Abstract EN:

In this thesis, we study the problem of hedging the volatility risk with consistent models and with volatility derivatives in two different cases : only the stock is traded and both the stock and vanilla options (European puts and calls) are traded. In the first case, we assume that the information content is the risk neutral moments of the logreturn. Then, we give a moment consistent model thanks to an extension for the local volatility model (Dupire 1994). We dynamically delta hedge the volatility risk. Moreover, we obtain closed-form formulas for the price of a European call similar to the Jarrow-Rudd formula (Jarrow and Rudd 1982) and for the smile of implied volatility. In the second case, we choose two different ways : the skew market model (SMM) and the volatility derivatives. The SMM describes the dynamics iod the smiles of implied volatility with the ATM (at-the-money) volatility, the skew and the curve, i. E. The convexity level. These processes are synthetically tradeable, i. E. Replicable, with vanilla options. They provide dynamic hedges, called vega hedges, against the main moves of the smile : the SMM dynamically delta-vega hedge the volatility risk. We give conditions of absence of arbitrage in this model. Finally, we introduce the volatility derivatives. They are used for statically hedging the volatility risk (static vega hedge), in addition to the dynamic delta hedge. After a presentation of the volatility derivatives, we will see that, except for a few volatility derivatives as the famous variance swap (Dupire 1992), their development strongly depends on an arbitrage free market model which provides efficient dynamic vega hedges.