## Financial Mathematics Courses

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### Courses for Financial Mathematics

#### BA5814 - Investment Management

Credit: 3(3-0); ECTS: 8.0

The purpose of this course is to introduce the student to the area of investment with emphasis upon why individuals and institutions invest and how they invest. Topics include measures of risk and return; capital and money markets; process and techniques of investment valuation; principles of fundamental analysis; technical analysis; analysis and management of bonds; analysis of alternative investments; portfolio theory and application.

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#### IAM500 - M.S. Thesis

Credit: 0(0-0); ECTS: 50.0

The Program of research leading to M.S. degree arranged between the student and a faculty member. Students register to this course in all semesters while the research program or write up of thesis is in progress. Student must start registering to this course no later than the second semester of his/her M.S. study.

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Credit: 3(3-0); ECTS: 8.0

Introduction to Derivative and Financial Markets. The Structure of Options Markets. Principles of Option Pricing. Option Pricing Models. Basic Option Strategies. Advanced Option Strategies. The Structure of Forward and Futures Markets. Principles of Spot Pricing. Principles of Forward and Futures Pricing. Futures Hedging Strategies. Advanced Futures Strategies. Options on Futures. Foreign Currency Derivatives. Swaps and Other Interest Rate Agreements.

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Credit: 3(3-0); ECTS: 8.0

Examination of special issues in finance incorporating advanced theory and practice with emphasis on investment and financing decisions of the firm. Special references to applications in Turkey. Outline of Topics: An Overview of Financial Management. Financial Statements, Cash Flow, and Taxes. Analysis of Financial Statements. The Financial Environment: Markets, Institutions, and Interest Rates. Risk and Return. Time Value of Money. Bonds and Their Valuation. Stocks and Their Valuation. The Cost of Capital. The Basics of Capital Budgeting. Cash Flow Estimation and Other Topics in Capital Budgeting. Capital Structure Decisions. Distribution to Shareholders: Dividends and Repurchases. Issuing Securities, Refunding, and Other Topics. Lease Financing. Current Asset Management. Mergers, LBOs, Divestitures, and Holding Companies.

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Credit: 3(3-0); ECTS: 8.0

The objective of this course is an introduction to the probabilistic techniques required for understanding the most widely used financial models. In the last few decades, financial quantitative analysts have used sophisticated mathematical concepts in order to describe the behavior of markets and derive computing methods. The course presents the martingales, the Brownian motion, the rules of stochastic calculus and the stochastic differential equations with their applications to finance. Outline of Topics: Discrete time models, Martingales and arbitrage opportunities, complete markets, European options, option pricing, stopping times, the Snell envelope, American options. Continuous time models: Brownian motion, stochastic integral with respect to the Brownian motion, the Itô Calculus, stochastic differential equations, change of probability, representation of martingales; pricing and hedging in the Black-Scholes model, American options in the Black-Scholes model; option pricing and partial differential equations; interest rate models; asset models with jumps.

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Credit: 3(3-0); ECTS: 8.0

The aim of this course is an initiation to the Itô Calculus (Stochastic Calculus by means of the Brownian motion) which constitutes one of the most important branches of stochastic processes because of their applications and extensions. Outline of Topics: Basic concepts of Probability Theory, stochastic processes, Brownian Motion, conditional expectation, martingales, stochastic integral, representation of martingales, Itô Lemma, change of probability, stochastic differential equations (existence and uniqueness of the solutions, approximations of the solutions), applications.

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Credit: 3(3-0); ECTS: 8.0

The focus of this course is on asset pricing. The topics that will be discussed can be summarized as follows: Individual investment decisions under uncertainty are analyzed and the optimal portfolio theory is discussed using both static and dynamic approach. Then the theory of capital market equilibrium and asset valuation is introduced. In this context several equilibrium models of asset markets are presented. These include the Arrow-Debreu model of complete markets, the Capital Asset Pricing Model (CAPM) and the Arbitrage Pricing Theory (APT). Besides mutual fund separation and aggregation theorems are analyzed. Finally, the financial decisions of firms are considered and the Modigliani-Miller theorems are analyzed.

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Credit: 3(3-0); ECTS: 8.0

Strategic games, Nash equilibrium, Bayesian Games, Mixed, Correlated, Evolutionary equilibrium, Extensive games with perfect information, Bargaining games, Repeated games, Extensive games with imperfect information, Sequential equilibrium, Coalition games, Core, Stable sets, Bargaining sets, Shapley value, Market games, Cooperation under uncertainty.

