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Module Title |
Probability and Finance II
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Module Code |
MS408
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School |
School of Mathematical Sciences
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Online Module Resources
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Level |
4
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Credit Rating |
7.5
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Pre-requisite |
MS407
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Co-requisite |
None
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Module Aims
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7 To provide a thorough introduction to Brownian motion and stochastic calculus.
7 To cover some of the most important topics in modern Mathematical Finance.
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Learning Outcomes
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As a result of this module, students should gain a working knowledge of Brownian motion
and of the tools of stochastic analysis used in Mathematical Finance. They will understand
how these tools are used in specific financial models.
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Indicative Time Allowances
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Hours
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Lectures |
3
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Tutorials |
12
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Laboratories |
0
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Seminars |
0
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Independent Learning Time |
97.5
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Total |
112.5
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Placements |
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Assignments |
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NOTE
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Assume that a 7.5 credit module load represents approximately 112.5 hours' work, which includes all teaching, in-course assignments, laboratory work or other specialised training and an estimated private learning time associated with the module.
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Indicative Syllabus
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· BROWNIAN MOTION: provisional definition, specification of a stochastic process through its finite order distributions, Daniell-Kolmogorov theorem; versions, difficulty withcontinuity, 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 A CONTINUOUS TIME: filtrations, adaptedness, Brownian martingales;stopping times, optional stopping, hitting times.
· ITÔ CALCULUS: Itô integral for simple adapted processes; Itô integral as an isometry; Itô processes, Itô 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.
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| Assessment | | Continuous Assessment | 25% | Examination Weight | 75% |
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Indicative Reading List
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Lamberton, D., and Lapeyre, B., Introduction to Stochastic Calculus with Financial
Applications, London: Chapman and Hall 1996.
Bjvrk, T.: Arbitrage Theory in Continuous Time, Oxford: University Press 1998.
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Programme or List of Programmes
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| BSSA | Study Abroad (DCU Business School) |
| BSSAO | Study Abroad (DCU Business School) |
| ECSA | Study Abroad (Engineering & Computing) |
| ECSAO | Study Abroad (Engineering & Computing) |
| FM | BSc in Financial & Actuarial Mathematics |
| HMSA | Study Abroad (Humanities & Soc Science) |
| HMSAO | Study Abroad (Humanities & Soc Science) |
| MS | BSc in Mathematical Sciences |
| SHSA | Study Abroad (Science & Health) |
| SHSAO | Study Abroad (Science & Health) |
| Archives: | |
