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Department of Mathematical Sciences

# MATH3301/4181 Mathematical Finance III/IV

Financial mathematics uses mathematical methods to solve problems in financial industry. It draws on tools from probability theory, statistics, partial differential equations, and scientific computing, such as numerical methods, Monte Carlo simulation and optimization.

The methods and tools in financial mathematics are widely used in investment banks, commercial banks, hedge funds, insurance companies, corporate treasuries, and regulatory agencies to solve such problems as derivative pricing, portfolio selection, risk management, and scenario simulation. It is one of the fastest developing areas of mathematics and it has brought efficiency and rigor to financial markets over the years. As such it is becoming increasingly important to the financial firms.

This is an introductory course and it covers important tools and ideas of financial mathematics. The first part of this course discusses discrete time financial models. Main concepts of financial mathematics such as forward and futures contracts, call and put options, completeness, self-financing condition, replicating strategies, and risk-neutral measures will be introduced. Pricing problems of European and American options under Cox-Ross-Rubinstein (CRR) Binomial model will be discussed. The Black-Scholes formula for the call price will be derived as a limit of the CRR model. Then the concepts of preferences, utility, and risk aversion will be introduced and the capital-asset pricing model will be discussed in detail.

The second part of the course will discuss continuous time financial models. A short introduction to the Ito's integral and ito's formula will be given. Pricing problems of path dependent options under the standard Black-Scholes model will be discussed. Then short rate models will be introduced and affine term structure of bond prices will be discussed. A short introduction to the Heath-Jarrow-Morton forward rate model will also be given.

## Outline of Course

Aim: To provide an introduction to the mathematical modelling of financial derivative products.

### Term 1

• Introduction to options and markets: Interest rates and present value analysis, asset price random walks, pricing contracts via arbitrage, risk neutral probabilities.
• The Black-Scholes formula: The Black-Scholes formula for the geometric Brownian price model and its derivation.
• More general models: Limitations of arbitrage pricing, volatility, pricing exotic options by tree methods and by Monte Carlo simulation.

### Term 2

• Introduction to stochastic calculus.
• The Black-Scholes model revisited: The Black-Scholes partial differential equation. The Black-Scholes model for American options.
• Finite difference methods.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook: MATH3301, MATH4181.