In 2 1 century, mathematical technology, like computer technology, has become an indispensable tool in any scientific development. American Citigroup

1On March 6th, 995, Collins, Vice President of the Bank, gave a speech at Newton Institute of Mathematical Sciences, Cambridge University, England.

"/kloc-At the beginning of the 8th century, Bernoulli, a famous mathematician of Newton's contemporaries, declared:' Engaged in physics research.

People who don't know mathematics actually deal with things that are of little significance. At that time, such a statement was harmful to physics.

That's true, but not necessarily for the banking industry. 18th century, you can do well without any math training.

Effectively operate the bank. What used to apply to physics also applies to banking now. So now you can

Say this:' People who work in a bank but don't know math are actually dealing with meaningless things'. "He is also.

Pointing out that 70% of Citibank's business depends on mathematics, he particularly emphasized that' without the tools developed by mathematics,

And technology, we can't do anything about many things ... we can't do without mathematics. "The bank is here.

The family described the importance of mathematics with his experience. After the cold war, thousands of Americans used to work in the military system.

Scientists entered Wall Street, and large fund management companies began to hire doctors in mathematics or physics.

. This is an important signal: the financial market is not a battlefield, but it is far better than the battlefield. But the market and the battlefield are inseparable.

Complex, difficult and fast calculation work.

But in China, we can't avoid that professionals with higher education can study domestic economics.

, the core journal of finance, but it is difficult for domestic financial undergraduates to understand the international core journal Jo.

In the financial field, few securities investment fund managers read A Journey of Portfolio Management.

"Ment", the reason lies not in the proficiency of foreign languages, but in the differences in content and research methods.

Most of them stay in descriptive analysis, focusing on describing the definition of finance, market division and financial organization, or description.

Describe finance; However, foreign academic and practical circles mainly focus on quantitative analysis, such as the pricing principles of capital assets and derivatives.

The method of copying assets, or analytical finance, even in the textbooks of domestic finance, involves the target.

Assets (basic assets) and derivative assets (derivative assets) are priced, but the original formula puts forward

This phenomenon is unreasonable for the following reasons: First,

According to the different research methods, China's financial discipline can be divided into China Philosophy and Social Science Planning Office or

In order to return to the Management Science Department of the National Natural Science Foundation of China, the former occupies the main position, and most of this team comes from classics.

Economic transition precedes philosophy and political science, so the research methods are mostly qualitative. On the contrary, in the west, finance

The research team has a good foundation in mathematics and physics. Secondly, it is determined by the actual environment of China's financial market. our country

The securities market has just started and there is no unified money market. The investor team is mainly composed of small and medium investors.

If speculation is high, there will be no demand for modern investment theory, and correspondingly, it is difficult for academic circles to produce it.

Passion for research.

However, with its precise description and strict deduction, mathematical technology has entered the financial field without dispute. Starting from 1952

In this paper, Markowitz proposed to describe the profitability of financial assets with the characteristic variables of random variables, which is inaccurate.

Because of the qualitative and liquidity, it is difficult to distinguish whether a world-class financial magazine is analyzing the financial market or writing one.

A math paper. Going back to Collins' speech, do we use statistics in the trend of financial securitization?

Methods Which is the best, analyzing historical data to find the law of price fluctuation or copying financial products through mathematical analysis?

Whoever discovers the internal law first can get high profits in the ever-changing financial market. Although due to the strict entrance

Entering the fortress, mathematics has been excluded and ignored in the financial field. However, in pursuit of profit, there are unknown fears.

It looks fragile.

Therefore, in the future, we can imagine such an industrial chain full of bright prospects: financial market-financial mathematics-

Computer technology. There are huge profits and high risks in the financial market, which need computer technology to help analyze, but the calculation

A machine can't be a descriptive language, such as approximate, left and right. It can only recognize the space consisting of 0 and 1 in essence. Financial mathematics is

In this process, it only plays an intermediary role, and can describe the randomly fluctuating market with accurate language. For example,

The risk-free discount factor is found through the yield state matrix, and there is no arbitrage. Therefore, financial mathematics can be helpful.

The IT industry extends to the financial industry and gains its own profit space.

