## Non-parametric calibration of jump-diffusion option pricing models (2004)

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@MISC{Cont04non-parametriccalibration,

author = {Rama Cont and Peter Tankov},

title = {Non-parametric calibration of jump-diffusion option pricing models},

year = {2004}

}

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2188 | Numerical recipes in C: The art of scientific computing - Press, Flannery, et al. - 1992 |

813 | Lévy Processes and Infinitely Divisible Distributions, Cambridge University - Sato - 1999 |

684 | Option pricing when the underlying stock returns are discontinuous
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- 1976
(Show Context)
Citation Context ...models based on Lévy processes (Andersen and Andreasen, 2000; Eberlein, 2001; Eberlein, Keller and Prause, 1998; Cont, Bouchaud and Potters, 1997; Kou, 2002; Madan, 2001; Madan, Carr and Chang, 1998;=-= Merton, 1976;-=- Schoutens, 2002). A widely studied class is that of exponential Lévy processes in which the price of the underlying asset is written as S t =exp(rt + X t ), where r is the discount rate and X is a L... |

504 | Lévy Processes - Bertoin - 1996 |

501 | Regularization of Inverse Problems - Engl, Hanke, et al. - 1996 |

370 |
Jumps and Stochastic Volatility: Exchange Rates
- Bates
- 1996
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Citation Context ...ption computed in an exponential Lévy model with volatility σ and Lévy measure ν. The optimization problem (27) is usually solved numerically by a gradient-based method (Andersen and Andreasen, 20=-=00; Bates, 1996-=-a). While, contrarily to (1), one can always find some solution, the minimization function is non-convex, so a gradient descent may not succeed in locating the global minimum. Owing to the non-convex ... |

342 |
Financial Modeling with Jump Processes
- Cont, Tankov
- 2004
(Show Context)
Citation Context ...velop a robust numerical method for solving it. The properties of the continuum version (29) and the convergence of the solutions of the discretized problem (31) are discussed in the companion paper (=-=Cont and Tankov, 2004-=-b). The following proposition shows that the use of entropy penalization makes our (discretized) problem well posed and hence numerically feasible. PROPOSITION 3 (WELL-POSEDNESS OF THE REGULARIZED PRO... |

317 | Pricing with a Smile - Dupire - 1994 |

316 | A Limited Memory Algorithm for Bound Constrained Optimization - Byrd, Lu, et al. - 1995 |

258 | The fine structure of asset returns: An empirical investigation
- Carr, Geman, et al.
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Citation Context ...e time-dependent jump or volatility parameters. Despite the fact that several empirical studies have shown that Lévy processes reproduce the implied volatility smile for a single maturity quite well =-=(Carr et al., 2002;-=- Madan, Carr and Chang, 1998), when it comes to calibrating several maturities at the same time the calibration by Lévy processes becomes much less precise. The reason is that, due to the stationary ... |

254 | The Variance Gamma process and option pricing, European Finance Review - Madan, Carr, et al. - 1998 |

253 | Option valuation using the Fast Fourier Transform - Carr, Madan - 1999 |

243 | Nonparametric estimation of state–price densities implicit in financial asset prices - Ait-Sahalia, Lo - 1998 |

148 | A jump diffusion model for option pricing
- Kou
(Show Context)
Citation Context ...velopment, in option pricing theory, of a variety of models based on Lévy processes (Andersen and Andreasen, 2000; Eberlein, 2001; Eberlein, Keller and Prause, 1998; Cont, Bouchaud and Potters, 1997;=-= Kou, 2002;-=- Madan, 2001; Madan, Carr and Chang, 1998; Merton, 1976; Schoutens, 2002). A widely studied class is that of exponential Lévy processes in which the price of the underlying asset is written as S t =e... |

100 | New insights into smile, mispricing, and value at risk: The hyperbolic model - Eberlein, Keller, et al. - 1998 |

85 |
Application of Generalized Hyperbolic Lévy Motions in Finance. Lévy Processes: Theory and Applications. Birkhäuser
- Eberlein
- 2001
(Show Context)
Citation Context ...ain certain empirical properties of asset returns and option prices has led to the development, in option pricing theory, of a variety of models based on Lévy processes (Andersen and Andreasen, 2000;=-= Eberlein, 2001-=-; Eberlein, Keller and Prause, 1998; Cont, Bouchaud and Potters, 1997; Kou, 2002; Madan, 2001; Madan, Carr and Chang, 1998; Merton, 1976; Schoutens, 2002). A widely studied class is that of exponentia... |

82 | The minimal entropy martingale measure and the valuation problem in incomplete markets - Frittelli - 2000 |

69 | Testing Option Pricing Models
- Bates, S
- 1996
(Show Context)
Citation Context ...ption computed in an exponential Lévy model with volatility σ and Lévy measure ν. The optimization problem (27) is usually solved numerically by a gradient-based method (Andersen and Andreasen, 20=-=00; Bates, 1996-=-a). While, contrarily to (1), one can always find some solution, the minimization function is non-convex, so a gradient descent may not succeed in locating the global minimum. Owing to the non-convex ... |