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Credit: 3(3-0); ECTS: 8.0

This course introduces time series methodology emphasizing the data analytic aspects related to financial applications. Topics that will be discussed are as follows: Univariate linear stochastic models: ARMA and ARIMA models building and forecasting using these models. Univariate non-linear stochastic models: Stochastic variance models, ARCH processes and other non-linear univariate models. Topics in the multivariate modeling of financial time series. Applications of these techniques to finance such as time series modeling of equity returns, trading day effects and volatility estimations will be discussed.

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Credit: 3(3-0); ECTS: 8.0

Review of linear algebra and multivariate calculus, sequences of series and functions, Riemann-Stieltjes and Lebesgue integration.

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Credit: 3(3-0); ECTS: 8.0

The course containing a strong emphasis on applications purports to give students the skills needed in dealing with the control of Markov Chains. The outline of Topics: Discrete-time Markov chains : Ordinary and strong Markov properties, classification of states, stationary probabilities, limit theorems. Continuous-time Markov chains (a survey). Discrete-time Markovian Decision Processes: Various policies, policy-iteration algorithm, linear programming formulation, value-iteration algorithm. Semi-Markov Decision Processes. Applications to inventory problems, to portfolio optimization and to communication systems.

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Credit: 3(3-0); ECTS: 8.0

Part I: Probability spaces, random variables, probability distributions and probability densities, conditional probability, Bayes formula, mathematical expectation, moments. Part II: Sampling distributions, decision theory, estimation (theory and applications), hypothesis testing (theory and applications), regression and correlation, analysis of variance, non-parametric tests.

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Credit: 3(3-0); ECTS: 8.0

Random variables and transformation of random variables. Common families of distributions. Multiple random variables. Properties of random sample and sampling methods. Principles of data reduction. Estimation and hypotheses testing. Asymptotic evaluations. Decision theory. Analysis of Variance. Linear and nonlinear regression.

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Credit: 3(3-0); ECTS: 8.0

Probability spaces. Independence. Conditional probability. Product probability spaces. Random variables and their distributions. Distribution functions. Mathematical expectation (Integration with respect to a probability measure.) Lp-spaces. Moments and generating functions. Conditional expectation. Linear estimation. Gaussian vectors. Various convergence concepts. Central Limit Theorem. Laws of large numbers.

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Credit: 3(3-0); ECTS: 8.0

Mean-Variance (Markowitz) analysis; continuous-time market model in finance; options and exotic options, pricing (valuation) of options; self-financing, optimal strategies, optimal portfolios (problems); martingale method; stochastic control and portfolio optimization.

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Credit: 3(3-0); ECTS: 8.0

Interest Rate Derivatives: Futures, Options on Bonds, and Options on Interest Rates such as Caps and Floors. Models of Arbitrage-Free pricing of Interest-Rate Derivatives: Arbitrage Pricing Theory for Derivative Securities. Basics for The Modeling of Interest-Rate movements. Dynamics of Interest-Rate movements. Short-Rate Models and the Heath-Jarrow-Morton Model of Forward Rates. Change of Numéraire Technique. Derivation of Formulae for the Pricing and Hedging of Certain Derivatives. Numerical Methods for the Actual Implementation of the Valuation of Term Structure Models.

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Credit: 3(3-0); ECTS: 8.0

Introduction to decision making, expected loss, decision rules and risk, decision principles, utility and loss, prior information and subjective probability, Bayesian analysis, posterior distribution, Bayesian inference, Bayesian Decision theory, minimax analysis, value of information, sequential decision procedures, multi decision problems.

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Credit: 3(3-0); ECTS: 8.0

Basic introduction to simulation concepts, generation of random variants from distributions, test for randomness, Monte Carlo Simulation, selecting input distribution, discrete event simulation, variance reduction techniques, statistical analysis of output.

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Credit: 3(3-0); ECTS: 8.0

Brief introduction to Statistical Learning: Regression versus Classification; Linear Regression: simple and multiple Linear Regression; Classification: Logistic Regression, Discriminant Analysis; Resampling Methods: Cross-Validation, the Bootstrap; Regularization: Subset Selection, Ridge Regression, the Lasso, Principle Components and Partial Least Squares Regression; Nonlinear Models: Polynomial; Splines; Generalized Additive Models; Tree-Based Models: Decision Trees, Random Forest, Boosting; Support Vector Machines; Unsupervised Learning: Principle Component Analysis, Clustering Methods.

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Credit: 3(3-0); ECTS: 8.0

Fundamentals of reinsurance including historical development, terminology and distribution systems. Treaty forms, facultative reinsurance, underwriting, rating, accounting and contract issues, analysis of annual statement, testing methods, advanced rating methods in property and casualty excess contracts, analysis accumulations, retention, contract wording and programming.

