Aims:

This module introduces the applied mathematical and computational aspects of Quantitative Finance.Learning outcomes:

On successful completion of the module, a student will be able to:

Apply the necessary probability and differential equation based approach to the pricing of financial derivatives; using both quantitative and numerical techniques.
Content:

Financial Products and Markets:


	Time value of money and applications. Equities, indices, foreign exchange, and commodities. Futures, Forwards and Options. Payoff, and P&L diagrams. Put-Call parity. The Binomial model and risk-neutrality.
	
Stochastic Calculus:


	Brownian motion and properties, Itô’s lemma and Itô integral. Stochastic Differential Equations – drift and diffusion; Geometric Brownian Motion and Vasicek model.
	

	Forward and Backward Kolmogorov equations for the transition density.
	
	
	Random number generation in Excel – RAND(), NORMSINV(), simulating random walks, correlations. Examining statistical properties of stock returns.
	
Black-Scholes Model:


	Assumptions, PDE and pricing formulae for European calls and puts. Extending to dividends, FX and commodities.
	
	
	The Greeks and risk management - theta, delta, gamma, vega, rho and their role in hedging. Two factor models and multi-asset options; Mathematics of early exercise:
	

	Computational Finance: Solving the pricing PDEs numerically using the Finite Difference Scheme. The Monte-Carlo method.
	
Fixed-Income world:


	Zero coupon bonds and coupon bearing bonds; yield curves, duration and convexity. Bond Pricing Equation (BPE). Popular models for the spot rate.
	
	
	Stochastic interest rate models - Vasicek, CIR, Ho and Lee, and Hull and White.
	
	
	Solutions of the BPE.
	
Introduction to Exotics:


	Basic features and classification of exotic options. Weak and strong path dependency
	
	
	Barriers, Asians and Lookbacks. Sampling continuous and discrete.
	
	
	Pricing using the PDE framework.
	
Requisites:

To be eligible to select this module as optional or elective, a student must: (1) be registered on a programme and year of study for which it is a formally available; (2) have a good understanding of basic probability and differential equations.