Articles de journaux
|
Giorgia Callegaro; Lucio Fiorin; Martino Grasselli Pricing via Quantization in Stochastic Volatility Models Article de journal Quantitative Finance, 17 (6), p. 855-872, 2017. Résumé | Liens | BibTeX @article{Callegaro2017,
title = {Pricing via Quantization in Stochastic Volatility Models},
author = {Giorgia Callegaro and Lucio Fiorin and Martino Grasselli},
url = {http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2669734},
year = {2017},
date = {2017-10-22},
journal = {Quantitative Finance},
volume = {17},
number = {6},
pages = {855-872},
abstract = {In this paper we apply a new methodology based on quantization to price options in stochastic volatility models. This method can be applied to any model for which an Euler scheme is available for the underlying process and it allows for pricing vanillas, as well as exotics, thanks to the knowledge of the transition probabilities for the discretized stock process. We apply the methodology to some celebrated stochastic volatility models, including the Stein and Stein (1991) model and the SABR model introduced in Hagan and Woodward (2002). A numerical exercise shows that the pricing of vanillas turns out to be accurate; in addition, when applied to some exotics like equity-volatility options, the quantization-based method overperforms by far the Monte Carlo simulation.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In this paper we apply a new methodology based on quantization to price options in stochastic volatility models. This method can be applied to any model for which an Euler scheme is available for the underlying process and it allows for pricing vanillas, as well as exotics, thanks to the knowledge of the transition probabilities for the discretized stock process. We apply the methodology to some celebrated stochastic volatility models, including the Stein and Stein (1991) model and the SABR model introduced in Hagan and Woodward (2002). A numerical exercise shows that the pricing of vanillas turns out to be accurate; in addition, when applied to some exotics like equity-volatility options, the quantization-based method overperforms by far the Monte Carlo simulation. |
Martino Grasselli The 4/2 stochastic volatility model Article de journal Mathematical Finance, 27 (4), p. 1013-1034, 2017. Résumé | Liens | BibTeX @article{Grassellibb,
title = {The 4/2 stochastic volatility model},
author = { Martino Grasselli},
url = {http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2523635},
doi = {DOI: 10.1111/mafi.12124},
year = {2017},
date = {2017-10-15},
journal = {Mathematical Finance},
volume = {27},
number = {4},
pages = {1013-1034},
abstract = {We introduce a new stochastic volatility model that includes as special cases the Heston (1993) and the 3/2 model of Heston (1997) and Platen (1997). The new model displays new features, namely the instantaneous volatility can be uniformly bounded away from zero, and it is highly tractable, meaning that the pricing procedure can be performed efficiently. This required the application of the Lie symmetries theory for PDEs, which allowed us to extend known results on Bessel processes. Finally, we provide an exact simulation and a second order discretization scheme for the model, which is useful in view of the numerical applications.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We introduce a new stochastic volatility model that includes as special cases the Heston (1993) and the 3/2 model of Heston (1997) and Platen (1997). The new model displays new features, namely the instantaneous volatility can be uniformly bounded away from zero, and it is highly tractable, meaning that the pricing procedure can be performed efficiently. This required the application of the Lie symmetries theory for PDEs, which allowed us to extend known results on Bessel processes. Finally, we provide an exact simulation and a second order discretization scheme for the model, which is useful in view of the numerical applications. |
Griselda Deelstra; Martino Grasselli; Christopher van Weverberg The Role of the Dependence between Mortality and Interest Rates when pricing Guaranteed Annuity Options Article de journal Insurance: Mathematics and Economics, 71 , p. 205-219, 2016, ISSN: 0167-6687. Résumé | Liens | BibTeX @article{Deelstra2015b,
title = {The Role of the Dependence between Mortality and Interest Rates when pricing Guaranteed Annuity Options},
author = {Griselda Deelstra and Martino Grasselli and Christopher van Weverberg},
url = {http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2667939
http://dx.doi.org/10.1016/j.insmatheco.2016.09.010},
doi = {http://dx.doi.org/10.1016/j.insmatheco.2016.09.010},
issn = {0167-6687},
year = {2016},
date = {2016-09-28},
journal = {Insurance: Mathematics and Economics},
volume = {71},
pages = {205-219},
institution = {ULB},
abstract = {In this paper we investigate the consequences on the pricing of insurance contingent claims when we relax the typical independence assumption made in the actuarial literature between mortality risk and interest rate risk. Starting from the Gaussian approach of Liu et al. (2014), we consider more general affine models where the mortality and interest rates remain positive and we derive pricing formulas for insurance contracts like Guaranteed Annuity Options (GAOs). In a Wishart affine model, which allows for a non-trivial dependence between the mortality and the interest rates, we go far beyond the results found in the Gaussian case by Liu et al. (2014), where the value of these insurance contracts can be explained only in terms of the initial pairwise linear correlation.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In this paper we investigate the consequences on the pricing of insurance contingent claims when we relax the typical independence assumption made in the actuarial literature between mortality risk and interest rate risk. Starting from the Gaussian approach of Liu et al. (2014), we consider more general affine models where the mortality and interest rates remain positive and we derive pricing formulas for insurance contracts like Guaranteed Annuity Options (GAOs). In a Wishart affine model, which allows for a non-trivial dependence between the mortality and the interest rates, we go far beyond the results found in the Gaussian case by Liu et al. (2014), where the value of these insurance contracts can be explained only in terms of the initial pairwise linear correlation. |
Martino Grasselli; Giulio Miglietta A Flexible Spot Multiple-Curve Model Article de journal Quantitative Finance, 16 (10), p. 1465-1477, 2016, ISSN: 1469-7688. Résumé | Liens | BibTeX @article{Grasselli2015,
title = {A Flexible Spot Multiple-Curve Model},
author = {Martino Grasselli and Giulio Miglietta},
url = {http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2424242
http://dx.doi.org/10.1080/14697688.2015.1108521},
doi = {10.1080/14697688.2015.1108521},
issn = {1469-7688},
year = {2016},
date = {2016-04-27},
journal = {Quantitative Finance},
volume = {16},
number = {10},
pages = {1465-1477},
abstract = {We propose a model for the instantaneous risk-free spot rate and for the spot LIBOR, driven by a time-homogeneous Markovian process. We introduce deterministic time-shifts in order to match any initial term-structure. By doing so, the model automatically becomes an exogenous term-structure model, in the spirit of Brigo and Mercurio (2001) who proposed this approach in the single curve case. A calibration exercise based on real data illustrates the flexibility of our approach for some typical specifications used in the literature and in the banking industry.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We propose a model for the instantaneous risk-free spot rate and for the spot LIBOR, driven by a time-homogeneous Markovian process. We introduce deterministic time-shifts in order to match any initial term-structure. By doing so, the model automatically becomes an exogenous term-structure model, in the spirit of Brigo and Mercurio (2001) who proposed this approach in the single curve case. A calibration exercise based on real data illustrates the flexibility of our approach for some typical specifications used in the literature and in the banking industry. |
Martino Grasselli; Jacinto Marabel Romo Stochastic Skew and Target Volatility Options Article de journal The Journal of Futures Markets, 36 (2), p. 174-193, 2016. Résumé | Liens | BibTeX @article{Grasselli,
title = {Stochastic Skew and Target Volatility Options},
author = { Martino Grasselli and Jacinto Marabel Romo},
url = {http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2419211},
doi = {DOI: 10.1002/fut.21720},
year = {2016},
date = {2016-01-01},
journal = {The Journal of Futures Markets},
volume = {36},
number = {2},
pages = {174-193},
