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Description:
Thesis by Simona Svoboda-Greenwood
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 significantly 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 quantified, 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.