VALUATION OF SOME NONLINEAR FINANCIAL CONTRACTS BY FINITE ELEMENT METHOD

Abstract

This thesis proposes a methodology for dealing with nonlinear financial derivative models using the finite element method (FEM). Financial engineering solutions are in high demand to mimic realistic market scenarios. Significantly, the nonlinear partial differential equations (PDE) seen in security pricing theory make it almost impossible to develop explicit solutions. Therefore, one resorts to numerical approximations. The literature contains articles dealing with nonlinear contracts using the finite difference method (FDM), which practitioners frequently use. This thesis aims to provide some computational gain in time and an accurate solution to nonlinear contracts in the derivative market. The generality of the approach is extendable to other types of American and European nonlinear contracts.

For nonlinear models, conventional FEM and Isogeometric analysis (IGA) are designed to be compared with benchmark results. The second-order P2-FEM performs better convergence properties than FDM and P1-FEM for convertible bond models. Moreover, the incorporation of an adaptive grid leads to the use of a few spatial discretizations.

Usually, PDE models seen in financial engineering consist of convection-dominated or degenerate terms. The naive approach relies on stabilization techniques, as they allow for mitigating spurious oscillations. Alternatively, we use a relatively new approach, demonstrated by IGA-NURBS-based finite element technology, where the monotonic convergence is achieved with uniform and non-uniform grids without any stabilization techniques and validated within the benchmark region. Numerical experiments were conducted among well-known conventional FEM and FDM methods. The presence of the IGA framework has showcased the classical results by using fitted curve approximation. IGA demonstrates notable results based on the linear case, where the exact solution was achieved using a lesser number of grids than those by FEM and/or FDM.

The post-processing Greek values are essential, as is the price of the contracts. The literature on computing the Greek values by FDM or finite volume methods (FVM) is vast. Specific models that consider frictionless markets may encounter challenges in accurately representing real-world scenarios. To satisfy the request of the derivative market, one shall consider the nonlinear pricing models that incorporate the specific request seen in financial derivative markets. The use of standard FDM or/and FEM leads to instability in the post-processing Greeks. In principle, a possible mitigation of such oscillations could be resolved using stabilization techniques. Employing NURBS basis functions with high compact support offers smoother Greek values, which may contribute to more reliable investment and trading strategies for hedging purposes.