Coupling Index and Stocks

In this paper, we are interested in continuous time models in which the index level induces some feedback on the dynamics of its composing stocks. More precisely, we propose a model in which the log-returns of each stock may be decomposed into a systemic part proportional to the log-returns of the index plus an idiosyncratic part. We show that, when the number of stocks in the index is large, this model may be approximated by a local volatility model for the index and a stochastic volatility model for each stock with volatility driven by the index. We address calibration of both the limit and the original models. Introduction From the early eighties, when trading on stock index was introduced, quantitative finance faced the problem of efficiently pricing and hedging index options along with their underlying components. Many advances have been made for single stock modeling and a variety of solutions to escape from the very restrictive Black & Scholes model has been deeply investigated (such as local volatility models, models with jumps or stochastic volatility models). However, when the number of underlyings is large, index option pricing, or more generally basket option pricing, remains a challenge unless one simply assumes constantly correlated dynamics for the stocks. The problem then is the impossibility of fitting both the stocks and the index smiles. We try to address this issue by making the dynamics of the stocks depend on the index. The natural fact that the volatility of the index is related to the volatilities of its underlying components has already been accounted for in the works of Avellaneda et al. [1] and Lee et al. [12]. In the first paper, the authors use a large deviation asymptotics to reconstruct the local volatility of the index from the local volatilities of the stocks. They express this dependence in terms of implied volatilities using the results of Berestecky et al.([4],[3]). In the second paper, the authors reconstruct the Gram-Charlier expansion of the probability density of the index from the stocks using a moments-matching technique. Both papers consider local volatility models for the stocks and a constant correlation matrix but the generalization to stochastic volatility models or to varying correlation coefficients is not straightforward. Another point of view is to say that the volatility of a composing stock should be related to the index level, or say to the volatility of the index, in some way. This is not astonishing since the index represents the move of the market and reflects the view of the investors on the state of the economy. Moreover, it Université Paris-Est, CERMICS, Projet MathFi ENPC-INRIA-UMLV. This research benefited from the support of the ”Chair Risques Financiers”, Fondation du Risque. Postal address : 6-8 av. Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2. E-mails : [email protected] and [email protected]