Financial mathematics, also known as mathematical finance, mathematical finance and analytical finance, uses mathematical tools to study finance, carries out quantitative analysis such as mathematical modeling, theoretical analysis and numerical calculation, so as to discover the internal laws of finance and guide practice. Financial mathematics can also be understood as the application of modern mathematics and computing technology in the financial field. Therefore, financial mathematics is a new interdisciplinary subject, which develops rapidly and is one of the most active frontier disciplines at present.

Financial mathematics is a new discipline and an important part of "financial high technology". Learning financial mathematics is of great significance. The overall research goal of financial mathematics is to make use of some advantages in China's mathematics field, deeply analyze the mathematical theory of financial market equilibrium and securities pricing, establish a mathematical model suitable for China's national conditions, write certain computer software, simulate the theoretical research results, conduct econometric analysis and research on actual data, and provide in-depth technical analysis and consultation for the actual financial sector.

The main research contents and problems to be solved in financial mathematics include:

(1) securities and portfolio pricing theory

Develop the pricing theory of securities (especially derivatives such as futures and options). The mathematical method used is mainly to put forward a suitable stochastic differential equation or stochastic difference equation model to form the corresponding backward equation. The corresponding nonlinear Feynman-Kac formula is established, from which the very general extended Black-Scho 1es pricing formula is derived. The backward equation will be a high-dimensional nonlinear singular equation with constraints.

This paper studies the pricing of portfolio with different maturities and yields. It is necessary to establish a mathematical model combining pricing and optimization. In the study of mathematical tools, it may be necessary to study stochastic programming, fuzzy programming and optimization algorithms.

Under the condition of incomplete market, the pricing theory related to preference is introduced.

(2) Incomplete market economy equilibrium theory (GEI)

It is planned to conduct research in the following aspects:

1. Infinite dimensional space, infinite horizontal space and infinite state.

2. Stochastic economy, no arbitrage equilibrium, change of economic structure parameters, nonlinear asset structure.

3. Innovation and design of asset securitization.

4. Friction economy

5. Corporate Behavior and Production, Bankruptcy and Bad Debt

6. Game in the securities market.

(3) The application of GEI's plate equilibrium algorithm and Monte Carlo method in the calculation of economic equilibrium point, the application of GEI theory in macro-control of finance, finance and economy, and the study of natural resource asset pricing and sustainable utilization under the framework of the theory of sustainable development under incomplete market conditions.

At present, there are Peking University, Fudan University, Zhejiang University, Shandong University, Nankai University, etc.

Later, I was very good at computer work. Financial Mathematics will be engaged in the research and analysis of banking, insurance, stocks, futures and other fields in the future, or do software development in these fields, with a good professional background, and these fields will be very important in the future.

There are few financial mathematics talents in China.

The Nobel Prize in Economics has been awarded at least three times to economists who use mathematics as a tool to analyze financial problems. Professor Wang Duo from the Department of Financial Mathematics in Peking University said, but unfortunately, the training of relevant talents in China has just started. Nowadays, compound talents who know both finance and mathematics are quite scarce.

Financial mathematics, a new interdisciplinary subject, has become a wonderful flower in the international financial community. The 2003 Nobel Prize in Economics, just announced, is to commend two new methods of analyzing economic time series by American economist Robert Engel and British economist clive granger, namely, "variability with time" and "* * * same trend", which have brought great influence to economic research and economic development.

Wang Duo introduced that the development of financial mathematics had twice triggered the "Wall Street Revolution". In the early 1950s, Markowitz put forward the portfolio theory of securities, and for the first time, he clearly gave out the investment methods of investing in various securities in different proportions at a certain risk level, thus obtaining the maximum possible return, which triggered the first "Wall Street Revolution". 1973, Blake and Scholes gave the option pricing formula by mathematical method, which promoted the development of option trading, and option trading soon became the main content of the world financial market and became the second "Wall Street Revolution".

Nowadays, financial mathematicians are one of the most sought-after talents on Wall Street. In the simplest example, the chief actuary may have the highest status and income among insurance companies. Paul kosslyn, vice chairman of Citibank in the United States, famously said, "A person who is engaged in banking business and does not know mathematics can only do trivial things."

