#### Course overview

This is an optional module introducing aspects of the fixed-income world in a financial engineering setting. The underlying of concern is primarily the spot interest-rate. Students wishing develop a quantitative finance basis for their MSc study will find this elective particularly beneficial. The theme of the course is to build the mathematical framework for pricing interest-rate securities. Derivation of important models and equations will be presented. The numerical and computational schemes for contract valuation will centre on the Monte-Carlo scheme.

#### Learning outcomes

- Understand the basic products traded in the fixed income markets
- Develop knowledge of mathematical models, their derivation and solution methods
- Be able to price more complex products computationally
- Understand what happens inside the ‘black-box’ rather than simply place blind faith in given mathematical formulae.

#### Topics covered

**Fixed-Income products and markets: **A brief overview of the fixed-income markets and the securities traded in them – zero coupon bonds and coupon bearing bonds; yield curves and forward rates; duration and convexity. Caps, Floors, Swaps.

**Mathematical review: **Stochastic Differential Equations for interest-rate models; drift, volatility and mean reversion. Itô’s lemma.

**Stochastic interest rate models**: Popular spot-rate models (Vasicek, Cox-Ingersoll-Ross). Bond pricing equation and similarities with the Black-Scholes equation; Market price of interest-rate risk and why it arises. Risk-neutral pricing. Tractable models and Affine solutions of the bond pricing equation. The need for multi-factor interest rate modelling: two-factor interest rate models (Brennan & Schwartz; Fong & Vasicek; Longstaff & Schwartz); and bond pricing equation. Stochastic market price of risk.

**Calibration**: Yield curve fitting – the importance of matching theoretical and market bond prices; time dependent one factor models (Ho & Lee, Hull & White).

**Heath, Jarrow and Morton (HJM)**: HJM model and evolution of the whole yield curve. Monte-Carlo simulations and implementing HJM. Computations in Excel.

**Market Models**: Dynamics of discrete forward rates – Libor Market Model. Brace, Gatarek, Musiela (BGM) model.

#### Assessment summary

Exam 80% Group Coursework 20%

Current students should refer to Moodle for specific details of the current year’s assessment.

#### Essential reading

· Wilmott, Paul. (2012) “Paul Wilmott Introduces Quantitative Finance; second edition reprinted,” (John Wiley & Sons). ISBN: 0470319585

· Brigo, Damiano; Mercurio, Fabio (2016), “Interest Rate Models – Theory and Practice: with Smile, Inflation and Credit” (Springer Finance). ISBN: 354034604X

#### Past versions of this module

*Last updated Tuesday, 21 May 2019*