Analytical approximation of the transition density in a local volatility model

Stefano Pagliarani 1  and Andrea Pascucci 2
  • 1 Università di Padova
  • 2 Università di Bologna

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.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [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

  • [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

  • [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

  • [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

  • [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

  • [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 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

  • [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 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

  • [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

  • [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 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

  • [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

  • [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

  • [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

  • [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 http://ssrn.com/paper=1413649

  • [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 http://dx.doi.org/10.1111/1467-9965.00034

  • [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

OPEN ACCESS

Journal + Issues

Search