New approaches to the valuation of American and barrier options

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This project concerns the valuation and hedging of certain financial derivatives, American and barrier options. A barrier option is a derivative that is activated or extinguished if a certain rate, asset price or index crosses a specified level. An American option on a stock is a contract that gives its holder the right to exercise it at any time before expiry and receive then a certain payment from the seller, which may depend on the price of the stock at that moment. As American and barrier options are widely traded in financal markets, accurate valuation and hedging of American and barrier options are important issues.By their nature the valuation of American and barrier options is generally more involved than that of standard European options such as calls and puts: for example there is no known closed form solution for the value of an American put even under the simplifying assumptions of the classical Black-Scholes model. For valuation and hedging of these contracts one therefore has to resort to approximations. The aim of the proposed research is to make a contribution to the development of new algorithms and to investigate their mathematical properties, especially in the context of more realistic pricing models than the classical Black-Scholes model.