An analytic valuation method for multivariate contingent claims with regime-switching volatilities

In this paper, we provide an analytic valuation method for European-type contingent claims written on multiple assets in a stochastic market environment. We employ a two-state Markov regime-switching volatility in order to reflect stochastically changing market conditions. The method is developed by exploiting the probability density of the occupation time for which the underlying asset processes are in a certain regime during a time period. In order to show its usefulness, we derive analytic valuation formulas for quanto options and exchange options with two underlying assets, as examples.     

Pricing and hedging of American contingent claims in incomplete markets

0.IntroductionandSummaryThecelebratedpapersof[2]and[3],pavedthewayforpricingoptionsonstocks,onthebasisofthefollowingprinciple:inacompletemarket(suchastheoneinSection1.5),everycontingentclaimcanbeattainedexactlybyinvestinginthemarketandstartingwithala...     

Solution of the elastic/visco-plastic constitutive equations: A finite volume approach

A novel two dimensional finite volume (FV) approach to the solution of elastic / visco-plastic solid mechanics problems is described. The numerical procedure is an extension of that developed by Fryer et al. for linear elastic materials and utilizes an unstructured mesh. The procedure is compared favorably with conventional finite element (FE) formulations with respect to both accuracy and CPU time. A number of test problems demonstrate the ability of the FV procedure to model a range of boundary constraints, in addition to mechanical and thermal loads. It might be argued from the work reported here that the FV approach shows the potential to be at least as effective as conventional FE formulations in representing nonlinear solid mechanics behavior.     

A convex duality approach for pricing contingent claims under partial information and short selling constraints

We consider the pricing problem facing a seller of a contingent claim. We assume that this seller has some general level of partial information, and that he is not allowed to sell short in certain assets. This pricing problem, which is our primal problem, is a constrained stochastic optimization problem. We derive a dual to this problem by using the conjugate duality theory introduced by Rockafellar. Furthermore, we give conditions for strong duality to hold. This gives a characterization of the price of the claim involving martingale- and super-martingale conditions on the optional projection of the price processes.     

The non-Markovian approach to the valuation and hedging of European contingent claims on power with scaling spikes

We present and further develop a new approach to modeling power prices with spikes proposed earlier by the author. In contrast to other approaches, we model power prices with spikes as a non-Markovian stochastic process that allows for modeling spikes directly as self-reversing jumps. We show how this approach can be used to value European contingent claims on power with spikes as well as to value and dynamically hedge European contingent claims on forwards on power for power with spikes in a practically important special case of the scaling probability distribution for the magnitude of spikes.     

A mixed finite volume method for elliptic problems

We derive a novel finite volume method for the elliptic equation, using the framework of mixed finite element methods to discretize the pressure and velocities on two different grids (covolumes), triangular (tetrahedral) mesh and control volume mesh. The new discretization is defined for tensor diffusion coefficient and well suited for heterogeneous media. When the control volumes are created by connecting the center of gravity of each triangle to the midpoints of its edges, we show that the discretization is stable and first order accurate for both scalar and vector unknowns. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A posteriori error estimator for finite volume methods

In this paper, first we discuss a technique to compare finite volume method and some well-known finite element methods, namely the dual mixed methods and nonconforming primal methods, for elliptic equations. These both equivalences are exploited to give us a posteriori error estimator for finite volume methods. This estimator is explicitly given, easy to compute and asymptotically exact without any regularity of the solution in unstructured grids.     

A superconvergent fitted finite volume method for Black–Scholes equations governing European and American option valuation

We develop a superconvergent fitted finite volume method for a degenerate nonlinear penalized Black–Scholes equation arising in the valuation of European and American options, based on the fitting idea in Wang [IMA J Numer Anal 24 (2004), 699–720]. Unlike conventional finite volume methods in which the dual mesh points are naively chosen to be the midpoints of the subintervals of the primal mesh, we construct the dual mesh judiciously using an error representation for the flux interpolation so that both the approximate flux and solution have the second‐order accuracy at the mesh points without any increase in computational costs. As the equation is degenerate, we also show that it is essential to refine the meshes locally near the degenerate point in order to maintain the second‐order accuracy. Numerical results for both European and American options with constant and nonconstant coefficients will be presented to demonstrate the superconvergence of the method. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1190–1208, 2015     

Weak Galerkin finite element method for valuation of American options

We introduce a weak Galerkin finite element method for the valuation of American options governed by the Black-Scholes equation. In order to implement, we need to solve the optimal exercise boundary and then introduce an artificial boundary to make the computational domain bounded. For the optimal exercise boundary, which satisfies a nonlinear Volterra integral equation, it is resolved by a higher-order collocation method based on graded meshes. With the computed optimal exercise boundary, the front-fixing technique is employed to transform the free boundary problem to a one- dimensional parabolic problem in a half infinite area. For the other spatial domain boundary, a perfectly matched layer is used to truncate the unbounded domain and carry out the computation. Finally, the resulting initial-boundary value problems are solved by weak Galerkin finite element method, and numerical examples are provided to illustrate the efficiency of the method.     

A machine learning approach for individual claims reserving in insurance

Accurate loss reserves are an important item in the financial statement of an insurance company and are mostly evaluated by macrolevel models with aggregate data in run‐off triangles. In recent years, a new set of literature has considered individual claims data and proposed parametric reserving models based on claim history profiles. In this paper, we present a nonparametric and flexible approach for estimating outstanding liabilities using all the covariates associated to the policy, its policyholder, and all the information received by the insurance company on the individual claims since its reporting date. We develop a machine learning–based method and explain how to build specific subsets of data for the machine learning algorithms to be trained and assessed on. The choice for a nonparametric model leads to new issues since the target variables (claim occurrence and claim severity) are right‐censored most of the time. The performance of our approach is evaluated by comparing the predictive values of the reserve estimates with their true values on simulated data. We compare our individual approach with the most used aggregate data method, namely, chain ladder, with respect to the bias and the variance of the estimates. We also provide a short real case study based on a Dutch loan insurance portfolio.     

