Module Code - Title:
MS4028
-
STOCHASTIC DIFFERENTIAL EQUATIONS FOR FINANCE
Year Last Offered:
2025/6
Hours Per Week:
Grading Type:
N
Prerequisite Modules:
MS4213
MS4217
Rationale and Purpose of the Module:
Methods of stochastic dynamics applied to finance, and with reference to problems involving stochastic differential equations from physics and engineering.
Syllabus:
Introduction to Monte Carlo simulation: Numerical simulation of paths; ensemble averaging and connections to partial differential equations. Examples from Finance and Physics. Stochastic differential equations and Langevin equations. Fokker-Planck/Kolmogorov equation and relation to Black-Scholes equation. Numerical methods for SDEs and Langevin equations: Euler-Maryuma method and higher-order schemes. Pricing barrier options and first-passage problems, including multiple stochastic factors.
Trinomial trees and finite difference methods: Pricing on trinomial trees. Analytical methods for partial differential equations. Explicit, implicit, Crank-Nicholson, and ADI implementations for numerical solution of partial differential equations, including options on multiple assets.
Modelling markets with stochastic differential equations: Comparison of modelling methods for stochastic dynamics problems in Finance, Physics, and Engineering. The Ito/Stratonovich dilemma. Non-Gaussian distributions and fat tails in the markets. Long-memory effects. Coloured noise and the Ornstein Uhlenbeck process. Autocorrelation functions and spectra of noise sources. Wiener-Khinchin theorem.
Learning Outcomes:
Cognitive (Knowledge, Understanding, Application, Analysis, Evaluation, Synthesis)
On successful completion of this module students will be able to:
1. Implement numerical solution methods for stochastic differential equations arising in financial mathematics.
2. Analyse the probability distributions resulting from systems of stochastic differential equations using partial differential equation methods.
3. Price options on trinomial trees and understand the connection to finite difference solution methods for partial differential equations.
4. Create models of continuous-time stochastic dynamical processes with non-Gaussian distributions and memory effects.
Affective (Attitudes and Values)
N/A
Psychomotor (Physical Skills)
N/A
How the Module will be Taught and what will be the Learning Experiences of the Students:
Normal mathematics lectures.
Research Findings Incorporated in to the Syllabus (If Relevant):
Recent research in applied stochastic dynamics will inform and provide examples used in lectures.
Prime Texts:
Clewlow, L. and Strickland, C. (1998)
Implementing Derivatives Models
, John Wiley & Sons
Risken, H. (1996)
The Fokker-Planck Equation: Methods of Solution and Applications, 2nd ed.
, Springer-Verlag Berlin
Other Relevant Texts:
Hull, J.C. (2006)
Options, Futures and other Derivative Securities, 6th ed.
, Pearson
Campbell, J.W. (1997)
The Econometrics of Financial Markets
, Princeton University Press
Programme(s) in which this Module is Offered:
Semester(s) Module is Offered:
Module Leader:
eberhard.mayerhofer@ul.ie