This course is an introduction to general equilibrium theory through the study of the Arrow –Debreu model.

Commodity space and price space- Preference relations and utility functions-Properties of preference relations and the equivalent properties of utility functions-

Budget sets and the consumer’s problem- Demand functions and demand correspondences- Exchange economies and the excess demand function- Equilibrium prices- Allocations of resources in exchange economies- Pareto optimal and weakly  Pareto optimal allocations- Core allocations and the Edgeworth box- First and Second Welfare theorem in exchange economies- Walras equilibrium and quasi –equilibium allocations – Taxes and the Second Welfare theorem- Proof of the existence of equilibrium in exchange economies.

Students who have successfully completed the course will have:

1.  understood in deep the essential notions of microeconomic theory through a broad mathematical presentation

2.  been introduced to essential principles of the contemporary economic science, such as the study of  the rational behavior of the economic units and the notions of the game and equilibrium

3.  received the required background required for the study of more specified mathematical models which are related to the interpretation of the individuals’ selections in investment and insurance market.

1.  C.D. Aliprantis, D.J. Brown, O. Burkinshaw, ‘Existence and Optimality of Competitive Equilibria’, Springer -Verlag (1990)

2.  A. Mas-Colell, M.D. Whinston, R.J. Green,  ‘Microeconomic Theory’, Oxford University Press (1995)

3.  W. Nicholson, ‘Microeconomic Theory: Basic Principles and Extensions, Parts A,B’ , Kritiki Scientific Library Publications (1998)

4.  I. A. Polyrakis, ‘Topics in Mathematical Analysis and Theory of General Equilibrium in Economics' Athens (2010)

 Stochastic Integration- Elementary stochastic processes and their Ito integral- The class of Ito integrable stochastic processes-Properties of the Ito integral (linearity, martingale property)- Multi-dimensional Ito integral- Ito processes- Ito’s Lemma for one-dimensional and multi-dimensional Ito processes- Calculation of Ito integrals by using Ito’s Lemma.

Financial markets and normalized financial markets- Portfolios and the value process of a portfolio process- Self-financing and admissible portfolios – The ‘numeraire invariance theorem’. The geometric Brownian motion and the Black –Scholes model The Ornstein –Uhlenbeck process.

Arbitrage in continuous –time finance models- Relation between arbitrage and general equilibrium in exchange economies with financial markets – Arbitrage and martingales- The Girsanov theorem and determination of equivalent martingale measures for a financial market- Equivalent measures and the Radon – Nikodym theorem- Expectation and conditional expectation under a change of measure- Girsanov’s theorem and the Black –Scholes model.

Elements of stochastic differential equations’ theory – Strong solutions and uniqueness – The Ito’s theorem- Solving one-dimensional linear and non-linear stochastic differential equations- Complete and incomplete financial markets –Characterization of completeness’ theorem and Harrison –Pliska’s theorem of completeness.

Arbitrage pricing of contingent claims in complete and incomplete markets – Perfect hedging of contingent claim from the side of the seller and from the side of the buyer in incomplete markets- Value process of a European contingent claim –Examples of arbitrage pricing of contingent claims in Black-Scholes model and in other models of markets.

The Markov property of the solutions of stochastic differential equations – Feynman-Kac formula and the probabilistic approach in partial differential equations’ initial value problems. The value function of a derivative claim as a solution of an initial value problem- The Black-Scholes’ partial differential equation- Determination of the portfolio which replicates a European contingent claim in complete markets in relation to the Feynman –Kac formuala- The Delta in Black-Scholes model.

Probabilities I

Probabilities II

Real  Analysis

Stochastic Processes II

Students who have successfully completed the course will have:

1.  been familiar with the essential notions of financial mathematics in continuous time, which are nowadays used in financial markets’ modeling.

2.  got a strong background needed for studying more advanced methods of financial markets’ modeling and contemporary methods of pricing of contingent claims in incomplete markets.

3.  got a strong mathematical background needed for dealing with practical aspects of markets’ modeling and pricing of contingent claims through arithmetic methods, methods of simulation, etc.

1) Α.Ν. Yannacopoulos, ‘Stochastic Analysis and Applications in Finance, Part I: Introduction to Stochastic Analysis’ (2003)

2) Α.Ν. Yannacopoulos,  ‘Stochastic Analysis and Applications in Finance, Part II: Applications in Finance’ (2004)

3) Ι. Spiliotis, ‘Stochastic Differential Equations with Applications in Finance’, Symeon Publications (2004)

4) P.C. Vasileiou, ‘Stochastic Finance’, Ziti Publications (2001)

5) B. Oksendal, ‘Stochastic Differential Equations’, Springer (2000)

1)    Uncertainty and information- Unfolding of information and information partitions of the state space- Event-tree and stochastic exchange economies- Stochastic exchange economy commodity space- Financial contracts in stochastic economies and sorts of them- Vectors of prices and payoff vectors of financial contracts, financial markets in the event-tree model- Positions of investors towards the financial contracts- Portfolios, payoff matrix and payoff subspace of a financial market- Self-financing portfolios, complete and incomplete markets, contingent claims and hedging portfolios for contingent claims, hedging portfolios and completeness of a market- Completion of a market with options and the theorem of Ross- Budget sets in stochastic exchange economies- The investor’s optimization problem in stochastic exchange economies- Arbitrage and its relation with the investor’s optimization problem- Characterization of the absence of arbitrage in financial markets and arbitrage pricing of financial contracts- Relation between completeness of the market and arbitrage pricing-Derivatives of European  type and their relation to the general form of  contingent claims in the event-tree model.

2)     Algebras of sets and partitions in finite probability spaces- Conditional expectation and finite-time  martingale process in finite probability spaces- Filtrations and event-trees- Stochastic processes being adapted on an event-tee- Equivalent martingale measures of probability and absence of arbitrage in the event-tree model of financial markets –Completeness of a market and equivalent martingale measures.

Applied Linear Algebra I

Applied Linear Algebra II

Calculus I

Calculus III

Probabilities II

Students who have successfully completed the course will have:

1.  been familiar with the essential notions of Financial Mathematics at a first level of generality.

2.  understood the relationship between the martingale stochastic processes and no-arbitrage pricing of contingent claims.

3.  received the required background in order to attend more advanced courses on Financial Mathematics

1) Μ. Magill, M. Quinzii, ‘Theory of Incomplete Markets : Volume I’, MIT Press (1996)

2) S. LeRoy , J. Werner, ‘Introduction to Financial Economics’,  Cambridge University Press (2001)

Semester B Course

(2  hours Theory)

2 Teaching Points

Calculus I

Calculus III

Real Analysis

(in the way that most of greek universities’ students of mathematical direction have been taught)

Students who have successfully completed the course will have:

1) been familiar with the mathematical aspects of the coherent and convex risk measures’ theory, which was developed during the past ten years as an attempt of finding new methods for measuring financial risk that overcome the disadvantages of older such methods, just like of the ones implied by the method of Value at Risk.

(for the second part of the course)

1. P. Artzner, F. Delbaen, J.M. Eber, D. Heath, ‘Coherent measures of risk’, Mathematical Finance 9, 203-228.

2. H. Föllmer, A. Schied, "Convex measures of risk and trading constraints", Finance and Stochastics 6 (2002), 429-447.