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.
@article{bwmeta1.element.doi-10_2478_s11533-011-0115-y,
author = {Stefano Pagliarani and Andrea Pascucci},
title = {Analytical approximation of the transition density in a local volatility model},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {250-270},
zbl = {1246.91137},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0115-y}
}
Stefano Pagliarani; Andrea Pascucci. Analytical approximation of the transition density in a local volatility model. Open Mathematics, Tome 10 (2012) pp. 250-270. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0115-y/
[1] Antonelli F., Scarlatti S., Pricing options under stochastic volatility: a power series approach, Finance Stoch., 2009, 13(2), 269–303 http://dx.doi.org/10.1007/s00780-008-0086-4 | Zbl 1199.91200
[2] Barjaktarevic J.P., Rebonato R., Approximate solutions for the SABR model: improving on the Hagan expansion, Talk at ICBI Global Derivatives Trading and Risk Management Conference, 2010
[3] Benhamou E., Gobet E., Miri M., Expansion formulas for European options in a local volatility model, Int. J. Theor. Appl. Finance, 2010, 13(4), 603–634 http://dx.doi.org/10.1142/S0219024910005887 | Zbl 1205.91153
[4] Berestycki H., Busca J., Florent I., Computing the implied volatility in stochastic volatility models, Comm. Pure Appl. Math., 2004, 57(10), 1352–1373 http://dx.doi.org/10.1002/cpa.20039 | Zbl 1181.91356
[5] Capriotti L., The exponent expansion: an effective approximation of transition probabilities of diffusion processes and pricing kernels of financial derivatives, Int. J. Theor. Appl. Finance, 2006, 9(7), 1179–1199 http://dx.doi.org/10.1142/S0219024906003925 | Zbl 1140.91380
[6] Cheng W., Costanzino N., Liechty J., Mazzucato A., Nistor V., Closed-form asymptotics and numerical approximations of 1D parabolic equations with applications to option pricing, SIAM J. Financial Math., 2011, 2, 901–934 http://dx.doi.org/10.1137/100796832 | Zbl 1242.35131
[7] Constantinescu R., Costanzino N., Mazzucato A.L., Nistor V., Approximate solutions to second order parabolic equations I: analytic estimates, J. Math. Phys., 2010, 51(10), #103502 | Zbl 1314.35063
[8] Corielli F., Foschi P., Pascucci A., Parametrix approximation of diffusion transition densities, SIAM J. Financial Math., 2010, 1, 833–867 http://dx.doi.org/10.1137/080742336
[9] Cox J.C., Notes on option pricing I: constant elasticity of variance diffusion, Stanford University, Stanford, 1975, manuscript
[10] Davydov D., Linetsky V., Pricing and hedging path-dependent options under the CEV process, Management Sci., 2001, 47(7), 949–965 http://dx.doi.org/10.1287/mnsc.47.7.949.9804 | Zbl 1232.91659
[11] Delbaen F., Shirakawa H., A note on option pricing for the constant elasticity of variance model, Financial Engineering and the Japanese Markets, 2002, 9(2), 85–99 | Zbl 1072.91020
[12] Doust P., No arbitrage SABR, 2010, manuscript
[13] Ekström E., Tysk J., Boundary behaviour of densities for non-negative diffusions, preprint available at www.math.uu.se/~johant/pq.pdf
[14] Foschi P., Pagliarani S., Pascucci A., Black-Scholes formulae for Asian options in local volatility models, preprint available at http://ssrn.com/paper=1898992 | Zbl 1260.91100
[15] Fouque J.-P., Papanicolaou G., Sircar R., Solna K., Singular perturbations in option pricing, SIAM J. Appl. Math., 2003, 63(5), 1648–1665 http://dx.doi.org/10.1137/S0036139902401550 | Zbl 1039.91024
