Module Handbook

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Module MAT-64-11A-M-7

Stochastic Differential Equations and Financial Mathematics (M, 13.5 LP)

Module Identification

Module Number Module Name CP (Effort)
MAT-64-11A-M-7 Stochastic Differential Equations and Financial Mathematics 13.5 CP (405 h)

Basedata

CP, Effort 13.5 CP = 405 h
Position of the semester 2 Sem. from irreg.
Level [7] Master (Advanced)
Language [EN] English
Module Manager
Ritter, Klaus, Prof. Dr. (PROF | DEPT: MAT)
Sass, Jörn, Prof. Dr. (PROF | DEPT: MAT)
Lecturers
Lecturers of the department Mathematics
Area of study [MAT-SPAS] Analysis and Stochastics
Reference course of study [MAT-88.105-SG] M.Sc. Mathematics
Livecycle-State [NORM] Active

Notice

This module is the union of the modules [MAT-61-11-M-7] Financial Mathematics and [MAT-64-11-M-7] Stochastic Differential Equations. Due to the large overlap in content, only 13.5 credit points are awarded in total.

Courses

Type/SWS Course Number Title Choice in
Module-Part 		Presence-Time /
Self-Study 		SL SL is
required for exa. 		PL CP Sem.
4V+2U MAT-64-11-K-7
Stochastic Differential Equations
P 84 h 186 h - - PL1 9.0 irreg. WiSe
4V+2U MAT-61-11-K-7
Financial Mathematics
P 84 h 186 h - - PL1 9.0 SuSe
  • About [MAT-64-11-K-7]: Title: "Stochastic Differential Equations"; Presence-Time: 84 h; Self-Study: 186 h
  • About [MAT-61-11-K-7]: Title: "Financial Mathematics"; Presence-Time: 84 h; Self-Study: 186 h

Examination achievement PL1

  • Form of examination: oral examination (30-45 Min.)
  • Examination Frequency: each semester
  • Examination number: 86431 ("Stochastic Differential Equations and Financial Mathematics")

Evaluation of grades

The grade of the module examination is also the module grade.


Contents

Stochastic Differential Equations (SDEs) help us to model continuous-time random phenomena and thus can be used for modeling financial markets in continuous time.

This module covers key elements of the theory of stochastic differential equations and of continuous-time financial mathematics. In addition, an introduction to the algorithmic approach to SDEs is given. The following topics are covered:

  • Brownian motion,
  • martingales theory,
  • stochastic integration (with respect to Brownian motion),
  • strong and weak solutions of SDEs,
  • martingale representation theorem,
  • Girsanov's theorem,
  • stochastic representation of the solution of partial differential equations,
  • diffusion model for stock prices and trading strategies,
  • complete markets,
  • option pricing by the replication principle, the Black-Scholes formula,
  • option pricing and partial differential equations,
  • exotic options,
  • arbitrage bounds (put-call parity, parity of prices for European and American calls),
  • classical approximations for SDEs,
  • stochastic multi-level algorithms.

Competencies / intended learning achievements

Upon successful completion of this module, the students have acquired an in-depth knowledge of the analysis of stochastic differential equations. In particular, they have studied and understand the basic structures and properties of stochastic integrals and stochastic differential equations. They are familiar with the Itô formula and Girsanov's theorem. They have been exposed to various financial market models, in particular, the Black Scholes model, to understand the different methods for the pricing of financial derivatives. They can critically assess the limits of modelling and the applicability of methods to different financial derivatives. Moreover, they have gained insight into modeling and numerical treatment of SDEs.

By completing exercises, students will have developed a skilled, precise and independent handling of the terms, propositions and techniques taught in the lecture. They have learnt how to apply the methods to new problems, analyze them and develop solution strategies. They understand the proofs presented in the lectures and are able to comprehend and explain them. In particular, they can outline the conditions and assumptions that are necessary for the validity of the statements and how these are to be interpreted in the context of actuarial and financial mathematics.

Literature

From [MAT-64-11-K-7] Stochastic Differential Equations:
  • I. Karatzas, S. E. Shreve: Brownian Motion and Stochastic Calculus,
  • P. E. Kloeden, E. Platen: Numerical Solution of Stochastic Differential Equations,
  • T. Müller-Gronbach, E. Novak, K. Ritter: Monte-Carlo-Algorithmen.
From [MAT-61-11-K-7] Financial Mathematics:
  • N.H. Bingham, R. Kiesel: Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives,
  • T. Björk: Arbitrage Theory in Continuous Time,
  • I. Karatzas, S.E. Shreve: Brownian Motion and Stochastic Calculus,
  • I. Karatzas, S.E. Shreve: Methods of Mathematical Finance,
  • R. Korn, E. Korn: Option Pricing and Portfolio Optimization – Modern Methods of Financial Mathematics.

Materials

Further literature will be announced in the lectures; Exercise material is provided.

Registration

Registration for the exercise classes via the online administration system URM (https://urm.mathematik.uni-kl.de).

Requirements for attendance (informal)

Basic knowledge in Functional Analysis, e.g. from the course [MAT-12-23-K-3].

Modules:

Requirements for attendance (formal)

None

References to Module / Module Number [MAT-64-11A-M-7]

Module-Pool Name
[MAT-61-MPOOL-7] Specialisation Financial Mathematics (M.Sc.)
[MAT-70-MPOOL-7] Specialisation Stochastic Analysis (M.Sc.)
[MAT-AM-MPOOL-7] Applied Mathematics (Advanced Modules M.Sc.)