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2014 | OriginalPaper | Buchkapitel

4. The LIBOR Market Model: A Markov-Switching Jump Diffusion Extension

verfasst von : Lea Steinrücke, Rudi Zagst, Anatoliy Swishchuk

Erschienen in: Hidden Markov Models in Finance

Verlag: Springer US

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Abstract

This paper demonstrates how the LIBOR Market Model of Brace et al. (Math Financ 7(2):127–147, 1997) and Miltersen et al. (J Financ 52(1):409–430, 1997) may be extended in a way that not only takes into account sudden market shocks without long-term effects, but also allows for structural breaks and changes in the overall economic climate. This is achieved by substituting the simple diffusion process of the original LIBOR Market model by a Markov-switching jump diffusion. Since interest rates of different maturities are modeled under different (forward) measures, we investigate the effects of changes between measures on all relevant quantities. Using the Fourier pricing technique, we derive pricing formula for the most important interest rate derivatives, caps/caplets, and calibrate the model to real data.

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Vorheriges Kapitel An Econometric Model of the Term Structure of Interest Rates Under Regime-Switching Risk

Nächstes Kapitel Exchange Rates and Net Portfolio Flows: A Markov-Switching Approach

Intuitively speaking, the $\mathcal{H}$-compensator measure ${\nu }^{\mathcal{H}}$ contains all the distributional information related to the random measure μ under $\mathcal{H}$. In more precise terms: Assuming that μ satisfies certain regularity conditions, ν is defined as the a.s. unique predictable random measure with the following property: For any predictable stochastic process $f:\varOmega \times \left [0,{T}^{{\ast}}\right ] \times E \rightarrow \mathbb{R}$ with | f | ∗μ an increasing, locally integrable process, the process M, $M\left (\omega,t\right )\,:=\,\int _{0}^{t}\int _{E}f(\omega,s,z)\mu (\omega,ds,dz) -\int _{0}^{t}\int _{E}f(\omega,s,z)\nu (\omega,ds,dz)$ is a (local) martingale with respect to $\mathcal{H}$ [38].

The introductory Sect. 4.3 is mainly based on Brigo and Mercurio [11], Filipovic [20], Rebonato [33] and Zagst [42].

Most authors substitute the accurate term “forward LIBOR rate” for $L_{i}\left (t\right ) = L\left (t,T_{i},T_{i+1}\right )$ by the more convenient shortened expression “LIBOR rate”. Strictly speaking, this is only appropriate when t = T _i, but since no great confusion should be expected, we will also follow this convention.

At least in the case, where no jumps are considered, this is a justifiable assumption, when it comes to caplet pricing. Similar to pricing formulas (4.10) and (4.11) developed in the log-normal LMM, our considerations will show that prices depend only on the norm of σ _i, that is $\Vert \sigma _{i}\left (t,X_{t-}\right )\Vert$, and not on $\sigma _{i}\left (t,X_{t-}\right )$ itself. As underlined, e.g., by Filipovic [20], p. 213, there is thus no gain in flexibility for caplet pricing by introducing additional dimension into the model. Note, nonetheless, that this is no longer true for swaption pricing.

Given these assumptions, the specified model can be seen as a simple, straight-forward generalization to the one-dimensional LIBOR model of Brace et al. [10], where not only the volatility, but also parameters characterizing jumps and structural breaks in the market can be derived in the calibration procedure. The reduction to one-dimensional spaces allows to significantly reduce the amount of parameters to be fitted. All challenges met and a more detailed explanation on the necessity of employing a multi-step iterative procedure may be found in Steinrücke et al. [40].

Observe that the bootstrapping procedure assumes constant volatilities for caplets in between caps of succeeding maturity. For example, for t = 0, the (identical) volatilities σ _1y for caplets with maturities 1y, 1.25y, 1.5y and 1.75y are derived based on the formula

$$\displaystyle\begin{array}{rcl} \mbox{ Cap}\left (0, 2\mbox{ y},K_{2\mbox{ y}}\right ) -\mbox{ Cap}\left (0, 1\mbox{ y},K_{1\mbox{ y}}\right )& =& \mbox{ Caplet}\left (0, 1\mbox{ y},\sigma _{1\mbox{ y}},K_{2\mbox{ y}}\right ) + \mbox{ Caplet}\left (0, 1.25\mbox{ y},\sigma _{1\mbox{ y}},K_{2\mbox{ y}}\right ) {}\\ & & +\mbox{ Caplet}\left (0, 1.5\mbox{ y},\sigma _{1\mbox{ y}},K_{2\mbox{ y}}\right ) + \mbox{ Caplet}\left (0, 1.75\mbox{ y},\sigma _{1\mbox{ y}},K_{2\mbox{ y}}\right )\,{}\\ \end{array}$$

where $\mbox{ Caplet}\left (0,T_{j},\sigma _{1\mbox{ y}},K_{2\mbox{ y}}\right )$ denotes the price of a caplet at t = 0 with maturity T _j, expiry T _j+1 and ATM cap strike K _2y for maturity 2y, evaluated by the Black [8]-formula for constant volatility σ _1y. For details, see, e.g., Filipovic [20], p.215 seqq.

The variance of the ergodic distribution π of the generated Markov chain X is given as ${\varSigma }^{2} = \mbox{ Var}_{\pi }\left (X_{1}\right ) + 2\sum _{t=2}^{\infty }\mbox{ Cov}_{\pi }\left (X_{1},X_{t}\right )$. For details on the estimation of the standard error, $\hat{\varSigma }/(N -\mbox{ burn-in})$ in the MCMC context, see, e.g., Flegal [21], p. 59 seqq.

In favor of computation time we only considered every tenth day in the available time series for the calibration of the remaining parameters.

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Titel: The LIBOR Market Model: A Markov-Switching Jump Diffusion Extension
verfasst von: Lea Steinrücke
Rudi Zagst
Anatoliy Swishchuk
Verlag: Springer US
Buch: Hidden Markov Models in Finance
Print ISBN: 978-1-4899-7441-9

Electronic ISBN: 978-1-4899-7442-6

Copyright-Jahr: 2014
DOI: https://doi.org/10.1007/978-1-4899-7442-6_4