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Credit: 3(3-0); ECTS: 8.0

Contents vary from year to year according to interest of students and instructor in charge.

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Credit: 3(3-0); ECTS: 8.0

Basic methods based on probability. Distorted probabilities. Decision and utility theories, state dependent utilities, risk taking agent types. Techniques for bounding the effects of missing information or the effects of incorrect information. Trading time and space resources with certainty. Fusing uncertain information of different kinds. Real-time inference algorithms.

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Credit: 2(2-0); ECTS: 6.0

LaTeX and Matlab; Basic Commands and Syntax of LaTeX and Matlab; Working within a Research Group via Subversion; Arrays and Matrices; Scripts and Function in Matlab; Commands and Environments in LaTeX; More on Matlab Functions; Toolboxes of Matlab; Packages in LaTeX; Graphics in Matlab; Handling Graphics and Plotting in LaTeX; Advanced Techniques in Matlab: memory allocation, vectoristaion, object orientation, scoping, structures, strings, file streams.

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Credit: 2(2-0); ECTS: 6.0

Review of Programming and Toolboxes, Packages, Modules; Iterative Linear Algebra Problems; Root Finding Programs; Recursive Functions and Algorithms; Optimisation Algorithms; Data Fitting and Interpolation; Extrapolation; Numerical Integration; Numerical Solutions of Differential Equations: IVPs and BVPs; Selected Topics (algorithms and coding in different fields).

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Credit: 0(0-0); ECTS: 130.0

Program of research leading to Ph.D. degree arranged between the student and a faculty member. Students register to this course in all semesters starting from the beginning of their second semester while the research program or write up of thesis is in progress.

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Credit: 3(3-0); ECTS: 8.0

Numerical methods for discrete time models: Algorithms for option prices, algorithms for discrete time optimal control problems. Reminders on continuous models: Stochastic Calculus, option pricing and partial differential equations, dynamic portfolio optimization. Monte-Carlo methods for options: Convergence results, variance reduction, simulation of stochastic processes, computing the hedge, Monte-Carlo methods for pricing American options. Finite difference methods for option prices: numerical analysis of elliptic and parabolic Kolmogrov equations, computation of European and American option prices in the lognormal model. Finite difference methods for stochastic control problems: Markov Chain approximation method, elliptic Hamilton-Jacobi-Bellman equations, computational methods.

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Credit: 3(3-0); ECTS: 8.0

Lévy processes. Building Lévy processes. Multidimensional models with jumps. Simulation of Lévy processes. Option pricing with jumps: Stochastic calculus for jump processes, measure transformations for Lévy processes, pricing and hedging in complete markets, risk-neutral modeling with exponential Lévy processes. Integro-differential equations and numerical methods. Inverse problems and model calibration.

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Credit: 3(3-0); ECTS: 8.0

Risk in insurance systems. Short, medium and long term financial structures of insurance funds. Modelling and valuation of insurance plans. Contribution and benefit schemes for insurance and reinsurance. Economic dynamics, financial markets and assets management for insurance systems. Multiple decrements, actuarial balance and fair premiums.

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Credit: 3(3-0); ECTS: 8.0

Numerical Methods for Discrete Time Models: binomial method for options; discrete time optimal control problems. Reminders on Continuous Models: Ito process and its applications in stock market, Black-Scholes equation and its solution; Hedging, Volatility smile. Monte Carlo Method for Options: generating random numbers, transformation of random variables and generating normal variates; Monte Carlo integration; pricing by Monte Carlo integration; variance reduction techniques, quasi-random numbers and quasi-Monte Carlo method. Finite Difference Methods for Options: explicit and implicit finite difference schemes, Crank-Nicolson method; Free-Boundary Problems for American options. Finite Difference Methods for Control Problems: Markov Chain approximation method, elliptic Hamiltion-Jacobi-Bellman equations, computational methods.

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Credit: 3(3-0); ECTS: 8.0

Financial modelling beyond Black-Scholes Model. Stochastic processes. Building Lévy processes. Option pricing with stochastic processes: Stochastic calculus for semimartingales, change of measure, exponential Lévy processes, stochastic volatility models, pricing with stochastic volatility models. Hedging in incomplete markets, risk-neutral modeling. Integro-partial differential equations. Further topics in numerical solutions, simulation and calibration of stochastic processes.

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Credit: 3(3-0); ECTS: 8.0

Globalization techniques, semidefinite and conic optimization, derivative free optimization, semi-infinite optimization methods, Newton Krylov methods, nonlinear parameter estimation and advanced spline regression, multi-objective optimization, nonsmooth optimization, optimization in support vector machines.

