This module aims at demonstrating the advanced mathematical techniques underlying financial markets and the practical use of financial derivative products to analyse various problems arising in financial markets. Emphases are on various option pricing formulae, hedging methods, and stochastic analysis.

At the end of the module students should be able to do the following things.

Use put-call parity to determine the relationship between prices of European put and call options and to identify arbitrage opportunities.

Calculate the value of European and American options using both the binomial model and the Black-Scholes option-pricing model.

Interpret option Greeks.

Explain the cash flow characteristics of the following exotic options: Asian, barrier, compound, gap and exchange.

Explain the properties of a lognormal distribution and explain the Black-Scholes formula as a limited expected value for a lognormal distribution.

Understand the principle of value derivatives by using numerical methods.

Understand the assumption that stock price follows geometric Brownian motion and be able to use Ito’s lemma to analyze how the option price changes in response to the stock price.

Understand Black-Scholes-Merton methodology. Be able to use this methodology to price virtually all derivatives.

Introduction to Derivatives: Futures, forwards and options; no arbitrage; commodity forwards and futures; swaps.

Parity and Other Options: Put-call parity; generalized parity and exchange options; comparing options with respect to style, maturity and strike.

Binomial Option Pricing: A one-period binomial tree; two or more periods binomial model; valuation of European and American Options using the binomial model; early exercise; risk-neutral pricing; estimating volatility; stocks paying discrete dividends.

The Black-Scholes Formula: Valuation of European and American options using the Black-Scholes option pricing model; the Greeks; implied volatility.

Market-Making and Delta-Hedging: the role of market-markers; delta-hedging

Exotic Options: Asian options; barrier options; compound options; gap options and exchange options.

The Lognormal Distribution: Lognormal model of stock prices; lognormal probability calculations; estimating the parameters of a lognormal distribution.

Numerical Methods in Option Pricing: Computing the option price as a discounted expected value; computing random numbers; simulating lognormal stock; Monte Carlo valuation; valuation of American options.

Brownian Motion and Ito’s Lemma: Brownian motion, geometric Brownian motion, Sharpe ratio, risk-neutral process.

The Black-Scholes Equation: Differential equations, Black-Scholes equation, risk-neutral pricing, changing the numeraire, option pricing when the stock price can jump.