## PhD Qualifying Exam in Financial Mathematics

##### Last Updated:

14/01/2021 - 17:38

## Rules and Regulations

- The exam has two parts: written and oral exams in English.
- The written exam consists of two parts:
- Part I. Financial Economics,
- Part II. Stochastic Processes and Probability with Applications to Finance.

Each part is over 100 points and the overall written exam grade is equal to the average of the points earned from each part.

- To pass the written exam, the student must receive at least 65 points from the overall written exam.
- The student takes the oral examination, if he/she passes the written exam.
- The oral examination is over 100. The grade of the oral examination is determined by the average of the grades of each jury member who gives a grade between 0 and 100.
- Overall grade is calculated by taking 70% of the written exam and 30% of the oral exam grade.

(overall grade= 0.7*written exam grade + 0.3*oral exam grade.) - The passing grade for the Ph.D. qualification exam is 70 over 100. A student obtaining a passing grade is declared to have passed the PhD Qualification Exam.
- Although the student might have passed the PhD Qualification Exam and finished his/her courses for the programme, the committee members can suggest the student to take a course (or courses) in addition to his/her studies.

## Content of the Examination

Here is the list of topics covered in the PhD Qualifying Exam.

### Part I. Financial Economics

#### Financial Economics

- Preference and choice
- Choice under uncertainty
- Portfolio theory, capital asset pricing model
- Arbitrage asset pricing theory
- General equilibrium under uncertainty

##### Suggested References

- Mas-Colell, A., Whinston, M. and Green, J. (1995): Microeconomics Theory, Oxford University Press.
- Huang, C. F. and Litzenberger, R. H. (1988): Foundations of Financial Economics, North Holland.
- Lengwiler Y. (2004), Microfundations of Financial Economics, Princeton University Press.
- LeRoy, S. F. and Werner, J. (2001): Principles of Financial Economics, Cambridge University Press.

### Part II. Stochastic Processes and Probability with Applications to Finance

#### Probability Theory

- Random variables, vectors and their distributions.
- Mathematical expectation (Integration with respect to a probability measure).
- Lp-spaces.
- Convergence (a.s. convergence, convergence in probability, etc.)
- Conditional expectation.
- Linear estimation. Gaussian vectors.

#### Stochastic Calculus

- Discrete-time models
- Martingales and arbitrage opportunities
- Complete markets and option pricing
- Optimal stopping problem and American options
- Stopping time, the Snell envelope
- Decomposition of supermartingales, submartingales and application to the American option
- Continuous-time processes and stochastic differential equations
- Brownian motion
- Continuous-time martingales and relevant properties
- Stochastic integral with respect to Brownian motion and Itô calculus
- Stochastic differential equations
- The Black-Scholes model
- Girsanov theorem
- Representation of martingales
- Pricing and hedging of options in the Black-Scholes model
- American options in the Black-Scholes model
- Option pricing and partial differential equations (Feynman-Kac theorem)
- European option pricing and diffusions

##### Suggested References

- Lamberton, D. and Lapeyre, B. (2007): Introduction to Stochastic Calculus Applied to Finance, Second Edition, Chapman & Hall/CRC Financial Mathematics Series.
- Shreve, S. E. (2004): Stochastic Calculus for Finance I & II, Springer-Verlag.

#### Advanced Stochastic Calculus for Finance

- Poisson Process
- Compound Poisson Process
- Jump processes and their integrals
- Stochastic Calculus for jump processes
- Ito-Doeblin formula
- Change of Measure
- Pricing European Call in a jump diffusion model
- Infinitely Divisible distributions and some of its properties
- Levy-Kintchine formula
- Poisson random measure and its construction
- Functionals of Poisson random measure
- Semimartingales
- Ito formula for semimartingales and some of its properties
- Pricing and Hedging in incomplete markets
- Merton’s approach
- Superhedging
- Quadratic hedging
- Non-Gaussian Ornstein-Uhlenbeck Process, its solution and evaluation of characteristic function

##### Suggested References

- Applebaum, D. (2004): Lévy Processes and Stochastic Calculus, Cambridge University Press.
- Cont, R. and Tankov, P. (2004): Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series.
- Kyprianou, A. E. (2006): Introductory lectures on Fluctuations of Lévy Processes with Applications, Springer.
- Protter P. E. (2005): Stochastic Integration and Differential Equations, Springer.
- Sato K. (1999): Levy Processes and Infinitely Divisible Distributions, Cambridge University Press.