Stefano, Pagliarani and Pascucci, Andrea and Candia, Riga (2011): Adjoint expansions in local Lévy models.
This is the latest version of this item.

PDF
MPRA_paper_35788.pdf Download (448kB)  Preview 
Abstract
We propose a novel method for the analytical approximation in local volatility models with Lèvy jumps. In the case of Gaussian jumps, we provide an explicit approximation of the transition density of the underlying process by a heat kernel expansion: the approximation is derived in two ways, using PIDE techniques and working in the Fourier space. Our second and main result is an expansion of the characteristic function for a local volatility model with general Lévy jumps. Combined with standard Fourier methods, such an expansion allows to obtain efficient and accurate pricing formulae. Numerical tests confirm the effectiveness of the method.
Item Type:  MPRA Paper 

Original Title:  Adjoint expansions in local Lévy models 
English Title:  Adjoint expansions in local Lévy models 
Language:  English 
Keywords:  Lévy process, local volatility, asymptotic expansion, partialintegro differential equation, Fourier methods 
Subjects:  G  Financial Economics > G1  General Financial Markets > G13  Contingent Pricing; Futures Pricing 
Item ID:  35788 
Depositing User:  Andrea Pascucci 
Date Deposited:  09. Jan 2012 04:41 
Last Modified:  15. Feb 2013 22:50 
References:  [1] L. Andersen and J. Andreasen, Jumpdiffusion processes: Volatil ity smile fitting and numerical methods for option pricing, Review of Derivatives Research, 4 (2000), pp. 231–262. [2] E. Benhamou, E. Gobet, and M. Miri, Smart expansion and fast calibration for jump diffusions, Finance Stoch., 13 (2009), pp. 563–589. [3] , Expansion formulas for European options in a local volatility model, Int. J. Theor. Appl. Finance, 13 (2010), pp. 603–634. [4] P. Carr, H. Geman, D. B. Madan, and M. Yor, From local volatil ity to local L´evy models, Quant. Finance, 4 (2004), pp. 581–588. [5] W. Cheng, N. Costanzino, J. Liechty, A. Mazzucato, and V. Nistor, Closedform asymptotics and numerical approximations of 1D parabolic equations with applications to option pricing, to appear in SIAM J. Fin. Math., (2011). [6] R. Cont, N. Lantos, and O. Pironneau, A reduced basis for option pricing, SIAM J. Financial Math., 2 (2011), pp. 287–316. [7] F. Corielli, P. Foschi, and A. Pascucci, Parametrix approxima tion of diffusion transition densities, SIAMJ. Financial Math., 1 (2010), pp. 833–867. [8] E. Ekstr¨om and J. Tysk, Boundary behaviour of densities for non negative diffusions, preprint, (2011). [9] F. Fang and C. W. Oosterlee, A novel pricing method for Euro pean options based on Fouriercosine series expansions, SIAM J. Sci. Comput., 31 (2008/09), pp. 826–848. [10] P. Foschi, S. Pagliarani, and A. Pascucci, BlackScholes formu lae for Asian options in local volatility models, SSRN eLibrary, (2011). [11] M. G. Garroni and J.L. Menaldi, Green functions for second or der parabolic integrodifferential problems, vol. 275 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1992. [12] J. Gatheral, E. P. Hsu, P. Laurence, C. Ouyang, and T.H. Wang, Asymptotics of implied volatility in local volatility models, to appear in Math. Finance, (2010). [13] P. Hagan and D. Woodward, Equivalent Black volatilities, Appl. Math. Finance, 6 (1999), pp. 147–159. [14] S. Howison, Matched asymptotic expansions in financial engineering, J. Engrg. Math., 53 (2005), pp. 385–406. [15] K. Itˆo and H. P. McKean, Jr., Diffusion processes and their sample paths, SpringerVerlag, Berlin, 1974. Second printing, corrected, Die Grundlehren der mathematischen Wissenschaften, Band 125. [16] D. Madan and E. Seneta, The variance gamma (VG) model for share market returns, Journal of Business, 63 (1990), pp. 511–524. [17] R. C. Merton, Option pricing when underlying stock returns are dis continuous., J. Financ. Econ., 3 (1976), pp. 125–144. [18] S. Pagliarani and A. Pascucci, Analytical approximation of the transition density in a local volatility model, to appear in Cent. Eur. J. Math., (2011). [19] A. Pascucci, PDE and martingale methods in option pricing, vol. 2 of Bocconi & Springer Series, Springer, Milan, 2011. [20] M. Widdicks, P. W. Duck, A. D. Andricopoulos, and D. P. Newton, The BlackScholes equation revisited: asymptotic expansions and singular perturbations, Math. Finance, 15 (2005), pp. 373–391. [21] G. Xu and H. Zheng, Basket options valuation for a local volatility jumpdiffusion model with the asymptotic expansion method, Insurance Math. Econom., 47 (2010), pp. 415–422. 29 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/35788 
Available Versions of this Item

Expansion formulae for local Lévy models. (deposited 07. Nov 2011 18:12)
 Adjoint expansions in local Lévy models. (deposited 09. Jan 2012 04:41) [Currently Displayed]