[16] Gatheral J., Hsu E.P., Laurence P., Ouyang C., Wang T.-H., Asymptotics of implied volatility in local volatility models, Math. Finance (in press), DOI: 10.1111/j.1467-9965.2010.00472.x | Zbl 1270.91093
[17] Gatheral J., Wang T.-H., The heat-kernel most-likely-path approximation, preprint available at http://ssrn.com/paper=1663318
[18] Hagan P.S., Kumar D., Lesniewski A.S., Woodward D.E., Managing smile risk, Wilmott Magazine, 2002, September, 84–108
[19] Hagan P., Lesniewski A., Woodward D., Probability distribution in the SABR model of stochastic volatility, 2005, preprint available at www.lesniewski.us/papers/working/ProbDistrForSABR.pdf
[20] Hagan P.S., Woodward D.E., Equivalent Black volatilities, Appl. Math. Finance, 1999, 6(3), 147–157 http://dx.doi.org/10.1080/135048699334500 | Zbl 1009.91033
[21] Henry-Labordère P., A geometric approach to the asymptotics of implied volatility, In: Frontiers in Quantitative Finance, Wiley Finance Ser., John Wiley & Sons, Hoboken, 2008, chapter 4, 89–128
[22] Henry-Labordère P., Analysis, Geometry, and Modeling in Finance, Chapman Hall/CRC Financ. Math. Ser., CRC Press, Boca Raton, 2009
[23] Heston S.L., Loewenstein M., Willard G.A., Options and bubbles, Review of Financial Studies, 2007, 20(2), 359–390 http://dx.doi.org/10.1093/rfs/hhl005
[24] Howison S., Matched asymptotic expansions in financial engineering, J. Engrg. Math., 2005, 53(3–4), 385–406 http://dx.doi.org/10.1007/s10665-005-7716-z | Zbl 1099.91061
[25] Janson S., Tysk J., Feynman-Kac formulas for Black-Scholes-type operators, Bull. London Math. Soc., 2006, 38(2), 269–282 http://dx.doi.org/10.1112/S0024609306018194 | Zbl 1110.35021
[26] Kristensen D., Mele A., Adding and subtracting Black-Scholes: a new approach to approximating derivative prices in continuous-time models, Journal of Financial Economics, 2011, 102(2), 390–415 http://dx.doi.org/10.1016/j.jfineco.2011.05.007
[27] Lesniewski A., Swaption smiles via the WKB method, Mathematical Finance Seminar, Courant Institute of Mathematical Sciences, 2002
[28] Lindsay A., Brecher D., Results on the CEV process, past and present, preprint available at http://ssrn.com/paper=1567864
[29] Pagliarani S., Pascucci A., Riga C., Expansion formulae for local Lévy models, preprint available at http://ssrn.com/paper=1937149 | Zbl 1285.60084
[30] Pascucci A., PDE and Martingale Methods in Option Pricing, Bocconi Springer Ser., 2, Springer, Milan, 2011 | Zbl 1214.91002
[31] Paulot L., Asymptotic implied volatility at the second order with application to the SABR model, preprint available at http://ssrn.com/paper=1413649
[32] Shaw W.T., Modelling Financial Derivatives with Mathematica, Cambridge University Press, Cambridge, 1998 | Zbl 0939.91002
[33] Taylor S., Perturbation and Symmetry Techniques Applied to Finance, PhD thesis, Frankfurt School of Finance & Management, Bankakademie HfB, 2011
[34] Whalley A.E., Wilmott P., An asymptotic analysis of an optimal hedging model for option pricing with transaction costs, Math. Finance, 1997, 7(3), 307–324 http://dx.doi.org/10.1111/1467-9965.00034 | Zbl 0885.90019
[35] Widdicks M., Duck P.W., Andricopoulos A.D., Newton D.P., The Black-Scholes equation revisited: asymptotic expansions and singular perturbations, Math. Finance, 2005, 15(2), 373–391 http://dx.doi.org/10.1111/j.0960-1627.2005.00224.x | Zbl 1124.91342