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Interest rate traders have been using the SABR model to price vanilla products for more than a decade. However this model suffers however from a severe limitation: its inability to value exotic products. A term structure model à la LIBOR Market Model (LMM) is often employed to value these more complex derivatives, however the LMM is unable to capture the volatility smile. A joint SABR LIBOR Market Model is the natural evolution towards a consistent pricing of vanilla and exotic products. Knowledge of these models is essential to all aspiring interest rate quants, traders and risk managers, as well an understanding of their failings and alternatives.
SABR and SABR Libor Market Models in Practice is an accessible guide to modern interest rate modelling. Rather than covering an array of models which are seldom used in practice, it focuses on the SABR model, the market standard for vanilla products, the LIBOR Market Model, the most commonly used model for exotic products and the extended SABR LIBOR Market Model. The book takes a hands-on approach, demonstrating simply how to implement and work with these models in a market setting. It bridges the gap between the understanding of the models from a conceptual and mathematical perspective and the actual implementation by supplementing the interest rate theory with modelling specific, practical code examples written in Python.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract
In the last two decades an extensive selection of books on interest rate or equity derivatives modelling became available through various publishers. We therefore take the opportunity to say a few words on the main texts out there and how our work complements the current literature.
Christian Crispoldi, Gérald Wigger, Peter Larkin

2. Interest Rate Derivatives Markets

Abstract
There are today, many different interest rates, and since this can be confusing for newcomers to the area, we will spend some time discussing the various types; we will concentrate only on a small number of them, which will form the building blocks for the derivatives markets. The money market has become the predominant source for providing liquidity funding for financial institutions, and allows them to manage their operational cash requirements up to a time horizon of one year. Various instruments exist, such as deposits, certificates of deposits, commercial papers, bills and repurchase agreements. To learn more about all these short-term instruments, the reader is referred to Choudhry (2010).
Christian Crispoldi, Gérald Wigger, Peter Larkin

3. Interest Rate Notions

Abstract
In this chapter we will lay the conceptual foundations on which to develop more advanced topics later in the book. Along with some fundamental quantities (zero-coupon bond, short rate, bank account, etc.) and concepts (measures and associated numeraires), we will be reviewing several basic instruments used for yield curve as well as volatility trading. We will also be presenting an introduction related to the fundamental curves needed for trading/risk-managing interest rate derivatives.
Christian Crispoldi, Gérald Wigger, Peter Larkin

4. Vanilla Models

Abstract
We have defined, in Section 3.4, how the value of caps/floors (as well as caplets/floorlets) depends on the future distribution of F k (t) under the T k -forward measure Q k associated with the numeraire P d (t,T k ). Equivalently, the swaption value depends on the future distribution of S m ,n(t) under the swap measure Qm.n associated with the numeraire A d ,m,n (t).
Christian Crispoldi, Gérald Wigger, Peter Larkin

5. SABR Model

Abstract
We have introduced in Chapter 4 the normal and lognormal models. They play an important role as they are intuitive, simple and their parameters can be adjusted quickly to obtain a price in agreement with the market. However, these simple models cannot be calibrated to more than one volatility per expiry. The authors of two papers, Derman and Kani (1994) and Dupire (1994), proposed a model, where the volatility is dependent on the current state of the underlying process. These so-called local volatility models could be calibrated, with some extra effort, to the entire volatility surface. One shortcoming of the local volatility models is that they predict a dynamic behaviour of the smile and the skew (the smile moves in the opposite direction as the underlying) that is different from what is observed in the market (where the smile moves in the same direction as the underlying).
Christian Crispoldi, Gérald Wigger, Peter Larkin

6. LIBOR Market Model

Abstract
In this chapter we draw our attention to the market models. We start by briefly discussing short rate models, with the goal of seeing where they fit in the story of interest rate modelling. We then make a natural transition from the more general short rate models to the Heath-Jarrow-Morton framework, then to the definition of the LIBOR Market Model (LMM).
Christian Crispoldi, Gérald Wigger, Peter Larkin

7. SABR LIBOR Market Model

Abstract
The extensive analysis carried out in Chapter 5 has shown how well the SABR model is able to capture the volatility dynamics characterizing the vanilla interest rate markets. Its major limitation is however the impossibility of modelling more than one forward rate at a time. This deficiency makes the SABR model impractical in the valuation of exotic interest rate derivatives, since these have payoffs which depend on a combination of several forward rates. While evaluating these payoffs, it is critical to simulate all relevant forward rates under a single measure to avoid arbitrage in the pricing model. Obviously, we have already achieved this by simulating multiple forward LIBOR rates in a market model as shown in Chapter 6. The market models are calibrated to the hedging vanilla instruments, one per expiry, whose payoff references the LIBOR forward rates or swap rates desired. As some of the complex products, such as range accruals or ratchets, depend on several strikes per expiry, we realize that these models suffer from an important shortcoming — they can only be calibrated to one strike per expiry and therefore cannot model the volatility smile characterizing today’s markets. Various attempts to overcome this limitation have been pursued by academics and practitioners, following two main directions.
Christian Crispoldi, Gérald Wigger, Peter Larkin

Backmatter

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