In the United States, famous universities such as the University of Chicago, the University of California, Berkeley, Stanford University, Carnegie Mellon University, new york University, etc. have all set up degree or professional certificate education related to financial mathematics.

Experts believe that the possible development of financial mathematics should be highlighted in Asia, especially in China, where the financial market is developing and has great potential. The Chinese University of Hong Kong, Hong Kong University of Science and Technology, City University of Science and Technology and other schools have launched relevant training courses and training plans, which have received enthusiastic response from banks and the financial industry. However, there are some difficulties in training such talents in Chinese mainland.

Wang Duo introduced that the research content of financial engineering was included in the Ninth Five-Year Plan by the National Natural Science Foundation, which can be said to have started the domestic financial mathematics research in an all-round way. But this is nearly half a century later than markowitz's research and application of financial mathematics.

Under the background that financial derivatives have become an important role in the international financial market, China's financial derivatives have just started, and financial derivatives are almost blank. "After joining W TO, international financiers will definitely bring this series of businesses to China. If there are no corresponding products and talents, how can we compete? " Wang Duo said anxiously.

He believes that the Mexican financial crisis in recent years and the collapse of Bahrain Bank in the past century have warned us that if we do not master modern financial technologies such as financial mathematics, financial engineering and financial management, and lack talents, we may suffer great losses in international financial competition. What we lack most now is senior compound talents who can master modern financial derivatives, quantitatively analyze financial risks and understand both finance and mathematics.

It is reported that at present, many colleges and universities in China have successively carried out teaching related to financial mathematics, but the graduates are far from meeting the needs of the whole market.

Wang Duo believes that there are still some insurmountable obstacles in cultivating such talents-financial mathematics will eventually be applied to practice, but at present, financial derivatives are not a climate, so it is difficult for students to have the opportunity to practice, and teaching and learning remain on paper. In addition, most of the colleges and universities train undergraduates and only a few graduate students, and high-end talents in this field are still rare in China. The state should pay more attention to the cultivation of compound talents combining finance and mathematics.

Wang Duo recalled that 1997, when Peking University established the first financial mathematics department in China, it wanted to run a school with some financiers. But quite a few people are obviously not interested in this: "what financial derivatives and financial mathematics are all things that the country should worry about."

Although some people thought it was too advanced when the Department of Financial Mathematics was set up, Wang Duo insisted that education should be ahead of industrial development and reserve talents for the market. If we don't pay attention to the cultivation of talents in related fields today, we may be at a disadvantage in international competition.

The reporter found that even today, on this issue, on the one hand, college teachers are worried about the scarcity of talents, on the other hand, some famous experts are indifferent to the cultivation of financial mathematics talents.

During the interview, the reporter repeatedly tried to contact several experts in financial mathematics or financial theory in China, but they were rejected many times. There is a simple reason. They think it's naive to talk about talent training, and some even say, "I don't know anything about talent training." Others said, "I have a lot of topics to do now, which is more important, and my topic is to discuss talent training." "I have no time and obligation to explain the Nobel Prize in Economics to the public, and whether people understand financial mathematics has nothing to do with me."

[Edit this paragraph] Data Mining in Finance

1. What is an association rule?

Before describing some details about association rules, let's look at an interesting story: "Diapers and Beer".

In a supermarket, there is an interesting phenomenon: diapers and beer are sold together. But this strange move has increased the sales of diapers and beer. This is not a joke, but a real case of Wal-Mart supermarket chain in the United States, which has always been talked about by merchants. Wal-Mart has the largest data warehouse system in the world. In order to accurately understand customers' buying habits in their stores, Wal-Mart analyzes customers' shopping behavior and wants to know what products customers often buy together. Wal-Mart's data warehouse centralizes the detailed raw transaction data of its stores. On the basis of these original transaction data, Wal-Mart uses data mining methods to analyze and mine these data. An unexpected discovery is: "The product most bought with diapers is beer!" After a lot of practical investigation and analysis, it reveals an American behavior pattern hidden behind "diapers and beer": in the United States, some young fathers often go to the supermarket to buy baby diapers after work, and 30% ~ 40% of them will buy some beer for themselves. The reason for this phenomenon is that American wives often tell their husbands to buy diapers for their children after work, and husbands will bring back their favorite beer after buying diapers.

