Abstract: The LMM is an effective framework for the pricing of interest rate derivatives, not least because it models observable market quantities. In its lognormal form, calibration to market implied volatilities is intuitive and fast. The amendments required to incorporate a monotonically decreasing implied volatility skew are fairly straightforward and do not signi¯cantly reduce the ease and speed of calibration. However, the incorporation of a full implied volatility smile is significantly more challenging,from both a mathematical and computational perspective.
There exist three main techniques for incorporating a volatility smile/skew in any modelling framework: allowing a local volatility function, stochastic volatility and jump dynamics. In this thesis various ways to incorporate smile/skew are studied, loosely based on the above three approaches.
Both the constant-elasticity-of-variance and displaced-diffusion processes give rise to an implied volatility skew. In fact it has been experimentally shown that, for a certain parameterisation, the two processes produce closely matching prices for European call options over a variety of strikes and maturities. Here, this similarity in prices is analytically quanti¯ed, not only via an asymptotic expansion of the call prices, but also via expansion of the conditional probability density functions and a comparison of the raw and central moments of the two distributions.
The regime shifting model of Rebonato and Kainth (2004) may be viewed as a reduced form of a full stochastic volatility model. Rather than attempting to replicate an implied volatility smile, Rebonato and Kainth propose this model to explain the dynamics of the matrix of swaption at-the-money volatilities. Their model is used as the basis for a two state, continuous time Markov Chain model, characterised by a time dependent volatility in each state. This model is calibrated to a time-series of at-the-money swaption implied volatilities to asses its ability to replicate the changes in swaption matrix shape.
Finally, the Levy LIBOR model is considered as a generalisation of jump processes. Using the Levy LIBOR framework of Eberlein and Ozkan (2005), a new model for the pricing of swaptions is developed. This model represents a significant improvement in tractability and computational speed to that suggested by Kluge (2005). Using the variance gamma process of Madan and Seneta (1990), a pure jump Levy process, the model is calibrated to market swaption smile data. The model has shortcomings in its ability to fit data across expiry and maturity, however this is a feature shared by all jump models; the results are consistent with results obtained by Das and Sundaram (1999) for the case of general jump models.

Abstract: In this paper, using a geometric method introduced in [12] and initiated by [4], we derive an asymptotic swaption implied volatility at the ¯rst-order for a general stochastic volatility Libor Market Model. This formula is useful to quickly calibrate a model to a full swaption matrix. We apply this formula to a speci¯c model where the forward rates are assumed to follow a multi-dimensional CEV process correlated to a SABR process. For a caplet, this model degenerates to the classical SABR model and our asymptotic swaption implied volatility reduces naturally to the Hagan-al formula [11]. The geometry underlying this model is the hyperbolic manifold Hn+1 with n the number of Libor forward rates.

Abstract: Does the World Need Securitization?
Yes, and Six Actions to Restart the Market
- About $8.7 trillion of assets are currently funded via securitization, with banks being the main issuers. Any new debt issuance will be minimal and short term. In the near term, banks will attempt to use government-backed debt to substitute for securitized debt.
- This still leaves many legacy assets stuck on balance sheets, with spreads at all-time wides. As this securitized leverage matures with no replacement, global economies will be forced to contract.
- To us, securitization remains a viable concept and should be a tool to stabilize this crisis. However, for securitization to come back, it will have to revert to basics, simplicity, and transparency.
- We discuss six necessary steps to see securitization start funding markets. Government actions such as TALF can play a key role. Re-instilling confidence in ratings will be part of the challenge, while several remaining accounting issues such as guidance on valuations in inactive markets will also be critical for the market.
- The securitized products recovery will be evolutionary, but if the economy deteriorates, we expect investors to eventually resume an appreciation for secured assets. Over time, banks and corporates will have fewer but stronger assets to securitize.