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Abstract:
This work focuses on the swaptions automatic cascade calibration algorithm (CCA) for the LIBOR Market Model (LMM) first appeared in Brigo and Mercurio (2001). This method induces a direct analytical correspondence between market swaption volatilities and LMM parameters, and allows for a perfect recovery of market quoted swaption volatilities if a common industry swaptions approximation is used.

We present explicitly an extension of the CCA to calibrate the entire swaption matrix rather than its upper triangular part. Then, while previous tests on earlier data showed the appearance of numerical problems, we present here different calibration cases leading to acceptable results. We analyze the characteristics of the configurations used and concentrate on the effects of different exogenous instantaneous historical or parametric correlation matrices.

We also investigate the influence of manipulations in input swaptions data for missing quotes, and devise a new algorithm maintaining all the positive characteristics of the CCA while relying only on directly quoted market data. Empirical results on a larger range of market situations and instantaneous covariance assumptions show this algorithm to be more robust and efficient than the previous version. Calibrated parameters are in general regular and financially satisfactory, as confirmed by the analysis of various diagnostics implied structures.

Finally we Monte Carlo investigate the reliability of the underlying LMM swaption analytical approximation in the new context, and present some possibilities to include information coming from the semi-annual tenor cap market.

Libor Market Model, swaptions, calibration, cascade calibration

Abstract:
The SABR closed-form formula (Hagan et. al 2002) is the standard for smile-consistent pricing in the swaption market. Here we address the issue of turning SABR assumptions into a consistent and arbitrage-free term structure model in the BGM/Libor Market Model framework. We compute the joint dynamics followed by Libor rates and stochastic volatility of SABR kind under the general pricing measures used for interest rate derivatives, and we observe that the volatility dynamics is non-standard. Based on the analysis of the equation found, we develop and justify theoretically a few approximations aimed at making these no-arbitrage dynamics compatible with the use of the SABR closed-form formula. Then the formulas developed above are confronted both with alternative numerical implementations and with market data. We verify that the formulas for no-arbitrage corrections are acceptably precise, maintain good fitting, and produce regular Libor parameters. Finally we verify that the no-arbitrage corrections to the volatility dynamics make the out-of-calibration-sample prices implied by the model closer to market quotations, compared to prices implied by a trivial multivariate SABR neglecting such corrections.

stochastic volatility, SABR, no-arbitrage,libor market model, BGM

Abstract:
In this work we consider three problems of the standard market approach to the pricing of credit index options: the definition of the index spread is not valid in general, the payoff considered leads to a pricing which is not always defined, and the candidate numeraire to define a pricing measure is not strictly positive, which would lead to an inequivalent pricing measure. We give to the three problems a general mathematical solution, based on a novel way of modelling the flow of information through the definition of a new subfiltration. Using this subfiltration, we take into account consistently the possibility of default of all names in the portfolio, that is neglected in the standard market approach. We show that, while this mispricing can be negligible for standard options in normal market conditions, it can become highly relevant for different options or in stressed market conditions.

In particular, we show on 2007 market data that after the subprime credit crisis the mispricing of the market formula compared to the no arbitrage formula we propose has become financially relevant even for the liquid Crossover Index Options.

credit option, subprime, correlation, market models, arbitrage

Abstract:
The behaviour of a smile model when applied to hedging should be consistent with market evidence that asset prices and market smiles move in the same direction (Hagan et al. 2002). Local volatility models are criticized because not consistent with this desired behaviour, and this has been an important driver towards the use of stochastic volatility models.

In this work we perform a simple analysis showing that, if we take into account explicitly the correlation between stochastic volatility and underlying asset which is typical of the most common stochastic volatility models, the hedging behaviour of stochastic volatility models does not always conform with the desired behaviour of a smile model in hedging.

With further simple tests we show that the behaviour of local volatility and stochastic volatility models calibrated to market skew is less different than assumed in current market wisdom. Both approaches, when used consistently with model assumptions, do not show the desired behaviour in hedging, while for both models the desired behaviour is obtained in market practice by hedging techniques which are not fully consistent with rigorous model assumptions.

hedging, local volatility, stochastic volatility, sabr

Abstract:
We present a simple methodology to guarantee that the total correlation structure in a Term Structure Model with one stochastic volatility factor remains positive semidefinite. We design the parameterization with the purpose of keeping as much freedom as possible for the correlation of interest rates and stochastic volatility, while letting the correlation among forward rates reproduce approximately the tendencies usually considered desirable in the market.

correlation, stochastic volatility, libor market model

Abstract:
Different anomalies have appeared in the interest rate market after the burst of the credit crunch. A wide wedge has opened between the market quotes of Forward Rate Agreements and their standard spot Libor replication, and large Basis Spreads have appeared for exchanging floating payments with different tenors. Here we tackle these issues under two aspects.

In Part 1 we focus on issues of direct interest to market practitioners. We show that the gap between FRA rates and their spot Libor replication can be explained by using the quoted Basis spreads. Then we explain the market patterns of the Basis spreads by modeling them as options on the credit worthiness of the counterparty. We also investigate analytically the FRA market payoff.

In Part 2 we study the mathematical representation of the interest rate market in the post-crisis reality. We introduce credit risk at market level, allowing for no-fault standard rule and collateralization. We use subfiltrations to model Libor rates, which now embed relevant credit risk although no default event is possible on Libor itself. We compute change of numeraire and convexity adjustments for collateralized derivatives tied to risky Libor.

basis swaps, FRA, credit crisis, counterparty risk, multicurve term structure modelling, liquidity