochastic differential equations sdes we construct finite variance unbiased estimators with a square root convergence rate for a general class of multidimensional sdes we then identify the optimal randomization distribution numerical experiments with various path functionals of continuous time processes that often arise in finance illustrate the effectiveness of our new approach

review of options for structural design sensitivity analysis part 1 linear systems the paper reviews options for structural design sensitivity analysis including global finite differences continuum derivatives discrete derivatives and computational or automated differentiation the objective is to put these different approaches to design sensitivity analysis in the context of accuracy and consistency computational cost and implementation options and effort linear static analysis and transient dynamic analysis are reviewed in a separate appendix special attention is paid to the semi analytical method a future paper will address design sensitivity analysis in nonlinear structural problems

lro in lattice systems of linear oscillators with strong bilinear pair nearest neighbour interaction for systems of one component interacting oscillators on a d dimensional lattice d 1 whose potential energy besides a large nearest neighbour ferromagnetic bilinear term contains a small general translation invariant term the existence of a ferromagnetic long range order for two valued lattice spins equal to the sign of the oscillator variables is established with the help of a peierls type contour bound the ruelle superstability bound is used to derive the contour bound

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

pricing barrier and bermudan style options under time changed levy processes fast hilbert transform approach we construct efficient and accurate numerical algorithms for pricing discretely monitored barrier and bermudan style options under time changed levy processes by applying the fast hilbert transform method to the log asset return dimension and quadrature rule to the dimension of log activity rate of stochastic time change some popular stochastic volatility models like the heston model can be nested in the class of time changed levy processes the computational advantages of the fast hilbert transform approach over the usual fast fourier transform method like exponential decay of errors with regard to the truncation level in the transform and avoidance of recovering option prices at the monitoring time instants can be extended to pricing barrier and bermudan style options under time changed levy processes we compute the fair value of a dividend ruin model with both embedded reflecting dividend barrier and absorbing ruined barrier we also consider pricing of bermudan options in conjunction with th

random generation of stochastic area integrals the authors describe a method of random generation of the integrals a 1 2 t t h int t t h int t s dw 1 r dw 2 s int t t h int t s dw 2 r dw 1 s together with the increments w 1 t h w 1 t and w 2 t h w 2 t of a two dimensional brownian path w 1 t w 2 t the method chosen is based on marsaglia s rectangle wedge tail method generalised to higher dimensions the motivation is the need for a numerical scheme for simulation of strong solutions of general multidimensional stochastic differential equations with an order of convergence o h where h is the stepsize previously no method has obtained an order of convergence better than o sqrt h in the general case

analysis of wiener functionals malliavin calculus and its applications to heat kernels on donne un expose facilement accessible du calcul de malliavin du calcul differentiel de dimension infinie sur l espace de wiener

a general control variate method for option pricing under levy processes abstract we present a general control variate method for simulating path dependent options under levy processes it is based on fast numerical inversion of the cumulative distribution functions and exploits the strong correlation of the payoff of the original option and the payoff of a similar option under geometric brownian motion the method is applicable for all types of levy processes for which the probability density function of the increments is available in closed form numerical experiments confirm that our method achieves considerable variance reduction for different options and levy processes we present the applications of our general approach for asian lookback and barrier options under variance gamma normal inverse gaussian generalized hyperbolic and meixner processes

exact simulation of the ornstein uhlenbeck driven stochastic volatility model abstract this paper proposes a novel exact simulation method for the ornstein uhlenbeck driven stochastic volatility model to accomplish this goal our task hinges on properly handling the ornstein uhlenbeck volatility process the major challenge involves conditionally sampling the integral of its square with respect to time given its marginal state as well as its integral with respect to time we thus derive a closed form laplace transform of this conditional distribution via the techniques of changing probability measure as well as analytical extension then we obtain the corresponding conditional cumulative distribution function via fourier transform inversion and finally sample the distribution via the inverse transform method we show that our method achieves a faster convergence rate of root mean square errors comparing with euler discretization method and apply it in the valuation of discretely monitored path dependent options

exact simulation of stochastic volatility and other affine jump diffusion processes the stochastic differential equations for affine jump diffusion models do not yield exact solutions that can be directly simulated discretization methods can be used for simulating security prices under these models however discretization introduces bias into the simulation results and a large number of time steps may be needed to reduce the discretization bias to an acceptable level this paper suggests a method for the exact simulation of the stock price and variance under hestons stochastic volatility model and other affine jump diffusion processes the sample stock price and variance from the exact distribution can then be used to generate an unbiased estimator of the price of a derivative security we compare our method with the more conventional euler discretization method and demonstrate the faster convergence rate of the error in our method specifically our method achieves an o s 1 2 convergence rate where s is the total computational budget the convergence rate for the euler discretization method is o s 1 3 or slower depending on the model coefficients and option payoff function

optimal importance sampling parameter search for levy processes via stochastic approximation the author proposes stochastic approximation methods of finding the optimal measure change by the exponential tilting for levy processes in monte carlo importance sampling variance reduction in accordance with the structure of the underlying levy measure either a constrained or unconstrained algorithm of the stochastic approximation is chosen for both cases the almost sure convergence to a unique stationary point is proved numerical examples are presented to illustrate the effectiveness of our method

the asymptotic efficiency of simulation estimators a decision theoretic framework is proposed for evaluating the efficiency of simulation estimators the framework includes the cost of obtaining the estimate as well as the cost of acting based on the estimate the cost of obtaining the estimate and the estimate itself are represented as realizations of jointly distributed stochastic processes in this context the efficiency of a simulation estimator based on a given computational budget is defined as the reciprocal of the risk the overall expected cost this framework is appealing philosophically but it is often difficult to apply in practice e g to compare the efficiency of two different estimators because only rarely can the efficiency associated with a given computational budget be calculated however a useful practical framework emerges in a large sample context when we consider the limiting behavior as the computational budget increases a limit theorem established for this model supports and extends a fairly well known efficiency principle

