Modern mathematical finance and quantitative analysis require a strong background in fields such as stochastic calculus, optimization, partial differential equations (PDEs) and numerical methods, or even infinite dimensional analysis. In addition, the emergence of new complex financial instruments on the markets makes it necessary to rely on increasingly sophisticated mathematical tools. Not all readers of this book will eventually work in quantitative financial analysis, nevertheless they may have to interact with quantitative analysts, and becoming familiar with the tools they employ will be an advantage. In addition, despite the availability of ready-made financial calculators it still makes sense to be able oneself to understand, design and implement such financial algorithms. This can be particularly useful under different types of conditions, including an eventual lack of trust in financial indicators, possible unreliability of expert advice such as buy/sell recommendations, or other factors such as market manipulation. To some extent we would like to have some form of control on the future behavior of random (risky) assets; however, since knowledge of the future is not possible, the time evolution of the prices of risky assets will be modeled by random variables and stochastic processes.

Historical Sketch