I. Definition of Financial Mathematics
Financial mathematics or mathematical finance, or mathematical finance, comes from mathematical finance and can be understood as a discipline that uses mathematics as a tool to solve financial problems. Financial mathematics is an applied subject that establishes a mathematical model suitable for the specific facts of the financial industry, compiles certain software, simulates theoretical research results, and conducts econometric analysis and research on actual data.
The biggest feature of financial mathematics is the wide application of modern mathematical tools, especially with the creative application of cybernetics and stochastic process research results in the financial field, financial mathematics, a new frontier discipline, has been formed, which is also called mathematical finance internationally. Financial mathematics originated from the study of financial problems. With the development of financial market, the relationship between finance and mathematics is getting closer and closer, and it has achieved rapid development.
Financial mathematics in a broad sense refers to a new discipline that applies mathematical theories and methods to study the laws of financial and economic operation. In a narrow sense, the main research content of financial mathematics is about portfolio selection and asset pricing theory under uncertain multi-period conditions, and arbitrage, optimality and equilibrium are the three most important concepts in this theory.
Financial mathematics is based on some financial or economic assumptions, using abstract mathematical methods to establish a mathematical model of financial mechanism. The scope of financial mathematics includes various applications of mathematical concepts and methods (or other natural science methods) in finance, especially in financial theory. The purpose of application is to express, reason and demonstrate financial principles by mathematical methods. Financial mathematics is a branch of finance, so financial mathematics is based on financial theory first, which does not mean that financial mathematics must have formal academic training in finance (this is really beneficial). Although finance is independent of it because of its sufficient characteristics, it is developed as an applied branch of economics, so financial mathematics is also based on economic principles and technologies. Because finance is closely related to classmates, mathematics and tax theory, financial mathematics needs to be based on accounting principles, financial technology and theory.
The theoretical basis of financial mathematics, of course, also includes modern mathematical theory and mathematical theory. The first step is mathematical or statistical modeling, that is, selecting key factors from complex finance, distinguishing relevant factors from irrelevant factors, then deducing various relationships from a series of hypothetical conditions, and finally getting a conclusion to explain the conclusion. This kind of modeling activity is not only very useful, but also extremely important, because in finance, a small mistake, a wrong deduction, a wrong conclusion or a wrong interpretation of the conclusion may even lead to a financial disaster. In addition, the application of computer technology also has a very prominent position in the research of financial mathematics.
To sum up, financial mathematics is an interdisciplinary subject of finance, mathematics, statistics, economics and computer science, and belongs to the level of applied science. Financial mathematics is also a higher-level quantitative and analytical discipline after the qualitative description stage of finance.
Second, the development of modern financial mathematics theory
1 stochastic optimal control theory
A more important application field of modern financial theory is to solve stochastic problems, and the important means to solve this problem is stochastic optimal control theory. Stochastic optimal control is a late stage in control theory. The application of Berman's optimization principle, measure theory and functional analysis method was an important contribution made by mathematicians to this new mathematical research field in the late 1960s and early 1970s, and financial economists quickly absorbed the theory and method of stochastic optimal control. In the early 1970s, several economic papers appeared, including Merton's discussion on consumption and portfolio with continuous time method, and Brock and millman's discussion on optimal economic growth under uncertain conditions with discrete time method. Since then, stochastic optimal control method has been applied to most financial fields, and young and middle-aged scholars represented by Peng Shige in China have also made outstanding contributions to it.
2 martingale theory
The latest research achievement of modern financial theory is the introduction of martingale theory. Assuming that F is efficient in the financial market, the price of securities can be equivalent to a martingale stochastic process. The martingale method advocated by Karatzas and Shreve directly introduces martingale theory into modern financial theory, and uses the concept of equivalent martingale measure to study the pricing of derivative securities. The results can not only profoundly reveal the operating rules of financial markets, but also provide an effective algorithm for solving the pricing and risk problems of complex derivative financial products. Another advantage of using martingale theory to study financial theory is that it can better solve the pricing problem of derivative securities when the financial market is incomplete, thus making a breakthrough in modern financial theory. At present, the pricing theory of derivative securities based on martingale method is dominant in modern financial theory, but it is still blank in China.