multilevel monte carlo path simulation we show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using monte carlo path simulations in the simplest case of a lipschitz payoff and a euler discretisation the computational cost to achieve an accuracy of o e is reduced from o e 3 to o e 2 log e 2 the analysis is supported by numerical results showing significant computational savings

an euler type method for the strong approximation of the cox ingersoll ross process we analyse the strong approximation of the cox ingersoll ross cir process in the regime where the process does not hit zero by a positivity preserving drift implicit euler type method as an error criterion we use the p th mean of the maximum distance between the cir process and its approximation on a finite time interval we show that under mild assumptions on the parameters of the cir process the proposed method attains up to a logarithmic term the convergence of order 1 2 this agrees with the standard rate of the strong convergence for global approximations of stochastic differential equations with lipschitz coefficients despite the fact that the cir process has a non lipschitz diffusion coefficient

exact sampling of jump diffusions this paper develops a method for the exact simulation of a skeleton a hitting time and other functionals of a one dimensional jump diffusion with state dependent drift volatility jump intensity and jump size the method requires the drift function to be c1 the volatility function to be c2 and the jump intensity function to be locally bounded no further structure is imposed on these functions the method leads to unbiased simulation estimators of security prices transition densities hitting probabilities and other quantities numerical results illustrate its features

quasi monte carlo methods in financial engineering an equivalence principle and dimension reduction quasi monte carlo qmc methods are playing an increasingly important role in the pricing of complex financial derivatives for models in which the prices of the underlying assets are driven by brownian motions the performance of qmc methods is known to depend crucially on the construction of brownian motions this paper focuses on the impact of various constructions although the brownian bridge bb construction often yields very good results as papageorgiou pointed out there are financial derivatives for which the bb construction performs badly papageorgiou a 2002 the brownian bridge does not offer a consistent advantage in quasi monte carlo integration j complexity18 1 171 186 in this paper we first extend papageorgiou s analysis to establish an equivalence principle if the bb construction or any other construction is the preferred construction for a particular financial derivative then for any other construction there is another financial derivative for which the latter construction is the preferred one in this sense all methods of construction are equivalent and no method is consistently superior to others it all depends on the particular financial derivative we then show how to find a good construction for a particular class of financial derivatives in practice our strategy is to find a good construction for an easy problem and then apply it to more complicated problems related to the easy one this strategy is applied to the arithmetic asian options including bermudan asian options based on the weighted average of the stock prices we do this by studying a simpler problem namely the geometric asian option for which the best construction is easily available and applying it to the arithmetic asian option numerical experiments confirm the success of this strategy whereas in qmc all the common methods the standard method bb and principal component analysis may lose their power in some situations the new method behaves very well in all cases further large variance reduction can be achieved in combination with a control variate the new method can be interpreted as a practical way of reducing the effective dimension for some class of functions

spitzer identity wiener hopf factorization and pricing of discretely monitored exotic options the wiener hopf factorization of a complex function arises in a variety of fields in applied mathematics such as probability finance insurance queuing theory radio engineering and fluid mechanics the factorization fully characterizes the distribution of functionals of a random walk or a levy process such as the maximum the minimum and hitting times here we propose a constructive procedure for the computation of the wiener hopf factors valid for both single and double barriers based on the combined use of the hilbert and the z transform the numerical implementation can be simply performed via the fast fourier transform and the euler summation given that the information in the wiener hopf factors is strictly related to the distributions of the first passage times as a concrete application in mathematical finance we consider the pricing of discretely monitored exotic options such as lookback and barrier options when the underlying price evolves according to an exponential levy process we show that the computational cost of our procedure is independent of the number of monitoring dates and the error decays exponentially with the number of grid points

sensitivity estimates from characteristic functions the likelihood ratio method lrm is a technique for estimating derivatives of expectations through simulation lrm estimators are constructed from the derivatives of probability densities of inputs to a simulation we investigate the application of the likelihood ratio method for sensitivity estimation when the relevant densities for the underlying model are known only through their characteristic functions or laplace transforms this problem arises in financial applications where sensitivities are used for managing risk and where a substantial class of models have transition densities known only through their transforms we quantify various sources of errors arising when numerical transform inversion is used to sample through the characteristic function and to evaluate the density and its derivative as required in lrm this analysis provides guidance for setting parameters in the method to accelerate convergence

testing transition state theory on kac zwanzig model a variant of the kac zwanzig model is used to test the prediction of transition state theory tst and variational transition state theory vtst the model describes the evolution of a distinguished particle moving in a double well external potential and coupled to n free particles through linear springs while the kac zwanzig model is deterministic under appropriate choice of the model parameters the evolution of the distinguished particle can be approximated by a two state markov chain whose transition rate constants can be computed exactly in suitable limit here these transition rate constants are compared with the predictions of tst and vtst it is shown that the application of tst with a naive albeit natural choice of dividing surface leads to the wrong prediction of the transition rate constants this is due to crossings of the dividing surface that do not correspond to actual transition events however optimizing over the dividing surface within vtst allows one to eliminate completely these spurious crossings and therefore derive the correct transition rate constants for the model the reasons why vtst is successful in this model are discussed which allows one to speculate on the reliability of vtst in more complicated systems

