ABSTRACT In this article we study the price of an American style option based on hedging the unde... more ABSTRACT In this article we study the price of an American style option based on hedging the underlying assets at discrete time. Like its European style analog, the value of the option is not given in general by an expectation with respect to an equivalent martingale measure. We provide the optimal solution that minimizes the hedging error variance. When the assets dynamics are Markovian or a component of a Markov process, the solution can be approximated easily by numerical methods already proposed for pricing American options. We proceed to a Monte Carlo experiment in which the hedging performance of the solution is evaluated. For assets returns that are either Gaussian or Variance Gamma, it is shown that the proposed solution results in lower root mean square hedging error than with traditional delta hedging.
ABSTRACT Wang & Wells ["J. Amer. Statist. Assoc." 95 (2000) 62] describ... more ABSTRACT Wang & Wells ["J. Amer. Statist. Assoc." 95 (2000) 62] describe a non-parametric approach for checking whether the dependence structure of a random sample of censored bivariate data is appropriately modelled by a given family of Archimedean copulas. Their procedure is based on a truncated version of the Kendall process introduced by Genest & Rivest ["J. Amer. Statist. Assoc." 88 (1993) 1034] and later studied by Barbe "et al". ["J. Multivariate Anal." 58 (1996) 197]. Although Wang & Wells (2000) determine the asymptotic behaviour of their truncated process, their model selection method is based exclusively on the observed value of its "L"-super-2-norm. This paper shows how to compute asymptotic "p"-values for various goodness-of-fit test statistics based on a non-truncated version of Kendall's process. Conditions for weak convergence are met in the most common copula models, whether Archimedean or not. The empirical behaviour of the proposed goodness-of-fit tests is studied by simulation, and power comparisons are made with a test proposed by Shih ["Biometrika" 85 (1998) 189] for the gamma frailty family. Copyright 2006 Board of the Foundation of the Scandinavian Journal of Statistics..
We consider several time series and for each of them, we fit an appropriate dynamic parametric mo... more We consider several time series and for each of them, we fit an appropriate dynamic parametric model. This produces serially independent error terms for each time series. The dependence between these error terms is then modeled by a regime-switching copula. The EM algorithm is used for estimating the parameters and a sequential goodness-of-fit procedure based on Cramer-von Mises statistics is proposed to select the appropriate number of regimes. Numerical experiments are performed to assess the validity of the proposed methodology. As an example of application, we evaluate a European put-on-max option on the returns of two assets. In order to facilitate the use of our methodology, we have built a R package HMMcopula available on CRAN.
ABSTRACT In view of applications to diagnostic tests of ARMA models, the asymptotic behavior of m... more ABSTRACT In view of applications to diagnostic tests of ARMA models, the asymptotic behavior of multivariate empirical and copula processes based on residuals of ARMA models is investigated. Multivariate empirical processes based on squared residuals and other functions of the residuals are also investigated. It is shown how these processes can be used to develop distribution free tests of change-point analysis and serial independence. It is also demonstrated that these empirical processes provide an easy mechanism for developing goodness-of-fit tests for the distribution of the innovations, and that the well-known Lilliefors test can be applied to the residuals of ARMA models without any change.
We consider several time series and for each of them, we fit an appropriate dynamic parametric mo... more We consider several time series and for each of them, we fit an appropriate dynamic parametric model. This produces serially independent error terms for each time series. The dependence between these error terms is then modeled by a regime-switching copula. The EM algorithm is used for estimating the parameters and a sequential goodness-of-fit procedure based on Cramér-von Mises statistics is proposed to select the appropriate number of regimes. Numerical experiments are performed to assess the validity of the proposed methodology. As an example of application, we evaluate a European put-on-max option on the returns of two assets. In order to facilitate the use of our methodology, we have built a R package HMMcopula available on CRAN.
ABSTRACT In this article we study the price of an American style option based on hedging the unde... more ABSTRACT In this article we study the price of an American style option based on hedging the underlying assets at discrete time. Like its European style analog, the value of the option is not given in general by an expectation with respect to an equivalent martingale measure. We provide the optimal solution that minimizes the hedging error variance. When the assets dynamics are Markovian or a component of a Markov process, the solution can be approximated easily by numerical methods already proposed for pricing American options. We proceed to a Monte Carlo experiment in which the hedging performance of the solution is evaluated. For assets returns that are either Gaussian or Variance Gamma, it is shown that the proposed solution results in lower root mean square hedging error than with traditional delta hedging.
ABSTRACT Wang & Wells ["J. Amer. Statist. Assoc." 95 (2000) 62] describ... more ABSTRACT Wang & Wells ["J. Amer. Statist. Assoc." 95 (2000) 62] describe a non-parametric approach for checking whether the dependence structure of a random sample of censored bivariate data is appropriately modelled by a given family of Archimedean copulas. Their procedure is based on a truncated version of the Kendall process introduced by Genest & Rivest ["J. Amer. Statist. Assoc." 88 (1993) 1034] and later studied by Barbe "et al". ["J. Multivariate Anal." 58 (1996) 197]. Although Wang & Wells (2000) determine the asymptotic behaviour of their truncated process, their model selection method is based exclusively on the observed value of its "L"-super-2-norm. This paper shows how to compute asymptotic "p"-values for various goodness-of-fit test statistics based on a non-truncated version of Kendall's process. Conditions for weak convergence are met in the most common copula models, whether Archimedean or not. The empirical behaviour of the proposed goodness-of-fit tests is studied by simulation, and power comparisons are made with a test proposed by Shih ["Biometrika" 85 (1998) 189] for the gamma frailty family. Copyright 2006 Board of the Foundation of the Scandinavian Journal of Statistics..
We consider several time series and for each of them, we fit an appropriate dynamic parametric mo... more We consider several time series and for each of them, we fit an appropriate dynamic parametric model. This produces serially independent error terms for each time series. The dependence between these error terms is then modeled by a regime-switching copula. The EM algorithm is used for estimating the parameters and a sequential goodness-of-fit procedure based on Cramer-von Mises statistics is proposed to select the appropriate number of regimes. Numerical experiments are performed to assess the validity of the proposed methodology. As an example of application, we evaluate a European put-on-max option on the returns of two assets. In order to facilitate the use of our methodology, we have built a R package HMMcopula available on CRAN.
ABSTRACT In view of applications to diagnostic tests of ARMA models, the asymptotic behavior of m... more ABSTRACT In view of applications to diagnostic tests of ARMA models, the asymptotic behavior of multivariate empirical and copula processes based on residuals of ARMA models is investigated. Multivariate empirical processes based on squared residuals and other functions of the residuals are also investigated. It is shown how these processes can be used to develop distribution free tests of change-point analysis and serial independence. It is also demonstrated that these empirical processes provide an easy mechanism for developing goodness-of-fit tests for the distribution of the innovations, and that the well-known Lilliefors test can be applied to the residuals of ARMA models without any change.
We consider several time series and for each of them, we fit an appropriate dynamic parametric mo... more We consider several time series and for each of them, we fit an appropriate dynamic parametric model. This produces serially independent error terms for each time series. The dependence between these error terms is then modeled by a regime-switching copula. The EM algorithm is used for estimating the parameters and a sequential goodness-of-fit procedure based on Cramér-von Mises statistics is proposed to select the appropriate number of regimes. Numerical experiments are performed to assess the validity of the proposed methodology. As an example of application, we evaluate a European put-on-max option on the returns of two assets. In order to facilitate the use of our methodology, we have built a R package HMMcopula available on CRAN.
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Papers by Bruno Remillard