The main purpose of this paper is to examine empirically the time series properties of the French Market Volatility Index (VX1). We also examine the VX1's ability to forecast future realized market volatility and finds a strong relationship. More importantly, we show how the index can be used to generate volatility forecasts over different horizons and that these forecasts are reasonably accurate predictors of future realized volatility. JEL classification codes: G14, C53, C13.

This paper introduces variants of strangles, called Euro-American or hybrid strangles, and it promotes a new numerical pricing technique. We highlight and compare the properties of European, American, and hybrid strangles with pricing and hedging in mind. The new quadrature approach we propose can account for systems of coupled integral equations that locate the early exercise boundaries of finite-lived contracts. We show that this method is efficient, accurate, and fast for pricing all types of early exercisable strangles. Other advantages of this technique are that it avoids the non-monotonic gradient problem faced by others and it allows users to control for errors. We then investigate the hedging of all strangles, we derive analytical expressions for some Greek parameters, and we stress how these parameters can differ (or not) from each other.

This paper develops a model of debt renegotiation in a structural framework that accounts for taxes, bankruptcy costs and renegotiation costs. To our knowledge, all the previous work on debt renegotiation implies an infinite number of renegotiations. This feature preempts the analysis of the optimal number of renegotiations. We address this drawback by incorporating fixed renegotiation costs in a model of multiple renegotiations, hence obtaining a small finite number of renegotiations. Simple analytical formulae are derived for debt and equity, as well as implicit formulae for the coupon reduction, as a result of a backward recursive technique. The results show that the optimal number of renegotiations, the size and the dynamics of the coupon reductions depend critically on the bargaining power of the claimants. Testable empirical implications regarding multiple costly renegotiations are drawn.