STOCHASTIC CALCULUS WITH APPLICATIONS TO FINANCIAL MARKETS
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- Versione italiana
- Academic year
- 2022/2023
- Teacher
- GIACOMO DIMARCO
- Credits
- 6
- Didactic period
- Secondo Semestre
- SSD
- MAT/07
Training objectives
- Aim of the course is providing to the students, in addition to a good knowledge of elementary financial mathematics, the mathematical methods and the basic concepts of the modern financial mathematics.
At the end of the course the main knowledge acquired will be
• basic concepts concerning bonds, stocks and derivative securities, financial markets, financial laws and arbitrage portfolio
• mathematical properties of options
• concept of stochastic process and some properties of particular stochastic processes
• integral and differential stochastic calculus
• models of stock prices
• Black-Scholes formula for the price of European call and put options, its extensions and applications.
At the end of the course the student will be able
• to recognize the risk of the securities
• to valuate the convenience of loans by means TAN and TAEG
• to evaluate the profitability of a project by using present net value (PNV) method
• to solve simple or linear stochastic differential equations
• to valuate the price of European call or put options before exercise time
• to use Delta-hedging strategy
• to evaluate stocks and bonds of a firm by means of Black-Scholes formula
• to expose the topics presented in the course by using the financial language. Prerequisites
- A very good knowledge of the differential and integral calculus for real functions with particular regard to Lebesgue integration theory. These topics are developed in the courses of Analisi Matematica 1 and Analisi Matematica 2 (bachelor's degree in Mathematics).
Basic concepts in Calculus of Probability Course programme
- The course is scheduled in 48 hours.
The programme is composed of three parts.
PART 1
FINANCIAL MARKETS: bonds, stocks, derivative securities, financial markets, Stock Exchange, financial laws, annuities, forward and futures, basic concepts about options, swap, warrant, convertible bonds, arbitrage portfolio (10 hours)
MATEMATICAL PROPERTIES OF THE OPTIONS: mathematical properties of European and American call options, parity put – call, use of the options, exotic options (4 hours).
PART 2
FUNDAMENTALS OF PROBABILITY: probability space, random variables, integration of a random variable with respect to a probability measure, independence of random variables, moments of a random variable, conditional expectations (8 hours)
STOCHASTIC PROCESSES: definition of stochastic process, martingales, Wiener process and its properties (4 hours)
CLASSICAL CALCULUS AND STOCHASTIC CALCULUS: Riemann-Stieltjes integral, definition of Ito stochastic integral, example of the calculation of an Ito integral, properties of Ito integral (7 hours)
STOCHASTIC DIFFERENTIAL CALCULUS: Ito process and definition of stochastic differential, Ito formula and its applications, stochastic differential equations, strong and weak solutions, strong solution existence and uniqueness theorem, linear stochastic differential equations, examples, geometric process and its properties (5 hours).
PART 3
BLACK-SCHOLES MODEL TO VALUATE THE PRICE OF CALL OPTIONS: stock prices mathematical models, valuation of the price of European call options: Black-Scholes equation and the solution to the problem of Black and Scholes with final value, calculations of the Greeks and other derivatives of the functions c of Black-Scholes formula (6 hours)
EXTENSIONS OF BLACK -SCHOLES FORMULA AND ITS APPLICATIONS: Black - Scholes formula for American call options and for European put options, perpetual American put, extensions of Black- Scholes model, applications of Black –Scholes formula: valuation of bonds and stocks of a firm, Delta-hedging strategy, real options (4 hours). Didactic methods
- The course is organized in theoretical lectures, exercises and examples about all topics of the programme.
Learning assessment procedures
- The aim of the final exam consists in verifying the level of knowledge of the formative objectives previously stated.
The final exam consists in an oral discussion on the topics of the course. First of all the student must present the resolution of some exercises assigned during the lessons. After the discussion about the exercises, the student will be invited to chose and to expose in a detailed way a topic developed in the course. The other questions on the remaining part of the programme have the objective to verify the comprehension of the basic concepts and the ability to link and compare different aspects discussed in the course. For this reason details are not required. Each part of the exam will contribute to the final vote.
Reference texts
- Lecture Notes are available at the course Web site.
Specific topics can be futher developed on
E. Agliardi, R. Agliardi: "Mercati finanziari. Analisi stocastica delle opzioni", Mc Graw-Hill, 2001.
A. Pascucci: Calcolo stocastico per la finanza, Springer, 2007.
B. Oksendal: Stochastic Differential Equations, Springer, 2005.
V. Capasso, D. Bakstein: "An Introduction to Continuous - Time Stochhastic Processes", Birkhauser, 2012.
