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STOCHASTIC CALCULUS AND MATHEMATICAL FINANCE
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STOCHASTIC CALCULUS AND MATHEMATICAL FINANCE
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Academic year 2023/2024
- Course ID
- SEM0174
- Teachers
- Bertrand Lods (Lecturer)
Tiziano De Angelis (Lecturer)
Andrea Bovo (Assistant technician) - Degree course
- Finance
Insurance and Statistics - Year
- 1st year
- Teaching period
- Second semester
- Type
- Distinctive
- Credits/Recognition
- 6
- Course disciplinary sector (SSD)
- SECS-S/06 - mathematical methods of economy, finance and actuarial sciences
- Delivery
- Formal authority
- Language
- English
- Attendance
- Obligatory
- Type of examination
- Written
- Prerequisites
- A good knowledge of basic calculus (Matematica Generale) and of advanced probability (Probability for Finance).
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Sommario del corso
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Course objectives
This course is aimed at introducing and developing many of the mathematical tools which are used in applied finance and insurance. ln this module, particular stress will be posed on the study of continuous time processes with emphasis on their applications to investment and insurance decisions. The introduction of stochastic processes and their properties is always motivated by the wish to develop models for observed phenomena.
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Results of learning outcomes
At the end of the course, the student is expected to be capable of:
- using the basic tools and results to pose, formalize and solve a probability problem of applied interest
- knowing the extent to which the results obtained in the previous step are dependent on the assumption that s/he has made about the behaviour of the economic agents
- being able to think about possible and useful generalizations of the model
- being able to communicate such findings using appropriate and clear mathematical notation and language
- applying the basic course knowledge to theoretical issues and situations
- approaching the subject in a critical manner through the examination of different approaches in the literature and practice of mathematical finance
- having gained communication skills, through class discussion
- having gained learning abilities, through a variety of learning tools (teaching material, class discussion, lab sessions, homeworks and tests)
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Program
- Basics of continuous time processes
- Brownian motion
- Stochastic calculus (Ito integral)
- Connection with PDEs
- Change of measure
- Introduction to jump processes
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Course delivery
The course is articulated in 48 hours of formal in-class lecture time, and in at least as many hours of at-home work solving practical exercises.
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Learning assessment methods
The course grade is determined solely on the basis of a written test. The test evaluates the student's ability to do the following:
- Present briefly the main ideas, concepts and results developed in the course, also explaining intuitively the meaning and scope of the definitions and the arguments behind the validity of the results
- Use effectively the concepts and the results to answer questions in stochastic process theory, e.g., computing the Ito integral of some given stochastic process, application of Girsanov Theorem.
The above is accomplished by asking the student to answer open questions. Questions can be essay questions or exercises. The minimum exam grade is 18/30, the maximum grade is 30/30 cum laude. More details on the exam can be found on Moodle.
The exam is an open-book exam lasting 2 hours. Use of calculators is not permitted.
The student can take the exam at most three times per academic year.
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Support activities
The course includes exercises classes; extra exercises are suggested as homework.
Suggested readings and bibliography
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- Shreve (2004). Stochastic calculus for finance II, Springer.
- Oksendal (2003). Stochastic differential equations: an introduction with applications, Springer
- Baldi (2017). Stochastic Calculus. An introduction through theory and exercises. Springer.
- Additional Lecture Notes will be made available to the students.
Books may be avaiable from the School library's online resources:
https://www.bem.unito.it/it/che-cosa-cerchi/ testi-desame-e-altri- materiali-didatticiopen_in_new - Enroll
- Open
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