Skip to main content

Analytical approximation of the transition density in a local volatility model

Central European Journal of Mathematics volume 10, pages 250–270 (2012)Cite this article

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.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price includes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Antonelli F., Scarlatti S., Pricing options under stochastic volatility: a power series approach, Finance Stoch., 2009, 13(2), 269–303

    MathSciNet  MATH  Article  Google Scholar 

  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

    MathSciNet  MATH  Article  Google Scholar 

  4. Berestycki H., Busca J., Florent I., Computing the implied volatility in stochastic volatility models, Comm. Pure Appl. Math., 2004, 57(10), 1352–1373

    MathSciNet  MATH  Article  Google Scholar 

  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

    MathSciNet  MATH  Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    MathSciNet  Article  Google Scholar 

  9. Cox J.C., Notes on option pricing I: constant elasticity of variance diffusion, Stanford University, Stanford, 1975, manuscript

    Google Scholar 

  10. Davydov D., Linetsky V., Pricing and hedging path-dependent options under the CEV process, Management Sci., 2001, 47(7), 949–965

    Article  Google Scholar 

  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

    MATH  Google Scholar 

  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

    MathSciNet  MATH  Article  Google Scholar 

  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

    MATH  Article  Google Scholar 

  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

    Google Scholar 

  22. Henry-Labordère P., Analysis, Geometry, and Modeling in Finance, Chapman Hall/CRC Financ. Math. Ser., CRC Press, Boca Raton, 2009

    Google Scholar 

  23. Heston S.L., Loewenstein M., Willard G.A., Options and bubbles, Review of Financial Studies, 2007, 20(2), 359–390

    Article  Google Scholar 

  24. Howison S., Matched asymptotic expansions in financial engineering, J. Engrg. Math., 2005, 53(3–4), 385–406

    MathSciNet  MATH  Article  Google Scholar 

  25. Janson S., Tysk J., Feynman-Kac formulas for Black-Scholes-type operators, Bull. London Math. Soc., 2006, 38(2), 269–282

    MathSciNet  MATH  Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Google Scholar 

  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

    MATH  Google Scholar 

  33. Taylor S., Perturbation and Symmetry Techniques Applied to Finance, PhD thesis, Frankfurt School of Finance & Management, Bankakademie HfB, 2011

    Google Scholar 

  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

    MathSciNet  MATH  Article  Google Scholar 

  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

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121, Padova, Italy

    Stefano Pagliarani

  2. Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126, Bologna, Italy

    Andrea Pascucci

Authors
  1. Stefano Pagliarani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrea Pascucci
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefano Pagliarani.

About this article

Cite this article

Pagliarani, S., Pascucci, A. Analytical approximation of the transition density in a local volatility model. centr.eur.j.math. 10, 250–270 (2012). https://doi.org/10.2478/s11533-011-0115-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s11533-011-0115-y

MSC

  • 91Gxx
  • 35K10
  • 60J70

Keywords

  • Local volatility
  • Analytical approximation
  • Heat kernel expansion
  • Black-Scholes formula
  • Transition density