According to conventional thinking, diapers have nothing to do with beer. If we don't use data mining technology to mine and analyze a large number of transaction data, it is impossible for Wal-Mart to discover this valuable law inside the data.

Data association is an important discovery knowledge in database. If there is some regularity between the values of two or more variables, it is called correlation. Correlation can be divided into simple correlation, time series correlation and causal correlation. The purpose of association analysis is to find out the hidden association network in the database. Sometimes we don't know the correlation function of the data in the database, and even if we do, it is uncertain, so the rules generated by correlation analysis are credible. Association rule mining finds interesting associations or related relationships between itemsets in a large number of data. Agrawal is equal to 1993. Firstly, the problem of mining association rules between itemsets in customer transaction database is proposed. Later, many researchers did a lot of research on mining association rules. Their work includes optimizing the original algorithm, such as introducing random sampling and parallel thinking to improve the efficiency of algorithm mining rules; Popularize the application of association rules. Mining association rules is an important topic in data mining, which has been widely studied by the industry in recent years.

2. Mining process, classification and related algorithms of association rules.

2. 1 association rule mining process

The process of mining association rules mainly includes two stages: the first stage, all high-frequency itemsets must be found from the data set, and the second stage, association rules are generated from these high-frequency itemsets.

In the first stage of association rule mining, all large itemsets must be found from the original data set. High frequency means that the frequency of a project group relative to all records must reach a certain level. The frequency with which teams appear is called support. Taking a 2- itemset containing two items A and B as an example, the support degree of the item group containing {A, B} can be obtained by the formula (1). If the support is greater than or equal to the set minimum support threshold, {A, B} is called high-frequency project group. K- itemsets that meet the minimum support are called frequent k- itemsets, which are generally expressed as big K or frequent K. The algorithm also generates big k+ 1 from the big K itemsets until no high-frequency itemsets can be found.

The second stage of association rule mining is to generate association rules. Generating association rules from high-frequency itemsets is to generate rules by using the high-frequency K itemsets in the previous step. Under the conditional threshold of minimum confidence, if the credibility obtained by a rule meets the minimum confidence, the rule is called association rule. For example, the reliability of rule AB generated by high-frequency k-item group {A, B} can be obtained by formula (2). If the reliability is greater than or equal to the minimum reliability, AB is called an association rule.

As far as the case of Vuormaa is concerned, using association rule mining technology to mine records in the transaction database, two thresholds of minimum support and minimum trust should be set at first, assuming that minimum support min_support=5% and minimum trust min_confidence=70%. Therefore, the association rules that meet the needs of this supermarket must meet the above two conditions at the same time. If the association rule "diaper, beer" found through the mining process meets the following conditions, the association rule of "diaper, beer" will be accepted. Support (diapers, beer) can be described by the formula >: =5%, confidence (diapers, beer) > =70%. Among them, support (diapers, beer) >: the significance of =5% in this application example is that at least 5% of all transaction records show that diapers and beer were purchased at the same time. In this application example, confidence (diapers, beer) > =70% means that at least 70% of all transaction records including diapers will buy beer at the same time. Therefore, if the consumer buys diapers in the future, the supermarket will be able to recommend the consumer to buy beer at the same time. This product recommendation behavior is based on the association rule of "diapers, beer", because the past transaction records of supermarkets support the consumption behavior that "most transactions that buy diapers will buy beer at the same time".

As can be seen from the above introduction, association rule mining is usually more suitable for the case that the indicators in the record take discrete values. If the index values in the original database are continuous data, the data should be discretized properly before mining association rules (in fact, the values in a certain interval correspond to a certain value). Data discretization is an important step before data mining, and whether the discretization process is reasonable will directly affect the mining results of association rules.

2.2 Classification of Association Rules

According to different situations, association rules can be classified as follows:

1. Association rules can be divided into Boolean and numerical types according to the types of variables handled in the rules.

