Finite element method: Meaning, Criticisms & Real-World Uses

What Is Finite Element Method?

The Finite Element Method (FEM) is a powerful numerical technique used to solve complex engineering and mathematical problems by dividing a large, intricate system into smaller, simpler, interconnected components called "finite elements." These elements are then analyzed individually, and their behaviors are combined to understand the overall system's response. While originating in structural engineering and physics, the finite element method has found significant application in computational finance, particularly for solving partial differential equations that describe financial phenomena. It allows for the approximation of solutions to problems that lack analytical closed-form solutions, providing a flexible framework for financial modeling and quantitative analysis.

History and Origin

The roots of the finite element method can be traced back to the early 20th century, with foundational mathematical work in variational methods. However, its modern form began to take shape in the 1940s, notably through the independent efforts of Russian-Canadian structural engineer Alexander Hrennikoff and German-American mathematician Richard Courant. Courant's approach involved the use of triangular elements for solving elasticity problems. The term "Finite Element Method" itself was coined by Ray Clough in 1960, a civil engineer at the University of California, Berkeley. FEA History ⁵. Over the subsequent decades, the method was rigorously developed and generalized by various researchers across the globe, including those at Cornell University, the University of Stuttgart, and the University of Swansea⁴. Its evolution was closely tied to the advent and increasing power of digital computers, which made the extensive numerical calculations required by the finite element method feasible.

Key Takeaways

The Finite Element Method (FEM) approximates solutions to complex problems by discretizing a continuous domain into smaller, interconnected "finite elements."
It is widely used in engineering for structural analysis and fluid dynamics, and increasingly in finance for pricing derivatives and managing risk.
FEM offers flexibility in handling complex geometries, boundary conditions, and material properties that analytical solutions cannot easily accommodate.
While powerful, the accuracy of FEM results depends heavily on proper model setup, element choice, and mesh refinement.
The method is a crucial numerical method in modern computational finance for solving partial differential equations.

Formula and Calculation

The finite element method typically involves converting a continuous problem, often described by partial differential equations, into a system of algebraic equations. While there isn't a single universal "formula" for FEM, the core process involves:

Discretization: Dividing the domain of the problem into a mesh of finite elements.
Element Formulation: Approximating the unknown field variable (e.g., temperature, displacement, option price) within each element using interpolation functions, often polynomials.
Assembly: Combining the equations from individual elements into a global system of equations for the entire domain.
Application of Boundary Conditions: Incorporating the known constraints or conditions at the boundaries of the system.
Solution: Solving the resulting system of algebraic equations to obtain the approximate values of the field variable at discrete points (nodes) within the mesh.

For a linear elastic problem, the general matrix equation derived from the finite element method can be represented as:

[K]{U} = {F}

Where:

([K]) = Global stiffness matrix, representing the assembled properties of all finite elements.
({U}) = Vector of unknown nodal displacements (or other field variables, such as option prices in financial applications).
({F}) = Vector of applied nodal forces (or boundary conditions, initial conditions, etc.).

This system is then solved for ({U}) using various linear algebra techniques.

Interpreting the Finite Element Method

Interpreting the results from the finite element method requires understanding that the output is an approximation of the real-world behavior of the system. In engineering, FEM outputs might include stress distributions, deformations, or heat transfer patterns. In finance, interpreting FEM results means understanding the calculated option pricing values or the sensitivities (Greeks) derived from the model.

For example, when used for derivative pricing, the finite element method provides a grid of option prices across different underlying asset values and times to maturity. Analysts interpret these results to understand how the option's value changes under varying market conditions. The method's ability to handle complex pay-off structures or dynamic volatility makes it particularly useful for interpreting exotic options where simple analytical models fall short. Users must also be mindful of the assumptions made during model construction, such as the chosen mesh density and element type, as these directly influence the accuracy of the interpretation.

Hypothetical Example

Consider a quantitative analyst who needs to price a complex derivative whose payoff structure makes it difficult to value using standard analytical formulas. The derivative's value depends on a stochastic process that can be modeled by a partial differential equation.

Instead of trying to find an elusive closed-form solution, the analyst employs the finite element method.

Define Domain: The analyst defines a two-dimensional domain representing the underlying asset's price range and time to maturity.
Mesh Generation: This domain is divided into a grid of small triangular finite elements. The mesh is made finer in areas where the option price is expected to change rapidly, such as near the strike price, to capture these dynamics more accurately.
Element Equations: For each small triangle, the analyst approximates the option price function using simple polynomial interpolation.
Assembly and Boundary Conditions: The equations for all triangles are combined, and the known payoff of the derivative at maturity (a boundary condition) is applied. For example, if it's a call option, the value at maturity is (\max(S_T - K, 0)), where (S_T) is the asset price at maturity and (K) is the strike price.
Solve: A computational algorithm solves the resulting large system of linear equations, yielding the approximate option price at each node in the grid.
Interpolation: The analyst can then interpolate these nodal values to find the option price for any specific underlying asset price and time to maturity within the defined grid.

This allows the analyst to obtain a valuation for the complex derivative, enabling further risk management or hedging strategies.