density approximations for multivariate affine jump diffusion processes we introduce closed form transition density expansions for multivariate affine jump diffusion processes the expansions rely on a general approximation theory which we develop in weighted hilbert spaces for random variables which possess all polynomial moments we establish parametric conditions which guarantee existence and differentiability of transition densities of affine models and show how they naturally fit into the approximation framework empirical applications in option pricing credit risk and likelihood inference highlight the usefulness of our expansions the approximations are extremely fast to evaluate and they perform very accurately and numerically stable

quasi monte carlo methods in numerical finance this paper introduces and illustrates a new version of the monte carlo method that has attractive properties for the numerical valuation of derivatives the traditional monte carlo method has proven to be a powerful and flexible tool for many types of derivatives calculations under the conventional approach pseudo random numbers are used to evaluate the expression of interest unfortunately the use of pseudo random numbers yields an error bound that is probabilistic which can be a disadvantage another drawback of the standard approach is that many simulations may be required to obtain a high level of accuracy there are several ways to improve the convergence of the standard method this paper suggests a new approach which promises to be very useful for applications in finance quasi monte carlo methods use sequences that are deterministic instead of random these sequences improve convergence and give rise to deterministic error bounds the method is explained and illustrated with several examples these examples include complex derivatives such as basket options asian options and energy swaps

handbook of brownian motion facts and formulae i theory i stochastic processes in general ii linear diffusions iii stochastic calculus iv brownian motion v local time as a markov process vi differential systems associated to brownian motion appendix 1 briefly on some diffusions ii tables of distributions of functionals of brownian motion and related processes 1 brownian motion 2 brownian motion with drift 3 reflecting brownian motion 4 bessel process of order 5 bessel process of order 1 2 6 bessel process of order zero 7 ornstein uhlenbeck process 8 radial ornstein uhlenbeck process 9 geometric brownian motion appendix 2 special functions appendix 3 inverse laplace transforms appendix 4 differential equations and their solutions appendix 5 formulae for n fold differentiation

an accelerating quasi monte carlo method for option pricing under the generalized hyperbolic levy process in this paper we develop a simple and yet practically efficient algorithm for simulating high dimensional exotic options our method is based on an extension of imai and tan s linear transformation method which is originally proposed in the context of simulating a gaussian process by generalizing this method to other stochastic processes and exploiting the numerical inversion method of hormann and leydold this method can be used to enhance quasi monte carlo method in a wide range of applications we demonstrate the relative efficiency of our proposed simulation technique using exotic option examples including asian lookback barrier and cliquet options for which the underlying asset price follows an exponential generalized hyperbolic levy process we also illustrate the impact of our proposed method on dimension reduction

a general control variate method for multi dimensional sdes an application to multi asset options under local stochastic volatility with jumps models in finance abstract this paper presents a new control variate method for general multi dimensional stochastic differential equations sdes including jumps in order to reduce the variance of monte carlo method our control variate method is based on an asymptotic expansion technique and does not require an explicit characteristic function of sdes this is an extension of previous researches using asymptotic expansions to obtain the control variates for such general models moreover in our control variate method the regression estimators can be chosen for each number of jump times with a stratified sampling and improve the efficiency of the variance reduction this paper also provides the asymptotic bias and variance of our method in terms of its terminal time and a small noise parameter used in an asymptotic expansion method for an application of our method we evaluate multi asset options whose payoffs are expressed as linear functions of the underlying asset price processes in general local stochastic volatility with jumps models and show a calculation scheme of control variates for greeks in numerical experiments we apply the new control variate method to pricing basket spread and average options and delta of basket options under a zabr type local stochastic volatility model with jumps and confirm that our method works very well

single transform formulas for pricing asian options in a general approximation framework under markov processes abstract recently cai song and kou 2015 proposed closed form double transform approximation formulas for prices of both discretely and continuously monitored asian options under the setting of a general continuous time markov chain in this note we analytically invert the z transform and the laplace transform involved in their final results respectively for the discretely and the continuously monitored cases and we obtain explicit single laplace transforms of option prices this reduction in the dimension of numerical integral has meaningful consequences both in computational efficiency and in practical implementation of the formulas extensive numerical experiments illustrate the improved performance of our results

on nonnegative unbiased estimators we study the existence of algorithms generating almost surely non negative unbiased estimators of some quantities of practical interest our work is motivated by recent exact approximation methods to compute integrals with respect to any target distribution up to any arbitrary precision these techniques can be implemented as long as non negative unbiased estimators of the target density evaluations are available in particular we show that given a non constant function f from r to r and real valued unbiased estimators xn of r there is no algorithm yielding almost surely non negative unbiased estimators of f even if xn is itself a s non negative then there is no algorithm yielding a s non negative unbiased estimators of f if f is not increasing however if the support of xn is a known interval of r then such algorithms exist as long as the function f satisfies some polynomial bound away from zero the results are discussed in the light of several typical situations corresponding to different functions f

path generation for quasi monte carlo simulation of mortgage backed securities monte carlo simulation is playing an increasingly important role in the pricing and hedging of complex path dependent financial instruments low discrepancy simulation methods offer the potential to provide faster rates of convergence than those of standard monte carlo methods however in high dimensional problems special methods are required to ensure that the faster convergence rates hold indeed ninomiya and tezuka 1996 have shown highdimensional examples in which low discrepancy methods perform worse than monte carlo methods the principal component construction introduced by acworth et al 1998 provides one solution to this problem however the computational effort required to generate each path grows quadratically with the dimension of the problem this article presents two new methods that offer accuracy equivalent in terms of explained variability to the principal components construction with computational requirements that are linearly related to the problem dimension one method is to take into account knowledge about the payoff function which makes it more flexible than the brownian bridge construction numerical results are presented that show the benefits of such adjustments the different methods are compared for the case of pricing mortgage backed securities using the hull white term structure model

