Piecewise Constant Policy Approximations to Hamilton-Jacobi-Bellman Equations

An advantageous feature of piecewise constant policy timestepping for Hamilton-Jacobi-Bellman (HJB) equations is that different linear approximation schemes, and indeed different meshes, can be used for the resulting linear equations for different control parameters. Standard convergence analysis suggests that monotone (i.e., linear) interpolation must be used to transfer data between meshes. Using the equivalence to a switching system and an adaptation of the usual arguments based on consistency, stability and monotonicity, we show that if limited, potentially higher order interpolation is used for the mesh transfer, convergence is guaranteed. We provide numerical tests for the mean-variance optimal investment problem and the uncertain volatility option pricing model, and compare the results to published test cases

A Penalty Method for the Numerical Solution of Hamilton-Jacobi-Bellman (HJB) Equations in Finance

We present a simple and easy to implement method for the numerical solution of a rather general class of Hamilton-Jacobi-Bellman (HJB) equations. In many cases, the considered problems have only a viscosity solution, to which, fortunately, many intuitive (e.g. finite difference based) discretisations can be shown to converge. However, especially when using fully implicit time stepping schemes with their desirable stability properties, one is still faced with the considerable task of solving the resulting nonlinear discrete system. In this paper, we introduce a penalty method which approximates the nonlinear discrete system to first order in the penalty parameter, and we show that an iterative scheme can be used to solve the penalised discrete problem in finitely many steps. We include a number of examples from mathematical finance for which the described approach yields a rigorous numerical scheme and present numerical results.Comment: 18 Pages, 4 Figures. This updated version has a slightly more detailed introduction. In the current form, the paper will appear in SIAM Journal on Numerical Analysi

Numerical Methods for Pricing a Guaranteed Minimum Withdrawal Benefit (GMWB) as a Singular Control Problem

Guaranteed Minimum Withdrawal Benefits(GMWB) have become popular riders on variable annuities. The pricing of a GMWB contract was originally formulated as a singular stochastic control problem which results in a Hamilton Jacobi Bellman (HJB) Variational Inequality (VI). A penalty method method can then be used to solve the HJB VI. We present a rigorous proof of convergence of the penalty method to the viscosity solution of the HJB VI assuming the underlying asset follows a Geometric Brownian Motion. A direct control method is an alternative formulation for the HJB VI. We also extend the HJB VI to the case of where the underlying asset follows a Poisson jump diffusion. The HJB VI is normally solved numerically by an implicit method, which gives rise to highly nonlinear discretized algebraic equations. The classic policy iteration approach works well for the Geometric Brownian Motion case. However it is not efficient in some circumstances such as when the underlying asset follows a Poisson jump diffusion process. We develop a combined fixed point policy iteration scheme which significantly increases the efficiency of solving the discretized equations. Sufficient conditions to ensure the convergence of the combined fixed point policy iteration scheme are derived both for the penalty method and direct control method. The GMWB formulated as a singular control problem has a special structure which results in a block matrix fixed point policy iteration converging about one order of magnitude faster than a full matrix fixed point policy iteration. Sufficient conditions for convergence of the block matrix fixed point policy iteration are derived. Estimates for bounds on the penalty parameter (penalty method) and scaling parameter (direct control method) are obtained so that convergence of the iteration can be expected in the presence of round-off error

Finite Difference Methods for the Non-Linear Black-Scholes-Barenblatt Equation

The Uncertain Volatility model is a non-linear generalisation of the Black-Scholes model in the sense that volatility and correlation can take arbitrary values in given intervals. The value of an option is then given by a non-linear partial differential equation of Hamilton-Jacobi-Bellman type. For this type of equation the concept of viscosity solution has to be considered since in general no smooth solutions in the classical sense exist. To assure the convergence of a discrete scheme it has to be consistent, stable and additionally monotone. Starting from a Finite Difference discretisation we first derive general and structural conditions to adequately price options in the Uncertain Volatility model. Additionally, an optimisation problem has to be solved which can either be done exactly or only approximatively where this choice depends on the discretisation. Finally, sufficient conditions are derived for the different discrete schemes to assure their convergence to the viscosity solution. The obtained theoretical results are finally tested in numerical experiments. The rates of convergence, the effort for excecuting the policy iteration, and the possible gain by using non-uniform grids are analysed

