Mathematics

What is a reasonable value for a derivative on the financial market? The course consists of two related parts. The first part looks at option theory in discrete time. The purpose is to introduce fundamental concepts of financial markets such as free of arbitrage and completeness as well as martingales and martingale measures. Tree structures to model time dynamics of stock prices and information flows are used. The second part studies models formulated in continuous time. The models used are formulated as stochastic differential equations (SDE:s). The theories behind Brownian motion, stochastic integrals, Ito-'s formula, measures changes, and numeraires are presented and applied to option theory both for the stock and the interest rate markets. Students derive e.g. the Black-Scholes formula and how to create a replicating portfolio for a derivative contract.

Linear algebra is the branch of mathematics that is primarily concerned with problems involving linearity of one kind or another. This is reflected by the three main themes around which this introductory course is centered. The first theme concerns how to solve a system of linear equations. For this problem, a complete solution procedure is developed which provides a way to deal with such problems systematically, regardless of the number of equations or the number of unknowns. The second theme addresses linear functions and mappings, which can be studied naturally from a geometric point of view. This involves geometric ‘primitives’ such as points, lines, and planes, and geometric ‘actions’ such as rotation, reflection, projection, and translation. One of the main tools of linear algebra is offered by matrices and vectors, for which a basic theory of matrix-vector computation is developed. This allows one to bring these two themes together in a common, exceptionally fruitful, framework. By introducing the notions of vector spaces, inner products, and orthogonality, a deeper understanding of the scope of these techniques is developed, opening up a large array of rather diverse application areas. The third theme shifts from the geometric point of view to the dynamic perspective, where the focus is on the effects of iteration (i.e., the repeated application of a linear mapping). This involves a basic theory of eigenvalues and eigenvectors. Examples and exercises are provided to clarify the issues and to develop practical computational skills. They also serve to demonstrate practical applications where the results of this course can be successfully employed. Prerequisites include Basic Mathematical Tools or substantial high school experience in Mathematics. 

This course gives an overview of quantitative finance and introduces mathematical concepts and data analytic tools used in finance. The topics include interest rate mathematics, bonds, mean-variance portfolio theory, risk diversification and hedging, forwards, futures and options, hedging strategies using futures, and trading strategies involving options.

This course covers the following topics: sets and mappings, complete induction; number representations, real numbers, complex numbers; number sequences, convergence, infinite series, power series, limits and continuity of functions; elementary rational and transcendental functions; differentiation, extreme values, mean value theorem and consequences; higher derivatives, Taylor polynomial and series; applications of differentiation; definite and indefinite integral, integration of rational and complex functions, improper integrals, Fourier series; matrices, linear systems of equations, Gauss algorithm; vectors and vector spaces; linear mappings; dimension and linear independence; matrix algebra; vector geometry; determinants, eigenvalues; linear differential equations.
 

Topics in this Mathematical Methods I course include: differential equations; systems of ordinary differential equations; functions of complex variables. 

This course is an introduction to finance. It starts by introducing the value of money, interest rates, and financial contracts, in particular, what are fair prices for contracts and why no one uses fair prices in real life. Then, there is a review of probability theory followed by an introduction to financial markets in discrete time. In discrete time, students learn how the ideas of fair pricing apply to price contracts commonly found in stock exchanges. The next block focuses on continuous time finance and contains an introduction to the basic ideas of Stochastic calculus.

This course provides individual research training for students in the Junior Year Engineering Program through the experience of belonging to a specific laboratory at Tohoku University. Students are assigned to a laboratory with the consent of the faculty member in charge. They participate in various group activities, including seminars, for the purposes of training in research methods and developing teamwork skills. The specific topic studied depends on the instructor in charge of the laboratory to which each student is assigned. The methods of assessment vary with the student's project and laboratory instructor. Students submit an abstract concerning the results of their individual research each semester and present the results near the end of this program.

This course examines core skills in the manipulation, statistical analysis, and communication of data. Using examples from the biological, earth, and environmental sciences and using the R programming language, students will examine the role of statistics in addressing scientific questions with different goals, including determining causes, describing variation, and predicting outcomes. 

This course offers an introduction to cryptography. Topics include: mathematical foundations of cryptography; classic cryptography; symmetric encryption; key distribution and asymmetric encryption; hash functions, MAC, and authenticated encryption; digital signatures schemes; public key infrastructure; user authentication.