Collocation method based on modified ‎cubic‎ B-spline ‎for option pricing ‎models

Collocation‎‎ ‎method ‎based ‎on ‎modified‎ cubic B-spline functions ‎has ‎been ‎developed‎ ‎for ‎the ‎valuation ‎‎‎of European‎, ‎American and Barrier options of single ‎asset. ‎The ‎new ‎approach ‎contains ‎‎discretizing ‎of‎ t‎‎emporal ‎derivative‎ ‎using ‎finite ‎difference ‎approximations ‎and ‎approximating‎ the option price with the ‎modified‎ B-spline functions‎. ‎Stability of this method has been discussed and shown that it is unconditionally stable‎. ‎The ‎efficiency ‎of ‎the‎ ‎proposed ‎method ‎is ‎tested ‎by ‎different ‎examples‎‎‎.

An efficient numerical method based on exponential B-splines for time-fractional Black-Scholes equation governing European options

In this paper a time-fractional Black-Scholes model (TFBSM) is considered to study the price change of the underlying fractal transmission system. We develop and analyze a numerical method to solve the TFBSM governing European options. The numerical method combines the exponential B-spline collocation to discretize in space and a finite difference method to discretize in time. The method is shown to be unconditionally stable using von-Neumann analysis. Also, the method is proved to be convergent of order two in space and $2-\mu$ is time, where $\mu$ is order of the fractional derivative. We implement the method on various numerical examples in order to illustrate the accuracy of the method, and validation of the theoretical findings. In addition, as an application, the method is used to price several different European options such as the European call option, European put option, and European double barrier knock-out call option.Comment: 34 pages, 12 figure

Numerical singular perturbation approaches based on spline approximation methods for solving problems in computational finance

Philosophiae Doctor - PhDOptions are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Bean. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature

Numerical singular perturbation approaches based on spline approximation methods for solving problems in computational finance

Philosophiae Doctor - PhDOptions are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.South Afric

The Seven-League scheme: Deep learning for large time step Monte Carlo simulations of stochastic differential equations

We propose an accurate data-driven numerical scheme to solve stochastic differential equations (SDEs), by taking large time steps. The SDE discretization is built up by means of the polynomial chaos expansion method, on the basis of accurately determined stochastic collocation (SC) points. By employing an artificial neural network to learn these SC points, we can perform Monte Carlo simulations with large time steps. Basic error analysis indicates that this data-driven scheme results in accurate SDE solutions in the sense of strong convergence, provided the learning methodology is robust and accurate. With a method variant called the compression–decompression collocation and interpolation technique, we can drastically reduce the number of neural network functions that have to be learned, so that computational speed is enhanced. As a proof of concept, 1D numerical experiments confirm a high-quality strong convergence error when using large time steps, and the novel scheme outperforms some classical numerical SDE discretizations. Some applications, here in financial option valuation, are also presented

Program: 2021 Undergraduate Mathematics Day

Schedule and general information about the event. 21st Annual Kenneth C. Schraut Memorial Lecture: One Health: Connecting Humans, Animals and the Environment (Suzanne Lenhart, University of Tennessee) Plenary talk: The Crossings of Art, History, and Mathematics (Jennifer White, St. Vincent College