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Credit: 3(3-0); ECTS: 8.0

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge.

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Credit: 3(3-0); ECTS: 8.0

Important markets such as commodities or credit derivatives are essentially incomplete. The recent financial crisis has increased even more the importance of pricing and hedging in incomplete markets. Therefore these lectures concentrate on advanced methods of stochastic finance required in the context of incomplete markets. We will consider both, process in discrete and continuous time.

The content of the course covers in particular the following topics: market efficiency, market incompleteness; perfect hedges; equivalent martingale measures; attainable payoffs; asset management; contingent claims; replicating portfolio; dynamical arbitrage theory; arbitrage-free pricing; geometric characterization of arbitrage; von Neumann representation; robust Savage representation; expected utility; fair value; certainty equivalent; risk premium; risk aversion; equilibrium pricing; relative entropy; convex risk measures; robust representation; coherent risk measures; VAR; average VAR; upper/lower hedging prices; superhedging duality; risk indifference pricing; HJB equations; dynamical programming.

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Credit: 3(3-0); ECTS: 8.0

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge.

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Credit: 3(3-0); ECTS: 8.0

Basic concepts of probability in connection with Risk Theory; introduction to risk processes (claim number process, claim amount process, total claim number process, total claim amount process, inter-occurance process); convolution and mixed type distributions; risk models (individual and collective risk models); numerical methods (simple methods for discrete distributions, Edgeworth approximation, Esscher approximation, normal power approximation); premium calculation principles; Credibility Theory; retentions and reinsurance; Ruin Theory; ordering of risks.

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Credit: 3(3-0); ECTS: 8.0

Pricing financial instruments, algorithms and programming. Econometric estimation of the stochastic processes.

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Credit: 3(2-2); ECTS: 8.0

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge.

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Credit: 3(3-0); ECTS: 8.0

The course will follow two main textbooks and selected articles from scientific journals. Course will be supported by the presentation and speeches of guest speakers. This course discusses financial risk management from the perspective of energy. Course focuses mostly on the energy market, energy trading and energy risk management through various instruments. The topics also cover the interest of hedge funds. A special attention is also given to behavior of energy-dominant sovereign wealth funds.

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Credit: 3(3-0); ECTS: 8.0

This course is an introduction to the mathematical formulation and treatment of problems arising from trade execution in financial markets. When there are costs and constraints imposed on the execution of trades, how to best execute them? The course studies mathematical formulations and solutions of these types of problems.

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Credit: 3(3-0); ECTS: 8.0

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Credit: 3(3-0); ECTS: 8.0

The goal of this course is to establish a well background in this field. The important subjects in this course are stationarity, autocorrelation, partial autocorrelation, ARIMA models, VAR models, cointegration, difference equation and unit roots. Thus, using these concepts, a student can study and analyze the matters in the area of Financial Econometrics.

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Credit: 3(3-0); ECTS: 8.0

Within the context of value maximization, this course focuses at length on a corporations financial risk management needs and techniques. While we pay particular attention to financial institutions, our coverage is general enough for extending its lessons to other corporate entities, including multinationals.

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Credit: 3(3-0); ECTS: 8.0

Generating Random Numbers; Basic Principles of Monte Carlo; Numerical Schemes for Stochastic Differential Equations; Simulating Financial Models; Jump-Diffusion and Levy Type Models; Simulating Actuarial Models; Markov Chain Monte Carlo Methods.

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Credit: 3(3-0); ECTS: 8.0

Convexity; Gradient Descent; Stochastic Gradient Methods; Noise Reduction Methods; Second-Order Methods; Adaptive Methods; Methods for Regularized Models; Introduction to Machine Learning: support machine vector, neural network.

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Credit: 3(0-0); ECTS: 8.0

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Credit: 3(3-0); ECTS: 8.0

Short review of basic concepts, terminology, definitions in life insurance and the types of life insurance. Product development, pricing strategy, preliminary and final product design, product implementation and management, pricing assumptions and life insurance cash flows, reserves, reinsurance, investment income, profit measurement, financial modeling, asset/liability modeling, and matching, stochastic modeling and financial management.

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Credit: 3(0-0); ECTS: 8.0

Types of pension systems, the nature of interrelationship between social security schemes, their demographic, economic and fiscal environments, the valuation of public pension systems, structural reform considerations, the valuation of short-term cash benefits.

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Credit: 3(3-0); ECTS: 8.0

Forms of survival models, survival distributions, parametric survival models, introduction to demography and life tables, force of mortality, estimation of parametric survival models, actuarial estimation with survival models.

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