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Description:
Thesis by Besiana Rexhepi
Abstract It is well-known that the fair value of options can be determined by using the Black-Scholes model. However, for liquidly traded options, i.e. instruments for which the market price is known, there is clear evidence that the Black-Scholes model is not correct. This is reflected in the existence of the volatility smile phenomenon which is one the most challenging problems in financial economics. Recently, a rigorous analysis of the time evolution of the empirically observed volatility smile, i.e. smile dynamics, has been reported by (CdF02) and (Fen05). However, the quantification of the volatility smile dynamics as implied by smile-consistent models has not been done rigorously, so far. People have addressed this by looking at the evolution of the smile, based on asymptotic analysis and qualitative investigations. In this work, we use similar statistical techniques as employed in the empirical studies, to quantify the smile dynamics that is implied by the following smile-consistent models: Displaced Diffusion, Constant Elasticity of Variance (CEV) and SABR Stochastic Volatility Models. We find that in markets where options exhibit extreme skew, e.g. equity options markets, the displaced diffusion and CEV models should be used with care, since these models have poor fitting capabilities to market prices and impose inaccurate smile dynamics. The SABR model on the other hand, was shown to be able to capture the smile dynamics very closely to the empirically observed dynamics.