Option Pricing When Correlations are Stochastic: An Analytical Framework

Fonseca, José Da; Grasselli, Martino; Tebaldi, Claudio

doi:10.2139/ssrn.982183

Cited by 61 publications

(61 citation statements)

References 27 publications

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“…The model belongs to the class of multidimensional stochastic volatility models. In the following, we follow the approach of Grasselli and Tebaldi (2008) and Da Fonseca et al (2007) and report their result on the joint Fourier transform of assets' returns, that we adapt to our setting.…”

Section: The Wasc Modelmentioning

confidence: 99%

General Closed-Form Basket Option Pricing Bounds

et al. 2014

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This is the accepted version of the paper.This version of the publication may differ from the final published version. This article presents lower and upper bounds on the prices of basket options for a general class of continuous-time financial models. The techniques we propose are applicable whenever the joint characteristic function of the vector of log-returns is known. Moreover, the basket value is not required to be positive. We test our new price approximations on different multivariate models, allowing for jumps and stochastic volatility. Numerical examples are discussed and benchmarked against Monte Carlo simulations. All bounds are general and do not require any additional assumption on the characteristic function, so our methods may be employed also to non-affine models. All bounds involve the computation of one-dimensional Fourier transforms, hence they do not suffer from the curse of dimensionality and can be applied also to high dimensional problems where most existing methods fail. In particular we study two kinds of price approximations: an accurate lower bound based on an approximating set and a fast bounded approximation based on the arithmetic-geometric mean inequality. We also show how to improve Monte Carlo accuracy by using one of our bounds as a control variate. Permanent repository link

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Section: The Wasc Modelmentioning

confidence: 99%

General Closed-Form Basket Option Pricing Bounds

et al. 2014

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“…In fact, a variant of this model is presently used in the calibration of implied volatility surfaces for single stocks and equity indices in the Bloomberg terminal [6], and in the subsequent pricing of European, American and path-dependent options on single assets and baskets of assets. The main advantage of the MVMD over other multidimensional models, such as e.g., the Wishart model ( [11] and [12]) is in its tractability and flexibility which allows the MVMD to calibrate index volatility smiles consistently with the univariate assets smiles. In addition, a full description of its dependence structure (terminal correlations, average correlations, copula functions) is available.…”

Section: Introduction To the Multivariate Mixture Dynamicsmentioning

confidence: 99%

The Multivariate Mixture Dynamics Model: Shifted Dynamics and Correlation Skew

2015

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The Multi Variate Mixture Dynamics model is a tractable, dynamical, arbitragefree multivariate model characterized by transparency on the dependence structure, since closed form formulae for terminal correlations, average correlations and copula function are available. It also allows for complete decorrelation between assets and instantaneous variances. Each single asset is modelled according to a lognormal mixture dynamics model, and this univariate version is widely used in the industry due to its flexibility and accuracy. The same property holds for the multivariate process of all assets, whose density is a mixture of multivariate basic densities. This allows for consistency of single asset and index/portfolio smile.In this paper, we generalize the MVMD model by introducing shifted dynamics and we propose a definition of implied correlation under this model. We investigate whether the model is able to consistently reproduce the implied volatility of FX cross rates once the single components are calibrated to univariate shifted lognormal mixture dynamics models. We consider in particular the case of the Chinese renminbi FX rate, showing that the shifted MVMD model correctly recovers the CNY/EUR smile given the EUR/USD smile and the USD/CNY smile, thus highlighting that the model can also work as an arbitrage free volatility smile extrapolation tool for cross currencies that may not be liquid or fully observable.We compare the performance of the shifted MVMD model in terms of implied correlation with those of the shifted Simply Correlated Mixture Dynamics model where the dynamics of the single assets are connected naively by introducing correlation among their Brownian motions. Finally, we introduce a model with uncertain volatilities and correlation. The Markovian projection of this model is a generalization of the shifted MVMD model.

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“…It is a direct multivariate extension of the Cox-Ingersoll-Ross model and has been extended and used for financial applications by e.g. Gourieroux & Sufana (2003Da Fonseca et al (2007, 2008 ;Buraschi et al (2010); Muhle-Karbe et al (2012). While these papers consider option pricing, hedging, credit risk and term structure models, we will investigate portfolio optimization problems.…”

Section: Introductionmentioning

confidence: 99%

Optimal Portfolios for Financial Markets with Wishart Volatility

Bäuerle¹,

Li²

2013

J. Appl. Probab.

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Abstract. We consider a multi asset financial market with stochastic volatility modeled by a Wishart process. This is an extension of the one-dimensional Heston model. Within this framework we study the problem of maximizing the expected utility of terminal wealth for power and logarithmic utility. We apply the usual stochastic control approach and obtain explicitly the optimal portfolio strategy and the value function in some parameter settings. In particular when the drift of the assets is a linear function of the volatility matrix. In this case the affine structure of the model can be exploited. In some cases we obtain a Feynman-Kac representation of the candidate value function. Though the approach we use is quite standard, the hard part is indeed to identify when the solution of the HJB equation is finite. This involves a couple of matrix analytic arguments. In a numerical study we discuss the influence of the investors' risk aversion on the hedging demand.

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Option Pricing When Correlations are Stochastic: An Analytical Framework

Cited by 61 publications

References 27 publications

General Closed-Form Basket Option Pricing Bounds

General Closed-Form Basket Option Pricing Bounds

The Multivariate Mixture Dynamics Model: Shifted Dynamics and Correlation Skew

Optimal Portfolios for Financial Markets with Wishart Volatility

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