# Module 5 - Fixed Income

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This module reviews the multitude of interest rate models used within the industry. You will learn the implementation and limitations of each of these models.

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Fixed Income Products and Analysis

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• Names and properties of the basic and most important fixed-income products
• Features commonly found in fixed-income products
• Simple ways to analyze the market value of the instruments: yield, duration and convexity
• How to construct yield curves and forward rates
• Swaps
• The relationship between swaps and zero-coupon bonds

Stochastic Interest Rate Modeling

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• Stochastic models for interest rates
• How to derive the pricing equation for many fixed-income products
• The structure of many popular one-factor interest rate models
• The theoretical framework for multi-factor interest rate modeling
• Popular two-factor models

Calibration and Data Analysis

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• How to choose time-dependent parameters in one-factor models so that
• Today’s yield curve is an output of the model
• How to analyze short-term interest rates to determine the best model for the volatility and the real drift
• How to analyze the slope of the yield curve to get information about the market price of risk

Probabilistic Methods For Interest Rates

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• The pricing of interest rate products in a probabilistic setting
• The equivalent martingale measures
• The fundamental asset pricing formula for bonds
• Application for popular interest rates models
• The dynamics of bond prices
• The forward measure
• The fundamental asset pricing formula for derivatives on bonds

Heath Jarrow and Morton Model

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• The Heath, Jarrow & Morton (HJM) forward rate model
• The relationship between HJM and spot rate models
• How to decompose the random movements of the forward rate curve into its principal components

The Libor Market Model

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• The Libor Market model
• The market view of the yield curve
• Yield curve discretisation
• Standard Libor market model dynamics
• Numéraire and measure
• The drift
• Factor reduction

Further Monte Carlo

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• The connection to statistics
• The basic Monte Carlo algorithm, standard error and uniform variates
• Non-uniform variates, efficiency ratio and yield
• Co-dependence in multiple dimensions
• Wiener path construction; Poisson path construction
• Numerical integration for solving SDEs
• Variance reduction techniques
• Sensitivity calculations
• Weighted Monte Carlo

Multiple Curve Interest Rate Modeling

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• LIBOR curve data and post-LIBOR world
• Multi-curve stripping to match swaps
• OIS discount & risk-free benchmarks
• Tenor basis & tenor volatility
• Optionality in fixed income products
• Industry uses of LMM, HJM, G2++

Fixed Income Market Practices

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• Basics: discount factors, FRAs, swaps, and other delta products
• Basic curve stripping, bucket deltas, and managing IR risks
• Interpolation methods
• Risk bleeding
• Scenarios-based risks and hedging (wave method)
• Current Market Practices

Volatility Smiles and the SABR Model

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• Vanilla options: European swaptions, caps, and floors
• Arbitrage Free SABR

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Lecture order and content may occasionally change due to circumstances beyond our control; however this will never affect the quality of the program.

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Data Science & Machine Learning
Credit Products & Risk