Topics in Stochastic Methods: Stochastic Analysis with Applications in Economics

ECO437 Topics in Stochastic Methods: Stochastic Analysis with Applications in Economics

  • Topics


    • Probabilistic set-up, information flow or filtration
    • Stochastic processes in continuous time, processes adapted to an information flow, processes predictable with respect to an information flow
    • Brownian motion, quadratic variation of Brownian motion
    • Stochastic Ito integral
    • Ito formula
    • Stochastic differential equations, particular indeepning in the linear stochastic differential equations, Ito diffusions, Markov properities of diffusions
    • Martingales and elements of semimartingales in continuous time
    • Girsanov theorem and change of measures
    • Continuous time trading financial market models, trading strategies, arbitrage
    • Equivalent martingale measures or risk neutral measures: fundamental theorem of mathematical finance, non-arbitrage conditions
    • (European) contingent claims and risk-neutral pricing
    • Black and Scholes market model
    • Sensitivity analysis and the Greeks
    • Delta hedging
    • Outlook into other possible models: stochastic volatility and Lévy based models


  • Learning outcome

    Learning outcome

    The course will provide the students with fundamental tools of stochastic processes and stochastic analysis in continuous time.

    The methods presented are central in the modeling of financial markets aimed at the pricing and hedging of contingent claims, in the analysis of risk, in the quantification of decisions under uncertainty.

    Besides the financial perspective, the methods presented find large application also in economics and insurance. The course will focus on the mathematical aspects discussing their application in most of the mentioned areas.

    The material presented provides solid grounds for those who wish to reinforce their quantitative skills and also for those who contemplate to continue with doctoral studies in this direction.

    Acquired competence at the end of the course:

    • The students will be able to set up a modeling oriented frame for quantitative situations where uncertain outcomes are involved
    • The students will be able to handle sofisticated stochastic methods in the area of application presented recognizing the power and the limitations of the methods according to the situation
    • The students will be able to use the stochastic methods introduced in other areas than the ones presented
    • The students will be able to read and use cutting-edges literature in quantitative aspects of finance, economics, and insurance
    • The students will be able to understand the methods presented in further education in finance, insurance, and economics

  • Teaching


    The course will be delivered through a combination of lectures and exercise classes in which a selection of the problems suggested will to be commented in auditorium.

    There will be 32 lectures grouped in in 7-8 days of semi-intensive work according to schedule. Exercises will be posted on the webpage along with their solutions. There will be 2 compulsory assignments during the course.

  • Required prerequisites

    Required prerequisites

    ECO437 is a course presenting mathematical tools applied to finance, economics, and risk analysis. A good mathematical background is important to progress better through the course.

    No strict prerequisite is compulsory, but we strongly suggest the students to have acquired the ground knowledge in analysis, linear systems, probability theory, and statistics.

    For example, refereeing to the basic and/or elective courses at NHH, we can mention the course MAT010/MAT10, MAT011/MAT11, MAT013/MAT12, MET020/MET1, MET040/MET2, VOA038/FOR10 where in whole or in parts, good knowledge is given of functions in one and several variables in the real and complex case, limits, derivatives, integrals, Taylor expansions, linear systems, matrices and operations with matrices, probabilities, sample spaces, random variables, random processes, probability distributions, expectations and moments, Gaussian or Normal distribution. Please note that other courses, at NHH or other institutions, may have introduced the same basic knowledge.

  • Requirements for course approval

    Requirements for course approval

    Submission of the 2 assignments. The assignments will be evaluated approved/not approved.

  • Assessment


    4 hour written exam.

  • Grading Scale

    Grading Scale

    Grading scale A - F.

  • Computer tools

    Computer tools


  • Semester


    Spring 2017. (offered biannually). Not offered spring 2018 

  • Literature


    Main reference:

    B. Øksendal: Stochastic Differential Equations, 6th edition. Springer 2007.

    Reference to chapters 1 - 5, section 7.1, 7.2 , section 8.6 and chapter 12.

    Alternative books:

    F. E. Benth: Option theory with stochastic analysis. Springer 2004

    Pensum refers to chapters 1, 3, 4. Interesting for a discussion on Gaussian models versus Levy type models in chapter 2

    N.H. Bingham and R. Kiesel: Risk-Neutral Valuation, 2nd edition. Springer 2004.

    Reference to Sessions 2.1-2.7, 5.1-5.3, 5.6-5.8, 5.10, 6.1, 6.2 Interesting for some mathematical aspects related to Levy processes.

    T. Mikosch: Elementary Stochastic Calculus with Finance in View. World Scientific 1998.


ECTS Credits
Teaching language

Course responsible

Giulia di Nunno, Department of Business and Management Science