Introduction: Some Aspects of Financial Mathematics

The conventional Financial Mathematics is, principally, financial calculi. Nevertheless, the financial phenomenon, the preference for liquidity, it would must be always present into the temporal economic studies. The financial value considers, joint the monetary amount, its temporal deferral, its liquidity. Both of them must be formalized in a binary vector (C, T), the financial capital, reclaiming different mathematic that financial conventional simply arithmetic (scalar). The financial analysis reclaims the vector Financial Mathematic that we propos. Financing financial operations (FFO) and investment financial operations (IFO) are very different financial operations (FO) by their purpose. FFO only pretends to get an interest, the financing price by its financial service. IFO intents to get an investment yield, its economic result. Interest and investment yield are different economic magnitudes. The interest is appointed by a financial market and it defines the financial equilibrium of FFO (finance equivalence). The investment yield is an economic result appointed by the financial disequilibrium of IFO. It is the reason why to Investigate an investment yield as a financial equilibrium, as an implicit interest, it is the grave financial mistake that IRR (Internal Rate of Return) commits, confusing investment yield with interest and an IFO with a FFO, with erroneous consequences to its definition and possible to investment decisions. The new methodological Financial Mathematics permits to revise conventional concepts as interest, investment yield, financial productivity, financial profitability, etc. also permitting to incorporate another unknown concepts as financial degeneration, financial immunity, etc, and to know other financial instruments, alternative to IRR, to investment decision.

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 mathematical treatment of models relying on pure jump processes is quite different from the treatment of models where a Brownian motion is present. For instance, risk management procedures, derivative pricing and portfolio optimization are all significantly altered, so there is interest from the mathematical finance side in finding out which model is more likely to have generated the data. This chapter provides explicit testing procedures to decide whether the Brownian motion is necessary to model the observed path, or whether the process is entirely driven by its jumps. The structural assumption is the same as in the previous two chapters, with the underlying process X being a one-dimensional Itô semimartingale, since in the multi-dimensional case we can again perform the test on each component separately.

Approximating payoffs and pricing formulas

Taylor series, a fundamental concept in mathematics, expresses a function as an infinite sum of terms which is calculated from the values of its derivatives at a single point. Its applications span various fields, notably physics and finance, where it simplifies complex functions for analysis and problem-solving. This study employs a qualitative approach, reviewing existing literature on Taylor series applications. It analyzes case studies in physics, such as approximating motion equations, and in finance, like option pricing models. The method includes mathematical derivations and simulations to demonstrate efficacy. Findings indicate that Taylor series significantly enhance computational efficiency and accuracy. In physics, it aids in predicting particle behavior under varying forces. In finance, it provides valuable insights into the pricing of derivatives and risk assessment. The Taylor series proves to be an indispensable tool in both physics and finance, facilitating complex calculations and improving analytical capabilities. Its versatility underscores the importance of mathematical foundations in real-world applications.

Simulation of Brownian motion at first-passage times

We introduce a suitable backward stochastic differential equation (BSDE) to represent the value of an optimal control problem with partial observation for a controlled stochastic equation driven by Brownian motion. Our model is general enough to include cases with latent factors in mathematical finance. By a standard reformulation based on the reference probability method, it also includes the classical model where the observation process is affected by a Brownian motion (even in presence of a correlated noise), a case where a BSDE representation of the value was not available so far. This approach based on BSDEs allows for greater generality beyond the Markovian case, in particular our model may include path-dependence in the coefficients (both with respect to the state and the control), and does not require any nondegeneracy condition on the controlled equation. We use a randomization method, previously adopted only for cases of full observation, and consisting in a first step, in replacing the control by an exogenous process independent of the driving noise and in formulating an auxiliary (“randomized”) control problem where optimization is performed over changes of equivalent probability measures affecting the characteristics of the exogenous process. Our first main result is to prove the equivalence between the original partially observed control problem and the randomized problem. In a second step, we prove that the latter can be associated by duality to a BSDE, which then characterizes the value of the original problem as well.

The study of exponential functionals of Brownian motion has recently attracted much attention, partly motivated by several problems in financial mathematics. Let be a linear Brownian motion starting from 0. Following Dufresne (1989), (1990), De Schepper and Goovaerts (1992) and De Schepper et al. (1992), we are interested in the process (for δ > 0), which stands for the discounted values of a continuous perpetuity payment. We characterize the upper class (in the sense of Paul Lévy) of X, as δ tends to zero, by an integral test. The law of the iterated logarithm is obtained as a straightforward consequence. The process exp(W(u))du is studied as well. The class of upper functions of Z is provided. An application to the lim inf behaviour of the winding clock of planar Brownian motion is presented.

Random walk is the basic concept in mathematics. Brownian motion is one of the most popular application. The core of the Brownian motion is the continuous path for small particles. After being developed by Wiener, it is used in many other studies, including biology, chemistry, and economics. Brownian motion is contemporarily used for predicting possible path. For animals scientists can know the probable range of activities for animals in order to keep people from bothering them and keep them out of danger. For chemical reactions, it is easy to predict every tiny particle about their movement so that people can directly understand the core of the reaction and one can predict the speed and the phenomenon of the reaction. For economics, economists can use the Brownian motion to calculate the probability for stock to up or down. This paper aims to present the theory of Brownian motion and illustrate the applications. This allows people to better choose the time to buy and sell in order to get the highest profit, and makes the economic market more controllable.