Introduction

In my eyes, interest rate modeling is one of the most interesting areas of mathematical finance. Interest rate (IR) models are more complex than their simple equity counterparts. In the IR world, the underlying variable is an entire term structure of interest rates, an unobservable quantity. Compare this to the equity world, where the underlying is often a single stock. IR models also lay claim to a wider set of instruments, both in the underlying and in the derivatives space. These instruments range from simple government bonds through to exotic derivatives such as target redemption notes (TARNs). Accordingly, IR modeling lays claim to the most abundant theoretical literature on pricing and hedging, and a fascinating history of how these models came into existence.

Beyond the theoretical merits of focusing on IR modeling, it's also the area in which I work. I sit on the interest rates derivative trading desk at Barclays, building and extending in-house models that the desk use for pricing and risk management. Accordingly, I feel comfortable weighing in on the existing literature and research from the perspective of a practitioner.

The literature

The interest rate modeling literature is vast. There have been thousands of academic papers written in the field, from the early work of Fischer Black on the Black model of 1976 and the Heath-Jarrow-Morton framework of 1987, to the emergence of the Libor Market Model published in the work of Brace, Gatarek and Musiela in 1997. More recent developments include the SABR volatility model, published in a paper by Hagan, Kumar, Lesniewski and Woodward in 2002, and post-financial-crisis work on interest rate modeling under a multi-curve framework.

There a few interest rate modeling books that really stand-out in the field from a practitioner's point of view. These are:

1. Interest Rate Option Models, Rebonato
2. Interest Rate Models - Theory and Practice, Brigo and Mercurio
3. Interest Rate Modeling I, II & III, Andersen and Piterbarg

I've listed these in increasing order of quality and usefulness. In my eyes, the Andersen and Piterbarg trilogy are all you could possibly need in order to start and maintain a successful interest rate derivatives quant operation at a professional level. In my work, I focus on these books, dipping into Brigo and Mercurio whenever a more detailed look on some of the more technical aspects is required.

Beyond these big three of the IR modeling literature, I highly recommend a couple of texts covering the more general volatility modeling space, these are:

1. The Volatility Surface, Gatheral
2. The Volatility Smile, Derman and Miller
3. Dynamic Hedging, Managing Vanilla and Exotic Options, Taleb

Tutorial structure

In the following, I will follow the structure of Andersen and Piterbarg's trilogy of interest rate modeling, assuming that a reader has a basic understanding of stochastic calculus, Monte Carlo methods and PDEs. My aim is to p present these topics in a concise and digestible way, while still maintaining the key concepts and ideas. The structure of the tutorials is as follows:

1. Fixed Income Notation
2. Fixed Income Probability Measures
3. Multi-Currency Markets
4. The HJM Framework
5. Basic Fixed Income Instruments
6. Caps, Floors and Swaptions
7. CMS Swaps, Caps and Floors
8. Bermudan Swaptions
9. Structured Notes and Exotic Swaps
10. Callable Libor Exotics and TARNs
11. Volatility Derivatives
12. Yield Curves and Curve Construction
13. Managing Yield Curve Risk
14. Vanilla Models With Local Volatility
15. Vanilla Models With Stochastic Volatility
16. Introduction to Term Structure Models
17. One-Factor Short Rate Models I
18. One-Factor Short Rate Models II
19. Multi-Factor Short Rate Models
20. The Libor Market Model I
21. The Libor Market Model II
22. Risk Management and Sensitivity Computations
23. P&L Attribution
24. Payoff Smoothing
25. Vegas in the Libor Market Model

Notation

• $\mathcal{F}(t)$: Filtration of a $\sigma$-algebra
• $\mathbb{P}$: Physical probability measure
• $\mathbb{Q}$: Risk-neutral probability measure
• $\mathbb{Q}^N$: Measure for numeraire $N$
• $\mathbb{Q}^T$ or $\mathbb{Q}^n$: Forward measure for date $T$ or $T_n$
• $\mathbb{E}_t^N[\cdot]$: Conditional expectation with respect to $\mathcal{F}(t)$ under measure $N$
• $T_0 < \cdots < T_N$: Tenor structure
• $\beta(t)$: Continuously compounded money market account
• $P(t,T)$: Zero-coupon bond (ZCB) price at time $t$ for maturity $T$
• $P(t,T,S)$: Forward bond price at time $t$ for delivery of $S$-maturity ZCB on date $T \in [t,S]$
• $y(t,T,S)$: Continuously compounded forward yield at time $t$ for the period $[T, S]$
• $f(t,T)$: Instantaneous forward rate at time $t$ for maturity $T$
• $r(t)$: Short rate at time $t$
• $L(t,T,S)$: Forward Libor rate at time $t$ for the period $[T,S]$
• $L_n(t)$: Forward Libor rate at time $t$ for the period $[T_n, T_{n+1}]$, i.e. $L_n(t) = L(t, T_n, T_{n+1})$
• $S_{n,m}(t)$: Forward swap rate at time $t$, starting at date $T_n$ with final payment on date $T_{n+m}$
• $A_{n,m}(t)$: Annuity at time $t$ with first payment date $T_{n+1}$ and final payment date at $T_{n+m}$
• $U_n(t)$: The $n$th exercise value of a Bermudan swaption or callable Libor Exotic
• $\sigma_B(t,S;T,K)$: Implied Black volatility smile, parameterised by the time $t$ spot $S$, strike $K$ and expiry $T$
• $c_B(t,S; T, K, \sigma)$: Price of a call option in the Black model with time $t$ spot $S$, strike $K$, expiry $T$ and Black volatility $\sigma$
• $c_N(r,S;T, K, \sigma)$: Price of a call option in the normal (Bachelier) model with time $t$ spot $S$, strike $K$, expiry $T$ and normal volatility $\sigma$