University of Twente Student Theses


Local volatility modelling

Kamp, Roel van der (2009) Local volatility modelling.

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Abstract:Many different models exist that describe the behaviour of stock prices and are used to value options on such an underlying asset. This report investigates the local volatility model in which the volatility of the underlying asset is assumed to be a deterministic function of both time and the underlying asset price. First the report considers how the local volatility surface can be extracted from market data for option prices. Theoretically this can be achieved by Dupire's formula, but it appears that in practice it is better to transform this equation so that the local volatility surface can be extracted from the implied volatilities. To fit the implied volatility surface to market data smoothed thin plate splines are used. Secondly a pricing mechanism has to be devised to value options using the local volatility surface. For this trinomial trees are used. The classical tree model is adjusted to make it work properly in the presence of local volatility, particularly to avoid the occurrence of negative transition probabilities. The method is quick and can easily incorporate discrete dividend payments. The accuracy of this method is verified for European and American options. Prices generated for European options are compared to Black-Scholes prices and prices for American options are compared to prices generated by Monte Carlo simulations. It is shown that the model works accurately for both European and American options. Finally the model is tested on real market data. The prices generated by the local volatility method are not always within the bid-ask spread of the market. Since the implied volatilities were extracted from the market data by inverting a different pricing mechanism, this shows there is non-neglible difference between these two methods. Also the stability of the local volatility surface and delta hedging are considered. On the basis of the analysis of the data used, no definitive statement can be made on the performance of the delta hedges suggested by the local volatility model compared to delta's suggested by the implied volatilities.
Item Type:Essay (Master)
AllOptions BV
Faculty:EEMCS: Electrical Engineering, Mathematics and Computer Science
Subject:31 mathematics
Programme:Applied Mathematics MSc (60348)
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