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The analysis of structured fi nancial products often leads to the pricing of exotic options. For instance, consider a re-callable convertible bond. The holder typically wants to recall the bond if ever the underlying stock has been traded above or below a given level for too long. Such a contract can be modelled with the help of Parisian options. Parisian options are barrier options that are activated or cancelled depending on the type of option if the underlying asset stays above or below the barrier long enough in a row. Parisian options are far less sensitive to infl uential agent on the market than standard barrier options. It is quite easy for an agent to push the price of a stock momentarily but not on a longer period so that it would affect the Parisian contract. In this work, we study the pricing of European style Parisian options using Laplace transforms. Some other methods have already been proposed. On path dependent options, crude Monte Carlo techniques do usually not perform well. An improvement of this strategy using sharp large deviation estimates was proposed by Baldi et al. (2000). Techniques using a two dimensional partial differential equation have also drawn much attention, see for instance the works of Avellaneda and Wu (1999), Haber et al. (1999), or Wilmott (1998). The PDE approach is quite fl exible and could even be used for American style Parisian option but the convergence is rather slow, which is badly suited for real time evaluation. In a quite similar state of mind, tree methods based on the framework of Cox et al. (1979) were investigated by Costabile (2002). An original concept of implied barrier was developed by Anderluh and van der Weide (2004), the idea is to replace the Parisian option by a standard barrier option with a suitably shifted barrier. The idea of using Laplace transforms to price Parisian options was introduced by Chesney et al. (1997).

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Extrait de la revue BMI 99


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