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Abstract:
Four papers introducing LIBOR market model (LMM) were published in 1997. They seemed to unify market practice with arbitrage-free framework - it came out that for one year only. The next year, after Russian crisis, cap and swaption markets started to show evident volatility smile and skew. Several attempts were made to capture that phenomenon into the arbitrage-free framework. Our note is strongly inspired by papers and conference talks by Mark Joshi and Riccardo Rebonato. We share their opinions that:

- Since smiles and skews are caused by different market features, it is more natural to model smile and skew separately, rather then to use unified framework of implied smile.

- Displaced Diffusion approach is easier in treatment then Constant Elasticity of Variance (CEV) approach for interest rate derivatives and gives the same modelling possibilities.

- Displaced Diffusion and Stochastic Volatility are perfectly suited to work together.

Since our attention is fixed more on swaptions then on caps/floors, we would like to opt for another version of the LIBOR market model with stochastic volatility and displaced diffusion (SVDDLMM) then Joshi and Rebonato:

- We use various random displacement factors for various LIBOR rates.

- For Stochastic Volatility we propose a new simple non mean reverting multi-lognormal model. We also try to convince the Reader that mean reversion in stochastic volatility models excludes correct modelling of long term options - swaptions are canonical example.

Easy closed form formulae are given for caps/floors and European swaptions what makes calibration procedure more effective and transparent - at least we are not "prisoners of Monte Carlo". We are able to calibrate model to various smile/skew shapes for caps/floors and swaptions with various length and of various maturities.

Libor market model, stochastic volatility, calibration

Abstract:
Constant maturity swaps can be regarded as generalizations of vanilla interest rate swaps. In a vanilla swap one exchanges the fixed swap rate against a floating LIBOR, which involves an interest rate relevant for that particular settlement period only. In a CMS swap this will be generalized. One will exchange the fixed legs against floating legs - usually the swap rate.

In this note we give a new (for our knowledge) approximate formula for convexity adjustment based on forward measure approach and LIBOR market model. This link is interesting itself - showing that convexity adjustment is model and calibration dependent.

Constant maturity swaps, forward measure, LIBOR market model

Abstract:
Pricing of European or even exotic (but without early exercise feature) interest rate and swap options in LIBOR market model can be easily performed in so called "Monsieur Jourdain approach". The general concept of "Monsieur Jourdain approach" is such, that all interest rate options without early exercise feature follow the Black-Merton-Scholes model. To this end we construct a set of "building blocks" consisting of volatilities of forward Libor rates and correlation parameters. A prescription for how to price a large class of instruments using forward Libor volatilities and the yield curve will be presented. Instantaneous volatilities are not used for model calibration and so the procedure is quite straightforward. In the paper we present three simple calibrations of the LIBOR market model useful in pricing.

Libor market model, calibration

Abstract:
There exist two classes of interest rate models. Short rate models (HW, CIR, BDT), easy in pricing and tough in calibration and forward rate models (HJM, BGM), easy in calibration and tough in pricing. Parameters in short rate models have no natural interpretation in terms of market volatility but many options can be priced on recombining trees. We find particularly inconvenient the procedure of fitting the initial yield curve - necessary for many short rate models. Parameters of forward rate models (especially BGM) have direct link to market volatility but there exists a common prejudice that recombining trees cannot be applied to forward rate models. This paper is an attempt to construct a model allowing both recombining trees and "calibration without programming". We would like to call both presented models "simplest possible term structure models" - at least we do not know any simpler model.

BGM model, HJM model, calibration, Bermudan swaptions, Brady bonds