The values processed by Boolean association rules are all discrete and classified, which shows the relationship between these variables. Numerical association rules can be combined with multi-dimensional association rules or multi-layer association rules to deal with numerical fields and dynamically divide them, or they can directly deal with original data. Of course, numeric association rules can also contain category variables. For example: gender = "female" => occupation = "secretary", which is a Boolean association rule; Gender = "female" => avg (income) = 2300, and the income involved is numerical, so it is a numerical association rule.

2. According to the abstract level of data in rules, it can be divided into single-layer association rules and multi-layer association rules.

In the single-level association rule, all variables do not take into account that there are many different levels of actual data; In multi-level association rules, the multi-level nature of data has been fully considered. For example, IBM desktop => Sony printer is a single-layer association rule for detailed data; Desktop => Sony printer is a multi-layer association rule between higher level and detail level.

3. According to the dimensions of data involved in rules, association rules can be divided into one-dimensional and multi-dimensional.

In one-dimensional association rules, we only involve one dimension of data, such as items purchased by users; In multidimensional association rules, the data to be processed will involve multiple dimensions. In other words, one-dimensional association rules deal with some relationships in a single attribute; Multidimensional association rules deal with some relationships among various attributes. For example: beer => diapers, this rule only involves the items purchased by users; Gender = "female" => occupation = "secretary". This rule involves information in two fields and is a two-dimensional association rule.

2.3 Association rule mining algorithm

1.Apriori algorithm: frequent itemsets are discovered by using candidate itemsets.

Apriori algorithm is the most influential algorithm for mining frequent itemsets of Boolean association rules. Its core is a recursive algorithm based on the idea of two-stage frequency set. This association rule belongs to single-dimensional, single-layer and Boolean association rules in classification. Here, all itemsets with support greater than the minimum support are called frequent itemsets, which is called frequency sets for short.

The basic idea of the algorithm is: first, find out all the frequency sets, and the frequencies of these itemsets are at least the same as the predefined minimum support. Then, strong association rules are generated from the frequency set, and these rules must meet the minimum support and minimum credibility. Then, the expected rules are generated by using the frequency set found in step 1, and all rules containing only set items are generated, in which there is only one item in the right half of each rule, and the definition of intermediate rules is adopted here. Once these rules are generated, only those rules that are greater than the minimum credibility given by the user are left behind. In order to generate all frequency sets, a recursive method is used.

There may be a large number of candidate sets, and it may be necessary to scan the database repeatedly, which are two major shortcomings of Apriori algorithm.

2. Algorithm based on partition

Savasere et al. designed an algorithm based on partition. Firstly, the algorithm logically divides the database into several disjoint blocks, considers one block at a time and generates all frequency sets for it, then generates all possible frequency sets, and finally calculates the support of these item sets. Here, the size of each block is selected so that each block can be put into the main memory and only needs to be scanned once in each stage. The fact that every possible frequency set is a frequency set in at least one block ensures the correctness of the algorithm. The algorithm can be highly parallel, and each block can be assigned to a processor to generate a frequency set. After each cycle of generating the frequency set, the processors communicate with each other to generate a global candidate set of k items. Usually the communication process here is the main bottleneck of algorithm execution time; On the other hand, the time for each independent processor to generate frequency sets is also a bottleneck.

3.FP- tree frequency set algorithm

Aiming at the inherent defects of Apriori algorithm, J. Han and others proposed a method of not generating candidate frequent itemsets: FP- tree frequency set algorithm. Take the strategy of divide and conquer. After the first scan, the frequency set in the database is compressed into a frequent pattern tree (FP-tree), and the relevant information is still preserved. Then the FP-tree is divided into several conditional bases, each of which is related to a frequency set with a length of 1, and then these conditional bases are mined separately. When the amount of original data is large, FP-tree can be put into main memory in combination with partition method. Experiments show that FP-growth has good adaptability to rules with different lengths, and its efficiency is much higher than that of Apriori algorithm.