Superconvergence of finite volume methods for the Stokes equations

A general superconvergence result of finite volume method for the Stokes equations is obtained by using a L² projection post‐processing technique. This superconvergence result can be applied to different finite volume methods and to general quasi‐uniform meshes.© 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009     

Pricing catastrophe swaps: A contingent claims approach

In this paper, we comprehensively analyze the catastrophe (cat) swap, a financial instrument which has attracted little scholarly attention to date. We begin with a discussion of the typical contract design, the current state of the market, as well as major areas of application. Subsequently, a two-stage contingent claims pricing approach is proposed, which distinguishes between the main risk drivers ex-ante as well as during the loss reestimation phase and additionally incorporates counterparty default risk. Catastrophe occurrence is modeled as a doubly stochastic Poisson process (Cox process) with mean-reverting Ornstein-Uhlenbeck intensity. In addition, we fit various parametric distributions to normalized historical loss data for hurricanes and earthquakes in the US and find the heavy-tailed Burr distribution to be the most adequate representation for loss severities. Applying our pricing model to market quotes for hurricane and earthquake contracts, we derive implied Poisson intensities which are subsequently condensed into a common factor for each peril by means of exploratory factor analysis. Further examining the resulting factor scores, we show that a first order autoregressive process provides a good fit. Hence, its continuous-time limit, the Ornstein-Uhlenbeck process should be well suited to represent the dynamics of the Poisson intensity in a cat swap pricing model.     

Convergence of a finite volume scheme for coagulation-fragmentation equations

This paper is devoted to the analysis of a numerical scheme for the coagulation and fragmentation equation. A time explicit finite volume scheme is developed, based on a conservative formulation of the equation. It is shown to converge under a stability condition on the time step, while a first order rate of convergence is established and an explicit error estimate is given. Finally, several numerical simulations are performed to investigate the gelation phenomenon and the long time behavior of the solution.

     

有限体积法定价美式期权

讨论美式期权定价的有限体积法.采用投影超松弛迭代法求解隐式欧拉和CrankNicolson有限体积格式离散Black-Scholes偏微分方程得到的线性互补问题.数值实验结果表明,两种有限体积格式都是有效的,而Crank-Nicolson格式的数值效果要优于隐式欧拉格式.     

A note on complete manifolds with finite volume

In this article, we concern on complete manifolds with finite volume. We prove that under some assumptions about scalar curvature and the Yamabe constant, the manifolds must be compact, and we also give the diameter estimates in terms of the scalar curvature and the Yamabe constant.     

A Power Penalty Approach to Numerical Solutions of Two-Asset American Options

This paper aims to develop a power penalty method for a linear parabolic variational inequality (Ⅵ) in two spatial dimensions governing the two-asset Ameri-can option valuation. This method yields a two-dimensional nonlinear parabolic PDE containing a power penalty term with penalty constant λ>1 and a power parameter k>0. We show that the nonlinear PDE is uniquely solvable and the solution of the PDE converges to that of the VI at the rate of order (λ<-k/2>). A fitted finite volume method is designed to solve the nonlinear PDE, and some numerical experiments are performed to illustrate the usefulness of this method.     

A posterior error estimate for finite volume methods of the second order elliptic problem

We establish a posteriori error analysis for finite volume methods of a second‐order elliptic problem based on the framework developed by Chou and Ye [SIAM Numer Anal, 45 (2007), 1639–1653]. This residual type estimators can be applied to different finite volume methods associated with different trial functions including conforming, nonconforming and totally discontinuous trial functions. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1165–1178, 2011     

美式利率期权定价的抛物型变分不等式

应用PDE方法对美式利率期权定价问题进行理论分析.在CIR利率模型下美式利率期权定价问题可归结为一个退化的一维抛物型变分不等式.通过引入惩罚函数证明了该变分不等式的解的存在唯一性,然后研究了自由边界的一些性质,如单调性,光滑性和自由边界在终止期的位置.     

A parallel finite volume scheme preserving positivity for diffusion equation on distorted meshes

Parallel domain decomposition methods are natural and efficient for solving the implicity schemes of diffusion equations on massive parallel computer systems. A finite volume scheme preserving positivity is essential for getting accurate numerical solutions of diffusion equations and ensuring the numerical solutions with physical meaning. We call their combination as a parallel finite volume scheme preserving positivity, and construct such a scheme for diffusion equation on distorted meshes. The basic procedure of constructing the parallel finite volume scheme is based on the domain decomposition method with the prediction‐correction technique at the interface of subdomains: First, we predict the values on each inner interface of subdomains partitioned by the domain decomposition. Second, we compute the values in each subdomain using a finite volume scheme preserving positivity. Third, we correct the values on each inner interface using the finite volume scheme preserving positivity. The resulting scheme has intrinsic parallelism, and needs only local communication among neighboring processors. Numerical results are presented to show the performance of our schemes, such as accuracy, stability, positivity, and parallel speedup.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2159–2178, 2017     

Symmetric finite volume schemes for eigenvalue problems in arbitrary dimensions

Based on a linear finite element space,two symmetric finite volume schemes for eigenvalue problems in arbitrary dimensions are constructed and analyzed.Some relationships between the finite element method and the finite difference method are addressed,too.