The Taylor Series and Its Application.

SUMMARY In this study, we developed a new method that can significantly accelerate the forward modelling of gravity fields generated by large-scale tesseroids while keeping the computational accuracy as high as possible. The cost of the high efficiency is that the method only works under the assumptions that (1) all tesseroids in the same latitude band have the same horizontal dimension, (2) the computation points are located at the same surface level and aligned with the horizontal centres of tesseroids and (3) each tesseroid has a constant or linearly varying density. The new method first integrates the kernel function of the Newton’s volume integral analytically in the radial direction to eliminate its dependence on the vertical dimension of the tesseroid, and then expands the integrated kernel function into a Taylor series up to a certain order. Because the Taylor series expansion term of the integrated kernel function is an odd or even function of the difference between the longitudes of the tesseroid and computation point, there exist shifting or swapping symmetry relations among the gravity field of tesseroids. Consequently, the shifting or swapping symmetry is extended to the tesseroids with unequal vertical dimensions. Numerical experiments using the spherical shell model are conducted to verify the effectiveness of the new method. The results show that the computational speed of the new method is about 30 times faster than that of the traditional method, which employs the Gauss–Legendre quadrature rule and a 2-D adaptive subdivision approach, while keeping almost the same computational accuracy. When applying the new method to an ice shell with unequal thicknesses, the results reveal that the relative errors of calculating V, Vz and Vzz are smaller than 10−8, 10−6 and 10−4, respectively if the Taylor series expansion is truncated at order 4, while the computational time consumed by the new method is about 7 times less than that of the traditional method. Finally, the influence of the truncation order on the computational accuracy and the strategies for dividing the latitude band into several parts to further improve the accuracy are discussed.

Financial mathematics plays a pivotal role in various aspects of modern economics and finance. This paper provides an introduction to the fundamental concepts, theories, and applications of financial mathematics. It begins by outlining the basic principles of financial mathematics, including the time value of money, interest rates, and compounding. Computational programs enhance these mathematical models, offering robust solutions and efficient computation for complex financial problems. This study explores the integration of computational programs with financial mathematics, their methodologies, and applications in the finance sector. The results underscore the significance of computational methods in improving the accuracy, speed, and scalability of financial models, ultimately contributing to better decision-making and risk management. We explore fundamental concepts, models, and techniques employed in financial mathematics, aiming to provide a comprehensive understanding of their applications and significance in real-world financial scenarios. This paper provides a comprehensive overview of the application of differential equations in financial mathematics, highlighting key models such as the Black-Scholes model, interest rate models, and optimal investment strategies.

In this study, we extend the multiscale stochastic volatility model of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254] by incorporating a slow varying factor of volatility. The resulting model can be viewed as a multifactor extension of the Heston model with two additional factors driving the volatility levels. An asymptotic analysis consisting of singular and regular perturbation expansions is developed to obtain an approximation to European option prices. We also find explicit expressions for some essential functions that are available only in integral formulas in the work of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254]. This finding basically leads to considerable reduction in computational time for numerical calculation as well as calibration problems. An accuracy result of the asymptotic approximation is also provided. For numerical illustration, the multifactor Heston model is calibrated to index options on the market, and we find that the resulting implied volatility surfaces fit the market data better than those produced by the multiscale stochastic volatility model of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254], particularly for long‐maturity call options.

A Framework for Extracting the Probability of Default from Listed Stock Option Prices

The aim of this paper is to integrate Fractals into the conventional High School curriculum through an interdisciplinary approach between Mathematics and Biology. The proposed activities aim to assist educators in delivering a comprehensive and captivating learning journey for students, encouraging deeper comprehension, creativity, and interconnections across various scientific disciplines. Fractals are intricate geometric shapes characterized by self-similarity, meaning they display similar patterns at multiple scales. They are formed through iterative processes, distinguishing them from traditional geometric figures and offer an enjoyable and stimulating method to introduce numerous mathematical and biological concepts. By connecting fractals to various mathematical concepts and real-world applications, junior High School students can enhance their problem-solving skills and gain a greater appreciation for the beauty and complexity of Mathematics.

