Michael D. Marcozzi

On the use of boundary conditions for variational formulations arising in financial mathematics

The general intractability of derivative security valuation models to present techniques, both analytic and numerical, arguably remains one of the preeminant problem of mathematical finance. It is the focus of this paper to examine the applicability of a promising recent development, namely Radial Basis Functions (RBF), to the problem of option valuation. A Black-Scholes framework is considered for American and European options written on a one and two risky assets. The performance of RBF and Finite-Differencing algorithms are examined with respect to artificial boundary conditions, computational domain, domain decomposition, and mesh scaling.

A numerical analysis of variational valuation techniques for derivative securities

In this paper, we consider the partial differential equations approach for valuing European and American style options on multiple assets. We use a method of lines finite element implementation available in the software package Femlab in order to solve the variational inequality that characterizes the American style option, as well as the partial differential equation that defines the European style option, for two and three state variables. A detailed study of the approximation error is provided, including a theoretical estimate, an asymptotic analysis, the space-time distribution, and the dependence on the size of the truncation domain.

The valuation of foreign currency options under stochastic interest rates

Abstract We consider the valuation of options written on a foreign currency when interest rates are stochastic and the matrix of the diffusion representing the global economy is strongly coercive. We solve the associated variational inequality for the value function numerically by the finite element method. In the European case, a comparison is made to the exact solution. The corresponding result for the American option is also presented.

A Numerical Approach to American Currency Option Valuation

Option theory has produced models of increasing richness that are capable of incorporating many sources of randomness (“stochastic state variables,” as theorists would say). For interest rate dependent instruments, the highly flexible Heath-Jarrow-Morton (HJM) family provides some of the most widely used models. However, implementation of HJM models typically runs into serious computational problems as the number of state variables increases. Pricing a currency option, for example, requires at least three state variables, one for each countrys interest rate and one for the exchange rate. American exercise makes the problem harder still. In this article, Choi and Marcozzi describe a numerical technique based on approximating the option value with radial basis functions that offers considerable efficiency improvement. They illustrate its use on HJM-style currency options. One large advantage of this approach is that the approximating functions are analytic, so that the Greek letter risk exposures can be obtained directly using calculus rather than requiring multiple runs through a pricing lattice to approximate them.

An adaptive extrapolation discontinuous Galerkin method for the valuation of Asian options

We consider the approximation of the optimal stopping problem associated with ultradiffusion processes in the context of mathematical finance and the valuation of Asian options. In particular, the value function is characterized as the solution of an ultraparabolic variational inequality. Employing the penalty method and a regularization of the state space, we develop higher-order adaptive approximation schemes which utilize the extrapolation discontinuous Galerkin method in temporal space. Numerical examples are provided in order to demonstrate the approach.

On the Approximation of Infinite Dimensional Optimal Stopping Problems with Application to Mathematical Finance

Abstract We consider the approximation of the optimal stopping problem for infinite dimensional processes by variational methods. To this end, we employ a Fourier-Legendre representation for the state space and exhaust an indexed family of regularized Hamilton-Jacobi characterizations. We implement our results utilizing penalization and a method-of-lines semi-implicit finite element method; application to term-structure valuation problems from mathematical finance demonstrate the applicability of the approach.

Chapter 7 Variational Methods in Derivatives Pricing

Abstract When underlying financial variables follow a Markov jump-diffusion process, the value function of a derivative security satisfies a partial integro-differential equation (PIDE) for European-style exercise or a partial integro-differential variational inequality (PIDVI) for American-style exercise. Unless the Markov process has a special structure, analytical solutions are generally not available, and it is necessary to solve the PIDE or the PIDVI numerically. In this chapter we briefly survey a computational method for the valuation of options in jump-diffusion models based on: (1) converting the PIDE or PIDVI to a variational (weak) form; (2) discretizing the weak formulation spatially by the Galerkin finite element method to obtain a system of ODEs; and (3) integrating the resulting system of ODEs in time. To introduce the method, we start with the basic examples of European, barrier, and American options in the Black–Scholes–Merton model, then describe the method in the general setting of multi-dimensional jump-diffusion processes, and conclude with a range of examples, including Mertons and Kous one-dimensional jump-diffusion models, Duffie–Pan–Singleton two-dimensional model with stochastic volatility and jumps in the asset price and its volatility, and multi-asset American options.

Optimal control of ultradiffusion processes with application to mathematical finance

We introduce the optimal control problem associated with ultradiffusion processes as a stochastic differential equation constrained optimization of the expected system performance over the set of feasible trajectories. The associated Bellman function is characterized as the solution to a Hamilton–Jacobi equation evaluated along an optimal process. For an important class of ultradiffusion processes, we define the value function in terms of the time and the natural state variables. Approximation solvability is shown and an application to mathematical finance demonstrates the applicability of the paradigm. In particular, we utilize a method-of-lines finite element method to approximate the value function of a European style call option in a market subject to asset liquidity risk (including limit orders) and brokerage fees.

Asset liquidity and the valuation of derivative securities

We consider the valuation of European-style derivative securities under limited asset liquidity through the dynamic management of a portfolio of assets effected through continuous transaction. The valuation arises from the optimal realization of a performance index relative to the set of all feasible portfolio trajectories. An approximation procedure based upon the method-of-lines finite element method is developed and analyzed; numerical examples are presented in order to demonstrate the viability of the approach.

On the approximation solvability of a class of strongly nonlinear elliptic problems on unbounded domains

A class of strongly nonlinear boundary value problems posed on unbounded regions is considered. A nonlocal coupling of the linearized far-field exterior to an auxiliary boundary allows for approximations to be defined on domains of finite extent. Constructive existence results for bounded domains are then extended by employing an exhausting sequence of approximating domains. In particular, well-posedness is seen to be equivalent to unique approximation solvability, with the rate of convergence dependent upon the radius of the auxiliary boundary. Application to a model of proteins immersed in an electrolyte solution is made.