abstract = {Target volatility options (TVO) are a new class of derivatives whose payoff depends on some measure of volatility. These options allow investors to take a joint exposure to the evolution of the underlying asset, as well as to its realized volatility. For instance, a target volatility call can be viewed as a European call whose notional amount depends on the ratio of the target volatility (a fixed quantity representing the investor's expectation of the future realized volatility) and the realized volatility of the underlying asset over the life of the option. In equity options markets the slope of the skew is largely independent of the volatility level. A single-factor Heston based volatility model can generate steep skew or flat skew at a given volatility level but cannot generate both for a given parameterization. Since the payoff corresponding to TVO is a function of the joint evolution of the underlying asset and its realized variance, the consideration of stochastic skew is a relevant question for the valuation of TVO. In this sense, this article studies the effect of considering a multifactor stochastic volatility specification in the valuation of the TVO. To this end, we consider the two-factor Heston-based model of Christoffersen et al. (2009) in order to investigate TVO, as well as forward-start TVO, that is, TVO where the strike is determined at a later date.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Target volatility options (TVO) are a new class of derivatives whose payoff depends on some measure of volatility. These options allow investors to take a joint exposure to the evolution of the underlying asset, as well as to its realized volatility. For instance, a target volatility call can be viewed as a European call whose notional amount depends on the ratio of the target volatility (a fixed quantity representing the investor's expectation of the future realized volatility) and the realized volatility of the underlying asset over the life of the option. In equity options markets the slope of the skew is largely independent of the volatility level. A single-factor Heston based volatility model can generate steep skew or flat skew at a given volatility level but cannot generate both for a given parameterization. Since the payoff corresponding to TVO is a function of the joint evolution of the underlying asset and its realized variance, the consideration of stochastic skew is a relevant question for the valuation of TVO. In this sense, this article studies the effect of considering a multifactor stochastic volatility specification in the valuation of the TVO. To this end, we consider the two-factor Heston-based model of Christoffersen et al. (2009) in order to investigate TVO, as well as forward-start TVO, that is, TVO where the strike is determined at a later date. |
José Da Fonseca; Alessandro Gnoatto; Martino Grasselli Analytic Pricing of Volatility-Equity Options within Affine Models: an Efficient Conditioning Technique Article de journal Operations Research Letters, 43 , p. 601-607, 2015, ISSN: 0167-6377. Résumé | Liens | BibTeX @article{Fonseca2015,
title = {Analytic Pricing of Volatility-Equity Options within Affine Models: an Efficient Conditioning Technique},
author = {José Da Fonseca and Alessandro Gnoatto and Martino Grasselli},
url = {http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2428628
http://dx.doi.org/10.1016/j.orl.2015.09.006},
doi = {http://dx.doi.org/10.1016/j.orl.2015.09.006},
issn = {0167-6377},
year = {2015},
date = {2015-09-25},
journal = {Operations Research Letters},
volume = {43},
pages = {601-607},
abstract = {We price for different affine stochastic volatility models some derivatives that recently appeared in the market. These products are characterised by payoffs depending on both stock and its volatility. Using a Fourier-analysis approach, we recover in a much simpler way some results already established in the literature for the single factor specification of the volatility and we push forward our methodology, which turns out to be independent of the dimension of the problem, thanks to a simple conditioning with respect to the subfiltration generated by the variance path. For each product we provide a closed form solution based on the Fast Fourier Transform and we illustrate the results for realistic model parameter values. Also, our results highlight the great flexibility and tractability of the Wishart based stochastic volatility models.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We price for different affine stochastic volatility models some derivatives that recently appeared in the market. These products are characterised by payoffs depending on both stock and its volatility. Using a Fourier-analysis approach, we recover in a much simpler way some results already established in the literature for the single factor specification of the volatility and we push forward our methodology, which turns out to be independent of the dimension of the problem, thanks to a simple conditioning with respect to the subfiltration generated by the variance path. For each product we provide a closed form solution based on the Fast Fourier Transform and we illustrate the results for realistic model parameter values. Also, our results highlight the great flexibility and tractability of the Wishart based stochastic volatility models. |
Martino Grasselli; Alessandro Gnoatto; Gianluca Fusai; Ruggero Caldana General Closed-From Basket Option Pricing Bounds Article de journal Quantitative Finance, 16 (4), p. 535-554, 2015, ISSN: 1469-7688. Résumé | Liens | BibTeX @article{Grassellib,
title = {General Closed-From Basket Option Pricing Bounds},
author = {Martino Grasselli and Alessandro Gnoatto and Gianluca Fusai and Ruggero Caldana},
url = {http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2376134
http://dx.doi.org/10.1080/14697688.2015.1073854},
doi = {http://dx.doi.org/10.1080/14697688.2015.1073854},
issn = {1469-7688},
year = {2015},
date = {2015-09-14},
journal = {Quantitative Finance},
volume = {16},
number = {4},
pages = {535-554},
abstract = {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 in closed-form. Moreover, the basket value is not required to be positive. We test our new bounds 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. In particular, our new lower bound turns out to be fast and accurate.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
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 in closed-form. Moreover, the basket value is not required to be positive. We test our new bounds 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. In particular, our new lower bound turns out to be fast and accurate. |
Jan Baldeaux; Martino Grasselli; Eckhard Platen Pricing currency derivatives under the benchmark approach Article de journal Journal of Banking and Finance, 53 , p. 34-48, 2015, ISSN: 0378-4266. Résumé | Liens | BibTeX @article{Grassellib,
title = {Pricing currency derivatives under the benchmark approach},
author = { Jan Baldeaux and Martino Grasselli and Eckhard Platen},
url = {http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2327316
http://dx.doi.org/10.1016/j.jbankfin.2014.11.018},
doi = {http://dx.doi.org/10.1016/j.jbankfin.2014.11.018},
issn = {0378-4266},
year = {2015},
date = {2015-04-01},
journal = {Journal of Banking and Finance},
volume = {53},
pages = {34-48},
abstract = {This paper considers the realistic modelling of derivative contracts on exchange rates. We propose a stochastic volatility model that recovers not only the typically observed implied volatility smiles and skews for short dated vanilla foreign exchange options but allows one also to price payoffs in foreign currencies, lower than possible under classical risk neutral pricing, in particular, for long dated derivatives. The main reason for this important feature is the strict supermartingale property of benchmarked savings accounts under the real world probability measure, which the calibrated parameters identify under the proposed model. Using a real dataset on vanilla option quotes, we calibrate our model on a triangle of currencies and find that the risk neutral approach fails for the calibrated model, while the benchmark approach still works.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
This paper considers the realistic modelling of derivative contracts on exchange rates. We propose a stochastic volatility model that recovers not only the typically observed implied volatility smiles and skews for short dated vanilla foreign exchange options but allows one also to price payoffs in foreign currencies, lower than possible under classical risk neutral pricing, in particular, for long dated derivatives. The main reason for this important feature is the strict supermartingale property of benchmarked savings accounts under the real world probability measure, which the calibrated parameters identify under the proposed model. Using a real dataset on vanilla option quotes, we calibrate our model on a triangle of currencies and find that the risk neutral approach fails for the calibrated model, while the benchmark approach still works. |
Giorgia Callegaro; Lucio Fiorin; Martino Grasselli Quantized calibration in local volatility models Article de journal Risk Magazine, 9 , p. 62-67, 2015, ISSN: 0952-8776. Résumé | Liens | BibTeX @article{Grassellibb,
title = {Quantized calibration in local volatility models},
author = { Giorgia Callegaro and Lucio Fiorin and Martino Grasselli},
url = {http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2495829},
doi = {10.2139/ssrn.2495829},
issn = {0952-8776},
year = {2015},
date = {2015-03-31},
journal = {Risk Magazine},
volume = {9},
pages = {62-67},
abstract = {In this paper we propose the first calibration exercise based on quantization methods. Pricing and calibration are typically difficult tasks to accomplish: pricing should be fast and accurate, otherwise calibration cannot be performed efficiently. We apply in a local volatility context the recursive marginal quantization methodology to the pricing of vanilla and barrier options. A successful calibration of the Quadratic Normal Volatility model is performed in order to show the potentiality of the method in a concrete example, while a numerical exercise on barrier options shows that quantization overcomes Monte-Carlo methods.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In this paper we propose the first calibration exercise based on quantization methods. Pricing and calibration are typically difficult tasks to accomplish: pricing should be fast and accurate, otherwise calibration cannot be performed efficiently. We apply in a local volatility context the recursive marginal quantization methodology to the pricing of vanilla and barrier options. A successful calibration of the Quadratic Normal Volatility model is performed in order to show the potentiality of the method in a concrete example, while a numerical exercise on barrier options shows that quantization overcomes Monte-Carlo methods. |
Alessandro Gnoatto; Martino Grasselli The Explicit Laplace Transform for the Wishart Process Article de journal Journal of Applied Probability, 51 (3), p. 640-656, 2014. Résumé | Liens | BibTeX @article{Grassellibb,
title = {The Explicit Laplace Transform for the Wishart Process},
author = { Alessandro Gnoatto and Martino Grasselli},
url = {http://arxiv.org/pdf/1107.2748.pdf},
doi = {doi:10.1239/jap/1409932664},
year = {2014},
