Pagliarani, Stefano and Pascucci, Andrea (2011): Analytical approximation of the transition density in a local volatility model.

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Abstract
We present a simplified approach to the analytical approximation of the transition density related to a general local volatility model. The methodology is sufficiently flexible to be extended to timedependent coefficients, multidimensional stochastic volatility models, degenerate parabolic PDEs related to Asian options and also to include jumps.
Item Type:  MPRA Paper 

Original Title:  Analytical approximation of the transition density in a local volatility model 
English Title:  Analytical approximation of the transition density in a local volatility model 
Language:  English 
Keywords:  option pricing, analytical approximation, local volatility 
Subjects:  G  Financial Economics > G1  General Financial Markets > G12  Asset Pricing ; Trading Volume ; Bond Interest Rates 
Item ID:  31107 
Depositing User:  Andrea Pascucci 
Date Deposited:  26. May 2011 09:37 
Last Modified:  13. Mar 2015 21:10 
References:  [1] F. Antonelli and S. Scarlatti, Pricing options under stochastic volatility: a power series approach, Finance Stoch., 13 (2009), pp. 269– 303. [2] J. P. Barjaktarevic and R. Rebonato, Approximate solutions for the SABR model: Improving on the Hagan expansion, Talk at ICBI Global Derivatives Trading and Risk Management Conference, (2010). [3] E. Benhamou, E. Gobet, and M. Miri, Expansion formulas for Eu ropean options in a local volatility model, Int. J. Theor. Appl. Finance, 13 (2010), pp. 603–634. [4] H. Berestycki, J. Busca, and I. Florent, Computing the implied volatility in stochastic volatility models, Comm. Pure Appl. Math., 57 (2004), pp. 1352–1373. [5] L. Capriotti, The exponent expansion: an effective approximation of transition probabilities of diffusion processes and pricing kernels of financial derivatives, Int. J. Theor. Appl. Finance, 9 (2006), pp. 1179– 1199. 22 [6] 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). [7] R. Constantinescu, N. Costanzino, A. L. Mazzucato, and V. Nistor, Approximate solutions to second order parabolic equations. I: analytic estimates, J. Math. Phys., 51 (2010), pp. 103502, 26. [8] F. Corielli, P. Foschi, and A. Pascucci, Parametrix approxima tion of diffusion transition densities, SIAMJ. Financial Math., 1 (2010), pp. 833–867. [9] J. C. Cox, Notes on option pricing I: constant elasticity of variance diffusion, Working paper, Stanford University, Stanford CA, (1975). [10] D. Davydov and V. Linetsky, Pricing and hedging pathdependent options under the CEV process, Management Science, 47 (2001), pp. 949–965. [11] F. Delbaen and H. Shirakawa, A note on option pricing for the constant elasticity of variance model, AsiaPacific Financial Markets, 9 (2002), pp. 85–99. 10.1023/A:1022269617674. [12] P. Doust, No arbitrage SABR, working paper, (2010). [13] E. Ekstr¨om and J. Tysk, Boundary behaviour of densities for non negative diffusions, preprint, (2011). [14] P. Foschi, S. Pagliarani, and A. Pascucci, BlackScholes formu lae for Asian options, Working paper, (2011). [15] J.P. Fouque, G. Papanicolaou, R. Sircar, and K. Solna, Sin gular perturbations in option pricing, SIAM J. Appl. Math., 63 (2003), pp. 1648–1665 (electronic). [16] 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). [17] P. Hagan, D. Kumar, A. Lesniewski, and D. Woodward, Man aging smile risk, Wilmott, September (2002), pp. 84–108. [18] P. Hagan, A. Lesniewski, and D. Woodward, Managing smile risk, Wilmott, September (2002), pp. 84–108. [19] P. Hagan and D. Woodward, Equivalent Black volatilities, Appl. Math. Finance, 6 (1999), pp. 147–159. [20] P. HenryLabord`ere, A general asymptotic implied volatility for stochastic volatility models, Frontiers in Quantitative Finance, Wiley, (2008). [21] , Analysis, geometry, and modeling in finance, Chapman & Hall/CRC Financial Mathematics Series, CRC Press, Boca Raton, FL, 2009. Advanced methods in option pricing. [22] S. L. Heston, M. Loewenstein, and G. A. Willard, Options and Bubbles, The Review of Financial Studies, Vol. 20, Issue 2, pp. 359390, (2007). [23] S. Howison, Matched asymptotic expansions in financial engineering, J. Engrg. Math., 53 (2005), pp. 385–406. [24] S. Janson and J. Tysk, FeynmanKac formulas for BlackScholes type operators, Bull. London Math. Soc., 38 (2006), pp. 269–282. [25] D. Kristensen and A. Mele, Adding and subtracting BlackScholes: A new approach to approximating derivative prices in continuous time models, to appear in Journal of Financial Economics, (2011). [26] A. Lesniewski, Swaption smiles via the WKB method, Mathematical Finance Seminar, Courant Institute of Mathematical Sciences, (2002). [27] A. Lindsay and D. Brecher, Results on the CEV Process, Past and Present, SSRN eLibrary, (2010). [28] A. Pascucci, PDE and martingale methods in option pricing, Bocconi& Springer Series, SpringerVerlag, New York, 2011. [29] L. Paulot, Asymptotic Implied Volatility at the Second Order with Application to the SABR Model, SSRN eLibrary, (2009). [30] W. T. Shaw, Modelling financial derivatives with Mathematica, Cambridge University Press, Cambridge, 1998. Mathematical models and benchmark algorithms, With 1 CDROM (Windows, Macintosh and UNIX). [31] S. Taylor, Perturbation and symmetry techniques applied to fi nance, Ph. D. thesis, Frankfurt School of Finance & Management. Bankakademie HfB, (2011). [32] A. E. Whalley and P. Wilmott, An asymptotic analysis of an optimal hedging model for option pricing with transaction costs, Math. Finance, 7 (1997), pp. 307–324. [33] 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. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/31107 