henrici s transformation and its application to the computation of derivatives of eigensystems in this paper we examine the application of henrici s transformation to a sequence of vectors generated by a simultaneous iteration method for computing the partial derivatives of eigenvalues and eigenvectors of parameter dependent matrices numerical results support our theoretical analysis and indicate that henrici s method compares favorably with other vector sequence transformation methods

general multilevel monte carlo methods for pricing discretely monitored asian options abstract we describe general multilevel monte carlo methods that estimate the price of an asian option monitored at m fixed dates for a variety of processes that can be simulated exactly we prove that for the same computational cost our method yields an unbiased estimator with variance lower than the variance of the standard monte carlo estimator by a factor of order m we show how to combine our approach with the milstein scheme for processes driven by scalar stochastic differential equations and with the euler scheme for processes driven by multidimensional stochastic differential equations numerical experiments confirm that our method outperforms the conventional monte carlo algorithm by a factor proportional to m

low discrepancy sequences in high dimensions how well are their projections distributed quasi monte carlo qmc methods have been successfully used to compute high dimensional integrals arising in many applications especially in finance to understand the success and the potential limitation of qmc this paper focuses on quality measures of point sets in high dimensions we introduce the email protected superposition and truncation discrepancies which measure the quality of selected projections of a point set on lower dimensional spaces these measures are more informative than the classical ones we study their relationships with the integration errors and study the tractability issues we present efficient algorithms to compute these discrepancies and perform computational investigations to compare the performance of the sobol nets with that of the sets of latin hypercube sampling and random points numerical results show that in high dimensions the superiority of the sobol nets mainly derives from the one dimensional projections and the projections associated with the earlier dimensions for order 2 and higher order projections all these point sets have similar behavior on the average in weighted cases with fast decaying weights the sobol nets have a better performance than the other two point sets the investigation enables us to better understand the properties of qmc and throws new light on when and why qmc can have a better or no better performance than monte carlo for multivariate integration in high dimensions

on the valuation of variance swaps with stochastic volatility abstract this paper is an extension to a recent paper by zhu and lian 2011 1 in which a closed form exact solution was presented for the price of variance swaps with a particular definition of the realized variance here we further demonstrate that our approach is quite versatile and can be used for other definitions of the realized variance as well in particular we present a closed form formula for the price of a variance swap with the realized variance in the payoff function being defined as a logarithmic return of the underlying asset at some pre specified discretely sampling points the simple formula presented here is a result of successfully finding an exact solution of the partial differential equation pde system based on the heston 1993 s 2 two factor stochastic volatility model a distinguishable feature of this new solution is that the computational time involved in pricing variance swaps with discretely sampling time has been substantially improved

mode tracking issues in structural optimization within the context of optimization of the structural dynamics properties of finite element models methodology is developed for the tracking of eigenpairs through changes in the structural eigenvalue problem the goal is to eliminate difficulties caused by mode switching i e frequency crossing out of several candidate methods two methods for mode tracking are successful the first method the higher order eigenpair perturbation algorithm is based on a perturbation expansion of the eigenproblem it iteratively computes changes in the eigenpairs due to parameter perturbations with the important feature of maintaining the correspondence between the baseline and perturbed eigenpairs the second method is a cross orthogonality check method which uses mass orthogonality to reestablish correspondence after a standard reanalysis modified eigenpair extraction routines lanczos subspace iteration inverse power were unsuccessful in tracking modes applications of mode tracking technology that are presented are frequency constrained optimization and optimization with mode shape constraints each application procedure is outlined and examples are given recommendations are made based on method efficiency and robustness in the example problems

exact simulation of point processes with stochastic intensities point processes with stochastic arrival intensities are ubiquitous in many areas including finance insurance reliability health care and queuing they can be simulated from a poisson process by time scaling with the cumulative intensity the paths of the cumulative intensity are often generated with a discretization method however discretization introduces bias into the simulation results the magnitude of the bias is difficult to quantify this paper develops a sampling method that eliminates the need to discretize the cumulative intensity the method is based on a projection argument and leads to unbiased simulation estimators it is exemplified for a point process whose intensity is a function of a jump diffusion process and the point process itself in this setting the method facilitates the exact sampling of both the point process and the driving jump diffusion process numerical experiments demonstrate the effectiveness of the method

exact simulation of the sabr model the stochastic alpha beta rho sabr model becomes popular in the financial industry because it is capable of providing good fits to various types of implied volatility curves observed in the marketplace however no analytical solution to the sabr model exists that can be simulated directly this paper explores the possibility of exact simulation for the sabr model our contribution is threefold i we propose an exact simulation method for the forward price and its volatility in two special but practically interesting cases i e when the elasticity 1 or when 1 and the price and volatility processes are instantaneously uncorrelated primary difficulties involved are how to simulate two random variables whose distributions can be expressed in terms of the hartman watson and the noncentral chi squared distribution functions respectively two novel simulation schemes are proposed to achieve numerical accuracy efficiency and stability one stems from numerical laplace inversion and asian option literature and the other is based on recent developments in evaluating the noncentral chi squared distribution functions in a robust way numerical examples demonstrate that our method is fast and accurate under various market environments ii when 1 but the price and volatility processes are correlated our simulation method becomes a semi exact one numerical results suggest that it is still quite accurate when the time horizon is not long e g no greater than one year for long time horizons a piecewise semi exact simulation scheme is developed that reduces the biases substantially iii for european option pricing under the sabr model we propose a conditional simulation method which reduces the variance of the plain simulation significantly e g by more than 99 the e companion is available at https doi org 10 1287 opre 2017 1617