In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. In practice, we are interested in approximating a function f(x) near x = a by a polynomial of degree M. Other issues involved in approximation are how good the approximation is and over what interval can we expect the approximation to be good. In this paper we present approximation that is realized as piecewise polynomial function where in each interval ƒ(x) is approximated by a Taylor polynomial of low order. The piecewise polynomial approximation can be effectively implemented using Farrow structure.

Option pricing is essential for market participants as it facilitates risk management and price discovery. This study provides a comprehensive elucidation of diverse option pricing models. The author delineates the development of option pricing models from the fundamental Black-Scholes-Merton model (BSM) to more intricate methodologies. The importance of comprehending option pricing for hedging strategies and risk management is crucial in financial mathematics. The study analyses several models, including the BS model, Jump Diffusion model (JDM), Binomial Tree model (BTM), Monte Carlo simulation (MCS), and stochastic volatility models (SVM), while addressing their applications, limitations, advantages, and disadvantages. This work delineates the history and distinctions between other models and the Black-Scholes-Merton model. The Black-Scholes-Merton model exerts the most substantial influence on financial markets. The alternative models exhibit greater flexibility and enhanced accuracy when confronting complicated and uncertain market trends. The trend of future development is bound to be the continuous updating of models to adapt to more complex situations.

This new and exciting book offers a fresh approach to quantitative finance and utilises novel features, including stereoscopic images which permit 3D visualisation of complex subjects without the need for additional tools. Offering an integrated approach to the subject, A First Course in Quantitative Finance introduces students to the architecture of complete financial markets before exploring the concepts and models of modern portfolio theory, derivative pricing and fixed income products in both complete and incomplete market settings. Subjects are organised throughout in a way that encourages a gradual and parallel learning process of both the economic concepts and their mathematical descriptions, framed by additional perspectives from classical utility theory, financial economics and behavioural finance. Suitable for postgraduate students studying courses in quantitative finance, financial engineering and financial econometrics as part of an economics, finance, econometric or mathematics program, this book contains all necessary theoretical and mathematical concepts and numerical methods, as well as the necessary programming code for porting algorithms onto a computer.

This new and exciting book offers a fresh approach to quantitative finance and utilises novel features, including stereoscopic images which permit 3D visualisation of complex subjects without the need for additional tools. Offering an integrated approach to the subject, A First Course in Quantitative Finance introduces students to the architecture of complete financial markets before exploring the concepts and models of modern portfolio theory, derivative pricing and fixed income products in both complete and incomplete market settings. Subjects are organised throughout in a way that encourages a gradual and parallel learning process of both the economic concepts and their mathematical descriptions, framed by additional perspectives from classical utility theory, financial economics and behavioural finance. Suitable for postgraduate students studying courses in quantitative finance, financial engineering and financial econometrics as part of an economics, finance, econometric or mathematics program, this book contains all necessary theoretical and mathematical concepts and numerical methods, as well as the necessary programming code for porting algorithms onto a computer.

This new and exciting book offers a fresh approach to quantitative finance and utilises novel features, including stereoscopic images which permit 3D visualisation of complex subjects without the need for additional tools. Offering an integrated approach to the subject, A First Course in Quantitative Finance introduces students to the architecture of complete financial markets before exploring the concepts and models of modern portfolio theory, derivative pricing and fixed income products in both complete and incomplete market settings. Subjects are organised throughout in a way that encourages a gradual and parallel learning process of both the economic concepts and their mathematical descriptions, framed by additional perspectives from classical utility theory, financial economics and behavioural finance. Suitable for postgraduate students studying courses in quantitative finance, financial engineering and financial econometrics as part of an economics, finance, econometric or mathematics program, this book contains all necessary theoretical and mathematical concepts and numerical methods, as well as the necessary programming code for porting algorithms onto a computer.

This new and exciting book offers a fresh approach to quantitative finance and utilises novel features, including stereoscopic images which permit 3D visualisation of complex subjects without the need for additional tools. Offering an integrated approach to the subject, A First Course in Quantitative Finance introduces students to the architecture of complete financial markets before exploring the concepts and models of modern portfolio theory, derivative pricing and fixed income products in both complete and incomplete market settings. Subjects are organised throughout in a way that encourages a gradual and parallel learning process of both the economic concepts and their mathematical descriptions, framed by additional perspectives from classical utility theory, financial economics and behavioural finance. Suitable for postgraduate students studying courses in quantitative finance, financial engineering and financial econometrics as part of an economics, finance, econometric or mathematics program, this book contains all necessary theoretical and mathematical concepts and numerical methods, as well as the necessary programming code for porting algorithms onto a computer.