3. Application in this field at home and abroad

3. 1 Application of Association Rules Mining Technology at Home and Abroad

At present, association rule mining technology has been widely used in western financial enterprises, and can successfully predict the needs of bank customers. Once this information is obtained, banks can improve their marketing. Now, banks are developing new ways to communicate with customers every day. Banks bind the information of their products that customers may be interested in to their ATM machines for users to understand. If the database shows that a customer with a high credit limit has changed his address, it is likely that this customer has recently bought a bigger house, so he may need a higher credit limit, a new high-end credit card, or a home improvement loan. These products can be mailed to customers through credit card bills. When customers call for advice, the database can effectively help telemarketing representatives. The computer screen of the sales representative can show the characteristics of the customer and what products the customer will be interested in.

At the same time, some well-known e-commerce websites also benefit from powerful association rules mining. These e-shopping websites use the rules in association rules to mine, and then set the bundles that users intend to buy together. There are also some shopping websites that use them to set up corresponding cross-selling, that is, customers who buy one product will see advertisements for another related product.

However, at present, in China, "massive data and lack of information" is a common embarrassment faced by commercial banks after data concentration. At present, most databases implemented in the financial industry can only realize the bottom-level functions such as data entry, query and statistics, but can't find all kinds of useful information in the data, such as analyzing these data, discovering their data patterns and characteristics, and then discovering the financial and commercial interests of a customer, consumer group or organization, and observing the changing trend of the financial market. It can be said that the research and application of association rule mining technology in China is not very extensive and deep.

3.2 Some research on association rule mining technology in recent years

Because many application problems are often more complicated than supermarket purchasing problems, a lot of research has expanded association rules from different angles, and integrated more factors into association rule mining methods, thus enriching the application fields of association rules and broadening the scope of supporting management decisions. For example, consider the hierarchical relationship between attributes, temporal relationship, multi-table mining and so on. In recent years, the research on association rules mainly focuses on two aspects, namely, expanding the scope that classical association rules can solve problems and improving the efficiency and interest of classical association rules mining algorithms.

Definition of financial engineering

There are many definitions of financial engineering, and jon John Finnerty, an American financial economist, puts forward the best definition: financial engineering includes the design, development and implementation of innovative financial instruments and financial means, as well as creative solutions to financial problems.

Financial engineering has two concepts: narrow sense and broad sense. Narrow financial engineering mainly refers to the use of advanced mathematics and communication tools, on the basis of existing basic financial products, to carry out different forms of combination decomposition, in order to design new financial products that meet customer needs and have specific profit and loss characteristics. Financial engineering in a broad sense refers to all technological developments that use engineering means to solve financial problems. It includes not only financial product design, but also financial product pricing, trading strategy design, financial risk management and other aspects. This paper adopts the broad concept of financial engineering.

[Edit this paragraph] The core content of financial engineering

In financial engineering, its core lies in developing and designing new financial products or businesses, and its essence lies in improving efficiency, including:

1. Create new financial instruments, such as creating the first zero coupon bond and the first swap contract;

2. Development and application of existing tools, such as applying futures trading to new fields and developing a large number of options and swaps;

3. Combine the existing financial tools and means with combinatorial decomposition technology to compound new financial products, such as forward swaps and futures options, and build a new financial structure.

[Edit this paragraph] Operating procedures of financial engineering

The operation of financial engineering has standardized procedures: diagnosis-analysis-development-pricing-delivery, and the basic process is programmed.

Among them, from the feasibility analysis of the project, the determination of the product performance target, the optimal design of the scheme, the product development, the determination of the pricing model, the simulation test of the simulation, the application and feedback correction in small batches, to the sales and promotion in large batches, all links are closely and orderly. Most innovative new financial products have become a tool to creatively solve other related financial problems by using financial engineering, that is, the basic unit of combined products.

Actuarial science

Actuarial science has a history of 300 years in the west. It is a subject that uses mathematical theories such as probability theory and various financial tools to study the quantitative methods and techniques of how the insurance industry and other financial industries deal with various risks. It is the theoretical basis for the development of modern insurance, financial investment and social security.

Actuarial science is a science that uses the theory of probability mathematics and various financial tools to analyze and predict economic activities. In western developed countries, actuarial science plays an important role in insurance, investment, financial supervision, social security and other fields related to risk management. Actuaries deal with "future uncertainty", and their purpose is to provide basis for financial decision-making.

actuaries