techniques for generating analytic covariance expressions for eigenvalues and eigenvectors one of the key problems in data fusion is the estimation of a parameter vector from a set of noisy measurements in many cases the optimal estimate of such a parameter vector can be solved through the least squares problem there are however a number of important problems whose solution instead takes the form of an eigenvalue eigenvector problem furthermore knowledge of the uncertainty of the optimal estimate is as important as the estimate itself for many practical applications and this uncertainty is typically captured in the form of a covariance matrix while the covariance matrix for a least squares estimate has been well studied there has been substantially less research on the covariance of an optimal estimate originating from an eigenvalue eigenvector problem in this paper we develop general expressions that determine the uncertainty in a vector estimate obtained from an eigenvalue eigenvector problem given the uncertainty of the matrix this includes developing expressions for the analytic derivatives of the eigenvalues and eigenvectors with respect to the matrix from which they come finally the techniques developed are numerically validated with forward finite differencing and a monte carlo analysis and then used to determine covariance expressions for an ellipse fitting technique and the estimation of attitude quaternions

closed form expansions of discretely monitored asian options in diffusion models in this paper we propose a closed form asymptotic expansion approach to pricing discretely monitored asian options in general one dimensional diffusion models our expansion is a small time expansion because the expansion parameter is selected to be the square root of the length of monitoring interval this expansion method is distinguished from many other pricing oriented expansion algorithms in the literature because of two appealing features first we illustrate that it is possible to explicitly calculate not only the first several expansion terms but also any general expansion term in a systematic way second the convergence of the expansion is proved rigorously under some regularity conditions numerical experiments suggest that the closed form expansion formula with only a few terms e g four terms up to the third order is accurate fast and easy to implement for a broad range of diffusion models even including those violating the regularity conditions

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

efficient estimation and filtering for multivariate jump diffusions abstract this paper develops estimators of the transition density filters and parameters of multivariate jump diffusion models the drift volatility jump intensity and jump magnitude are allowed to be state dependent and non affine it is not necessary to diagonalize the volatility matrix our density and filter estimators converge at the canonical rate typically associated with exact monte carlo estimation our parameter estimators have the same asymptotic distribution as maximum likelihood estimators which are often intractable for the class of models we consider the results of this paper enable the empirical analysis of previously intractable models of asset prices and economic time series

on multiple eigenvalues of matrices depending on several parameters the eigenvalues and eigenvectors as well as their derivatives of matrices depending on one or several parameters have been studied by sun j comput math 3 1985 pp 351 364 using the implicit function theorems and others but not for the more general multiple eigenvalues in this paper the mean of the multiple eigenvalues in terms of the trace operator tr the corresponding invariant subspaces and their derivatives are studied using a generalization of sun s approach the numerical aspects concerning the computation of such derivatives by direct and iterative methods e g simultaneous and inverse simultaneous iterations will be discussed briefly the main results in this paper apply to clusters of nonmultiple eigenvalues as well the implications e g on simultaneous iteration will be explored

universal recurrence algorithm for computing nuttall generalized marcum and incomplete toronto functions and moments of a noncentral χ 2 random variable abstract this article proposes a novel recurrence algorithm for computing nuttall generalized marcum and incomplete toronto functions as well as truncated and raw moments of any real order for a noncentral 2 random variable the numerical results highlight that the proposed recurrence method offers an excellent speed accuracy trade off for a wide variety of meaningful and relevant applications ranging from the operations management and industrial engineering fields to problems encountered in the financial engineering industry

efficient simulation of clustering jumps with cir intensity we introduce a broad family of generalised self exciting point processes with cir type intensities and we develop associated algorithms for their exact simulation the underlying models are extensions of the classical hawkes process which already has numerous applications in modelling the arrival of events with clustering or contagion effect in finance economics and many other fields interestingly we find that the cir type intensity together with its point process can be sequentially decomposed into simple random variables which immediately leads to a very efficient simulation scheme our algorithms are also pretty accurate and flexible they can be easily extended to further incorporate externally excited jumps or to a multidimensional framework some typical numerical examples and comparisons with other well known schemes are reported in detail in addition a simple application for modelling a portfolio loss process is presented the online appendix is available at https doi org 10 1287 op

a closed form solution for the exercise strategy in a real options model with a jump diffusion process an investment strategy in a real options model with a jump diffusion process is considered using an integro differential equation approach the optimal stopping problem is converted to a free boundary problem an analytical formula for the option value and a closed form solution for the investment exercise threshold are obtained furthermore both upper and lower bounds of the threshold are derived and an approximation for the threshold is developed finally numerical examples are presented

boundary preserving semianalytic numerical algorithms for stochastic differential equations construction of splitting step methods and properties of related nonnegativity and boundary preserving semianalytic numerical algorithms for solving stochastic differential equations sdes of ito type are discussed as the crucial assumption we oppose conditions such that one can decompose the original system of sdes into subsystems for which one knows either the exact solution or its conditional transition probability we present convergence proofs for a newly designed splitting step algorithm and simulation studies for numerous well known numerical examples ranging from stochastic dynamics occurring in asset pricing in mathematical finance cox ingersoll ross cir and constant elasticity of variance cev models to measure valued diffusion and super brownian motion stochastic pdes spdes as met in biology and physics

fast numerical pricing of barrier options under stochastic volatility and jumps in this paper we prove the existence of an integral closed form solution for pricing barrier options in both heston and bates frameworks the option value depends on time on the price and on the volatility of the underlying asset and it can be computed as the solution of a two dimensional pricing partial integro differential equation the integral representation formula of the solution is derived by projection of the differential equation and exploiting the properties of the adjoint operator we derive the expression of the fundamental solution green s function necessary for the integral representation formula the computation is based on the interpretation of the fundamental solution as the joint transition probability density function of the underlying asset price and variance and is obtained through fourier inverse transform of a suitable conditional characteristic function we propose a numerical scheme to approximate the option price based on the classical boundary element method and we provide two numerical examples showing the computational efficiency and accuracy of the proposed new method the algorithm can be modified to compute greeks as well