date = {2014-09-05},
journal = {Journal of Applied Probability},
volume = {51},
number = {3},
pages = {640-656},
abstract = {We derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral, which extends the original approach of Bru (1991). We compare our methodology with the alternative results given by the variation-of-constants method, the linearization of the matrix Riccati ordinary differential equation, and the Runge-Kutta algorithm. The new formula turns out to be fast and accurate.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral, which extends the original approach of Bru (1991). We compare our methodology with the alternative results given by the variation-of-constants method, the linearization of the matrix Riccati ordinary differential equation, and the Runge-Kutta algorithm. The new formula turns out to be fast and accurate. |
Alessandro Gnoatto; Martino Grasselli A tractable multi-currency model with stochastic volatility and stochastic interest rates Article de journal SIAM Journal of Financial Mathematics, 5 (1), p. 493-531, 2014, ISSN: 1945-497X. Résumé | Liens | BibTeX @article{Grassellibb,
title = {A tractable multi-currency model with stochastic volatility and stochastic interest rates},
author = { Alessandro Gnoatto and Martino Grasselli},
url = {http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2226499},
doi = {10.1137/130922902},
issn = {1945-497X},
year = {2014},
date = {2014-08-28},
journal = {SIAM Journal of Financial Mathematics},
volume = {5},
number = {1},
pages = {493-531},
abstract = {We introduce a tractable multicurrency model with stochastic volatility and correlated stochastic interest rates that takes into account the smile in the foreign exchange (FX) market and the evolution of yield curves. The pricing of vanilla options on FX rates can be efficiently performed through the FFT methodology thanks to the affine property of the model. Our framework is also able to describe many nontrivial links between FX rates and interest rates: a calibration exercise highlights the ability of the model to simultaneously fit FX implied volatilities while being coherent with interest rate products.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We introduce a tractable multicurrency model with stochastic volatility and correlated stochastic interest rates that takes into account the smile in the foreign exchange (FX) market and the evolution of yield curves. The pricing of vanilla options on FX rates can be efficiently performed through the FFT methodology thanks to the affine property of the model. Our framework is also able to describe many nontrivial links between FX rates and interest rates: a calibration exercise highlights the ability of the model to simultaneously fit FX implied volatilities while being coherent with interest rate products. |
José Da Fonseca, Martino Grasselli, Florian Ielpo Estimating the Wishart Affine Stochastic Correlation Model using the Empirical Characteristic Function Article de journal Studies in Nonlinear Dynamics & Econometrics, 18 (3), p. 253-289, 2014. Résumé | Liens | BibTeX @article{Grassellibb,
title = {Estimating the Wishart Affine Stochastic Correlation Model using the Empirical Characteristic Function},
author = { José Da Fonseca, Martino Grasselli, Florian Ielpo},
url = {http://www.researchgate.net/publication/228266805_Estimating_the_Wishart_Affine_Stochastic_Correlation_Model_Using_the_Empirical_Characteristic_Function},
doi = {10.2139/ssrn.1054721},
year = {2014},
date = {2014-06-10},
journal = {Studies in Nonlinear Dynamics & Econometrics},
volume = {18},
number = {3},
pages = {253-289},
abstract = {In this paper, we present and discuss the estimation of the Wishart Affine Stochastic Correlation (WASC) model introduced in Da Fonseca et al. (2006) under the historical measure. We review the main estimation possibilities for this continuous time process and provide elements to show that the utilization of empirical characteristic function-based estimates is advisable as this function is exponential affine in the WASC case. We thus propose to use the estimation strategy presented in Carrasco et al. (2003), using a continuum of moment conditions based on the characteristic function. We investigate the behavior of the estimates through Monte Carlo simulations. Then, we present the estimation results obtained using a dataset of equity indexes: SP500, FTSE, DAX and CAC. On the basis of these results, we show that the WASC captures many of the known stylized facts associated with financial markets, including the negative correlation between stock returns and volatility. It also helps reveal interesting patterns in the studied indexes'covariances and their correlation dynamics.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In this paper, we present and discuss the estimation of the Wishart Affine Stochastic Correlation (WASC) model introduced in Da Fonseca et al. (2006) under the historical measure. We review the main estimation possibilities for this continuous time process and provide elements to show that the utilization of empirical characteristic function-based estimates is advisable as this function is exponential affine in the WASC case. We thus propose to use the estimation strategy presented in Carrasco et al. (2003), using a continuum of moment conditions based on the characteristic function. We investigate the behavior of the estimates through Monte Carlo simulations. Then, we present the estimation results obtained using a dataset of equity indexes: SP500, FTSE, DAX and CAC. On the basis of these results, we show that the WASC captures many of the known stylized facts associated with financial markets, including the negative correlation between stock returns and volatility. It also helps reveal interesting patterns in the studied indexes'covariances and their correlation dynamics. |
Carl Chiarella; José Da Fonseca; Martino Grasselli Pricing Range Notes within Wishart Affine Models Article de journal Insurance: Mathematics and Economics, 58 , p. 774-793, 2014, ISSN: 0167-6687. Résumé | Liens | BibTeX @article{Grassellibb,
title = {Pricing Range Notes within Wishart Affine Models},
author = { Carl Chiarella and José Da Fonseca and Martino Grasselli},
url = {http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2308636},
doi = {http://dx.doi.org/10.1016/j.insmatheco.2014.07.008},
issn = {0167-6687},
year = {2014},
date = {2014-01-03},
journal = {Insurance: Mathematics and Economics},
volume = {58},
pages = {774-793},
abstract = {We provide analytic pricing formulas for Fixed and Floating Range Accrual Notes within the multi-factor Wishart affine framework which extends significantly the standard affine model. Using estimates for three short rate models, two of which are based on the Wishart process whilst the third one belongs to the standard affine framework, we price these structured products using the FFT methodology. Thanks to the Wishart tractability the hedge ratios are also easily computed. As the models are estimated on the same data-set, our results illustrate how the fit discrepancies (meaning differences in the likelihood functions) between models translate in terms of derivatives pricing errors, and we show that the models can produce different price evolutions for the Range Accrual Notes. The differences can be substantial and underline the importance of model risk both from a static and dynamic perspective. These results are confirmed by an analysis performed at the hedge ratios level.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We provide analytic pricing formulas for Fixed and Floating Range Accrual Notes within the multi-factor Wishart affine framework which extends significantly the standard affine model. Using estimates for three short rate models, two of which are based on the Wishart process whilst the third one belongs to the standard affine framework, we price these structured products using the FFT methodology. Thanks to the Wishart tractability the hedge ratios are also easily computed. As the models are estimated on the same data-set, our results illustrate how the fit discrepancies (meaning differences in the likelihood functions) between models translate in terms of derivatives pricing errors, and we show that the models can produce different price evolutions for the Range Accrual Notes. The differences can be substantial and underline the importance of model risk both from a static and dynamic perspective. These results are confirmed by an analysis performed at the hedge ratios level. |
Alvise De Col, Alessandro Gnoatto, Martino Grasselli Smiles all around: FX joint calibration in a multi-Heston model Article de journal Journal of Banking & Finance, 37 (10), p. 3799-3818, 2013, ISSN: 0378-4266. Résumé | Liens | BibTeX @article{Grassellibb,
title = {Smiles all around: FX joint calibration in a multi-Heston model},
author = { Alvise De Col, Alessandro Gnoatto, Martino Grasselli},
url = {http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1982092},
doi = {http://dx.doi.org/10.1016/j.jbankfin.2013.05.031},
issn = {0378-4266},
year = {2013},
date = {2013-10-01},
journal = {Journal of Banking & Finance},
volume = {37},
number = {10},
pages = {3799-3818},
abstract = {We introduce a novel multi-factor Heston-based stochastic volatility model, which is able to reproduce consistently typical multi-dimensional FX vanilla markets, while retaining the (semi)-analytical tractability typical of affine models and relying on a reasonable number of parameters. A successful joint calibration to real market data is presented together with various in- and out-of-sample calibration exercises to highlight the robustness of the parameters estimation. The proposed model preserves the natural inversion and triangulation symmetries of FX spot rates and its functional form, irrespective of choice of the risk-free currency. That is, all currencies are treated in the same way.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We introduce a novel multi-factor Heston-based stochastic volatility model, which is able to reproduce consistently typical multi-dimensional FX vanilla markets, while retaining the (semi)-analytical tractability typical of affine models and relying on a reasonable number of parameters. A successful joint calibration to real market data is presented together with various in- and out-of-sample calibration exercises to highlight the robustness of the parameters estimation. The proposed model preserves the natural inversion and triangulation symmetries of FX spot rates and its functional form, irrespective of choice of the risk-free currency. That is, all currencies are treated in the same way. |
