Deterministic shift extension of affine models for variance derivatives

The growing demand for volatility trading and hedging has lead today to a liquid market for derivative securities written on it, which made these instruments a widely accepted asset class for trading, diversifying and hedging. This growing market has consistently driven the interest of both practitioner and academic researchers, which can find in VIX and derivatives written on it a valuable source of informations on S&P500 dynamics, over and above vanilla options. Their popularity stems from the negative correlation between VIX and SPX index, which make these instruments ideal to take a pure position on the volatility of the S&P500 without necessarily taking a position on its direction. In this respect futures on VIX enable the trader to express a vision of the markets future volatility and call options on VIX offer protection from market downturns in a clear-cut way. From the theoretical point of view, this has lead to the need of a framework for consistently pricing volatility derivatives and derivatives on the underlying, that is the need to design models able to fit the observed cross-section of option prices of both markets and properly price and hedge exotic products. The consistent pricing of vanilla options on S&P500 and futures and options on VIX is a requirement of primary importance for a model to provide an accurate description of the volatility dynamics. Since equity and volatility markets are deeply related, but at the same time show striking differences, the academic debate around the relevant features should a model incorporate in order to be coherent with both markets is still ongoing. In this thesis we leverage on the growing literature concerning the developing of models for consistently pricing volatility derivatives and derivatives on the underlying and propose the Heston++ model, which is an affine model belonging to the class of models analyzed by Duffie et al. (2000) with a multi-factor volatility dynamics and a rich jumps structure both for price and volatility. The multi-factor Heston (1993) structure enables the model to better capture VIX futures term structures along with maturity-dependent smiles of options. Moreover, both correlated and idiosyncratic jumps in price and volatility factors help in reproducing the positive sloping skew of options on VIX, thanks to an increased level of the skewness of VIX distribution subsumed by the model. The key feature of our approach is to impose an additive displacement, in the spirit of Brigo and Mercurio (2001), on the instantaneous volatility dynamics which, acting as lower bound for its dynamics, noticeably helps in capturing the term structure of volatility. Both increasing the fit to the at-the-money term structure of vanilla options, as already pointed out in Pacati et al. (2014), and remarkably capturing the different shapes experienced by the term structure of futures on VIX. Moreover, we propose a general affine framework which extends the affine volatility frameworks of Leippold et al. (2007), Egloff et al. (2010) and Branger et al. (2014) in which the risk-neutral dynamics of the S&P500 index features several diffusive and jump risk sources and two general forms of displacement characterize the dynamics of the instantaneous variance process, which is affine in the state vector of volatility factors. The instantaneous volatility is modified according to a general affine transformation in which both an additive and a multiplicative displacement are imposed, the first supporting its dynamics, the second modulating its amplitude. We calibrate the Heston++ model jointly and consistently on the three markets over a sample period of two years, with overall absolute (relative) estimation error below 2.2% (4%). We analyze the different contributions of jumps in volatility. We add two sources of exponential upward jumps in one of the two volatility factors. We first add them separately as an idiosyncratic source of discontinuity (as in the SVVJ model of Sepp (2008b)) and then correlated and synchronized with jumps in price (as in the SVCJ model of Duffie et al. (2000)). Finally, we let the two discontinuity sources act together in the full-specified model. For any model considered, we analyze the impact of acting a displacement transformation on the volatility dynamics. In addition, we perform the analysis restricting factor parameters freedom to satisfy the Feller condition. Our empirical results show a decisive improvement in the pricing performance over non-displaced models, and also provide strong empirical support for the presence of both price-volatility co-jumps and idiosyncratic jumps in the volatility dynamic