#### DMCA

## Jump-diffusion models: a practitioner’s guide

Citations: | 1 - 0 self |

### Citations

1019 | Non-Uniform Random Variate Generation
- Devroye
- 1986
(Show Context)
Citation Context ...isson distribution1 with parameter λT . 1The random number generator from the Poisson distribution is available in most MATLAB-like scientific computing environments. If you need to implement it, see =-=[13]-=-. 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −7 −6 −5 −4 −3 −2 −1 0 1 0 0.2 0.4 0.6 0.8 1 −1.5 −1 −0.5 0 0.5 1 Figure 2: Left: sample path of a compound Poisson process with Gaussian distribution of ju... |

1000 | Option pricing when underlying stock returns are discontinuous
- Merton
- 1976
(Show Context)
Citation Context ...r European option pricing, partial differential equations for barrier and American options, and the existing approaches to calibration and hedging. 1 Introduction Starting with Merton’s seminal paper =-=[21]-=- and up to the present date, various aspects of jump-diffusion models have been studied in the academic finance community (see [8] for a list of almost 400 references on the subject). In the last deca... |

710 | Transform analysis and asset pricing for affine jump diffusions,
- Duffie, Pan, et al.
- 2000
(Show Context)
Citation Context ...e characteristic function of logstock price is known or easy to compute. This is the case for exponential Lévy models (see, e.g., Table 1) but also holds for a more general class of affine processes =-=[14, 15]-=-, which includes in particular the Bates model mentioned in section 2. 4 Integro-differential equations for barriers and American options The Fourier-transform based algorithm of the preceding section... |

569 | A limited memory algorithm for bound constrained optimization
- Byrd, Lu, et al.
- 1995
(Show Context)
Citation Context ...ed simultaneously for all strikes present in the data using the FFT-based algorithm described in section 3. The functional in (28) was then minimized using a quasi-newton method (LBFGS-B described in =-=[6]-=-). In the case of Merton model, the calibration functional is sufficiently well behaved, and can be minimized using this convex optimization algorithm. In more complex jumpdiffusion models, in particu... |

547 | Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche mark options
- Bates
- 1996
(Show Context)
Citation Context ...ys coincide with the empirically observed time dependence of these quantities [3]. For these reasons, several models combining jumps and stochastic volatility appeared in the literature. In the Bates =-=[2]-=- model, one of the most popular examples of the class, an independent jump component is added to the Heston stochastic volatility model: dXt = µdt+ √ VtdW X t + dZt, St = S0e Xt , (8) dVt = ξ(η − Vt)d... |

409 | Option valuation using the fast Fourier transform.
- Carr, Madan
- 1999
(Show Context)
Citation Context ...the Black-Scholes formula. In the case of Lévy jump-diffusions, closed formulas are no longer available but a fast deterministic algorithm, based on Fourier transform, was proposed by Carr and Madan =-=[7]-=-. Here we present a slightly improved version of their method, due to [22, 8]. Let {Xt}t≥0 be a Lévy process and, for simplicity, take S0 = 1. We would like to compute the price of a European call wi... |

365 | The variance gamma process and option pricing.
- Madan, Carr, et al.
- 1998
(Show Context)
Citation Context ... called subordinator). Its characteristic function has a very simple form: E[eiuXt ] = (1− iu/λ)−ct . The gamma process is the building block for a very popular jump model, the variance gamma process =-=[20, 19]-=-, which is constructed by taking a Brownian motion with drift and changing its time scale with a gamma process: Yt = µXt + σBXt . Using Yt to model the logarithm of stock prices can be justified by sa... |

147 | Financial Modelling with
- Cont, Tankov
- 2004
(Show Context)
Citation Context ...ion and hedging. 1 Introduction Starting with Merton’s seminal paper [21] and up to the present date, various aspects of jump-diffusion models have been studied in the academic finance community (see =-=[8]-=- for a list of almost 400 references on the subject). In the last decade, also the research departments of major banks started to accept jump-diffusions as a valuable tool in their day-to-day modeling... |

101 | Affine processes and applications in finance.
- Duffie, Filipović, et al.
- 2003
(Show Context)
Citation Context ...e characteristic function of logstock price is known or easy to compute. This is the case for exponential Lévy models (see, e.g., Table 1) but also holds for a more general class of affine processes =-=[14, 15]-=-, which includes in particular the Bates model mentioned in section 2. 4 Integro-differential equations for barriers and American options The Fourier-transform based algorithm of the preceding section... |

95 | Option pricing under a double exponential jump diusion model.
- Kou, Wang
- 2004
(Show Context)
Citation Context ...tion ∂P ∂t + 1 2 σ2S2 ∂2C ∂S2 = rC − rS ∂C ∂S (10) 4Fourier-transform based methods for pricing single-barrier options can be found in the literature [24, 5, 18] but except for some particular models =-=[17]-=-, the numerical complexity of the resulting formulae is prohibitive. 11 with appropriate boundary conditions. In this section, we show how this method can be generalized to models with jumps by introd... |

83 | Testing Option Pricing Models
- Bates
- 1996
(Show Context)
Citation Context ... determined by the law of X1. Therefore, moments and cumulants depend on time in a well-defined manner which does not always coincide with the empirically observed time dependence of these quantities =-=[3]-=-. For these reasons, several models combining jumps and stochastic volatility appeared in the literature. In the Bates [2] model, one of the most popular examples of the class, an independent jump com... |