Alessandro Gnoatto; Martino Grasselli; José Da Fonseca A flexible matrix Libor model with smiles Article de journal Journal of Economic Dynamics and Control, 37 (4), p. 774-793, 2013, ISSN: 0165-1889. Résumé | Liens | BibTeX @article{Grassellibb,
title = {A flexible matrix Libor model with smiles},
author = { Alessandro Gnoatto and Martino Grasselli and José Da Fonseca},
url = {http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2027034},
doi = {http://dx.doi.org/10.1016/j.jedc.2012.11.006},
issn = {0165-1889},
year = {2013},
date = {2013-04-01},
journal = {Journal of Economic Dynamics and Control},
volume = {37},
number = {4},
pages = {774-793},
abstract = {We present a flexible approach for the valuation of interest rate derivatives based on affine processes. We extend the methodology proposed in Keller-Ressel et al. (in press) by changing the choice of the state space. We provide semi-closed-form solutions for the pricing of caps and floors. We then show that it is possible to price swaptions in this multifactor setting with a good degree of analytical tractability. This is done via the Edgeworth expansion approach developed in Collin-Dufresne and Goldstein (2002). A numerical exercise illustrates the flexibility of Wishart Libor model in describing the movements of the implied volatility surface.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We present a flexible approach for the valuation of interest rate derivatives based on affine processes. We extend the methodology proposed in Keller-Ressel et al. (in press) by changing the choice of the state space. We provide semi-closed-form solutions for the pricing of caps and floors. We then show that it is possible to price swaptions in this multifactor setting with a good degree of analytical tractability. This is done via the Edgeworth expansion approach developed in Collin-Dufresne and Goldstein (2002). A numerical exercise illustrates the flexibility of Wishart Libor model in describing the movements of the implied volatility surface. |
Daniel Gabay; Martino Grasselli Fair Demographic Risk Sharing in Defined Contribution Pension Systems Article de journal Journal of Economic Dynamics and Control, 36 (4), p. 657-669, 2012, ISSN: 0165-1889. Résumé | Liens | BibTeX @article{Grassellibb,
title = {Fair Demographic Risk Sharing in Defined Contribution Pension Systems},
author = { Daniel Gabay and Martino Grasselli},
url = {http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1667579},
doi = {doi:10.1016/j.jedc.2011.12.002},
issn = {0165-1889},
year = {2012},
date = {2012-04-01},
journal = {Journal of Economic Dynamics and Control},
volume = {36},
number = {4},
pages = {657-669},
abstract = {In this article we formulate and solve the optimal design problem of a defined contribution public pension fund, in a highly stylized but still rather general non-stationary framework. We adopt the viewpoint of a benevolent social planner who aims at treating in a fair manner the successive overlapping generations participating to such a long-term mandatory system. Using the approach of El Karoui and Jeanblanc (1998) for the optimal consumption and portfolio choice problem with random income in a complete market, we exhibit a solution to our intertemporal stochastic control problem where each generation receives a fair (lumpsum) retirement benefit: it is proportional to the contributions she has paid during her active worklife and follows a fixed common rule (although her pension value itself may depend on variables only observable at her retirement time). We next relax the assumption that the collective pension system is mandatory and investigate the performance of individual investment plans in the market. Comparing the outcomes of both alternatives, we derive a condition under which the collective fund can be expected to overperform the individual plan. In the special case of a stationary economy, such a possibility has been pointed out by Gollier (2008). In fact this effect results from the possibility for the collective fund to borrow today against contributions of future generations, which allows to implement riskier strategies and may improve its performance.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In this article we formulate and solve the optimal design problem of a defined contribution public pension fund, in a highly stylized but still rather general non-stationary framework. We adopt the viewpoint of a benevolent social planner who aims at treating in a fair manner the successive overlapping generations participating to such a long-term mandatory system. Using the approach of El Karoui and Jeanblanc (1998) for the optimal consumption and portfolio choice problem with random income in a complete market, we exhibit a solution to our intertemporal stochastic control problem where each generation receives a fair (lumpsum) retirement benefit: it is proportional to the contributions she has paid during her active worklife and follows a fixed common rule (although her pension value itself may depend on variables only observable at her retirement time). We next relax the assumption that the collective pension system is mandatory and investigate the performance of individual investment plans in the market. Comparing the outcomes of both alternatives, we derive a condition under which the collective fund can be expected to overperform the individual plan. In the special case of a stationary economy, such a possibility has been pointed out by Gollier (2008). In fact this effect results from the possibility for the collective fund to borrow today against contributions of future generations, which allows to implement riskier strategies and may improve its performance. |
Martino Grasselli; Lucas Grosset Allocazione ottimale per il Private Banking Article de journal Matematica e Impresa, 2 , 2012. BibTeX @article{Grassellibb,
title = {Allocazione ottimale per il Private Banking},
author = { Martino Grasselli and Lucas Grosset},
year = {2012},
date = {2012-01-01},
journal = {Matematica e Impresa},
volume = {2},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
|
José Da Fonseca; Martino Grasselli Riding on the Smiles Article de journal Quantitative Finance, 11 (11), p. 1609-1632, 2011, ISSN: 1469–7688. Résumé | Liens | BibTeX @article{Grassellibb,
title = {Riding on the Smiles},
author = { José Da Fonseca and Martino Grasselli},
url = {http://papers.ssrn.com/sol3/papers.cfm?abstractid=1663448},
doi = {http://dx.doi.org/10.1080/14697688.2011.615218},
issn = {1469–7688},
year = {2011},
date = {2011-10-20},
journal = {Quantitative Finance},
volume = {11},
number = {11},
pages = {1609-1632},
abstract = {Using a data set of vanilla options on the major indexes we investigate the calibration properties of several multi-factor stochastic volatility models by adopting the fast Fourier transform as the pricing methodology. We study the impact of the penalizing function on the calibration performance and how it affects the calibrated parameters. We consider single-asset as well as multiple-asset models, with particular emphasis on the single-asset Wishart Multidimensional Stochastic Volatility model and the Wishart Affine Stochastic Correlation model, which provides a natural framework for pricing basket options while keeping the stylized smile–skew effects on single-name vanillas. For all models we give some option price approximations that are very useful for speeding up the pricing process. In addition, these approximations allow us to compare different models by conveniently aggregating the parameters, and they highlight the ability of the Wishart-based models to control separately the smile and the skew effects. This is extremely important from a risk-management perspective of a book of derivatives that includes exotic as well as basket options.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Using a data set of vanilla options on the major indexes we investigate the calibration properties of several multi-factor stochastic volatility models by adopting the fast Fourier transform as the pricing methodology. We study the impact of the penalizing function on the calibration performance and how it affects the calibrated parameters. We consider single-asset as well as multiple-asset models, with particular emphasis on the single-asset Wishart Multidimensional Stochastic Volatility model and the Wishart Affine Stochastic Correlation model, which provides a natural framework for pricing basket options while keeping the stylized smile–skew effects on single-name vanillas. For all models we give some option price approximations that are very useful for speeding up the pricing process. In addition, these approximations allow us to compare different models by conveniently aggregating the parameters, and they highlight the ability of the Wishart-based models to control separately the smile and the skew effects. This is extremely important from a risk-management perspective of a book of derivatives that includes exotic as well as basket options. |
José Da Fonseca; Martino Grasselli; Florian Ielpo Hedging (Co)Variance Risk with Variance Swaps Article de journal International Journal of Theoretical and Applied Finance, 14 (6), p. 899-943, 2011, ISSN: 0219-0249. Résumé | Liens | BibTeX @article{Grassellibb,
title = {Hedging (Co)Variance Risk with Variance Swaps},
author = { José Da Fonseca and Martino Grasselli and Florian Ielpo},
url = {http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1102521},
doi = {DOI: 10.1142/S0219024911006784},
issn = {0219-0249},
year = {2011},
date = {2011-05-03},
journal = {International Journal of Theoretical and Applied Finance},
