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## Über dieses Buch

The Libor Market Model (LMM) is a mathematical model for pricing and risk management of interest rate derivatives and has been built on the framework of modelling forward rates. For the conceptual understanding of the model a strong background in the fields of mathematics, statistics, finance and especially for implementation, computer science is necessary. The book provides the ne cessary groundwork to understand the LMM and delivers a framework to implement a working model where possible calibration and parameterization methods for volatility and correlation are explained. Special emphasis lies also on the trade off of speed and correctness where differences in choosing random number generators and the advantages of factor reduction are shown.

## Inhaltsverzeichnis

### 1. Introduction

Quantitative Finance is a topic which has become quite popular in the last decade, which combines techniques out of the disciplines of mathematics, finance, statistics and computer science. Risk Management and its quantitative applications in financial institutions has become a very important topic which is enforced through regulatory topics especially Basel III for the banking industry and Solvency II for the insurance industry.
Christoph Hackl

### 2. Foundations of Mathematical Finance and Stochastic Calculus

This chapter presents the necessary foundations to understand the mathematical, financial and computational aspects behind this model. In the first section we start with simple interest rate necessities and go on to financial Derivatives which are necessary to understand to correctly calibrate and use the model for pricing. The section 2.2 starts with the most important aspects in stochastic calculus which is the key step to understand and work with stochastic differential equations. One additional topic in this section is the no-arbitrage Pricing which are a prerequisite to understand modern option pricing theory. The final section in this chapter gives the reader an overview about the computational aspects which are important to build this model.
Christoph Hackl

### 3. The Libor Market Model

L j (t) is a martingale , as stated in Definition 2.2.8, under the T j forward measure $${\mathbb{Q}^{{T_j}}}$$, where we use the shorthand notation L j (t) for L(t, T j−1, T j ), j=1,...,N. We assume that the differential of L j (t) follows the following driftless SDE:
$${{d}}{{{L}}_j}{{(t) = }}{\sigma _{j}}{(t)}{{L}_{j}}{(t)d}{{W}^{j}}{(t)\; for\; t } \le {{T}_{{{j - 1}}}}$$
(3.1)
Christoph Hackl

### 4. Volatility and Correlation in the Libor Market Model

In this chapter, two calibration possibilities for the LMM are presented, where the first is the direct calibration to market data and the second is a Rebonato's popular linear exponential parametric function, see Brigo and Mercurio [2006]. To calibrate the LMM directly to market data, the volatility curve has to be "bootstraped", as L j (t) is modeled and therefore each caplet on its own.
Christoph Hackl

### 5. Applications and Results

The first Figure 5.1 shows the market cap volatility structure with the characteristic hump at the beginning where cubic spline interpolation has been used between the market volatility points. The dashed line is the stripped caplet volatility which is used to calibrate the libor market model for pricing caps.
Christoph Hackl

### 6. Conclusion

The aim of the first chapters was to provide the building blocks which are necessary to build up a working state of the art interest rate derivatives pricing engine. Especially for deriving the LMM drifts, the stochastic calculus part is a necessity. The intention of the following chapters was to provide good calibration and parameterization methods to "bring the model to life". In the chapter where we have presented the calibration and finally the pricing results, all major outcomes starting with the calculated volatility values, volatility model parameters, factor reduced correlation matrices up to the validated cap and swaption prices with its market price differences are provided. The presented calibration and parameterization methods for the LMM deliver valid results for interest rate derivatives pricing.
Christoph Hackl

### Backmatter

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