The use of implied methodologies in mathematical finance
This thesis has as an objective to explore the uses of implied methodologies in the area of Mathematical Finance. The existing literature broadly separates the ways that implied methodologies can be exploited in to two different categories; for purposes of recovery of the market sentiment and for consistent pricing of exotic and derivatives. I explore and exploit the first use by examining the possibility of using an implied distribution of a mixture of two lognormal distributions in order to predict the macroeconomic event of the sterling pound's exit from the ERM in 1992. Market evidence presented, indicate that the market was indeed expecting a sterling devaluation a few days prior to the exit. Furthermore, the two component lognormal mixtures distribution is proven to be a powerful tool for assessing the market sentiment, especially when there are two possible scenarios for the future movement of the underlying asset. Subsequently, I am using a jump diffusion stochastic process with a Bernoulli distributed jump component, the parameters of which are implicitly derived from observed option prices, for the pricing of exotic options. Closed form valuation expressions are provided within this generalised approach for Asian and Basket options. Furthermore, analytical formulae for the hedging parameters of those exotic products are derived. Monte Carlo simulation confirms the validity of all the results presented in this thesis.