Numerical Methods for Continuous Time Mean Variance Type Asset Allocation

Many optimal stochastic control problems in finance can be formulated in the form of Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs). In this thesis, a general framework for solutions of HJB PDEs in finance is developed, with application to asset allocation. The numerical scheme has the following properties: it is unconditionally stable; convergence to the viscosity solution is guaranteed; there are no restrictions on the underlying stochastic process; it can be easily extended to include features as needed such as uncertain volatility and transaction costs; and central differencing is used as much as possible so that use of a locally second order method is maximized. In this thesis, continuous time mean variance type strategies for dynamic asset allocation problems are studied. Three mean variance type strategies: pre-commitment mean variance, time-consistent mean variance, and mean quadratic variation, are investigated. The numerical method can handle various constraints on the control policy. The following cases are studied: allowing bankruptcy (unconstrained case), no bankruptcy, and bounded control. In some special cases where analytic solutions are available, the numerical results agree with the analytic solutions. These three mean variance type strategies are compared. For the allowing bankruptcy case, analytic solutions exist for all strategies. However, when additional constraints are applied to the control policy, analytic solutions do not exist for all strategies. After realistic constraints are applied, the efficient frontiers for all three strategies are very similar. However, the investment policies are quite different. These results show that, in deciding which objective function is appropriate for a given economic problem, it is not sufficient to simply examine the efficient frontiers. Instead, the actual investment policies need to be studied in order to determine if a particular strategy is applicable to specific investment problem

Numerical Solutions of Two-factor Hamilton-Jacobi-Bellman Equations in Finance

In this thesis, we focus on solving multidimensional HJB equations which are derived from optimal stochastic control problems in the financial market. We develop a fully implicit, unconditionally monotone finite difference numerical scheme. Consequently, there are no time step restrictions due to stability considerations, and the fully implicit method has essentially the same complexity per step as the explicit method. The main difficulty in designing a discretization scheme is development of a monotonicity preserving approximation of cross derivative terms in the PDE. We primarily use a wide stencil based on a local coordination rotation. The analysis rigorously show that our numerical scheme is $l_\infty$ stable, consistent in the viscosity sense, and monotone. Therefore, our numerical scheme guarantees convergence to the viscosity solution. Firstly, our numerical schemes are applied to pricing two factor options under an uncertain volatility model. For this application, a hybrid scheme which uses the fixed point stencil as much as possible is developed to take advantage of its accuracy and computational efficiency. Secondly, using our numerical method, we study the problem of optimal asset allocation where the risky asset follows stochastic volatility. Finally, we utilize our numerical scheme to carry out an optimal static hedge, in the case of an uncertain local volatility model

Numerical approximation of a cash-constrained firm value with investment opportunities

We consider a singular control problem with regime switching that arises in problems of optimal investment decisions of cash-constrained firms. The value function is proved to be the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation. Moreover, we give regularity properties of the value function as well as a description of the shape of the control regions. Based on these theoretical results, a numerical deterministic approximation of the related HJB variational inequality is provided. We finally show that this numerical approximation converges to the value function. This allows us to describe the investment and dividend optimal policies.Comment: 30 pages, 10 figure

Monotone Numerical Methods for Nonlinear Systems and Second Order Partial Differential Equations

Multigrid methods are numerical solvers for partial differential equations (PDEs) that systematically exploit the relationship between approximate solutions on multiple grids to arrive at a solution whose accuracy is consistent with the finest grid but for considerably less work. These methods converge in a small number of constant iterations independent of the grid size and hence, are often dramatically more efficient than others. In this thesis, we develop multigrid methods for three different classes of PDEs. In addition, we also develop discretization schemes for two model problems. First, we propose multigrid methods based on upwind interpolation and restriction techniques for computing the steady state solutions for systems of one and two-dimensional nonlinear hyperbolic conservation laws. We prove that the two-grid method is total variation diminishing and the multigrid methods are consistent and convergent for one-dimensional linear systems. Second, we propose a fully implicit, positive coefficient discretization that converges to the viscosity solution for a two-dimensional system of Hamilton-Jacobi-Bellman (HJB) PDEs resulting from dynamic Bertrand duopoly. Furthermore, we develop fast multigrid methods for solving the systems of discrete nonlinear HJB PDEs. The new multigrid methods are general and can be applied to other systems of HJB and HJB-Isaacs (HJBI) PDEs resulting from American options under regime switching and American options with unequal lending/borrowing rates and stock borrowing fees under regime switching, respectively. We provide a theoretical analysis for the smoother, restriction and interpolation operators of the multigrid methods. Finally, we develop a fully implicit, unconditionally monotone finite difference numerical scheme, that converges to the viscosity solution of the three-dimensional PDE to price European options under a two-factor stochastic volatility model. The presence of cross derivative terms in high dimensional PDEs makes the construction of monotone discretization schemes challenging. We develop a wide stencil discretization based on a local coordinate transformation to eliminate the cross derivative terms. But, wide stencil discretization is first order accurate and computationally expensive compared to the second order fixed stencil discretization. Therefore, we use a hybrid stencil in which fixed stencil is used as much as possible and a wide stencil when the fixed stencil discretization does not satisfy the positive coefficient condition. We also develop fast multigrid methods to solve the discrete linear system