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

calculation of eigenvector derivatives for structures with repeated eigenvalues in structural optimization and system identification eigenvector derivatives provide important information for updating design model parameters when the current parameter values yield repeated eigenvalues it has not been possible previously to calculate unique eigenvector derivatives recent work has provided a method for determining unique eigenvalue derivatives for this case but methods for calculating the eigenvector sensitivities have been incomplete in this work a complete method for calculation of repeated root eigenvector derivatives is shown for the real symmetric structural eigenproblem the derivation is completed by using information from the second derivative of the eigen problem and is limited to the case of distinct eigenvalue sensitivities as an example the repeated root eigenvector sensitivities are calculated for a simple three degree of freedom beam grillage comparisons of linear approximations using these derivatives to the calculated eigenvectors support the accuracy of the formulation

the law of the euler scheme for stochastic differential equations i convergence rate of the distribution function we study the approximation problem ofef x t byef x t n where x t is the solution of a stochastic differential equation x t n is defined by the euler discretization scheme with stept n andf is a given function for smoothf s talay and tubaro have shown that the erroref x t f x t n can be expanded in powers of 1 n which permits to construct romberg extrapolation precedures to accelerate the convergence rate here we prove that the expansion exists also whenf is only supposed measurable and bounded under an additional nondegeneracy condition of hormander type for the infinitesimal generator of x t to obtain this result we use the stochastic variations calculus in the second part of this work we will consider the density of the law ofx t n and compare it to the density of the law ofx t

efficient monte carlo and quasi monte carlo option pricing under the variance gamma model we develop and study efficient monte carlo algorithms for pricing path dependent options with the variance gamma model the key ingredient is difference of gamma bridge sampling based on the representation of a variance gamma process as the difference of two increasing gamma processes for typical payoffs we obtain a pair of estimators named low and high with expectations that 1 are monotone along any such bridge sampler and 2 contain the continuous time price these estimators provide pathwise bounds on unbiased estimators that would be more expensive infinitely expensive in some situations to compute by using these bounds with extrapolation techniques we obtain significant efficiency improvements by work reduction we then combine the gamma bridge sampling with randomized quasi monte carlo to reduce the variance and thus further improve the efficiency we illustrate the large efficiency improvements on numerical examples for asian lookback and barrier options

dirichlet bridge sampling for the variance gamma process pricing path dependent options the authors develop a new monte carlo based method for pricing path dependent options under the variance gamma vg model the gamma bridge sampling method proposed by avramidis et al avramidis a n p l ecuyer p a tremblay 2003 efficient simulation of gamma and variance gamma processes proc 2003 winter simulation conf ieee press piscataway nj 319 326 and ribeiro and webber ribeiro c n webber 2004 valuing path dependent options in the variance gamma model by monte carlo with a gamma bridge j computational finance7 2 81 100 is generalized to a multivariate dirichlet construction bridging simultaneously over all time partition points of the trajectory of a gamma process the generation of the increments of the gamma process given its value at the terminal point is interpreted as a dirichlet partition of the unit interval the increments are generated in a decreasing stochastic order and under the kingman limit have a known distribution thus simulation of a trajectory from the gamma process requires generating only a small number of uniforms avoiding the expensive simulation of beta variates via numerical probability integral inversion the proposed method is then applied in simulating the trajectory of a vg process using its difference of gammas representation it has been implemented in both plain monte carlo and quasi monte carlo environments it is tested in pricing lookback barrier and asian options and is shown to provide consistent efficiency gains compared to the sequential method and the difference of gammas bridge sampling proposed by avramidis and l ecuyer avramidis a n p l ecuyer 2006 efficient monte carlo and quasi monte carlo option pricing under the variance gamma model management sci 52 12 1930 1944

high order discretization schemes for the cir process application to affine term structure and heston models this paper presents weak second and third order schemes for the cox ingersoll ross cir process without any restriction on its parameters at the same time it gives a general recursive construction method to get weak second order schemes that extends the one introduced by ninomiya and victoir cite nv combining these both results this allows to propose a second order scheme for more general affine diffusions simulation examples are given to illustrate the convergence of these schemes on cir and heston models algorithms are stated in a pseudocode language

superreplication of financial derivatives via convex programming we give a method based on convex programming to calculate the optimal superreplicating and subreplicating prices and corresponding hedging portfolios of a financial derivative in terms of other financial derivatives in a discrete time setting our method produces a model that matches the superreplicating or subreplicating price within an arbitrary precision and is consistent with the other financial derivatives prices applications include robust replication in terms of call prices with various strikes and maturities of forward start options volatility and variance swaps and derivatives cliquet calls barrier options and lookback and asian options numerical examples show that in some cases the best superreplicating and or subreplicating prices are within 10 of the price obtained by a standard model but considerably differ from it in other cases our method can incorporate bid ask spreads interest rates and dividends and various limitations to the diffusion model the supplementary material is a

