New approaches to the valuation of American and barrier options

该项目涉及某些金融衍生品、美式期权和障碍期权的估值和对冲。障碍期权是一种衍生品，如果某一利率、资产价格或指数超过指定水平，该期权就会被激活或终止。股票的美式期权是一种合同，它赋予持有者在到期前的任何时候行使期权的权利，然后从卖方那里获得一定的付款，这可能取决于当时股票的价格。随着美式期权和障碍期权在金融市场上的广泛交易，美式期权和障碍期权的准确估值和套期保值是一个重要的问题。就其性质而言，美式期权和障碍期权的估值通常比标准欧式期权(如看涨期权和看跌期权)的估值更复杂：例如，即使在经典Black-Scholes模型的简化假设下，也没有已知的美式看跌期权价值的封闭形式解。因此，对于这些合约的估值和对冲，人们不得不求助于近似值。这项研究的目的是为新算法的发展做出贡献，并研究它们的数学性质，特别是在比经典的Black-Scholes模型更现实的定价模型的背景下。

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.