This new and exciting book offers a fresh approach to quantitative finance and utilises novel features, including stereoscopic images which permit 3D visualisation of complex subjects without the need for additional tools. Offering an integrated approach to the subject, A First Course in Quantitative Finance introduces students to the architecture of complete financial markets before exploring the concepts and models of modern portfolio theory, derivative pricing and fixed income products in both complete and incomplete market settings. Subjects are organised throughout in a way that encourages a gradual and parallel learning process of both the economic concepts and their mathematical descriptions, framed by additional perspectives from classical utility theory, financial economics and behavioural finance. Suitable for postgraduate students studying courses in quantitative finance, financial engineering and financial econometrics as part of an economics, finance, econometric or mathematics program, this book contains all necessary theoretical and mathematical concepts and numerical methods, as well as the necessary programming code for porting algorithms onto a computer.

This new and exciting book offers a fresh approach to quantitative finance and utilises novel features, including stereoscopic images which permit 3D visualisation of complex subjects without the need for additional tools. Offering an integrated approach to the subject, A First Course in Quantitative Finance introduces students to the architecture of complete financial markets before exploring the concepts and models of modern portfolio theory, derivative pricing and fixed income products in both complete and incomplete market settings. Subjects are organised throughout in a way that encourages a gradual and parallel learning process of both the economic concepts and their mathematical descriptions, framed by additional perspectives from classical utility theory, financial economics and behavioural finance. Suitable for postgraduate students studying courses in quantitative finance, financial engineering and financial econometrics as part of an economics, finance, econometric or mathematics program, this book contains all necessary theoretical and mathematical concepts and numerical methods, as well as the necessary programming code for porting algorithms onto a computer.

This new and exciting book offers a fresh approach to quantitative finance and utilises novel features, including stereoscopic images which permit 3D visualisation of complex subjects without the need for additional tools. Offering an integrated approach to the subject, A First Course in Quantitative Finance introduces students to the architecture of complete financial markets before exploring the concepts and models of modern portfolio theory, derivative pricing and fixed income products in both complete and incomplete market settings. Subjects are organised throughout in a way that encourages a gradual and parallel learning process of both the economic concepts and their mathematical descriptions, framed by additional perspectives from classical utility theory, financial economics and behavioural finance. Suitable for postgraduate students studying courses in quantitative finance, financial engineering and financial econometrics as part of an economics, finance, econometric or mathematics program, this book contains all necessary theoretical and mathematical concepts and numerical methods, as well as the necessary programming code for porting algorithms onto a computer.

The articles in this volume summarize the lectures given by the authors on February 1, 2005, at an open conference held at the Academie des Sciences, quai Conti, on Financial Mathematics, a field in constant development since the end of the 1970s. The aim is to make these articles accessible to a wide audience belonging to scientific horizons; indeed, this half-day conference was attended by many young postgraduates in disciplines such as economics, management, and applied mathematics, along with many members of the different sections of the Academie. Among the reasons to get interested in financial mathematics, the following is one: who has never wondered, looking at the financial pages of a newspaper, displaying the erratic evolutions of quotations on the Stock Exchange, if these were not “governed” by some models, likely to be probabilistic? This question was at the heart of the studies conducted by Louis Bachelier, particularly in his famous thesis (1900), and he answered the above question in terms of Brownian Motion. His remarkable results, however, remained in a kind of scientific limbo for almost 75 years, until Samuelson “corrected” Bachelier (in 1965) by replacing the Brownian Motion by its exponential, and the famous Black-Scholes formula began (in 1973) to play an essential role in the computation of option prices. Since the 1980s, we have witnessed the explosion of probabilistic models, along with financial products, each in turn becoming more and more complex. All this technology, which now forms an important part of financial engineering, exists only because some mathematical concepts both simple and universal allow building a “theory of the laws of markets,” based on principles such as the prices across time of an uncertain asset having the probabilistic structure of a fair game, that is to say of a martingale. From this concept, little by little was built the entire theory of stochastic processes, which is a pillar on which the mathematical theory of “arbitrage” was developed by Delbaen and Schachermayer. The six articles in this volume present the following main topics.