volume = {14},
number = {6},
pages = {899-943},
abstract = {In this paper, we quantify the impact on the representative agent's welfare of the presence of derivative products spanning covariance risk. In an asset allocation framework with stochastic (co)variances, we allow the agent to invest not only in the stocks but also in the associated variance swaps. We solve this optimal portfolio allocation program using the Wishart Affine Stochastic Correlation framework, as introduced in Da Fonseca, Grasselli and Tebaldi (2007): it shares the analytical tractability of the single-asset counterpart represented by the [36] model and it seems to be the natural framework for studying multivariate problems when volatilities as well as correlations are stochastic. What is more, this framework shows how variance swaps can implicitly span the covariance risk. We provide the explicit solution to the portfolio optimization problem and we discuss the structure of the portfolio loadings with respect to model parameters. Using real data on major indexes, we find that the impact of covariance risk on the optimal strategy is huge. It first leads to a portfolio that is mostly driven by the market price of volatility-covolatility risks. It is then strongly leveraged through variance swaps, thus leading to a much higher utility, when compared to the case when investing in such derivatives is not possible.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In this paper, we quantify the impact on the representative agent's welfare of the presence of derivative products spanning covariance risk. In an asset allocation framework with stochastic (co)variances, we allow the agent to invest not only in the stocks but also in the associated variance swaps. We solve this optimal portfolio allocation program using the Wishart Affine Stochastic Correlation framework, as introduced in Da Fonseca, Grasselli and Tebaldi (2007): it shares the analytical tractability of the single-asset counterpart represented by the [36] model and it seems to be the natural framework for studying multivariate problems when volatilities as well as correlations are stochastic. What is more, this framework shows how variance swaps can implicitly span the covariance risk. We provide the explicit solution to the portfolio optimization problem and we discuss the structure of the portfolio loadings with respect to model parameters. Using real data on major indexes, we find that the impact of covariance risk on the optimal strategy is huge. It first leads to a portfolio that is mostly driven by the market price of volatility-covolatility risks. It is then strongly leveraged through variance swaps, thus leading to a much higher utility, when compared to the case when investing in such derivatives is not possible. |
Alberto Lanzavecchia; Martino Grasselli I nuovi piani di incentivazione Azionaria Article de journal Contabilità, Finanza e Controllo, 34 (2), p. 130-139, 2011. Liens | BibTeX @article{Grassellibb,
title = {I nuovi piani di incentivazione Azionaria},
author = { Alberto Lanzavecchia and Martino Grasselli},
url = {http://www.biblio.liuc.it/scripts/essper/ricerca.asp?tipo=scheda&codice=2253839},
year = {2011},
date = {2011-01-01},
journal = {Contabilità, Finanza e Controllo},
volume = {34},
number = {2},
pages = {130-139},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
|
José Da Fonseca; Martino Grasselli; Claudio Tebaldi A Multifactor Volatility Heston Model Article de journal Quantitative Finance, 8 (6), p. 591-604, 2008, ISSN: 1469–7688. Résumé | Liens | BibTeX @article{Grassellibb,
title = {A Multifactor Volatility Heston Model},
author = { José Da Fonseca and Martino Grasselli and Claudio Tebaldi},
url = {http://www.dse.univr.it/documenti/RapportoRicercaEst/allegato/allegato975769.pdf},
doi = {DOI: 10.1080/14697680701668418},
issn = {1469–7688},
year = {2008},
date = {2008-09-01},
journal = {Quantitative Finance},
volume = {8},
number = {6},
pages = {591-604},
abstract = {We model the volatility of a single risky asset using a multifactor (matrix) Wishart affine process, recently introduced in finance by Gourieroux and Sufana. As in standard Duffie and Kan affine models the pricing problem can be solved through the Fast Fourier Transform of Carr and Madan. A numerical illustration shows that this specification provides a separate fit of the long-term and short-term implied volatility surface and, differently from previous diffusive stochastic volatility models, it is possible to identify a specific factor accounting for the stochastic leverage effect, a well-known stylized fact of the FX option markets analysed by Carr and Wu. A multifactor volatility Heston model - ResearchGate. Available from: http://www.researchgate.net/publication/24086234_A_multifactor_volatility_Heston_model [accessed Jul 3, 2015].},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We model the volatility of a single risky asset using a multifactor (matrix) Wishart affine process, recently introduced in finance by Gourieroux and Sufana. As in standard Duffie and Kan affine models the pricing problem can be solved through the Fast Fourier Transform of Carr and Madan. A numerical illustration shows that this specification provides a separate fit of the long-term and short-term implied volatility surface and, differently from previous diffusive stochastic volatility models, it is possible to identify a specific factor accounting for the stochastic leverage effect, a well-known stylized fact of the FX option markets analysed by Carr and Wu. A multifactor volatility Heston model - ResearchGate. Available from: http://www.researchgate.net/publication/24086234_A_multifactor_volatility_Heston_model [accessed Jul 3, 2015]. |
Martino Grasselli; Claudio Tebaldi Solvable Affine Term Structure Models Article de journal Mathematical Finance, 18 (1), p. 135-153, 2008, ISSN: 0960-1627. Résumé | Liens | BibTeX @article{Grassellibb,
title = {Solvable Affine Term Structure Models},
author = { Martino Grasselli and Claudio Tebaldi},
url = {http://www.researchgate.net/profile/Claudio_Tebaldi/publication/227611192_SOLVABLE_AFFINE_TERM_STRUCTURE_MODELS/links/00b7d5263a7a59cca8000000.pdf
http://dse.univr.it/safe/Papers/GrasselliTebaldi04b.pdf},
doi = {10.1111/j.1467-9965.2007.00325.x},
issn = {0960-1627},
year = {2008},
date = {2008-01-01},
journal = {Mathematical Finance},
volume = {18},
number = {1},
pages = {135-153},
abstract = {An Affine Term Structure Model (ATSM) is said to be solvable if the pricing problem has an explicit solution, i.e., the corresponding Riccati ordinary differential equations have a regular globally integrable flow. We identify the parametric restrictions which are necessary and sufficient for an ATSM with continuous paths, to be solvable in a state spaceD+ × Rn-m, whereD+, the domain of positive factors, has the geometry of a symmetric cone. This class of state spaces includes as special cases those introduced by Duffie andKan (1996), andWishart termstructure processes discussed by Gourieroux and Sufana (2003). For all solvable models we provide the procedure to find the explicit solution of the Riccati ODE.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
An Affine Term Structure Model (ATSM) is said to be solvable if the pricing problem has an explicit solution, i.e., the corresponding Riccati ordinary differential equations have a regular globally integrable flow. We identify the parametric restrictions which are necessary and sufficient for an ATSM with continuous paths, to be solvable in a state spaceD+ × Rn-m, whereD+, the domain of positive factors, has the geometry of a symmetric cone. This class of state spaces includes as special cases those introduced by Duffie andKan (1996), andWishart termstructure processes discussed by Gourieroux and Sufana (2003). For all solvable models we provide the procedure to find the explicit solution of the Riccati ODE. |
Martino Grasselli; Claudio Tebaldi Stochastic Jacobian and Riccati ODE in Affine Term Structure Models Article de journal Decisions in Economics and Finance, 30 (2), p. 95-018, 2007. Résumé | Liens | BibTeX @article{Grassellibb,
title = {Stochastic Jacobian and Riccati ODE in Affine Term Structure Models},
author = { Martino Grasselli and Claudio Tebaldi},
editor = {Springer},
url = {http://www.researchgate.net/publication/225396707_Stochastic_Jacobian_and_Riccati_ODE_in_affine_term_structure_models},
doi = {10.1007/s10203-007-0071-y},
year = {2007},
date = {2007-11-01},
journal = {Decisions in Economics and Finance},
volume = {30},
number = {2},
pages = {95-018},
abstract = {In affine term structure models (ATSM) the stochastic Jacobian under the forward measure plays a crucial role for pricing, as discussed in Elliott and van der Hoek (Finance Stoch 5:511–525, 2001). Their approach leads to a deterministic integro-differential equation which, apparently, has the advantage of by-passing the solution to the Riccati ODE obtained by the standard Feynman-Kac argument. In the generic multi-dimensional case, we find a procedure to reduce such integro-differential equation to a non linear matrix ODE. We prove that its solution does necessarily require the solution of the vector Riccati ODE. This result is obtained proving an extension of the celebrated Radon Lemma, which allows us to highlight a deep relation between the geometry of the Riccati flow and the stochastic calculus of variations for an ATSM.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In affine term structure models (ATSM) the stochastic Jacobian under the forward measure plays a crucial role for pricing, as discussed in Elliott and van der Hoek (Finance Stoch 5:511–525, 2001). Their approach leads to a deterministic integro-differential equation which, apparently, has the advantage of by-passing the solution to the Riccati ODE obtained by the standard Feynman-Kac argument. In the generic multi-dimensional case, we find a procedure to reduce such integro-differential equation to a non linear matrix ODE. We prove that its solution does necessarily require the solution of the vector Riccati ODE. This result is obtained proving an extension of the celebrated Radon Lemma, which allows us to highlight a deep relation between the geometry of the Riccati flow and the stochastic calculus of variations for an ATSM. |
