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Aims:

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

Intended learning outcomes:

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

1. Apply the necessary probability and differential equation-based approach to the pricing of financial derivatives, using both quantitative and numerical techniques.

Indicative content:

The following are indicative of the topics the module will typically cover:

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.