Module Specifications..
Current Academic Year 2023  2024
Please note that this information is subject to change.
 
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Description This module provides a thorough introduction to Brownian motion, stochastic calculus and their application to finance. It builds on Probability and Finance I in that it deals with the problem of extending to continuous time the ideas first encountered in a discretetime setup. The BlackScholes model is covered in detail. The end of semester examination is of two hoursâ€™ duration and a choice of questions is available for students.  
Learning Outcomes 1. Demonstrate an understanding of the fundamental concepts of the theory of stochastic processes in continuous time through examples and counterexamples 2. Use the Optional Stopping Theorem to establish properties of various hitting times 3. Solve simple stochastic differential equations 4. Prove the basic results of utility theory and solve Merton's problem for CRRA utilities 5. Prove Girsanov's Theorem an apply it to selected problems in continuoustime finance 6. Derive the BlackScholes formula and apply the method to a variety of extensions of the basic problem  
All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml 

Indicative Content and Learning Activities
The Limit Theorems of Probability Theory The BorelCantelli lemmas; modes of convergence; Weak and Strong Laws of Large Numbers; the Central Limit Theorem. Brownian Motion Provisional definition, specification of a stochastic process through its finite order distributions, DaniellKolmogorov theorem; versions, difficulty with continuity, completion of the probability space, Kolmogorov's continuity criterion, modification of a process; properties of Brownian motion: scaling, nowhere differentiability of sample paths. Martingales in Continuous Time Filtrations, adaptedness, Brownian martingales; stopping times, optional stopping, hitting times. Ito Calculus Ito integral for simple adapted processes; Ito integral as an isometry; Ito processes, Ito's lemma, stochastic differential equations. Optimal Portfolio Theory The stochastic differential equation of stock prices; utility, Merton's problem. Option Pricing Girsanov's theorem and the equivalent martingale measure approach to option pricing; the arbitrage approach.  
 
Indicative Reading List
 
Other Resources None  
Programme or List of Programmes  
Date of Last Revision  18JUN08  
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