José Da Fonseca; Martino Grasselli; Claudio Tebaldi Option pricing when correlations are stochastic: an analytical framework Article de journal Review of Derivatives Research, 10 , p. 151-180, 2007. Résumé | Liens | BibTeX @article{Grassellibb,
title = {Option pricing when correlations are stochastic: an analytical framework},
author = { José Da Fonseca and Martino Grasselli and Claudio Tebaldi},
url = {http://www.researchgate.net/profile/Claudio_Tebaldi/publication/5156698_Option_pricing_when_correlations_are_stochastic_an_analytical_framework/links/00b7d5263a7a679530000000.pdf},
doi = {10.2139/ssrn.982183},
year = {2007},
date = {2007-02-01},
journal = {Review of Derivatives Research},
volume = {10},
pages = {151-180},
abstract = {In this paper we develop a novel market model where asset variancescovariances evolve stochastically. In addition shocks on asset return dynamics are assumed to be linearly correlated with shocks driving the variance-covariance matrix. Analytical tractability is preserved since the model is linear-affine and the conditional characteristic function can be determined explicitly. Quite remarkably, the model provides prices of vanilla options consistent with the smile and skew effects observed, while making possible to detect and quantify the correlation risk in multiple asset derivatives like basket options. In particular it can reproduce the asymmetric conditional correlations effect documented in Ang and Chen (2002) for equity markets. We exemplify analytical tractability providing explicit pricing formulas for rainbow ”Best-of” options.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In this paper we develop a novel market model where asset variancescovariances evolve stochastically. In addition shocks on asset return dynamics are assumed to be linearly correlated with shocks driving the variance-covariance matrix. Analytical tractability is preserved since the model is linear-affine and the conditional characteristic function can be determined explicitly. Quite remarkably, the model provides prices of vanilla options consistent with the smile and skew effects observed, while making possible to detect and quantify the correlation risk in multiple asset derivatives like basket options. In particular it can reproduce the asymmetric conditional correlations effect documented in Ang and Chen (2002) for equity markets. We exemplify analytical tractability providing explicit pricing formulas for rainbow ”Best-of” options. |
Martino Grasselli Sup-Convolution of HARA utility functions in the affine term structure Article de journal Decisions in Economics and Finance, 28 (1), p. 67-78, 2005. Résumé | Liens | BibTeX @article{Grassellibb,
title = {Sup-Convolution of HARA utility functions in the affine term structure},
author = { Martino Grasselli},
editor = {Springer},
url = {http://www.researchgate.net/publication/24054133_Notes_and_Comments_Sup-convolutions_of_HARA_utilities_in_the_affine_term_structure},
doi = {10.1007/s10203-005-0054-9},
year = {2005},
date = {2005-02-01},
journal = {Decisions in Economics and Finance},
volume = {28},
number = {1},
pages = {67-78},
abstract = {In the financial literature, the problem of maximizing the expected utility of the terminal wealth has been investigated extensively (for a survey, see, e.g., Karatzas and Shreve (1998), p. 153, and references therein) by using different approaches. In this paper, we extend the existing literature in two directions. First, we let the utility function U(.) of the financial agent (who is a price taker) be implicitly defined through I(.)=(U ' (.))–1, which is assumed to be additively separable, i.e., I(.)=? k=1 N I k (.). Second, we solve the investment problem in the general affine term structure model proposed by Duffie and Kan (1996) in which the functions I k (.)},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In the financial literature, the problem of maximizing the expected utility of the terminal wealth has been investigated extensively (for a survey, see, e.g., Karatzas and Shreve (1998), p. 153, and references therein) by using different approaches. In this paper, we extend the existing literature in two directions. First, we let the utility function U(.) of the financial agent (who is a price taker) be implicitly defined through I(.)=(U ' (.))–1, which is assumed to be additively separable, i.e., I(.)=? k=1 N I k (.). Second, we solve the investment problem in the general affine term structure model proposed by Duffie and Kan (1996) in which the functions I k (.) |
Martino Grasselli; Claudio Tebaldi Bond Price and Impulse Response Function in the Balduzzi, Das, Foresi and Sundaram (1996) Model Article de journal Economic Notes, 33 (3), p. 359-374, 2004. Résumé | Liens | BibTeX @article{Grassellibb,
title = {Bond Price and Impulse Response Function in the Balduzzi, Das, Foresi and Sundaram (1996) Model},
author = { Martino Grasselli and Claudio Tebaldi},
url = {http://www.researchgate.net/publication/4755273_Bond_Price_and_Impulse_Response_Function_for_the_Balduzzi_Das_Foresi_and_Sundaram_%281996%29_Model},
doi = {10.1111/j.0391-5026.2004.00136.x},
year = {2004},
date = {2004-11-01},
journal = {Economic Notes},
volume = {33},
number = {3},
pages = {359-374},
abstract = {In this paper, we analyse the Affine Term Structure Model (ATSM) proposed by Balduzzi, Das, Foresi and Sundaram (BDFS, 1996) and provide the closed-form expression of the bond price. In addition, we extend the notion of Impulse Response Function to the class of ATSM. We show that it is closely related to the duration measure, and we compute it explicitly in the BDFS model.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In this paper, we analyse the Affine Term Structure Model (ATSM) proposed by Balduzzi, Das, Foresi and Sundaram (BDFS, 1996) and provide the closed-form expression of the bond price. In addition, we extend the notion of Impulse Response Function to the class of ATSM. We show that it is closely related to the duration measure, and we compute it explicitly in the BDFS model. |
Griselda Deelstra; Martino Grasselli; Pierre-François Koehl Optimal Design of the Guarantee for Defined Contribution Funds Article de journal Journal of Economic Dynamics and Control, 28 (11), p. 2239-2260, 2004. Résumé | Liens | BibTeX @article{Grassellibb,
title = {Optimal Design of the Guarantee for Defined Contribution Funds},
author = { Griselda Deelstra and Martino Grasselli and Pierre-François Koehl},
url = {http://www.sciencedirect.com/science/article/pii/S0165188903002057},
doi = {10.1016/j.jedc.2003.10.003},
year = {2004},
date = {2004-10-01},
journal = {Journal of Economic Dynamics and Control},
volume = {28},
number = {11},
pages = {2239-2260},
abstract = {The question we solve is the optimal design of the minimum guarantee in a Defined Contribution Pension Fund Scheme. We study the investment in the financial market by assuming that the pension fund optimizes its retribution which is a part of the surplus, that is the difference between the pension fund value and the guarantee. Then we define the optimal guarantee as the solution of the contributor's optimization program and find the solution explicitly. Finally, we analyze the impact of the main parameters, and particularly the sharing rule between the contributor and the pension fund. We find that favorable sharing rules for the pension fund lead to conservative guarantees for the contributor: the sharing rule is a way to create a continuum between two extreme pension funding methods that are Defined Benefit and Defined Contribution Pension Schemes, and the sharing rule allows partial risk transfer between the contributor and the pension fund manager.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
The question we solve is the optimal design of the minimum guarantee in a Defined Contribution Pension Fund Scheme. We study the investment in the financial market by assuming that the pension fund optimizes its retribution which is a part of the surplus, that is the difference between the pension fund value and the guarantee. Then we define the optimal guarantee as the solution of the contributor's optimization program and find the solution explicitly. Finally, we analyze the impact of the main parameters, and particularly the sharing rule between the contributor and the pension fund. We find that favorable sharing rules for the pension fund lead to conservative guarantees for the contributor: the sharing rule is a way to create a continuum between two extreme pension funding methods that are Defined Benefit and Defined Contribution Pension Schemes, and the sharing rule allows partial risk transfer between the contributor and the pension fund manager. |
Martino Grasselli Méthodes récentes de gestion des fonds de retraite Article de journal Banque & Marchés, 72 , p. 34-46, 2004. BibTeX @article{Grassellibb,
title = {Méthodes récentes de gestion des fonds de retraite},
author = { Martino Grasselli},
year = {2004},
date = {2004-01-01},
journal = {Banque & Marchés},
volume = {72},
pages = {34-46},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
|
Martino Grasselli A Stability Result for the HARA Class with Stochastic Interest Rates Article de journal Insurance : Mathematics and Economics, 33 (3), p. 611-627, 2003. Résumé | Liens | BibTeX @article{Grassellibb,
title = {A Stability Result for the HARA Class with Stochastic Interest Rates},
author = { Martino Grasselli},
url = {http://www.sciencedirect.com/science/article/pii/S0167668703001938},
doi = {10.1016/j.insmatheco.2003.09.003},
year = {2003},
date = {2003-12-19},
journal = {Insurance : Mathematics and Economics},
volume = {33},
number = {3},
pages = {611-627},
