Mathematical analysis and numerical methods for pricing pension plans allowing early retirement

In this paper, we address the mathematical analysis and numerical solution of a model for pricing a defined benefit pension plan. More precisely, the benefits received by the member of the plan depend on the average salary and early retirement is allowed. Thus, the mathematical model is posed as an obstacle problem associated to a Kolmogorov equation in the time region where the salary is being averaged. Previously to the initial averaging date, a nonhomogeneous one factor Black-Scholes equation is posed. After stating the model, existence and regularity of solutions are studied. Moreover, appropriate numerical methods based on a Lagrange-Galerkin discretization and an augmented Lagrangian active set method are proposed. Finally, some numerical examples illustrate the performance of the numerical techniques and the properties of the solution and the free boundary.retirement plans, options pricing, Kolmogorov equations, complementarity problem, numerical methods, augmented Lagrangian formulation

PDE Models and Numerical Methods for Total Value Adjustment in European and American Options with Counterparty Risk

The final publication is available: https://doi.org/10.1016/j.amc.2017.03.008[Abstract] Since the last financial crisis, a relevant effort in quantitative finance research concerns the consideration of counterparty risk in financial contracts, specially in the pricing of derivatives. As a consequence of this new ingredient, new models, mathematical tools and numerical methods are required. In the present paper, we mainly consider the problem formulation in terms of partial differential equations (PDEs) models to price the total credit value adjustment (XVA) to be added to the price of the derivative without counterparty risk. Thus, in the case of European options and forward contracts different linear and nonlinear PDEs arise. In the present paper we propose suitable boundary conditions and original numerical methods to solve these PDEs problems. Moreover, for the first time in the literature, we consider XVA associated to American options by the introduction of complementarity problems associated to PDEs, as well as numerical methods to be added in order to solve them. Finally, numerical examples are presented to illustrate the behavior of the models and numerical method to recover the expected qualitative and quantitative properties of the XVA adjustments in different cases. Also, the first order convergence of the numerical method is illustrated when applied to particular cases in which the analytical expression for the XVA is available.Ministerio de Economía y Competitividad; MTM2013-47800-C2-1-PXunta de Galicia; GRC2014/044Ministerio de Economía y Competitividad; BES-2014-070782Ministerio de Economía y Competitividad; BES-2014-070782

Projection methods in conic optimization

There exist efficient algorithms to project a point onto the intersection of a convex cone and an affine subspace. Those conic projections are in turn the work-horse of a range of algorithms in conic optimization, having a variety of applications in science, finance and engineering. This chapter reviews some of these algorithms, emphasizing the so-called regularization algorithms for linear conic optimization, and applications in polynomial optimization. This is a presentation of the material of several recent research articles; we aim here at clarifying the ideas, presenting them in a general framework, and pointing out important techniques

Primjena integro-diferencijalne varijacijske razlomne zadaće i razlomnog pristupa integralima po putevima stohastičkom modeliranju

The fractional path integral approach is applied to stochastic models, in particular the financial derivatives and options pricing formulated within the framework of the fractional action-like variational approach recently introduced by the author. Many interesting features and consequences are revealed in some details.Proučavaju se stohastički modeli primjenom integrala po putevima, a posebno se razlažu novčane izvodnice i mogućnosti u određivanju cijena u okviru razlomnog djelotvornog varijacijskog pristupa nedavno uvedenog autorom. Mnoge se zanimljive odlike i posljedice otkrivaju djelomično

Primjena integro-diferencijalne varijacijske razlomne zadaće i razlomnog pristupa integralima po putevima stohastičkom modeliranju

The fractional path integral approach is applied to stochastic models, in particular the financial derivatives and options pricing formulated within the framework of the fractional action-like variational approach recently introduced by the author. Many interesting features and consequences are revealed in some details.Proučavaju se stohastički modeli primjenom integrala po putevima, a posebno se razlažu novčane izvodnice i mogućnosti u određivanju cijena u okviru razlomnog djelotvornog varijacijskog pristupa nedavno uvedenog autorom. Mnoge se zanimljive odlike i posljedice otkrivaju djelomično

The mathematical modelling and numerical solution of options pricing problems

Accurate and efficient numerical solutions have been described for a selection of financial options pricing problems. The methods are based on finite difference discretisation coupled with optimal solvers of the resulting discrete systems. Regular Cartesian meshes have been combined with orthogonal co-ordinate transformations chosen for numerical accuracy rather than reduction of the differential operator to constant coefficient form. They allow detailed resolution in the regions of interest where accuracy is most desired, and grid coarsening where there is least interest. These transformations are shown to be effective in producing accurate solutions on modest computational grids. The spatial discretisation strategy is chosen to meet accuracy requirements as sell as to produce coefficient matrices with favourable sparsity and stability properties. In the case of single factor European options, a modified Crank-Nicolson, second order accurate finite difference scheme is presented, which uses adaptive upwind differences when the mesh Peclet conditions are violated. The resulting tridiagonal system of equations is solved using a direct solver. A careful study of grid refinement displays convergence towards the true solution and demonstrates a high level of accuracy can be obtained with this approach. Laplace inversion methods are also implemented as an alternative solution approach for the one-factor European option. Results are compared to those produced by the direct solver algorithm and are shown to be favourable. It is shown how Semi-Lagrange time-integration can solve the path-dependent Asian pricing problem, by integrating out the average price term and simplifying the finite difference equations into a parameterised Black-Scholes form. The implicit equations that result are unconditionally stable, second order accurate and can be solved using standard tridiagonal solvers. The Semi-Lagrange method is shown to be easily used in conjunction with co-ordinate transformations applied in both spatial directions. A variable time-stepping scheme is implemented in the algorithm. Early exercise is also easily incorporated, the resulting linear complementarity problem can be solved using a projection or penalty method (the penalty method is shown to be slightly more efficient). Second order accuracy has been confirmed for Asian options that must be held to maturity. A comparison with published results for continuous-average-rate put and call options, with and without early exercise, shows that the method achieves basis point accuracy and that Richardson extrapolation can also be applied