# Module 4: Fixed Income

### Fixed Income Products and Analysis

- 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

- 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

- How to choose time-dependent parameters in one-factor models so that
- Today’s yield curve is an output of the model
- The advantages and disadvantages of yield curve fitting
- 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

- 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

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

### Fixed Income Market Practices

- 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
- Advanced stripping

### Volatility Smiles and the SABR Model

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

### The Libor Market Model

- 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

- 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

### Energy Derivatives

- Participants' risk exposure in electricity markets
- Price volatility
- Fluctuation of electricity production costs
- Hedging with electricity futures for generators, marketers and end-users
- Risks of hedging with electricity futures
- "Stack and Roll" hedging for the long-term periods
- "Hedging with electricity options and crack spreads
- Speculation (views taking) using electricity futures
- Politics of Speculation. Dodd-Frank Act
- Restrictions to oil speculation and high frequency commodity trading
- Energy trading case study. Sparks spread