Coupling Index and Stocks

TL;DR

This paper introduces a coupled model where index levels influence stock dynamics, approximating to local and stochastic volatility models for large stock sets, aiding calibration processes.

Contribution

It proposes a novel coupled model linking index and stock dynamics, with a practical approximation for large portfolios to facilitate calibration.

Findings

Model approximates to local volatility for index and stochastic volatility for stocks.

Calibration strategy involves sequentially calibrating index then stocks.

Provides methods for calibration in both simplified and original models.

Abstract

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. This result is useful in a calibration perspective : it suggests that one should first calibrate the local volatility of the index and then calibrate the dynamics of each stock. We explain how to do so in the limiting simplified model and in the original model.