Computing Financial Derivatives: A Finite-Difference Approach

This volume provides accurate and efficient numerical solutions to the options pricing problem, covering state-of-the-art developments, discretization techniques, numerical algorithms, distributed algorithms, and practical applications with mathematical modeling and implementation.

Format: Hardback
Length: 268 pages
Publication date: 01 January 2021
Publisher: Taylor & Francis Ltd

Numerical Solutions to the Options Pricing Problem: A Comprehensive Guide
Numerical solutions to the options pricing problem have become increasingly important in recent years, as the financial industry has become more complex and dynamic. This volume provides a comprehensive guide to accurate and efficient numerical solutions to the options pricing problem, covering state-of-the-art developments in option pricing along with discretization techniques, numerical algorithms, distributed algorithms, and practical applications of these methods to real-world examples.

The book begins with a basic introduction to the options pricing problem, including the definition of options, their characteristics, and their valuation. It then discusses the various discretization techniques used to solve the options pricing problem, including finite difference methods, finite element methods, and Monte Carlo methods. These techniques are used to approximate the value of an option at a given point in time, taking into account the underlying asset's price, volatility, and other factors.

One of the key challenges in numerical solutions to the options pricing problem is the discretization of the underlying asset's price. This is because the price of an asset can change rapidly, and it can be difficult to accurately represent its value using a finite number of points. Finite difference methods are one of the most popular discretization techniques, and they involve dividing the time interval into a series of small time steps and calculating the value of the option at each step. However, finite difference methods can be computationally expensive, and they can produce inaccurate results if the underlying asset's price is not smooth.

Finite element methods are another discretization technique that can be used to solve the options pricing problem. These methods involve dividing the underlying asset's domain into a series of small elements and solving the equations governing the asset's behavior at each element. Finite element methods can be more accurate than finite difference methods, but they can be more computationally expensive.

Monte Carlo methods are a third discretization technique that can be used to solve the options pricing problem. These methods involve generating a large number of random samples of the underlying asset's price and calculating the value of the option at each sample. Monte Carlo methods can be very accurate, but they can be computationally expensive.

In addition to discretization techniques, numerical algorithms are also used to solve the options pricing problem. These algorithms include the Black-Scholes model, the binomial model, and the trinomial model. The Black-Scholes model is a widely used model that assumes that the underlying asset's price follows a random walk, and it calculates the value of an option using a series of equations. The binomial model is a more complex model that assumes that the underlying asset's price follows a binomial distribution, and it calculates the value of an option using a series of equations. The trinomial model is a more complex model that assumes that the underlying asset's price follows a trinomial distribution, and it calculates the value of an option using a series of equations.

Numerical algorithms can be implemented in a variety of programming languages, including C++, Java, and Python. These algorithms can be run on a variety of computing platforms, including desktop computers, laptops, and servers.

Practical applications of numerical solutions to the options pricing problem include portfolio optimization, risk management, and hedging. Portfolio optimization involves selecting a portfolio of assets that will maximize the return while minimizing the risk. Risk management involves identifying and managing the risks associated with a portfolio of assets. Hedging involves using derivatives to reduce the risk associated with a particular asset.

In addition to numerical solutions to the options pricing problem, this volume also covers Cartesian meshes, non-uniform time-stepping routines, and semi-Lagrangian time integration schemes. Cartesian meshes are a type of discretization technique that involves dividing the underlying asset's domain into a series of rectangular cells. Non-uniform time-stepping routines are a type of discretization technique that involves adjusting the time step size based on the underlying asset's price. Semi-Lagrangian time integration schemes are a type of discretization technique that involves solving the equations governing the asset's behavior at each time step.

One of the challenges associated with numerical solutions to the options pricing problem is the accuracy of the results. Numerical algorithms can produce inaccurate results if the underlying asset's price is not smooth, or if the discretization technique is not appropriate for the problem. To address this challenge, researchers have developed a variety of techniques for improving the accuracy of numerical solutions. These techniques include adaptive mesh refinement, adaptive time-stepping, and adaptive discretization.

Adaptive mesh refinement involves dividing the underlying asset's domain into a series of smaller cells, and then adjusting the time step size based on the size of the cells. Adaptive time-stepping involves adjusting the time step size based on the underlying asset's price. Adaptive discretization involves adjusting the discretization technique based on the underlying asset's price.

In addition to improving the accuracy of numerical solutions, researchers have also developed a variety of techniques for improving the efficiency of numerical solutions. These techniques include parallel computing, distributed computing, and GPU computing. Parallel computing involves dividing the problem into a series of smaller problems and then solving each problem on a separate computer. Distributed computing involves dividing the problem into a series of smaller problems and then solving each problem on a network of computers. GPU computing involves using a graphics processing unit (GPU) to solve the problem.

In conclusion, numerical solutions to the options pricing problem have become increasingly important in recent years, as the financial industry has become more complex and dynamic. This volume provides a comprehensive guide to accurate and efficient numerical solutions to the options pricing problem, covering state-of-the-art developments in option pricing along with discretization techniques, numerical algorithms, distributed algorithms, and practical applications of these methods to real-world examples. By improving the accuracy and efficiency of numerical solutions, researchers can help to improve the performance of the financial industry and to reduce the risk associated with investing in options.

Dimension: 235 x 156 (mm)
ISBN-13: 9781420082647