Mathematics in Finance

Mathematical finance, also known as quantitative finance, is a field of applied mathematics, concerned with financial markets. Generally, mathematical finance will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. Thus, for example, while a financial economist might study the structural reasons why a company may have a certain share price, a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the corresponding value of derivatives of the stock. The fundamental theorem of arbitrage-free pricing is one of the key theorems in mathematical finance, while the Black-Scholes equation and formula are amongst the key results.Mathematical finance also overlaps heavily with the fields of computational finance and financial engineering. The latter focuses on applications and modeling, often by help of stochastic asset models, while the former focuses, in addition to analysis, on building tools of implementation for the models. In general, there exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and risk and portfolio management on the other. This Book caters specifically for mathematic students by covering topics in a mathematically rigorous way. It is suited for independent study, and should prepare the reader to study more advanced texts, assuming only basic knowledge of probability and calculus, it presents three major areas of mathematical finance, namely option pricing based on the no-arbitrage principle in discrete and continuous time setting providing alongside an introduction to Financial Engineering to it's readers.