Practical Applications

The finite element method has evolved significantly beyond its engineering origins, finding critical applications in various financial sectors:

Derivative Pricing: One of the most prominent uses is in pricing complex options and other derivatives, especially those that do not fit the assumptions of simpler models like the Black-Scholes model. FEM can handle options with path-dependent payoffs, multiple underlying assets, or transaction costs, which are typically governed by high-dimensional partial differential equations. This is a well-established area of application, as detailed in academic works like "Option Pricing with Finite Elements" by Jürgen Topper.³
Risk Management: FEM can be used in advanced risk models to simulate the behavior of portfolios under various market conditions. It helps in calculating risk measures like Value at Risk (VaR) or Conditional Value at Risk (CVaR) for complex financial instruments.
Financial Modeling: Beyond derivatives, FEM assists in general financial modeling, particularly in scenarios involving continuous time dynamics, such as modeling interest rate curves or credit default probabilities, where the underlying processes are described by partial differential equations.
Computational Finance Research: Researchers utilize the finite element method to develop and test new financial theories and models, especially when exploring the implications of non-standard market dynamics or complex interactions between financial variables.

Limitations and Criticisms

While a powerful numerical method, the finite element method is not without its limitations, particularly when applied to complex financial problems:

Computational Cost: FEM can be computationally intensive, especially for problems with many dimensions (e.g., options dependent on multiple underlying assets) or requiring very fine meshes for accuracy. This can lead to long calculation times, which might be impractical for real-time derivative pricing or trading strategies.
Model Dependence and Assumptions: The accuracy of FEM results heavily relies on the quality of the underlying mathematical model, the chosen element types, and the boundary conditions applied. Poorly defined assumptions or incorrect model setup can lead to inaccurate or misleading results, as highlighted by critiques in engineering applications.²
Meshing Challenges: Generating an optimal mesh for complex financial domains can be challenging. An overly coarse mesh will yield inaccurate results, while an excessively fine mesh increases computational burden unnecessarily. Issues like mesh distortion can also negatively impact solution quality.
Approximation Errors: The finite element method inherently provides an approximation, not an exact solution. While techniques exist to estimate and control errors, completely eliminating them is impossible. Understanding the magnitude of these approximation errors is crucial for reliable interpretation.
Complexity for Non-Specialists: Implementing and properly utilizing FEM requires a strong background in numerical analysis, partial differential equations, and the specific domain (finance or engineering). It is not a "push-button" solution, and misuse by unqualified individuals can lead to significant errors.¹

Finite Element Method vs. Finite Difference Method

The Finite Element Method (FEM) and the Finite Difference Method (FDM) are both widely used numerical methods for solving partial differential equations, including those encountered in computational finance. While they share the goal of approximating solutions, their approaches differ significantly.

Feature	Finite Element Method (FEM)	Finite Difference Method (FDM)
Discretization	Divides the problem domain into a mesh of elements (e.g., triangles, quadrilaterals).	Discretizes the domain into a grid of discrete points or nodes.
Approximation	Approximates the solution within each element using basis or shape functions (often polynomials).	Approximates derivatives using finite differences at grid points.
Flexibility	Highly flexible for complex geometries and irregular domains. Allows for local mesh refinement.	Less flexible for complex geometries; typically requires structured, rectangular grids.
Equation Type	Leads to a system of algebraic equations, often solved using variational principles.	Directly replaces derivatives with difference approximations, leading to a system of algebraic equations.
Boundary Conditions	Easier to incorporate complex or natural boundary conditions directly into the formulation.	Boundary conditions must be explicitly approximated at grid points, which can be challenging for irregular boundaries.
Computational Cost	Can be more computationally expensive for simple, regular problems, but efficient for complex ones.	Generally simpler and faster for regular grids, but can struggle with complex geometries.

The main confusion often arises because both methods convert differential equations into systems of algebraic equations. However, FEM's strength lies in its ability to handle irregularly shaped domains and complex boundary conditions with relative ease, making it a preferred choice for problems where the geometry or material properties are non-uniform. In contrast, the finite difference method is often simpler to implement for problems on regular grids.

FAQs

What types of financial problems can the finite element method solve?

The finite element method is particularly effective for solving partial differential equations that arise in derivative pricing, especially for complex options that do not have simple analytical solutions. It can handle features like multiple underlying assets, changing volatility, transaction costs, and early exercise options. It's also used in advanced risk management and financial modeling.

Is the finite element method an exact solution?

No, the finite element method provides an approximate solution. It breaks down a continuous problem into smaller, simpler pieces and solves for these. The accuracy of the approximation depends on factors like the fineness of the mesh (how small the "finite elements" are) and the quality of the interpolation functions used within each element.

Why is FEM used in finance if it's from engineering?

Many financial problems, particularly in quantitative analysis and option pricing, can be formulated as partial differential equations. Since the finite element method is a robust numerical technique for solving such equations, its application naturally extended from engineering and physics into computational finance. Its ability to handle complex problems and boundary conditions makes it valuable in financial contexts.

What is "meshing" in the context of FEM?

Meshing refers to the process of dividing the continuous domain of a problem into a discrete set of finite elements. This creates a "mesh" or grid of interconnected elements. The quality and density of the mesh are crucial for the accuracy and efficiency of the finite element method. A finer mesh generally leads to more accurate results but requires more computational resources.