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2012 | 10 | 1 | 250-270
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Analytical approximation of the transition density in a local volatility model

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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.
  • [1] Antonelli F., Scarlatti S., Pricing options under stochastic volatility: a power series approach, Finance Stoch., 2009, 13(2), 269–303
  • [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
  • [4] Berestycki H., Busca J., Florent I., Computing the implied volatility in stochastic volatility models, Comm. Pure Appl. Math., 2004, 57(10), 1352–1373
  • [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
  • [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
  • [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
  • [8] Corielli F., Foschi P., Pascucci A., Parametrix approximation of diffusion transition densities, SIAM J. Financial Math., 2010, 1, 833–867
  • [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
  • [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
  • [12] Doust P., No arbitrage SABR, 2010, manuscript
  • [13] Ekström E., Tysk J., Boundary behaviour of densities for non-negative diffusions, preprint available at
  • [14] Foschi P., Pagliarani S., Pascucci A., Black-Scholes formulae for Asian options in local volatility models, preprint available at
  • [15] Fouque J.-P., Papanicolaou G., Sircar R., Solna K., Singular perturbations in option pricing, SIAM J. Appl. Math., 2003, 63(5), 1648–1665
  • [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
  • [17] Gatheral J., Wang T.-H., The heat-kernel most-likely-path approximation, preprint available at
  • [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
  • [20] Hagan P.S., Woodward D.E., Equivalent Black volatilities, Appl. Math. Finance, 1999, 6(3), 147–157
  • [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
  • [24] Howison S., Matched asymptotic expansions in financial engineering, J. Engrg. Math., 2005, 53(3–4), 385–406
  • [25] Janson S., Tysk J., Feynman-Kac formulas for Black-Scholes-type operators, Bull. London Math. Soc., 2006, 38(2), 269–282
  • [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
  • [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
  • [29] Pagliarani S., Pascucci A., Riga C., Expansion formulae for local Lévy models, preprint available at
  • [30] Pascucci A., PDE and Martingale Methods in Option Pricing, Bocconi Springer Ser., 2, Springer, Milan, 2011
  • [31] Paulot L., Asymptotic implied volatility at the second order with application to the SABR model, preprint available at
  • [32] Shaw W.T., Modelling Financial Derivatives with Mathematica, Cambridge University Press, Cambridge, 1998
  • [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
  • [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
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