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 and particular emphasis is placed on learning to adapt it to new situations. The knowledge of proofs of certain deep theorems, the emphasis on interpretation of theoretical results in a financial context and on financial model building, as well as the appraisal of the strengths and weaknesses of this financial model are among the features which distinguish this module from a corresponding undergraduate module (MS408). The end of semester examination is of three hoursâ€™ duration and will require students to answer all questions on the paper.  
Learning Outcomes 1. State and interpret the fundamental concepts of the theory of stochastic processes in continuoustime and demonstrate a mastery of these concepts through examples and counterexamples 2. Employ the Optional Stopping Theorem to establish properties of various hitting times 3. Apply stopping time methods to solve problems in continuous time financial models 4. Solve simple stochastic differential equations using Ito calculus 5. Model financial phenomena using stochastic differential equations and interpret the solutions of these equations 6. Prove and interpret the basic results of utility theory 7. Solve Merton's problem for constant relative risk aversion (CRRA) utilities and interpret and appraise the results 8. Prove Girsanov's Theorem an apply it to selected problems in continuoustime finance 9. Derive the BlackScholes formula and apply the method to a variety of extensions of the basic problem 10. Critique the assumptions underlying simple continuous time stock price models  
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
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. Necessity of nondifferentiable paths of stock prices in arbitrage free markets. Martingales in Continuous Time Filtrations, adaptedness, Brownian martingales. Stopping times, optional stopping, hitting times. Applications of hitting times to exchange rate bands, timing of buying or selling risky assets. Ito Calculus Ito integral for simple adapted processes; Ito integral as an isometry; Ito processes, Ito's lemma, stochastic differential equations. Applications to the evolution of stock prices and interest rates. Optimal Portfolio Theory The BlackScholes stochastic differential equation of stock prices. Significance of the equation. Utility theory. Merton's optimal investment problem. Option Pricing Girsanov's theorem and the equivalent martingale measure approach to option pricing; the arbitrage approach. Derivation of the BlackScholes Formula. Extensions of the formula. Interpretation and significance of the BlackScholes formula. The strengths and weaknesses of the BlackScholes model.  
 
Indicative Reading List
 
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