Registry
Module Specifications
Archived Version 2012 - 2013
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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 discrete-time set-up . The Black-Scholes model is covered in detail and particular emphasis is placed on learning to adapt it to new situations . This model building provides a "know-how and skill" element to a module which is otherwise mostly of "knowledge" type. The final exam will be of two hours duration and will require students to answer three out of four questions . | |||||||||||||||||||||||||||||||||||||
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Learning Outcomes 1. Demonstrate an understanding of the fundamental concepts of the theory of stochastic processes in discrete time through examples and counter-examples 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 continuous-time finance 6. Derive the Black-Scholes 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 |
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Indicative Content and
Learning Activities BROWNIAN MOTIONprovisional definition, specification of a stochastic process through its finite order distributions, Daniell-Kolmogorov 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 TIMEfiltrations, adaptedness, Brownian martingales;stopping times, optional stopping, hitting times.ITO CALCULUSIto integral for simple adapted processes; Ito integral as an isometry; Ito processes, Ito's lemma, stochastic differential equations.OPTIMAL PORTFOLIO THEORYthe stochastic differential equation of stock prices; utility, Merton's problem.OPTION PRICINGGirsanov's theorem and the equivalent martingale measure approach to option pricing; the arbitrage approach. | |||||||||||||||||||||||||||||||||||||
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Indicative Reading List
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Other Resources None | |||||||||||||||||||||||||||||||||||||
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