Undergraduate - UnitMTH3251 - Financial mathematics

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Synopsis

Random variables, application to models of random payoffs. Conditional expectation. Normal distribution and multivariate normal distribution. Best predictors. Stochastic (random) processes. Random walk. Limit theorems. Brownian motion. Ito integral and Ito's formula. Black-Scholes, Ornstein-Uhlenbeck process and Vasicek's stochastic differential equations. Martingales. Gambler's ruin. Fundamental theorems of Mathematical Finance. Binomial and Black-Scholes models. Models for Interest Rates. Risk models in insurance. Ruin probability bound. Principles of simulation. Use of Excel package.

Outcomes

On the completion of this unit students will gain an understanding of the methods of modern probability and random processes, and develop skills for modelling of random systems. Students will be able to apply this knowledge and skills in the context of financial and insurance modelling. Specifically, on the completion of this unit, students will:

• understand the modern approach to evaluation of uncertain future payoffs;
• understand the concepts of arbitrage and fair games and their relevance to finance and insurance;
• understand the concept of conditional expectation, martingales, and stopping times, as well as the Optional Stopping Theorem;
• understand models of random processes such as Random Walk, Brownian Motion and Diffusions, and Stochastic Differential equations;
• be able to use Ito's formula, and basic stochastic calculus, and be able to solve some stochastic differential equations by using rules of stochastic calculus;
• know Fundamental theorems of asset pricing, being able to apply them to the Binomial and Black-Scholes models, as well as models for bonds and options on bonds;
• be able to formulate discrete time Risk Model in Insurance and use the Optional Stopping Theorem to control probabilities of ruin;
• be able to simulate stochastic processes and solutions of stochastic differential equations, and obtain prices by simulations.

Assessment

Assignments: 20%
Weekly exercises: 10%
Final examination (three hours): 70%

Chief examiner(s)

Professor Fima Klebaner

Contact hours

Three 1-hour lectures and one 1-hour support class per week

Prerequisites

One of MTH2010, MTH2015, MTH2032, or MTH2222. MTH2222 is highly recommended.

Prohibitions

ETC3510, ETC5351