67 | Pricing via utility maximization and entropy - Rouge, Karoui - 2000 |

67 |
Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process
- Koponen
- 1995
(Show Context)
Citation Context ... 2002): ν(x) =pα 1 e –α 1x 1x>0 + (1 – p) × α 2 e α 2x 1x<0 . ❏ Infinite activity models. ❏ Variance gamma (Madan, Carr and Chang, 1998): ν(x) = A⏐x⏐ –1 × exp(–η ±⏐x⏐). �=-=�� Tempered stable 2 processes (Koponen, 1995; Cont, Bouchaud and -=-Potters, 1997): ν(x) =A ±⏐x⏐ –(1+α) exp(–η ±⏐x⏐). ❏ Normal inverse gaussian process (Barndorff-Nielsen, 1998). ❏ Hyperbolic and generalized hyperbolic processes (Eberlein, 2001; E... |

65 |
On the solution of functional equations by the method of regularization
- Morozov
- 1966
(Show Context)
Citation Context ...therefore plausible that the right value of α should depend on the data at hand and should not be determined a priori. One way to achieve this trade-off is by using the Morozov discrepancy principle =-=(Morozov, 1966). Fir-=-st, we need to estimate the “intrinsic error”, � 0 , present in the data, that is, the lower bound on the possible or desirable quadratic Volume 7/Number 3, Spring 2004 www.thejournalofcomputati... |

35 | Minimax and minimal distance martingale measures and their relationship to portfolio optimization
- Goll, Rüschendorf
- 2001
(Show Context)
Citation Context ...ingale measures The concept of relative entropy has been used in several contexts as a criterion for selecting pricing measures (Avellaneda, 1998; El Karoui and Rouge, 2000; Föllmer and Schied, 2002;=-= Goll and Rüschendorf, 2001-=-; Kallsen, 2001; Fritelli, 2000; Miyahara and Fujiwara, 2003). We briefly recall them here in relation to the present work. In the absence of calibration constraints, the problem studied above reduces... |

35 |
The Minimal Entropy Martingale Measures for Geometric
- Fujiwara, Miyahara
- 2003
(Show Context)
Citation Context ...ed in several contexts as a criterion for selecting pricing measures (Avellaneda, 1998; El Karoui and Rouge, 2000; Föllmer and Schied, 2002; Goll and Rüschendorf, 2001; Kallsen, 2001; Fritelli, 2000=-=; Miyahara and Fujiwara, 2003-=-). We briefly recall them here in relation to the present work. In the absence of calibration constraints, the problem studied above reduces to that of identifying the equivalent martingale measure wi... |

28 | Dynamics of implied volatility surfaces, Quantitative Finance 2(1 - Cont, Fonseca - 2002 |

23 | Weighted Monte Carlo: A new Technique for Calibrating Asset-Pricing Models - Avellaneda, Buff, et al. - 2001 |

19 |
Stochastic finance
- Föllmer, Schied
- 2002
(Show Context)
Citation Context ...on to minimal entropy martingale measures The concept of relative entropy has been used in several contexts as a criterion for selecting pricing measures (Avellaneda, 1998; El Karoui and Rouge, 2000; =-=Föllmer and Schied, 2002;-=- Goll and Rüschendorf, 2001; Kallsen, 2001; Fritelli, 2000; Miyahara and Fujiwara, 2003). We briefly recall them here in relation to the present work. In the absence of calibration constraints, the p... |

12 |
Jump-diffusion models: volatility smile fitting and numerical methods for pricing
- Anderson, Andreasen
- 2000
(Show Context)
Citation Context ...ty of diffusion models to explain certain empirical properties of asset returns and option prices has led to the development, in option pricing theory, of a variety of models based on Lévy processes =-=(Andersen and Andreasen, 2000-=-; Eberlein, 2001; Eberlein, Keller and Prause, 1998; Cont, Bouchaud and Potters, 1997; Kou, 2002; Madan, 2001; Madan, Carr and Chang, 1998; Merton, 1976; Schoutens, 2002). A widely studied class is th... |

11 | Scaling in financial data: stable laws and beyond - Cont, Bouchaud, et al. - 1997 |

8 | eds): Lévy processes: theory and applications - Barndorff-Nielsen, Mikosch, et al. - 2001 |

8 |
The Meixner Process: Theory and Applications in Finance
- Schoutens
- 2002
(Show Context)
Citation Context ...n Lévy processes (Andersen and Andreasen, 2000; Eberlein, 2001; Eberlein, Keller and Prause, 1998; Cont, Bouchaud and Potters, 1997; Kou, 2002; Madan, 2001; Madan, Carr and Chang, 1998; Merton, 1976;=-= Schoutens, 2002).-=- A widely studied class is that of exponential Lévy processes in which the price of the underlying asset is written as S t =exp(rt + X t ), where r is the discount rate and X is a Lévy process defin... |

