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In their paper titled "Hilbert transform, spectral filters and option pricing," authors Carolyn E. Phelan, Daniele Marazzina, Gianluca Fusai, and Guido Germano explore the use of spectral filters to improve the convergence of numerical schemes utilizing discrete Hilbert transforms based on a sinc function expansion and fast Fourier transform. This enhancement is particularly valuable for computing fluctuation identities that determine the distribution of maximum or minimum values along a random path, as well as joint distributions at maturity with extrema remaining below or above barriers. The study focuses on methods proposed by Feng and Linetsky (2008) and Fusai, Germano, and Marazzina (2016) for pricing discretely monitored barrier options within the framework of an exponential Lévy process modeling the underlying asset price. While both methods exhibit exponential convergence in most scenarios, they are constrained to polynomial convergence under specific conditions. The authors establish a connection between these convergence rates and the Gibbs phenomenon observed in Fourier transforms. By introducing spectral filtering techniques, the researchers achieve improved results in terms of convergence rates. Their findings highlight the significance of incorporating spectral filters in numerical schemes involving Hilbert transforms for more accurate option pricing calculations. The study contributes valuable insights into enhancing computational efficiency and accuracy in financial modeling applications related to barrier options.
SummaryAuthors are looking at using special filters to make math problems work better. These filters help with finding certain patterns and values in numbers. The study is about figuring out the prices of certain options using a specific type of math process. They found a link between how fast things come together in math and a pattern they see in another type of math. By using these special filters, they can make the math problems about pricing options work faster.
Definitions
- Spectral filters: Special tools that help make certain calculations in math more accurate.
- Convergence: When numbers or values come together or get closer to each other.
- Numerical schemes: Different ways of solving problems using numbers and calculations.
- Discrete Hilbert transforms: A specific mathematical process used to analyze data.
- Barrier options: Financial contracts where payouts depend on whether an asset's price reaches a certain level.
Introduction:
The pricing of financial derivatives, such as options, is a crucial aspect of modern finance. Accurate and efficient option pricing is essential for risk management and investment decision-making. In recent years, there has been an increasing interest in using exponential Lévy processes to model the underlying asset price in option pricing models. However, these models often involve complex mathematical calculations that require numerical methods for their solution.
One such method is the use of discrete Hilbert transforms based on a sinc function expansion and fast Fourier transform (FFT). While this approach has shown promising results in terms of convergence rates, it also has its limitations. In their paper titled "Hilbert transform, spectral filters and option pricing," Phelan et al. explore the use of spectral filters to improve the convergence of numerical schemes utilizing discrete Hilbert transforms.
Background:
The authors begin by providing a brief overview of previous research on option pricing using exponential Lévy processes. They highlight two existing methods proposed by Feng and Linetsky (2008) and Fusai et al. (2016) that exhibit exponential convergence rates but are limited to polynomial convergence under certain conditions.
The Gibbs Phenomenon:
To understand the significance of incorporating spectral filters in numerical schemes involving Hilbert transforms, the authors introduce the concept of Gibbs phenomenon observed in Fourier transforms. This phenomenon refers to oscillations or overshoots near discontinuities when approximating functions with finite sums of trigonometric functions.
Spectral Filters:
Phelan et al. propose using spectral filters to reduce these oscillations and improve convergence rates in numerical schemes involving Hilbert transforms. The researchers demonstrate how this can be achieved through various examples involving barrier options with different payoff structures.
Results:
Through their experiments, Phelan et al. show that incorporating spectral filters leads to significant improvements in terms of both accuracy and efficiency compared to existing methods without filtering techniques. The authors also provide theoretical justifications for these results by establishing a connection between the convergence rates and the Gibbs phenomenon.
Conclusion:
The study concludes that incorporating spectral filters in numerical schemes involving Hilbert transforms can significantly enhance computational efficiency and accuracy in financial modeling applications related to barrier options. The findings of this research have important implications for option pricing and risk management in the financial industry.
Implications:
The use of exponential Lévy processes has gained popularity in recent years due to their ability to model complex market dynamics. However, accurate and efficient option pricing using these models remains a challenge. The research by Phelan et al. provides valuable insights into improving convergence rates through the use of spectral filters, which can have significant implications for risk management and investment decision-making.
Future Research:
While this study focuses on barrier options, there is potential for further exploration of spectral filtering techniques in other types of financial derivatives as well. Additionally, future research could also investigate the application of these methods in different asset pricing models beyond exponential Lévy processes.
Conclusion:
In conclusion, Phelan et al.'s paper "Hilbert transform, spectral filters and option pricing" highlights the importance of incorporating spectral filters in numerical schemes involving Hilbert transforms for more accurate option pricing calculations. Their findings contribute valuable insights into enhancing computational efficiency and accuracy in financial modeling applications related to barrier options. This research has significant implications for risk management and investment decision-making in the finance industry.
We show how spectral filters can improve the convergence of numerical schemes which use discrete Hilbert transforms based on a sinc function expansion, and thus ultimately on the fast Fourier transform. This is relevant, for example, for the computation of fluctuation identities, which give the distribution of the maximum or the minimum of a random path, or the joint distribution at maturity with the extrema staying below or above barriers. We use as examples the methods by Feng and Linetsky (2008) and Fusai, Germano and Marazzina (2016) to price discretely monitored barrier options where the underlying asset price is modelled by an exponential L\'evy process. Both methods show exponential convergence with respect to the number of grid points in most cases, but are limited to polynomial convergence under certain conditions. We relate these rates of convergence to the Gibbs phenomenon for Fourier transforms and achieve improved results with spectral filtering.
