Spatial Approximation of Nondivergent Type Parabolic PDEs with Unbounded Coefficients Related to Finance

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 801059, 11 pages http://dx.doi.org/10.1155/2014/801059 Research Article Spatial Approximation of Nondivergent Type Parabolic PDEs with Unbounded Coefficients Related to Finance Fernando F. Gonçalves1,2 and Maria Rosário Grossinho1 1 2 CEMAPRE & Departmento de Matemática, ISEG, Universidade de Lisboa, Rua do Quelhas 6, 1200-781 Lisboa, Portugal Universidade Europeia, Estrada da Correia 53, 1500-210 Lisboa, Portugal Correspondence should be addressed to Fernando F. Gonçalves; Received 26 July 2013; Accepted 5 January 2014; Published 6 March 2014 Academic Editor: István Györi Copyright © 2014 F. F. Gonçalves and M. R. Grossinho. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the spatial discretisation of the Cauchy problem for a multidimensional linear parabolic PDE of second order, with nondivergent operator and unbounded time- and space-dependent coefficients. The equation free term and the initial data are also allowed to grow. Under a nondegeneracy assumption, we consider the PDE solvability in the framework of the variational approach and approximate in space the PDE problem’s generalised solution, with the use of finite-difference methods. The rate of convergence is estimated. 1. Introduction In this paper, we study the discretisation in space of the Cauchy problem 𝜕𝑢 = 𝐿𝑢 + 𝑓 𝜕𝑡 in [0, 𝑇] × R𝑑 , 𝑢 (0, 𝑥) = 𝑔 (𝑥) (1) 𝑑 on R , where 𝐿 is the second-order partial differential operator in the nondivergence form 𝐿 (𝑡, 𝑥) = 𝑎𝑖𝑗 (𝑡, 𝑥) 𝜕2 𝜕 + 𝑏𝑖 (𝑡, 𝑥) 𝑖 + 𝑐 (𝑡, 𝑥) , 𝜕𝑥𝑖 𝜕𝑥𝑗 𝜕𝑥 𝑖, 𝑗 = 1, . . . , 𝑑, (2) with real coefficients (written with the usual summation convention), 𝑓 and 𝑔 are given real-valued functions, and 𝑇 ∈ (0, ∞) is a constant. We assume that operator 𝜕/𝜕𝑡 − 𝐿 is uniformly parabolic and allows the growth in the spatial variables of the first- and second-order coefficients in 𝐿 (linear and quadratic growth, resp.) and of the data 𝑓 and 𝑔 (polynomial growth). Multidimensional partial differential equation (PDE) problems arise in Financial Mathematics and in Mathematical Physics. We are mainly motivated by the application to a large class of stochastic models in Financial Mathematics comprising the non-path-dependent options, with fixed exercise, written on multiple assets (basket options, exchange options, compound options, European options on future contracts and foreign-exchange, and others) and also to a particular type of path-dependent options: the Asian options (see, e.g., [1]). Let us consider the stochastic modelling of a multiasset option of European type under the framework of a general version of Black-Scholes model, where the vector of asset appreciation rates and the volatility matrix are taken to be time- and space-dependent, and the riskless interest rate is a function of time. Owing to a Feynman-Kač type formula, pricing this option can be reduced to solving the Cauchy problem (with terminal condition) for the degenerate second-order linear parabolic PDE of nondivergent type, with null term and unbounded coefficients (see, e.g., [1]), 𝜕2 𝑉 𝜕𝑉 𝜕𝑉 1 𝑖𝑗 + 𝜎 (𝑡, 𝑆) 𝑆𝑖 𝑆𝑗 𝑖 𝑗 + 𝑟 (𝑡) 𝑆𝑖 𝑖 − 𝑟 (𝑡) 𝑉 = 0 𝜕𝑡 2 𝜕𝑆 𝜕𝑆 𝜕𝑆 in [0, 𝑇] × R𝑑+ , 𝑉 (𝑇, 𝑆) = 𝜙 (𝑆) on R𝑑+ , (3) 2 where R𝑑+ ≡ {𝑥 ∈ R𝑑 : 𝑥𝑖 > 0, 𝑖 = 1, . . . , 𝑑}, 𝑉 is the (unknown) option value, 𝑆𝑖 the price of the 𝑖th underlying asset, (𝜎𝑖𝑗 ) the volatility matrix, 𝑟 the risk-free interest rate, and 𝜙 the pay-off function. Therefore, as an alternative to approximating the option price with probabilistic numerical methods, we can approximate the solution of the corresponding PDE problem (3) with the use of nonprobabilistic techniques. When problem (1) is considered in connection with option pricing, we see that the growth of the Black-Scholes PDE coefficients is appropriately matched. Also, the general case where the asset appreciation rate vector, the volatility matrix, and the risk-free interest rate are variable is covered. Finally, by imposing weak conditions on the initial data 𝑔, we will allow the financial derivative pay-off to be specified in a large class of functions. The free term 𝑓 is included to further improve generality. In this paper, we study the approximation in space (for the time approximation, we refer to [2–4], where a general evolution equation problem of parabolic type is discretised) of the second-order parabolic problem (1), in the challenging case where the coefficients are unbounded (as well as the free data 𝑓 and 𝑔). The results are obtained under the strong assumption that the PDE does not degenerate but by imposing weak regularity assumptions. In order to facilitate the approach, we avoid any numerical methods’ sophistication and make use of basic one-step finite-difference schemes. Also, an estimate for the rate of convergence of the discretised problem’s generalised solution to the exact problem’s generalised solution is given. The numerical methods and possible approximation results are strongly linked to the theory on the solvability of the PDEs. We make use of the 𝐿2 theory of solvability of linear PDEs in weighted Sobolev spaces. In particular, we consider the PDE solvability in a class of weighted Sobolev spaces used by O. G. Purtukhia (the references for Purtukhia’s works can be found in [5]) for the treatment of linear stochastic partial differential equations (SPDEs) and further generalised by Gyöngy and Krylov (see [5]), the so-called well-weighted Sobolev spaces. By constructing discrete versions of these spaces, we set a suitable discretised framework and investigate the spatial approximation to the PDE generalised solution with the use of standard variational techniques. We emphasize some points. Firstly, we note that many PDE problems related to finance are Cauchy problems: initial-boundary value problems arise mostly after a localisation procedure for the purpose of obtaining implementable numerical schemes. Therefore, we do not find in many of these problems the complex domain geometries which are one important reason to favour other numerical methods (e.g., finite-element methods). Also, although the finite-difference method for approximating PDEs is a well-developed area, and the theory could be considered reasonably complete since three decades ago, some important research is still currently pursued (see, e.g., the recent works [8–10]). (We refer to [6] for a brief summary Abstract and Applied Analysis of the method’s history, and also for the references of the seminal works by R. Courant, K. O. Friedrichs, and H. Lewy, and further major contributions by many others. For the application of the finite-difference method to option pricing, we refer to the review paper [7] for the references of the original publications by M. Brennan and E. S. Schwartz and further major (...truncated)