7 |
Utility-Based Derivative Pricing
- Kallsen
(Show Context)
Citation Context ...of relative entropy has been used in several contexts as a criterion for selecting pricing measures (Avellaneda, 1998; El Karoui and Rouge, 2000; Föllmer and Schied, 2002; Goll and Rüschendorf, 2001=-=; Kallsen, 2001-=-; Fritelli, 2000; Miyahara and Fujiwara, 2003). We briefly recall them here in relation to the present work. In the absence of calibration constraints, the problem studied above reduces to that of ide... |

7 | Option Pricing Using Variance GammaMarkov Chains - Konikov, Madan |

7 |
Calibrating a diffusion pricing model with uncertain volatility: regularization and stability
- Samperi
(Show Context)
Citation Context ...e problem 1 in stable way. Our approach leads to a non-parametric method for calibrating exponential Lévy models to option prices, extending similar methods previously developed for diffusion models =-=(Samperi, 2002-=-). However, the use of jump processes enables us to formulate the problem in a way that makes sense in a continuous-time framework without giving rise to singularities as in the diffusion calibration ... |

6 |
Financial modeling with discontinuous price processes
- Madan
- 2001
(Show Context)
Citation Context ...in option pricing theory, of a variety of models based on Lévy processes (Andersen and Andreasen, 2000; Eberlein, 2001; Eberlein, Keller and Prause, 1998; Cont, Bouchaud and Potters, 1997; Kou, 2002;=-= Madan, 2001;-=- Madan, Carr and Chang, 1998; Merton, 1976; Schoutens, 2002). A widely studied class is that of exponential Lévy processes in which the price of the underlying asset is written as S t =exp(rt + X t )... |

4 |
Minimum-entropy Calibration of Asset Pricing Models
- Avellaneda
- 1998
(Show Context)
Citation Context ... Relation to previous literature 3.3.1 Relation to minimal entropy martingale measures The concept of relative entropy has been used in several contexts as a criterion for selecting pricing measures (=-=Avellaneda, 1998; -=-El Karoui and Rouge, 2000; Föllmer and Schied, 2002; Goll and Rüschendorf, 2001; Kallsen, 2001; Fritelli, 2000; Miyahara and Fujiwara, 2003). We briefly recall them here in relation to the present w... |

4 |
Estimating exponential Lévy models from option prices via Tikhonov regularization, Working Paper
- Cont, Rouis
(Show Context)
Citation Context ...velop a robust numerical method for solving it. The properties of the continuum version (29) and the convergence of the solutions of the discretized problem (31) are discussed in the companion paper (=-=Cont and Tankov, 2004-=-b). The following proposition shows that the use of entropy penalization makes our (discretized) problem well posed and hence numerically feasible. PROPOSITION 3 (WELL-POSEDNESS OF THE REGULARIZED PRO... |

2 |
A note on estimating parameters of a jump-diffusion process of stock returns
- Beckers
- 1981
(Show Context)
Citation Context ...s region: the intensity of small jumps cannot be retrieved accurately. The redundancy of small jumps and the diffusion component is well known in the context of statistical estimation on time series (=-=Beckers, 1981; -=-FIGURE 7 Calibrated vs. simulated (true) implied volatilities corresponding to Figure 6 for the Kou (2002) model. Implied volatility 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 Prior True Calibrated 0 –0.5 ... |

2 |
Disentangling the jumps from the diffusion in a geometric jumping Brownian motion, Giornale dell’Istituto Italiano degli Attuari, Vol LXIV
- Mancini
- 2001
(Show Context)
Citation Context ....1 0.2 0.3 0.4 0.5 Simulated Calibrated 0.1 6 7 8 9 10 Strike 11 12 13 14 www.thejournalofcomputationalfinance.com Journal of Computational Finance 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 Prior True Calibrateds=-=Mancini, 2001-=-). Here we retrieve another version of this redundancy in a context of calibration to a cross-sectional data set of options. Comparison of the left and right graphs in Figure 6 further illustrates the... |

1 | Volume 7/Number 3, Spring 2004 www.thejournalofcomputationalfinance.com Rama Cont and Peter Tankov - E - 1998 |

1 | www.thejournalofcomputationalfinance.com Journal of Computational Finance Non-parametric calibration of jump–diffusion option pricing models Jacod - J, Shiryaev - 2003 |

1 | Volume 7/Number 3, Spring 2004 Article 1 24/3/04 3:05 pm Page 48 Rama Cont and Peter Tankov - E - 1998 |

1 | Finance Article 1 24/3/04 3:05 pm Page 49 Non-parametric calibration of jump–diffusion option pricing models Jacod - J, Shiryaev - 2003 |