Term Structure Modelling and Interest Rate Option Pricing

The objective of this advanced level seminar is to give you a good understanding of interest rate models and their uses in option pricing and risk management. We first present and explain important concepts such as the term structure of interest rates and the term structure of volatility.

We then take at closer look at various processes for interest rate evolvement over time, and we explain how interest rate volatility can be modelled in these processes using models such as GARCH and EWMA (Exponentially Weighted Moving Average). We also explain various approaches to modelling, including the use of partial differential equations and “Martingales”.

Next, we present and explain a number of models for interest rate processes, including “Equilibrium” models such as the Rendleman-Barter and Cox-Ingersoll-Ross and “No-arbitrage” models - with and without mean reversion features. This class of models includes single-factor models such as the Ho-Lee, Vasicek, Hull-White, Black-Derman-Toy as well as multi-factors models such as Longstaff-Schwartz. We also present the popular “Libor Market”, or BGM (Brace-Gatarek-Muselia) model, which is widely used by practitioners. We discuss the important characteristics and parameters of these models, and we demonstrate how they can be implemented in practice.

Finally, we explain and illustrate how these models can be used for pricing and risk assessment of interest rate options such as Caps, Floors, Swaptions, Delivery Options, Prepayment Options and Defaultable Bonds. We also demonstrate the pricing and hedging of more advanced structures such as “Constant Maturity Swaps” (convexity adjustment) and of path-dependent option structures (using Monte Carlo simulation).

 

Day One

09.00 - 09.15 Welcome and Introduction

09.15 - 12.00 The Term Structure of Interest Rates

• The Term Structure and its Applications
• Factors Explaining Shape of Term Structure
• Brief Review of Term Structure Estimation Techniques

Modelling Volatility

• Definition of Volatility
• Types of Volatility
• Standard Approach to Estimating Volatility
• Check for Normality of the Returns
• Possible Explanations for the Fat Tails
• Moving Averages
  • Simple moving average
  • EWMA
  • Attractions of EWMA
  • The EWMA as special case of the GARCH process
• Examples and exercises

12.00 - 13.00 Lunch

13.00 - 16.30 Modelling Volatility (Continued)

• GARCH Modelling
  • Introduction
  • General model appearance
  • Conditional and unconditional variance
  • GARCH (1,1)
  • Higher order GARCH
• Implementing GARCH
  • Variance targeting
  • Maximum Likelihood Methods
• Forecasting Future Volatility
• Volatility Term Structures
• Combining GARCH with EVT-theory to Capture Fat Tails
• Modelling Correlations
• Volatility Clustering and Correlations
• Examples and exercises

Day Two

09.00 - 09.15 Recap

09.15 - 12.00 Term Structure Models

• Introduction to Interest Rate Models
• Features of Interest Rate Models
  • One or two factors
  • No-arbitrage
  • Mean reversion
  • Spot or forward rates
• Equilibrium Models
  • Rendleman and Barter
  • Vasicek
  • Mean reversion in the Vasicek model
  • Term structures in the Vasicek Model
  • Cox, Ingersoll, & Ross (CIR)
  • General form of CIR
  • Mean reversion in the CIR model
  • Term structures in the CIR model
• Disadvantage of equilibrium models
• Examples and Exercises

12.00 - 13.00 Lunch

13.00 - 16.30 Term Structure Models (Continued)

• No-arbitrage Models
• Markov vs. non-Markov Models
• The Ho and Lee Model
• The BDT Model
  • General form
  • Deriving the model from zero curve and volatility structure
• The Hull-White Model
  • A general tree-building procedure
  • Building the tree – stage one
  • Calculating branching probabilities
  • Building the tree – stage two
• The Libor Market (BGM) Model
• Using Monte Carlo Simulation with Interest Rate Models
• Examples and Exercises

Day Three

09.00 - 09.15 Recap

09.15 - 12.00 Pricing Interest Rate Options Using Term Structure Models

• Pricing Options on Zero Coupon Bonds
• Pricing Options on Coupon-Bearing Bonds
• Pricing Libor Options
  • Interest Rate Guarantees
  • Caps and Floors
  • Swaptions
  • “Cancellation Swaps“
• Pricing Structured Interest Rate Products
  • “Capped FRNs“
  • “Inverse Floaters“
  • “Fairway Bonds“
• Pricing Exotic Structures
  • Captions, floptions and other compounds
  • Ratchet caps, sticky caps, and flexi caps etc.
• Examples and Exercises

12.00 - 13.00 Lunch

13.00 - 16.00 Pricing Callable and Defaultable Bonds

• General Characteristics of Callable Bonds
• Pricing Callable Bonds
  • Single-call and multiple-call bonds
• Prepayment Models
• Integrating Prepayment Models into an Interest Rate Model
• Option-adjusted analysis
  • OAS, OAY, Option-adjusted duration, Z-Spread
• Pricing Mortgage Bonds
  • MBS, CMO, IO, PO
• Pricing Defaultable Bonds
  • Incorporating credit spreads into term structure models
• Examples and Exercises

Evaluation and Termination of the Seminar