The study of financial derivatives, their underlying science and the fair pricing of European and American put and call options.

PHY1012 – Mathematics Module

The aims are to expose the students to a range of financial derivatives, their underlying science, and to show, by various means, how the fair price of options may be determined.

At the end of the module the student should;

i.                understand the mathematics and models that underpin the analysis of financial data, including the properties of random variables, probability distributions (including the Levy distribution) and share price models; 

ii.              know about a range of common financial derivatives, be able to explain financial terminology and produce pay-off and profit diagrams for forward contracts, put and call options;

iii.             understand and be able to derive and use the Black-Scholes-Merton equation and be able to determine the prices of European options through its solution or through binomial models;

iv.            explain the role of quantities known as the “greeks” in financial analysis.

i.                 Introduction to various forms of investment.  Risk and return.

ii.               Interest rates: risk-free, cumulative, compound.  Depreciation.

iii.              Share prices as quantities with a deterministic and random component. Drift, volatility and interest rate. 

iv.             Random variables; density and distribution functions.  Some common density functions such as Normal , binomial & Poisson.  The Levy distribution as a generalised distribution with improved representation of rare events.

v.               Methods for calculating the volatility (risk) of a stock.

vi.             European put and call options and their use to reduce risk or as a speculative investment.  Introduction to American options and covered warrants.  Put-call parity. Interpreting information in the Financial Times.

vii.            The binomial tree as a model for pricing European and American options.  Implementation of the one-step, two-step and multi-step models in practice. 

viii.          Ito’s Lemma ( Taylor ’s theorem for random variables). The lognormal density function.

ix.             Derivation of the Black-Scholes equation for the value of an option.  Arbitrage and delta hedging. Reduction to a diffusion equation. Solution of the Black-Scholes equation for European calls and puts.  Implied volatility.

x.               The perpetual American put.

xi.             The Greeks.

24 hours of lecture classes and computer-based problem-solving.

i.                John C Hull, Options, Futures and Other Derivatives, Prentice-Hall.

ii.              P Wilmott, S Howison and J Dewynne, The Mathematics of Financial Derivatives, Cambridge University Press.

iii.             S N Nefti , Mathematics of Financial Derivatives, Academic Press.

iv.            M Anthony and N Biggs, Mathematics for Economics and Finance, Cambridge University Press.

v.              D Blake, Financial Market Analysis, Wiley.

vi.            P Wilmott, Derivatives, Wiley.