We present a closed pricing formula for European options under the Black-Scholes model and formulas for its partial derivatives. The formulas are developed making use of Taylor series expansions and by expressing the spatial derivatives as expectations under special measures, as in Carr,...

In Foreign Exchange Markets vanilla and barrier options are traded frequently. The market standard is a cutoff time of 10:00 a.m. in New York for the strike of vanillas and a knock-out event based on a continuously observed barrier in the inter bank market. However, many clients, particularly...

In Foreign Exchange Markets Compound options (options on options) are traded frequently. Instalment options generalize the concept of Compound options as they allow the holder to prolong a Vanilla Call or Put option by paying instalments of a discrete payment plan. We derive a closed-form...

No front-office software can survive without providing derivatives of option prices with respect to underlying market or model parameters, the so called Greeks. If a closed form solution for an option exists, Greeks can be computed analytically and they are numerically stable. However, for...

The foreign exchange options market is one of the largest and most liquid OTC derivative markets in the world. Surprisingly, very little is known in the academic literature about the construction of the most important object in this market: The implied volatility smile. The smile construction...

Bei der Altersvorsorge von Privatanlegern ergibt sich in der derzeitigen Marktsituation die Frage nach der Einführung von Anlageprodukten mit Garantien. Garantien können sich auf das Kapital oder auf eine Mindestrendite beziehen. Produktvarianten können Fonds oder Zertifikate sein. Aus Furcht...

Bei der Altersvorsorge von Privatanlegern boomt in der derzeitigen Marktsituation die Vermarktung von Riesterverträgen. Verschiedene Anbieter versuchen, sich diesen Markt zu erschließen. Neben den Versicherungen haben auch Banken und Investmentgesellschaften Angebote auf den Markt gebracht. In...

The vanna-volga method, also called the traders' rule of thumb is an empirical procedure that can be used to infer an implied-volatility smile from three available quotes for a given maturity. It is based on the construction of locally replicating portfolios whose associated hedging costs are...

Bei der Altersvorsorge von Privatanlegern boomt in der derzeitigen Marktsituation die Vermarktung von Riesterverträgen. Verschiedene Anbieter versuchen, sich diesen Markt zu erschließen. Neben den Versicherungen haben auch Banken und Investmentgesellschaften Angebote auf den Markt gebracht. In...