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Rev Deriv Res (2007) 10:151180 DOI 10.1007/s11147-008-9018-x
Option pricing when correlations are stochastic: an analytical framework
Jos Da Fonseca Martino Grasselli Claudio Tebaldi
Published online: 31 January 2008 Springer Science+Business Media, LLC 2008
Abstract In this paper we develop a novel market model where asset variances covariances evolve stochastically. In addition shocks on asset return dynamics are assumed to be linearly correlated with shocks driving the variancecovariance matrix. Analytical tractability is preserved since the model is linear-afne and the conditional characteristic function can be determined explicitly. Quite remarkably, the model provides prices for vanilla options consistent with observed smile and skew effects, while making it possible to detect and quantify the correlation risk in multiple-asset derivatives like basket options. In particular, it can reproduce and quantify the asymmetric conditional correlations observed on historical data for equity markets. As an illustrative example, we provide explicit pricing formulas for rainbow Best-of options.
Keywords Wishart processes Best-of basket option Stochastic correlation FFT
J. Da Fonseca (B) M. Grasselli Ecole Suprieure dIngnieurs Lonard de Vinci, Departement Mathmatiques et Ingnierie Financire, Paris La Defense 92916, France e-mail: [email protected]
J. Da FonsecaZeliade Systems, 56, Rue Jean-Jacques Rousseau, Paris 75001, France
M. GrasselliDipartimento di Matematica Pura ed Applicata, Universit degli Studi di Padova, via Trieste 63, Padova, Italye-mail: [email protected]
C. TebaldiIMQ, Universit Bocconi Milano, Viale Isonzo 25, Milano 20139, Italy e-mail: [email protected]
C. Tebaldi Universit degli Studi di Verona, Verona, Italy
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1 Introduction
This paper presents an analytically tractable model for capturing the joint behavior of prices of an underlying set of assets, options on the individual underlying assets, and derivatives on baskets of the underlying assets. We believe this is the rst tractable model that allows for non-trivial stochastic volatility of asset returns and stochastic correlation of cross-sectional asset returns in a manner that is fully consistent with the sorts of smile and skew effects that are apparent in typical market pricing of plain vanilla option prices.
In this model, prices (conditional to volatilities) evolve according to a lognormal diffusion while the stochastic variancecovariance matrix follows a Wishart process as proposed by Gourieroux and Sufana (2004b). Originally introduced by Bru (1991), the Wishart process is the natural matrix counterpart of the Feller (1951)...