an auto realignment method in quasi monte carlo for pricing financial derivatives with jump structures abstract discontinuities are common in the pricing of financial derivatives and have a tremendous impact on the accuracy of quasi monte carlo qmc method while if the discontinuities are parallel to the axes good efficiency of the qmc method can still be expected by realigning the discontinuities to be axes parallel wang tan 2013 succeeded in recovering the high efficiency of the qmc method for a special class of functions motivated by this work we propose an auto realignment method to deal with more general discontinuous functions the k means clustering algorithm a classical algorithm of machine learning is used to select the most representative normal vectors of the discontinuity surface by applying this new method the discontinuities of the resulting function are realigned to be friendly for the qmc method numerical experiments demonstrate that the proposed method significantly improves the performance of the qmc method

randomized dimension reduction for monte carlo simulations we present a new unbiased algorithm that estimates the expected value of f u via monte carlo simulation where u is a vector of d independent random variables and f is a function of d variables

two order parameters in quantum xz spin models with gibbsian ground states we describe a family of quantum spin models which are generators of a discrete markovian process we show that there exists an explicit expression for the ground state of such models and give a simple argument for the existence of two types of long range order in such systems two special examples of these systems are analysed in detail

the importance of strictly local martingales applications to radial ornstein uhlenbeck processes for a wide class of local martingales m t there is a default function which is not identically zero only when m t is strictly local i e not a true martingale this default in the martingale property allows us to characterize the integrability of functions of sup s t m s in terms of the integrability of the function itself we describe some paradoxical mean decreasing local sub martingales and the default functions for bessel processes and radial ornstein uhlenbeck processes in relation to their first hitting and last exit times

unbiased estimators and multilevel monte carlo multilevel monte carlo mlmc and recently proposed unbiased estimators are closely related this connection is elaborated by presenting a new general class of unbiased estimators which admits previous debiasing schemes as special cases new lower variance estimators are proposed which are stratified versions of earlier unbiased schemes under general conditions essentially when mlmc admits the canonical square root monte carlo error rate the proposed new schemes are shown to be asymptotically as efficient as mlmc both in terms of variance and cost the experiments demonstrate that the variance reduction provided by the new schemes can be substantial

exact simulation of the wishart multidimensional stochastic volatility model in this article we propose an exact simulation method of the wishart multidimensional stochastic volatility wmsv model a single asset model with a multidimensional wishart variance process our method is based on analysis of the conditional characteristic function of the log price given a terminal volatility level in particular we found an explicit expression for the conditional characteristic function for the heston model numerical experiments demonstrate that our new method is much faster and reliable than the euler discretization method

general optimized lower and upper bounds for discrete and continuous arithmetic asian options we propose an accurate method for pricing arithmetic asian options on the discrete or continuous average in a general model setting by means of a lower bound approximation in particular we derive analytical expressions for the lower bound in the fourier domain this is then recovered by a single univariate inversion and sharpened using an optimization technique in addition we derive an upper bound to the error from the lower bound price approximation our proposed method can be applied to computing the prices and price sensitivities of asian options with fixed or floating strike price discrete or continuous averaging under a wide range of stochastic dynamic models including exponential levy models stochastic volatility models and the constant elasticity of variance diffusion our extensive numerical experiments highlight the notable performance and robustness of our optimized lower bound for different test cases

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

alternative models for stock price dynamics abstract this paper evaluates the role of various volatility specifications such as multiple stochastic volatility sv factors and jump components in appropriate modeling of equity return distributions we use estimation technology that facilitates nonnested model comparisons and use a long data set which provides rich information about the conditional and unconditional distribution of returns we consider two broad families of models 1 the multifactor loglinear family and 2 the affine jump family both classes of models have attracted much attention in the derivatives and econometrics literatures there are various tradeoffs in considering such diverse specifications if pure diffusion sv models are chosen over jump diffusions it has important implications for hedging strategies if logarithmic models are chosen over affine ones it may seriously complicate option pricing comparing many different specifications of pure diffusion multifactor models and jump diffusion models we find that 1 log linear models have to be extended to two factors with feedback in the mean reverting factor 2 affine models have to have a jump in returns stochastic volatility or probably both models 1 and 2 are observationally equivalent on the data set in hand in either 1 or 2 the key is that the volatility can move violently as we obtain models with comparable empirical fit one must make a choice based on arguments other than statistical goodness of fit criteria the considerations include facility to price options to hedge and parsimony the affine specification with jumps in volatility might therefore be preferred because of the closed form derivatives prices

long range order in nonequilibrium systems of interacting brownian linear oscillators long range order lro is established with the help of a generalized peierls argument for non equilibrium lattice systems of one dimensional linear interacting oscillators whose equation of motion for a finite number of them is the smolouchowski equation for the density of a probability distribution interaction is mediated through the pair nearest neighbor quadratic translation invariant potential the initial density is gibbsian with a potential energy satisfying the ruelle superstability and regularity conditions

estimating option greeks under the stochastic volatility using simulation abstract as the black scholes bs equation being widely used to price options which is based on a hypothesis that the underlying bonds stocks volatility is constant many scholars proposed the extended version of this formula to predict the behavior of the volatility so stochastic volatility model is the improved version in which fixed volatility is replaced the purpose of this study is to adopt one of the famous stochastic volatility models heston model 1993 to price european call options put option values can easily obtained by call put parity if it is needed simulation has proved to be a valuable tool for estimating options price derivatives i e greeks this paper proposes the method for the simulation of stock prices and variance under the heston stochastic volatility model we consider three different models based on the heston model we present two direct methods a path wise method and likelihood ratio method for estimating the derivatives of options then we compare it with black scholes equation and make a sensitivity analysis for its parameters by using estimator s approaches

