Parisian Options - The Implied Barrier Concept

摘要

Research into the direction of specific exotic options - like the Parisians - is often driven by the analysis of structured products. These products contain features that are similar to exotic options. Exchange-trading of the pure exotics is very rare. In the period of rising stock markets, investors were less interested in buying bonds. In order to regain their interest, firms added extra features to the bonds they wanted to issue. One of these features is the right of the bond holder to convert the bond into a given number of stocks under certain conditions. Bonds with this feature are called convertible bonds and are nowadays very common. Most convertible bonds can be re-called by the issuer when the convertible trades above some level for some period. Modelling this feature corresponds to valuation of a Parisian option. In this paper we will point out how we quickly can approximate the Parisian option price by using a standard barrier option with a modified barrier. This is common practice for traders; they increase or decrease the barrier a bit. Here we want to argue what that bit should be. First we will introduce the Parisian contract. Thereafter we list the methods of valuing the Parisian, followed by a section about the implied barrier method. Here we will use concepts from the theory on Brownian excursions and exploit them to derive prices for Parisians that are already in the excursion. We will conclude with a numerical example.