Using an explicit scheme for an application of finite difference methods may lead to stability issues. If one wants to increase the accuracy by raising the number of spatial grid points, the number of time intervals have to be increased to a certain extent in order to sustain a converging behavior.
As for quite accurate results ridiculously many grid points in time are needed, the practical use of the explicit scheme is rather limited due to high computational effort. Implicit methods for finite difference methods are designed to overcome these stability limitations imposed by the already mentioned convergence restrictions. Since such methods are unconditionally stable, both accuracy and limited computational effort can be combined.
This text offers an introductory treatment of Finite Difference Methods employing an implicit scheme. It includes a theoretical derivation of the implicit scheme and the Crank-Nicolson scheme, a numerical application to European puts as well as a theoretical discussion and comparison of the truncation error for both schemes. Finally, Richard-Extrapolation is introduced as a nice tool for lowering the truncation error.
Table of Contents
1 Theoretical Aspects of Implicit Finite Difference Methods
1.1 Derivation of a Fully Implicit Scheme
1.1.1 Exemplary Boundary Conditions for a European Put
1.1.2 The Fully Implicit Scheme in Matrix Notation
1.2 Solving Tridiagonal System of Linear Equations
1.3 Derivation of the Crank-Nicolson Scheme
1.3.1 Exemplary Boundary Conditions for a European Put
1.3.2 The Crank-Nicolson Scheme in Matrix Notation
2 Numerical Application for a European Put
2.1 Results and Interpretations
2.2 Technical Implementation
3 Truncation Error and Rate of Convergence
3.1 Richardson Extrapolation for the Fully Implicit Scheme
3.2 Richardson Extrapolation for Crank-Nicolson Scheme
3.3 Preliminary Efficiency Analysis
3.4 Technical Implementation
Research Objectives & Topics
The primary objective of this work is to evaluate the performance and efficiency of implicit finite difference methods, specifically the Fully Implicit scheme and the Crank-Nicolson (CN) scheme, for pricing financial derivatives such as European puts. The study investigates how these numerical schemes, combined with series acceleration methods like Richardson Extrapolation, balance computational effort against the accuracy of the derivative pricing.
- Numerical derivation and implementation of the Fully Implicit and Crank-Nicolson schemes.
- Analysis of truncation errors and convergence rates for different discretization approaches.
- Efficiency comparison between standard implicit schemes and those enhanced with Richardson Extrapolation.
- Identification of numerical challenges, such as spurious oscillations and the impact of non-smooth boundary conditions.
Excerpt from the Book
1.1 Derivation of a Fully Implicit Scheme
As a starting point, assume that the price of an asset follows a geometric Brownian motion: dSt = (r − q)Stdt + σStdWt . (1)
According to Hirsa (2013, p. 115), the value of a derivative on this financial asset specified in (1) must satisfy the Black-Scholes-Merton PDE subject to a given terminal condition at every point in time: 1/2 η^2 S^2 FSS(S, t) + (r − q)SFS(S, t) − rF(S, t) + Ft(S, t) = 0 s.t. F(S, T) = f(S, T) . (2)
In the PDE, η is the so-called diffusion coefficient of the underlying Brownian motion, S the price of the underlying asset, i.e. a stock, r the risk-free interest rate and q the continuously paid dividend rate.
In order to obtain a numerical solution for the above PDE, the initial variables should be transformed. Furthermore, the continuous pricing problem has to be converted into a discrete one by dividing the feasible time and space domain into a rectangular grid and substituting the partial derivatives with differential quotients. Following this idea, a new set of independent variables is introduced: x = S = ih ∀ i ∈ [0, M], τ = T − t = nk ∀ n ∈ [0, N], where i and n are index numbers for the discretized grid points, and h and k are step sizes in space and time direction, respectively. Consequently, the rectangular grid reads: x ∈ [0, xmax] with xmax = M · h, τ ∈ [0, T] with T = N · k.
Summary of Chapters
1 Theoretical Aspects of Implicit Finite Difference Methods: This chapter introduces the mathematical foundations for pricing financial derivatives using the Fully Implicit and Crank-Nicolson schemes, including their derivations and matrix representations.
2 Numerical Application for a European Put: This chapter presents a practical implementation in MATLAB, comparing the accuracy and computational runtime of the proposed schemes for pricing a European put option.
3 Truncation Error and Rate of Convergence: This chapter provides a theoretical analysis of truncation errors and demonstrates how Richardson Extrapolation can enhance convergence rates and improve accuracy with optimal grid selection.
Keywords
Finite Difference Methods, Implicit Scheme, Crank-Nicolson, Richardson Extrapolation, Derivative Pricing, Black-Scholes-Merton, Truncation Error, Convergence, Numerical Analysis, Computational Efficiency, MATLAB, European Put, Stability, Grid Points, Stochastic Calculus.
Frequently Asked Questions
What is the core subject of this research paper?
The paper focuses on the numerical solution of the Black-Scholes-Merton partial differential equation using implicit finite difference methods to price financial derivatives.
What are the primary methods discussed?
The work primarily covers the Fully Implicit scheme and the Crank-Nicolson scheme, comparing their stability, accuracy, and computational requirements.
What is the main objective of the study?
The goal is to determine the most efficient numerical implementation, particularly by utilizing Richardson Extrapolation to achieve higher accuracy without prohibitively increasing computational cost.
Which mathematical framework is used for error analysis?
The analysis relies on calculating the leading truncation error terms derived from Taylor series expansions to determine the order of accuracy and rate of convergence for each scheme.
What does the numerical implementation cover?
The implementation section details how these methods are translated into MATLAB code, optimizing performance through matrix decomposition and solving tridiagonal equation systems.
Which keywords characterize this work?
Key terms include Finite Difference Methods, Crank-Nicolson, Richardson Extrapolation, Derivative Pricing, and Numerical Convergence.
Why does the Crank-Nicolson method sometimes face stability issues?
Although the CN method is unconditionally stable, the paper notes that spurious oscillations can occur if the time step is not small enough, particularly due to non-smooth initial conditions like the payoff of a European put.
How does grid positioning affect the results?
The paper references Tavella & Randall (2000), noting that if the option's strike price coincides exactly with a grid point, the approximation by differential quotients yields suboptimal results; thus, positioning the strike halfway between grid points is recommended.
What is the trade-off between the two schemes?
There is a fundamental trade-off between the Fully Implicit scheme and the Crank-Nicolson scheme regarding computational time and absolute error, which varies depending on the refinement of the space and time grids.
- Quote paper
- Pascal Sturm (Author), 2016, Finite difference methods with an implicit scheme, Munich, GRIN Verlag, https://www.grin.com/document/373359
