Option prices in the Black-Scholes model can usually be expressed as solutions of partial differential equations (PDE). In general exponential Levy models an additional integral term has to be added and the prices can be expressed as solutions of partial integro-differential equations (PIDE). The sensitivity of a price function to changes in its arguments is given by its derivatives, in finance known as greeks. The greeks can be obtained as a solution to a PDE or PIDE which is obtained by differentiating the equation and side conditions of the price function. We call the method of simultaneously solving the equations for the price function and the greeks the dynamic partial (-integro) differential approach. So far this approach has been analysed for a few contracts in the Black-Scholes model and in a Markov Chain model. In this thesis, we extend the use of the dynamic approach in the Black-Scholes model and apply it to a financial market where the underlying stock prices are driven by Levy processes. We derive and solve systems of equations that determine the price and the greeks both for vanilla and for exotic options. In particular we are interested in options whose prices depend only on time and one state variable. Furthermore, we calculate sensitivities of option prices with respect to changes in the stochastic model of the underlying price process. Such sensitivities can again be expressed as solutions to PIDE. The occurring systems of PIDE are solved numerically via a finite difference approach and the results are compared with simulation and numerical integration methods to compute prices and sensitivities. We show that the dynamic approach in many cases outperforms its competitors. Finally, we investigate the smoothness of the price functions and give conditions for the existence of solutions of the PIDE.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:645751 |
| Date | January 2008 |
| Creators | Gfeller, Adrian Urs |
| Publisher | London School of Economics and Political Science (University of London) |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://etheses.lse.ac.uk/2575/ |
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