abstract = {We study an investment problem where the interest rates follow the Cox–Ingersoll–Ross dynamics. The optimal investment strategy is obtained in explicit form under the hypotheses that financial markets are complete and that the utility functions belong to the HARA, exponential and logarithmic classes. We show that the solution for the HARA utility is stable when the parameters vary in a suitable way: more precisely, we find that the optimal investment strategy corresponding to the HARA function converges almost surely to the one corresponding to the exponential and logarithmic utility functions.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We study an investment problem where the interest rates follow the Cox–Ingersoll–Ross dynamics. The optimal investment strategy is obtained in explicit form under the hypotheses that financial markets are complete and that the utility functions belong to the HARA, exponential and logarithmic classes. We show that the solution for the HARA utility is stable when the parameters vary in a suitable way: more precisely, we find that the optimal investment strategy corresponding to the HARA function converges almost surely to the one corresponding to the exponential and logarithmic utility functions. |
Griselda Deelstra; Martino Grasselli; Pierre-François Koehl Optimal Investment Strategies in the presence of a minimum guarantee Article de journal Insurance: Mathematics and Economics, 33 (1), p. 189-207, 2003. Résumé | Liens | BibTeX @article{Grassellibb,
title = {Optimal Investment Strategies in the presence of a minimum guarantee},
author = { Griselda Deelstra and Martino Grasselli and Pierre-François Koehl},
url = {http://www.sciencedirect.com/science/article/pii/S0167668703001537},
doi = {10.1016/S0167-6687(03)00153-7},
year = {2003},
date = {2003-08-08},
journal = {Insurance: Mathematics and Economics},
volume = {33},
number = {1},
pages = {189-207},
abstract = {In a continuous-time framework, we consider the problem of a Defined Contribution Pension Fund in the presence of a minimum guarantee. The problem of the fund manager is to invest the initial wealth and the (stochastic) contribution flow into the financial market, in order to maximize the expected utility function of the terminal wealth under the constraint that the terminal wealth must exceed the minimum guarantee. We assume that the stochastic interest rates follow the affine dynamics, including the Cox–Ingersoll–Ross (CIR) model [Econometrica 53 (1985) 385] and the Vasiček model. The optimal investment strategies are obtained by assuming the completeness of financial markets and a CRRA utility function. Explicit formulae for the optimal investment strategies are included for different examples of guarantees and contributions.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In a continuous-time framework, we consider the problem of a Defined Contribution Pension Fund in the presence of a minimum guarantee. The problem of the fund manager is to invest the initial wealth and the (stochastic) contribution flow into the financial market, in order to maximize the expected utility function of the terminal wealth under the constraint that the terminal wealth must exceed the minimum guarantee. We assume that the stochastic interest rates follow the affine dynamics, including the Cox–Ingersoll–Ross (CIR) model [Econometrica 53 (1985) 385] and the Vasiček model. The optimal investment strategies are obtained by assuming the completeness of financial markets and a CRRA utility function. Explicit formulae for the optimal investment strategies are included for different examples of guarantees and contributions. |
Griselda Deelstra; Martino Grasselli Optimal Investment Strategies in a CIR Framework Article de journal Journal of Applied Probability, 1 , p. 1-15, 2000. Résumé | Liens | BibTeX @article{Grassellibb,
title = {Optimal Investment Strategies in a CIR Framework},
author = { Griselda Deelstra and Martino Grasselli},
url = {http://www.researchgate.net/publication/38350231_Optimal_investment_strategies_in_a_CIR_framework},
doi = {10.1239/jap/1014843074},
year = {2000},
date = {2000-01-01},
journal = {Journal of Applied Probability},
volume = {1},
pages = {1-15},
abstract = {We study an optimal investment problem in a continuous-time framework where the interest rates follow Cox-Ingersoll-Ross dynamics. Closed form formulae for the optimal investment strategy are obtained by assuming the completeness of financial markets and the CRRA utility function. In particular, we study the behaviour of the solution when time approaches the terminal date.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We study an optimal investment problem in a continuous-time framework where the interest rates follow Cox-Ingersoll-Ross dynamics. Closed form formulae for the optimal investment strategy are obtained by assuming the completeness of financial markets and the CRRA utility function. In particular, we study the behaviour of the solution when time approaches the terminal date. |
Griselda Deelstra; Martino Grasselli; Pierre-François Koehl Conditional Dominance Criteria: Definition and Application to Risk-Management Article de journal Insurance: Mathematics and Economics, 25 (3), p. 295-306, 1999. Résumé | Liens | BibTeX @article{Grassellibb,
title = {Conditional Dominance Criteria: Definition and Application to Risk-Management},
author = { Griselda Deelstra and Martino Grasselli and Pierre-François Koehl},
url = {http://www.sciencedirect.com/science/article/pii/S016766879900013X},
doi = {10.1016/S0167-6687(99)00013-X},
year = {1999},
date = {1999-12-10},
journal = {Insurance: Mathematics and Economics},
volume = {25},
number = {3},
pages = {295-306},
abstract = {We define the concept of conditional dominance and use it for obtaining bounds on the hedging prices of random variables. These bounds depend only on the characteristics of the financial market and the random variables to hedge. Moreover, they are coherent with the equilibrium and tighter than the ones obtained by the classical super-replication approach, significantly in some cases. This approach can be applied in static as well as dynamic frameworks.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We define the concept of conditional dominance and use it for obtaining bounds on the hedging prices of random variables. These bounds depend only on the characteristics of the financial market and the random variables to hedge. Moreover, they are coherent with the equilibrium and tighter than the ones obtained by the classical super-replication approach, significantly in some cases. This approach can be applied in static as well as dynamic frameworks. |
Martino Grasselli; Bruno Viscolani On a financial project valuation model proposed by De Giuli and Magnani Article de journal Rendiconti per gli Studi Economici e Quantitativi, p. 92-107, 1999. BibTeX @article{Grassellibb,
title = {On a financial project valuation model proposed by De Giuli and Magnani},
author = { Martino Grasselli and Bruno Viscolani},
year = {1999},
date = {1999-01-01},
journal = {Rendiconti per gli Studi Economici e Quantitativi},
pages = {92-107},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
|
grasselli The 4/2 stochastic volatility model Article de journal Mathematical Finance, 27 (4), p. 1013-1034, 0000. BibTeX @article{Grassellinov2017,
title = {The 4/2 stochastic volatility model},
author = {grasselli},
journal = {Mathematical Finance},
volume = {27},
number = {4},
pages = {1013-1034},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
|
Livres
|
Martino Grasselli; Alessandra Buratto; Luca Grosset; Bruno Viscolani Matematica Generale Livre Progetto Libreria, Padova, Italy, 2010, ISBN: 9788896477120. Liens | BibTeX @book{Grassellibb,
title = {Matematica Generale},
author = { Martino Grasselli and Alessandra Buratto and Luca Grosset and Bruno Viscolani},
editor = {Libreria Progetto Padova},
url = {http://www.didattica.unipd.it/offerta/2015/EP/EP2093/2014/000ZZ/1122988},
isbn = {9788896477120},
year = {2010},
date = {2010-01-01},
publisher = {Progetto Libreria},
address = {Padova, Italy},
keywords = {},
pubstate = {published},
tppubtype = {book}
}
|
Conférences
|
Giorgia Callegaro; Lucio Fiorin; Martino Grasselli Pricing via Quantization in stochastic volatility models 5500 - 5599 Conférence QMF Quantitative Methods in Finance, Sydney, UTS, 15-17 december, 2015. BibTeX @conference{grasselli2015confb,
title = {Pricing via Quantization in stochastic volatility models},
author = {Giorgia Callegaro and Lucio Fiorin and Martino Grasselli},
year = {2015},
date = {2015-12-15},
publisher = {QMF Quantitative Methods in Finance, Sydney, UTS, 15-17 december},
keywords = {},
pubstate = {published},
tppubtype = {conference}
}
|
Martino Grasselli; Mark Craddock Lie Symmetries Methods in Local Volatility Models 5500 - 5599 Conférence Workshop in Quantitative Finance, Florence (Italy), 29-30 january, 2015. BibTeX @conference{grasselli2015conf,
title = {Lie Symmetries Methods in Local Volatility Models},
author = {Martino Grasselli and Mark Craddock},
year = {2015},
date = {2015-01-30},
publisher = {Workshop in Quantitative Finance, Florence (Italy), 29-30 january},
keywords = {},
pubstate = {published},
tppubtype = {conference}
}
|
Inproceedings
|
Grasselli, M.; Fiorin, L.; Callegaro, G. Quantization meets Fourier: A New methodology for pricing options inproceedings Quantitative Methods in Finance QMF2017, Sydney, Australie, 12-15 december., 2017. BibTeX @inproceedings{grasselli2017b,
title = {Quantization meets Fourier: A New methodology for pricing options},
author = {Grasselli, M. and Fiorin, L. and Callegaro, G. },
year = {2017},
date = {2017-12-30},
booktitle = {Quantitative Methods in Finance QMF2017, Sydney, Australie, 12-15 december.},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Grasselli, M.; Fiorin, L.; Callegaro, G. Quantization meets Fourier: A New methodology for pricing options inproceedings Mathematics in Finance 2017 International Conference, Cape Town, South Africa, 2-3 november, 2017. BibTeX @inproceedings{grasselli2017c,
title = {Quantization meets Fourier: A New methodology for pricing options},
author = {Grasselli, M. and Fiorin, L. and Callegaro, G. },
year = {2017},
date = {2017-11-10},