accuracy of maximum likelihood parameter estimators for heston stochastic volatility sde we study approximate maximum likelihood estimators mles for the parameters of the widely used heston stock price and volatility stochastic differential equations sdes we compute explicit closed form estimators maximizing the discretized log likelihood of n observations recorded at times t 2t ldots nt we compute the asymptotic biases of these parameter estimators for t fixed and n rightarrow infty as well as the rate at which these biases vanish when t rightarrow 0 we determine asymptotically consistent explicit modifications of these mles for the heston volatility sde we identify a canonical form determined by two canonical parameters 0 1 2 which are explicit functions of the original sde parameters we analyze theoretically the asymptotic distribution of the mles and of their consistent modifications and we outline their concrete speeds of convergence by numerical simulations we clarify in terms of zeta the precise dichotomy between asymptotic normality and attraction by stable like distributions with heavy tails we illustrate numerical model fitting for heston sdes by two concrete examples one for daily data and one for intraday data both with moderate values of n

filtered likelihood for point processes abstract point processes are widely used in finance and economics to model the timing of defaults market transactions unemployment spells births and a range of other events we develop and analyze likelihood estimators for the parameters of a marked point process and incompletely observed explanatory factors that influence the arrival intensity and mark distribution we establish an approximation to the likelihood and analyze the convergence and large sample properties of the associated estimators numerical results illustrate the behavior of our estimators

quasi monte carlo method for infinitely divisible random vectors via series representations an infinitely divisible random vector without gaussian component admits representations of shot noise series due to possible slow convergence of the series they have not been investigated as a device for monte carlo simulation in this paper we investigate the structure of shot noise series representations from a simulation point of view and discuss the effectiveness of quasi monte carlo methods applied to series representations the structure of series representations in nature tends to decrease their effective dimension and thus increase the efficiency of quasi monte carlo methods thanks to the greater uniformity of low discrepancy sequence in the lower dimension we illustrate the effectiveness of our approach through numerical results of moment and tail probability estimations for stable and gamma random variables

optimal unbiased estimation for expected cumulative discounted cost abstract we consider estimating an expected infinite horizon cumulative discounted cost reward contingent on an underlying stochastic process by monte carlo simulation an unbiased estimator based on truncating the cumulative cost at a random horizon is proposed explicit forms for the optimal distributions of the random horizon are given and explicit expressions for the optimal random truncation level are obtained leading to a full analysis of the bias variance tradeoff when comparing this new class of randomized estimators with traditional fixed truncation estimators moreover we characterize when the optimal randomized estimator is preferred over a fixed truncation estimator by considering the tradeoff between bias and variance this comparison provides guidance on when to choose randomized estimators over fixed truncation estimators in practice numerical experiments substantiate the theoretical results

a cluster expansion for stochastic lattice fields a langevin equation of landau ginzburg type for the stochastic dynamics of a scalar field on a lattice is studied a cluster expansion is developed for this problem which converges for large mass as a consequence one establishes uniformly in the volume a exponential decay of correlations in space and time and b exponential approach to equilibrium for a class of nearby initial distributions

statistical properties and economic implications of jump diffusion processes with shot noise effects abstract the shot noise jump diffusion snjd model aims to reflect how economic variables respond to the arrival of sudden information this paper analyzes the snjd model providing its statistical distribution and closed form expressions for the characteristic function and moments we also analyze the dynamics of the model concluding that the degree of serial autocorrelation is related to the occurrence and magnitude of abnormal information in addition we provide some useful approximations in a particular case that considers exponential type decay empirically we propose a gmm approach to estimate the parameters of the model and present an empirical application for the stocks included in the dow jones averaged index our findings seem to confirm the presence of shot noise effects in 73 of the stocks and a strong relationship between the shot noise process and the autocorrelation pattern embedded in data

lro in lattice systems of linear classical and quantum oscillators strong nearest neighbor pair quadratic interaction for systems of one component interacting oscillators on the d dimensional lattice d 1 whose potential energy besides a large nearest neighbour n n ferromagnetic translation invariant quadratic term contains small non nearest neighbour translation invariant term an existence of a ferromagnetic long range order for two valued lattice spins equal to a sign of oscillator variables is established for sufficiently large magnitude g of the n n interaction with the help of the peierls type contour bound the ruelle superstability bound is used for a derivation of the contour bound

computation of derivatives of repeated eigenvalues and corresponding eigenvectors by simultaneous iteration an earlier simultaneous iteration method for computing derivatives of eigenvalues and the corresponding eigenvectors of nonsymmetric matrices depending smoothly on some real parameters is extended to the case of repeated eigenvalues

robust sensitivity analysis for stochastic systems we study a worst case approach to measure the sensitivity to model misspecification in the performance analysis of stochastic systems the situation of interest is when only minimal parametric information is available on the form of the true model under this setting we post optimization programs that compute the worst case performance measures subject to constraints on the amount of model misspecification measured by kullback leibler divergence our main contribution is the development of infinitesimal approximations for these programs resulting in asymptotic expansions of their optimal values as the divergence shrinks to zero the coefficients of these expansions can be computed via simulation and are mathematically derived from the representation of the worst case models as changes of measure that satisfy a well defined class of functional fixed point equations

localization and exact simulation of brownian motion driven stochastic differential equations generating sample paths of stochastic differential equations sde using the monte carlo method finds wide applications in financial engineering discretization is a popular approximate approach to generating those paths it is easy to implement but prone to simulation bias this paper presents a new simulation scheme to exactly generate samples for sdes the key observation is that the law of a general sde can be decomposed into a product of the law of standard brownian motion and the law of a doubly stochastic poisson process an acceptance rejection algorithm is devised based on the combination of this decomposition and a localization technique the numerical results corroborates that the mean square error of the proposed method is in the order of ot 1 2 which is superior to the conventional discretization schemes furthermore the proposed method also can generate exact samples for sde with boundaries which the discretization schemes usually find difficulty in dealing with