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

Preview |
PDF
MPRA_paper_31107.pdf Download (5MB) | Preview |

## 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 time-dependent coefficients, multi-dimensional 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: | 27 Sep 2019 13:54 |

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, Closed-form 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 path-dependent 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, Asia-Pacific 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, Black-Scholes 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. Henry-Labord`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. 359-390, (2007). [23] S. Howison, Matched asymptotic expansions in financial engineering, J. Engrg. Math., 53 (2005), pp. 385–406. [24] S. Janson and J. Tysk, Feynman-Kac formulas for Black-Scholes- type operators, Bull. London Math. Soc., 38 (2006), pp. 269–282. [25] D. Kristensen and A. Mele, Adding and subtracting Black-Scholes: 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, Springer-Verlag, 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 CD-ROM (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 Black-Scholes equation revisited: asymptotic expansions and singular perturbations, Math. Finance, 15 (2005), pp. 373–391. |

URI: | https://mpra.ub.uni-muenchen.de/id/eprint/31107 |