booktitle = {Mathematics in Finance 2017 International Conference, Cape Town, South Africa, 2-3 november},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Grasselli, M.; Wagalath, L. VIX versus VXX: a joint analytical framework inproceedings Mathematics and Finance: Research in Options, Rio de Janeiro, Brazil, 25-30 november, 2017. BibTeX @inproceedings{grasselli2017e,
title = {VIX versus VXX: a joint analytical framework},
author = {Grasselli, M. and Wagalath, L. },
year = {2017},
date = {2017-11-10},
booktitle = {Mathematics and Finance: Research in Options, Rio de Janeiro, Brazil, 25-30 november},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Martino Grasselli Organizer of a mini-symposium on Quantization inproceedings 8th General AMaMeF Conference, Amsterdam, June 19-23, 2017. BibTeX @inproceedings{grasselli2017a,
title = {Organizer of a mini-symposium on Quantization},
author = {Martino Grasselli},
year = {2017},
date = {2017-06-14},
booktitle = {8th General AMaMeF Conference, Amsterdam, June 19-23},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Grasselli, M.; Mesias, A.; Schlogl, E A consistent stochastic model of the term structure of interest rates for multiple tenors inproceedings Second Paris-Asia Conference in Quantitative Finance, Suzhou, China, 26-27 mai, 2017. BibTeX @inproceedings{grasselli2017d,
title = {A consistent stochastic model of the term structure of interest rates for multiple tenors},
author = {Grasselli, M. and Mesias, A. and Schlogl, E},
year = {2017},
date = {2017-05-15},
booktitle = {Second Paris-Asia Conference in Quantitative Finance, Suzhou, China, 26-27 mai},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Grasselli Martino; Callegaro G.; Fiorin L. Quantized stochastic volatility inproceedings Quantitative Methods in Finance QMF 2016 , Sydney, Australia, 2016. BibTeX @inproceedings{GrasseliiConf1,
title = {Quantized stochastic volatility},
author = {Grasselli Martino and Callegaro G. and Fiorin L. },
year = {2016},
date = {2016-12-13},
booktitle = {Quantitative Methods in Finance QMF 2016 , Sydney, Australia},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Grasselli M.; Craddock M. Lie symmetry methods for local volatility models inproceedings Mathematics and Finance: Research in Options, IMPA, Rio de Janeiro, Brazil, 2016. BibTeX @inproceedings{GrasselliConf2,
title = {Lie symmetry methods for local volatility models},
author = {Grasselli M. and Craddock M. },
year = {2016},
date = {2016-11-25},
booktitle = {Mathematics and Finance: Research in Options, IMPA, Rio de Janeiro, Brazil},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Grasselli M.; Callegaro G.; Fiorin L. Pricing via Recursive Quantization in Stochastic Volatility Models inproceedings Workshop in Quantitative Finance, Pisa, Italy, 2016. BibTeX @inproceedings{Grasselliconf3,
title = {Pricing via Recursive Quantization in Stochastic Volatility Models},
author = {Grasselli M. and Callegaro G. and Fiorin L. },
year = {2016},
date = {2016-04-12},
booktitle = {Workshop in Quantitative Finance, Pisa, Italy},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Thèses
|
Martino Grasselli La Gestion de Portefeuille à Long Terme : une Approche de Finance Mathématique Thèse de doctorat Université de Paris I Panthéon Sorbonne, 2001. Résumé | Liens | BibTeX @phdthesis{Grassellibb,
title = {La Gestion de Portefeuille à Long Terme : une Approche de Finance Mathématique},
author = { Martino Grasselli},
url = {http://www.theses.fr/2001PA010023},
year = {2001},
date = {2001-01-01},
address = {Paris, France},
school = {Université de Paris I Panthéon Sorbonne},
abstract = {Nous proposons des modèles mathématiques appliqués à la gestion des fonds de pension, et de façon plus générale à celle des institutions financières dont l 'horizon temporel est assez long (30 ou 40 ans). Dans la première partie nous explicitons la stratégie optimale d'investissement du problème de consommation-investissement lorsque les taux d'intérêt suivent la dynamique (stochastique) proposée par Cox, lngersoll et Ross ( 1985) et lorsque la fonction d'utilité est du type HARA ou exponentiel. L'approche utilisée est l'approche par martingale développée par Karatzas et al. ( 1987) dans le cadre de marchés complets. Nous étendons alors les résultats dans deux directions: structure des taux multifactoriels et fonctions d'utilité plus générales. Comme application, nous considérons le cas des fonds de pension à contributions définies et minimum garanti. Dans la deuxième partie, on s'intéresse à l'incomplétude du marché. Nous introduisons un critère, que nous appelons Dominance Conditionnelle, qui nous permet d'obtenir des bornes sur la valeur de tout actif contingent. Ces bornes ne dépendent pas des caractéristiques des agents et dans certains cas elles sont strictement meilleures que celles obtenues par sur-réplication. Comme application, nous considérons le cas d'un fond de pension à prestation définie. En utilisant le critère de Dominance Conditionnelle, nous pouvons trouver explicitement la borne supérieure pour la valeur de prestation définie, ainsi que la stratégie de couverture.},
keywords = {},
pubstate = {published},
tppubtype = {phdthesis}
}
Nous proposons des modèles mathématiques appliqués à la gestion des fonds de pension, et de façon plus générale à celle des institutions financières dont l 'horizon temporel est assez long (30 ou 40 ans). Dans la première partie nous explicitons la stratégie optimale d'investissement du problème de consommation-investissement lorsque les taux d'intérêt suivent la dynamique (stochastique) proposée par Cox, lngersoll et Ross ( 1985) et lorsque la fonction d'utilité est du type HARA ou exponentiel. L'approche utilisée est l'approche par martingale développée par Karatzas et al. ( 1987) dans le cadre de marchés complets. Nous étendons alors les résultats dans deux directions: structure des taux multifactoriels et fonctions d'utilité plus générales. Comme application, nous considérons le cas des fonds de pension à contributions définies et minimum garanti. Dans la deuxième partie, on s'intéresse à l'incomplétude du marché. Nous introduisons un critère, que nous appelons Dominance Conditionnelle, qui nous permet d'obtenir des bornes sur la valeur de tout actif contingent. Ces bornes ne dépendent pas des caractéristiques des agents et dans certains cas elles sont strictement meilleures que celles obtenues par sur-réplication. Comme application, nous considérons le cas d'un fond de pension à prestation définie. En utilisant le critère de Dominance Conditionnelle, nous pouvons trouver explicitement la borne supérieure pour la valeur de prestation définie, ainsi que la stratégie de couverture. |
Martino Grasselli Pension Funds: Deterministic and Stochastic Approaches Thèse de doctorat Università degli Studi di Trieste, 1999. Liens | BibTeX @phdthesis{Grassellibb,
title = {Pension Funds: Deterministic and Stochastic Approaches},
author = { Martino Grasselli},
url = {http://opac.bncf.firenze.sbn.it/opac/controller.jsp;jsessionid=45688DAF69DF0671877B2222EF30283C?action=notizia_view&notizia_idn=tsi9900709},
year = {1999},
date = {1999-01-01},
address = {Trieste, Italy},
school = {Università degli Studi di Trieste},
keywords = {},
pubstate = {published},
tppubtype = {phdthesis}
}
|
Technical Reports
|
Griselda Deelstra; Martino Grasselli; Christopher van Weverberg Explosion time for some Wishart transforms Technical Report ULB 2015. Résumé | Liens | BibTeX @techreport{Deelstra2015,
title = {Explosion time for some Wishart transforms},
author = {Griselda Deelstra and Martino Grasselli and Christopher van Weverberg},
url = {http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2542185},
year = {2015},
date = {2015-10-22},
institution = {ULB},
abstract = {We consider non mean-reverting Wishart processes and we study the problem of determining the smallest time such that the Laplace transforms of the process and its integral become infinite. Thanks to the remarkable property of (affine) Wishart processes to reproduce non-trivial dependence among the positive factors, we are able to explain the behavior of the explosion times in terms of the relative importance of the involved factors and their correlations. In this way, we go far beyond the known results that can be recovered in the classical affine framework.},
keywords = {},
pubstate = {published},
tppubtype = {techreport}
}
We consider non mean-reverting Wishart processes and we study the problem of determining the smallest time such that the Laplace transforms of the process and its integral become infinite. Thanks to the remarkable property of (affine) Wishart processes to reproduce non-trivial dependence among the positive factors, we are able to explain the behavior of the explosion times in terms of the relative importance of the involved factors and their correlations. In this way, we go far beyond the known results that can be recovered in the classical affine framework. |
Dépubli
|
Bruno Bouchard; Martino Grasselli Super-Hedging under the Real World Measure with Stochastic Volatility Dépubli 2013. BibTeX @unpublished{Grassellibb,
title = {Super-Hedging under the Real World Measure with Stochastic Volatility},
author = { Bruno Bouchard and Martino Grasselli},
year = {2013},
date = {2013-01-01},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
|
Carlo Bertolazzi; Bruno Giacomello; Martino Grasselli; Alberto Lanzavecchia Stock Option Plans with Incentives Dépubli 2012. Liens | BibTeX @unpublished{Grassellibb,
title = {Stock Option Plans with Incentives},
author = { Carlo Bertolazzi and Bruno Giacomello and Martino Grasselli and Alberto Lanzavecchia},
url = {https://iris.univr.it/handle/11562/345874},
year = {2012},
date = {2012-01-01},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
|
Organisés au sein de Léosquale, les nageurs du Pôle Léonard de Vinci ont obtenu 5 médailles dont 4 titres nationaux au Champion…
Pour sa première participation à l’Altigliss Challenge, retour sur le bilan de Slide Session sur cette édition 2018 de la c…
Au terme d’une campagne animée qui a opposé deux listes hautes en couleurs, c’est le BDE Pulvwatch qui prend la relève…
La Joute De Vinci, c’est l’association débats du Pôle Léonard de Vinci. Débats et … belles paroles ! Pour la pr…