66 | A finite difference scheme for option pricing in jump diffusion and exponential Levy models
- Cont, Voltchkova
- 2003
(Show Context)
Citation Context ...o models with jumps by introducing partial integro-differential equations (PIDEs). A complete presentation with proofs, as well as the general case of possibly infinite Lévy measure, can be found in =-=[23, 12]-=-. Barrier “out” options We start with up-and-out, down-and-out, and double barrier options which have, respectively, an upper barrier U > S0, a lower barrier L < S0, or both of them. If the stock pric... |

55 | Non-parametric calibration of jump diffusion option pricing models.
- Cont, Tankov
- 2004
(Show Context)
Citation Context ...when no parametric shape of the Lévy measure is assumed, a penalty term must be added to the distance functional in (28) to ensure convergence and stability. This procedure is described in detail in =-=[9, 10, 22]-=-. The calibration for each individual maturity is quite good, however, although the options of different maturities correspond to the same trading day and the same underlying, the parameter values for... |

26 | Asset prices are Brownian motion: Only in buisness time, Quantitativr Analysis
- Geman, Yor
(Show Context)
Citation Context ...= µXt + σBXt . Using Yt to model the logarithm of stock prices can be justified by saying that the price is a geometric Brownian motion if viewed on a stochastic time scale given by the gamma process =-=[16]-=-. The variance gamma process is another example of a Lévy process with infinitely many jumps and has characteristic function E[eiuYt ] = ( 1 + σ2u2 2 − iµκu )−κt . The parameters have the following (... |

24 | Tankov Retrieving Levy processes from option prices: Regularization of an ill-posed inverse problem.
- Cont, P
- 2006
(Show Context)
Citation Context ...when no parametric shape of the Lévy measure is assumed, a penalty term must be added to the distance functional in (28) to ensure convergence and stability. This procedure is described in detail in =-=[9, 10, 22]-=-. The calibration for each individual maturity is quite good, however, although the options of different maturities correspond to the same trading day and the same underlying, the parameter values for... |

15 | Hedging with options in models with jumps
- Cont, Tankov, et al.
- 2007
(Show Context)
Citation Context ...+ ∫ T 0 φtdS ∗ t −HT )2 with HT the option’s payoff. Using the Itô formula for jump processes and the isometry relation for stochastic integrals (both are out of scope of the present paper but see =-=[11]-=- for details), the residual hedging error can be expressed as E[(VT −HT )2] = ( erTV0 − E[HT ] )2 + E ∫ T 0 dt(S∗t ) 2σ2 { φt − ∂C ∂S }2 + E ∫ T 0 ∫ R ν(dz)e2r(T−t) {C(t, St(1 + z))− C(t, St)− Stφtz}2... |

14 |
Option pricing with variance gamma martingale components,”Mathematical Finance
- Madan, Milne
- 1991
(Show Context)
Citation Context ... called subordinator). Its characteristic function has a very simple form: E[eiuXt ] = (1− iu/λ)−ct . The gamma process is the building block for a very popular jump model, the variance gamma process =-=[20, 19]-=-, which is constructed by taking a Brownian motion with drift and changing its time scale with a gamma process: Yt = µXt + σBXt . Using Yt to model the logarithm of stock prices can be justified by sa... |

7 | Numerical valuation of American options under the CGMY process
- Almendral
- 2005
(Show Context)
Citation Context ...satisfy the constraint∫ R (1 ∧ x2)ν(dx) <∞ 6 and describes the jumps of X in the following sense: for every closed set A ⊂ R with 0 /∈ A, ν(A) is the average number of jumps of X in the time interval =-=[0, 1]-=-, whose sizes fall in A. To keep the discussion simple, in the rest of this paper we will only consider Lévy jump-diffusions, that is, Lévy processes with finite jump intensity of the form (2), but ... |

4 |
Lookback and barrier options under general Lévy processes
- Yor, Nguyen-Ngoc
- 2007
(Show Context)
Citation Context ...ons of the Black-Scholes partial differential equation ∂P ∂t + 1 2 σ2S2 ∂2C ∂S2 = rC − rS ∂C ∂S (10) 4Fourier-transform based methods for pricing single-barrier options can be found in the literature =-=[24, 5, 18]-=- but except for some particular models [17], the numerical complexity of the resulting formulae is prohibitive. 11 with appropriate boundary conditions. In this section, we show how this method can be... |

2 |
Haj Yedder, Calibration of stochastic volatility model with jumps. A computer program, part of Premia software. See www.premia.fr
- Ben
- 2004
(Show Context)
Citation Context ...ton jump-diffusion model. The results of calibration of the Merton model to S&P index options are presented in figure 8. The calibration was carried out separately for each maturity using the routine =-=[4]-=- from Premia software. In this program, the vector of unknown parameters θ is found by minimizing numerically the squared norm of the difference between market and model prices: θ∗ = arg inf ‖P obs − ... |

2 |
Integro-differential evolution equations: numerical methods and applications
- VOLTCHKOVA
- 2005
(Show Context)
Citation Context ...o models with jumps by introducing partial integro-differential equations (PIDEs). A complete presentation with proofs, as well as the general case of possibly infinite Lévy measure, can be found in =-=[23, 12]-=-. Barrier “out” options We start with up-and-out, down-and-out, and double barrier options which have, respectively, an upper barrier U > S0, a lower barrier L < S0, or both of them. If the stock pric... |