Nonlinear Parabolic Equations arising in Mathematical Finance

This survey paper is focused on qualitative and numerical analyses of fully nonlinear partial differential equations of parabolic type arising in financial mathematics. The main purpose is to review various non-linear extensions of the classical Black-Scholes theory for pricing financial instruments, as well as models of stochastic dynamic portfolio optimization leading to the Hamilton-Jacobi-Bellman (HJB) equation. After suitable transformations, both problems can be represented by solutions to nonlinear parabolic equations. Qualitative analysis will be focused on issues concerning the existence and uniqueness of solutions. In the numerical part we discuss a stable finite-volume and finite difference schemes for solving fully nonlinear parabolic equations.Comment: arXiv admin note: substantial text overlap with arXiv:1603.0387

Randomized Optimal Stopping Problem in Continuous time and Reinforcement Learning Algorithm

In this paper, we study the optimal stopping problem in the so-called exploratory framework, in which the agent takes actions randomly conditioning on current state and an entropy-regularized term is added to the reward functional. Such a transformation reduces the optimal stopping problem to a standard optimal control problem. We derive the related HJB equation and prove its solvability. Furthermore, we give a convergence rate of policy iteration and the comparison to classical optimal stopping problem. Based on the theoretical analysis, a reinforcement learning algorithm is designed and numerical results are demonstrated for several models

A Multidimensional Exponential Utility Indifference Pricing Model with Applications to Counterparty Risk

This paper considers exponential utility indifference pricing for a multidimensional non-traded assets model subject to inter-temporal default risk, and provides a semigroup approximation for the utility indifference price. The key tool is the splitting method, whose convergence is proved based on the Barles-Souganidis monotone scheme, and the convergence rate is derived based on Krylov's shaking the coefficients technique. We apply our methodology to study the counterparty risk of derivatives in incomplete markets.Comment: 29 pages, 5 figure

An efficient stochastic particle method for high-dimensional nonlinear PDEs

Numerical resolution of high-dimensional nonlinear PDEs remains a huge challenge due to the curse of dimensionality. Starting from the weak formulation of the Lawson-Euler scheme, this paper proposes a stochastic particle method (SPM) by tracking the deterministic motion, random jump, resampling and reweighting of particles. Real-valued weighted particles are adopted by SPM to approximate the high-dimensional solution, which automatically adjusts the point distribution to intimate the relevant feature of the solution. A piecewise constant reconstruction with virtual uniform grid is employed to evaluate the nonlinear terms, which fully exploits the intrinsic adaptive characteristic of SPM. Combining both can SPM achieve the goal of adaptive sampling in time. Numerical experiments on the 6-D Allen-Cahn equation and the 7-D Hamiltonian-Jacobi-Bellman equation demonstrate the potential of SPM in solving high-dimensional nonlinear PDEs efficiently while maintaining an acceptable accuracy

Finite element methods for Bellman and Isaacs Equations

This work concerns the numerical analysis of the Partial Differential Equations (PDEs) with a particular focus on fully nonlinear PDEs. More specifically, the main goal is to provide a finite element method to approximate solutions of Isaacs equations, which come from game theory and can be thought of as generalisation of Hamilton-Jacobi-Bellman (HJB) equations. Both of these classes of problems arise from the stochastic optimal control problems. Is is widely known that nonlinear PDEs do not in general admit classical solutions. A way to circumvent this issue is to use a relaxed definition of derivative leading to the notion of a generalised solution. One such notion is that of viscosity solution introduced in 1980s by Crandall and Lions. The main idea is to regularise non-smooth functions by using comparison principles and subtractive testing. The theory of viscosity solutions gave rise to novel numerical methods. A general framework of formulating convergent numerical schemes for (possibly degenerate) elliptic PDEs was formulated by Barles and Souganidis in 1991. The main result states that, given a comparison principle depending on the application at hand, a monotone, stable and consistent numerical scheme converges to the unique viscosity solution of a fully nonlinear problem. This framework is used throughout this work to formulate convergent numerical schemes. The main three contributions of the thesis are as follows. First we present a Finite Element Method to approximate solutions of isotropic parabolic problems of Isaacs type with possibly degenerate diffusions. Second we design a method of numerically approximating isotropic parabolic Hamilton-Jacobi-Bellman equations with nonlinear, mixed boundary conditions where Robin type boundary conditions are imposed via one-sided Dini derivatives. In both cases we prove the convergence of the numerical solution to the unique viscosity solution. The uniqueness of numerical solution is guaranteed by Howard’s algorithm. The analysis of the HJB equations with mixed boundary conditions is motivated by option pricing in a financial setting, which leads to our third contribution. We extend the Heston model of mathematical finance to permit the uncertain market price of volatility risk and we interpret it as an HJB equation. Finally, we present a case study investigating the effects of the market price of volatility risk on the option value and its derivatives

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

Multi-scale Volatility in Option Pricing

This PhD thesis investigated the influence of kaolin and bentonite clays in the ore on flotation, filtration and centrifugal concentration. The results showed that the presence of particularly bentonite in the ore had a detrimental effect on flotation and filtration. The information generated from this work will advance our knowledge as well as provide important information for plant metallurgists. The project, therefore, is essential for the mineral industry that process clay-containing ores