Efficient and accurate log-Lévy approximations of Lévy-driven LIBOR models

Papapantoleon, Antonis; Schoenmakers, John; Skovmand, David

doi:10.21314/jcf.2012.250

Cited by 7 publications

(12 citation statements)

References 31 publications

(42 reference statements)

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“…This approximation is widely known to be unreliable in many realistic settings; cf. Papapantoleon, Schoenmakers, and Skovmand (2012) and the references therein.…”

Section: The Multiple Curve Affine Libor Modelmentioning

confidence: 99%

Affine LIBOR Models with Multiple Curves: Theory, Examples and Calibration

Grbac¹,

Papapantoleon²,

Schoenmakers³

et al. 2015

SIAM J. Finan. Math.

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Abstract. We introduce a multiple curve framework that combines tractable dynamics and semi-analytic pricing formulas with positive interest rates and basis spreads. Negatives rates and positive spreads can also be accommodated in this framework. The dynamics of OIS and LIBOR rates are specified following the methodology of the affine LIBOR models and are driven by the wide and flexible class of affine processes. The affine property is preserved under forward measures, which allows us to derive Fourier pricing formulas for caps, swaptions and basis swaptions. A model specification with dependent LIBOR rates is developed, that allows for an efficient and accurate calibration to a system of caplet prices.

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“…This approximation is widely known to be unreliable in many realistic settings; cf. Papapantoleon, Schoenmakers, and Skovmand (2012) and the references therein.…”

Section: The Multiple Curve Affine Libor Modelmentioning

confidence: 99%

Affine LIBOR Models with Multiple Curves: Theory, Examples and Calibration

Grbac¹,

Papapantoleon²,

Schoenmakers³

et al. 2015

SIAM J. Finan. Math.

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“…Moreover, they offer a theoretical justification for the “frozen drift” approximation as the zero‐order Taylor expansion. In related work, Papapantoleon, Schoenmakers, and Skovmand (2012) have developed log‐Lévy approximations for the Lévy LIBOR model, thus allowing for accurate very long stepping in the Monte Carlo simulation.…”

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confidence: 99%

The Affine Libor Models

Keller‐Ressel

Papapantoleon

Teichmann

2012

Mathematical Finance

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We provide a general and flexible approach to LIBOR modeling based on the class of affine factor processes. Our approach respects the basic economic requirement that LIBOR rates are nonnegative, and the basic requirement from mathematical finance that LIBOR rates are analytically tractable martingales with respect to their own forward measure. Additionally, and most importantly, our approach also leads to analytically tractable expressions of multi-LIBOR payoffs. This approach unifies therefore the advantages of well-known forward price models with those of classical LIBOR rate models. Several examples are added and prototypical volatility smiles are shown. We believe that the CIR process-based LIBOR model might be of particular interest for applications, since closed form valuation formulas for caps and swaptions are derived. KEY WORDS: LIBOR rate models, forward price models, affine processes, analytically tractable models. 627 628 M. KELLER-RESSEL, A. PAPAPANTOLEON, AND J. TEICHMANN THE AFFINE LIBOR MODELS 629 AXIOMSLet us denote by L(t, T) the time-t forward LIBOR rate that is settled at time T and received at time T + δ; here T denotes some finite time horizon. The LIBOR rate is related to the prices of zero coupon bonds, denoted by B(t, T), and the forward price, denoted by F(t, T, T + δ), by the following equations:One postulates that the LIBOR rate should satisfy the following axioms, motivated by economic theory, arbitrage pricing theory, and applications.AXIOM 2.1. The LIBOR rate should be nonnegative, i.e., L(t, T) ≥ 0 for all 0 ≤ t ≤ T.AXIOM 2.2. The LIBOR rate process should be a martingale under the corresponding forward measure, i.e., L(·, T) ∈ M(P T+δ ). AXIOM 2.3. The LIBOR rate process, i.e., the (multivariate) collection of all LIBOR rates, should be analytically tractable with respect to as many forward measures as possible. Minimally, closed-form or semi-analytic valuation formulas should be available for the most liquid interest rate derivatives, i.e., caps and swaptions, so that the model can be calibrated to market data in reasonable time.Furthermore we wish to have rich structural properties: that is, the model should be able to reproduce the observed phenomena in interest rate markets, e.g., the shape of the implied volatility surface in cap markets or the implied correlation structure in swaption markets. EXISTING APPROACHESThere are several approaches to LIBOR modeling developed in the literature attempting to fulfill the axioms and practical requirements discussed in the previous section. We describe later the two main approaches, namely the LIBOR market models and the forward price model, and comment on their ability to fulfill them. We also briefly discuss Markov-functional models. forward LIBOR rate is modeled as an exponential Brownian motion under its corresponding forward measure. This model provides a theoretical justification for the common market practice of pricing caplets according to Black's futures formula (Black 1976), i.e., assuming that the forward LIBOR rate is log-no...

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“…where ϕ disp j+1 is given by (21) and we recall that v j (0) = θ j . The integrand in (22) decays with order z −2 if |z| → ∞, which is rather slow from a numerical point of view.…”

Section: Carr and Madan Inversion Formulamentioning

confidence: 99%

Libor model with expiry-wise stochastic volatility and displacement

Ladkau

Schoenmakers

Zhang

2013

IJPAM

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We develop a multi-factor stochastic volatility Libor model with displacement, where each individual forward Libor is driven by its own square-root stochastic volatility process. The main advantage of this approach is that, maturity-wise, each square-root process can be calibrated to the corresponding cap(let)vola-strike panel at the market. However, since even after freezing the Libors in the drift of this model, the Libor dynamics are not affine, new affine approximations have to be developed in order to obtain Fourier-based (approximate) pricing procedures for caps and swaptions. As a result, we end up with a Libor modelling package that allows for efficient calibration to a complete system of cap/swaption market quotes that performs well even in crises times, where structural breaks in vola-strike-maturity panels are typically observed.Reference to this paper should be made as follows: Ladkau, M., Schoenmakers, J. and Zhang, J. (2013) 'Libor model with expiry-wise stochastic volatility and displacement', Int.

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Efficient and accurate log-Lévy approximations of Lévy-driven LIBOR models

Cited by 7 publications

References 31 publications

Affine LIBOR Models with Multiple Curves: Theory, Examples and Calibration

Affine LIBOR Models with Multiple Curves: Theory, Examples and Calibration

The Affine Libor Models

Libor model with expiry-wise stochastic volatility and displacement

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