Advertisement

Methodology and Computing in Applied Probability

, Volume 18, Issue 3, pp 847–867 | Cite as

Explicit Density Approximations for Local Volatility Models Using Heat Kernel Expansions

  • Stephen TaylorEmail author
  • Scott Glasgow
  • James Taylor
  • Jan Vecer
Article
  • 80 Downloads

Abstract

Heat kernel perturbation theory is a tool for constructing explicit approximation formulas for the solutions of linear parabolic equations. We review the crux of this perturbative formalism and then apply it to differential equations which govern the transition densities of several local volatility processes. In particular, we compute all the heat kernel coefficients for the CEV and quadratic local volatility models; in the later case, we are able to use these to construct an exact explicit formula for the processes’ transition density. We then derive low order approximation formulas for the cubic local volatility model, an affine-affine short rate model, and a generalized mean reverting CEV model. We finally demonstrate that the approximation formulas are accurate in certain model parameter regimes via comparison to Monte Carlo simulations.

Keywords

Heat kernel expansion Local volatility CEV model Short rate models 

Keywords

35K08 91G20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andersen LBG, Piterbarg VV (2010) Interest rate modeling volumes 1,2, and 3. Atlantic Financial PressGoogle Scholar
  2. Andersen LBG (2011) Option pricing with quadratic volatility: a revisit. Finance Stoch 15:191–219. Available at SSRN: http://ssrn.com/abstract=1118399 MathSciNetCrossRefzbMATHGoogle Scholar
  3. Avramidi IG (2007) Analytic and geometric methods for heat kernel applications in finance. Available at http://infohost.nmt.edu/iavramid/notes/hkt/hktutorial13.pdf
  4. Brecher DR, Lindsay AE (2010) Results on the CEV Process, Past and Present. Working Paper. Available at http://ssrn.com/abstract=1567864
  5. Brigo D, Mercurio F (2006) Interest Rate Models? Theory and Practice: With Smile, Inflation and Credit, 2nd. Springer FinanceGoogle Scholar
  6. Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Political Econ 81(3):637–654CrossRefzbMATHGoogle Scholar
  7. Dupire B (1994) Pricing with a smile. Risk 7(1):18–20Google Scholar
  8. Feller W (1951) Two singular diffusion problems. Ann Math, Second Ser 54 (1):173–182MathSciNetCrossRefzbMATHGoogle Scholar
  9. Forde M (2011) Exact pricing and large-time asymptotics for the modified sabr model and the brownian exponential functional. Int J Theor Appl Finan 14:559MathSciNetCrossRefzbMATHGoogle Scholar
  10. Forde M (2013) The large maturity smile for the SABR and CEV-Heston models. Int J Theor Appl Finan 16(8)Google Scholar
  11. Gatheral J, Hsu EP, Laurence PM, Ouyang C, Wang TH (2012) Asymptotics of implied volatility in local volatility models. Math Finance 22(4):591–620MathSciNetCrossRefzbMATHGoogle Scholar
  12. Hagan P, Lesniewski A, Woodward D (2005) Probability Distribution in the SABR Model of Stochastic Volatility. Working Paper available at http://lesniewski.us/papers/working/ProbDistrForSABR.pdf
  13. Labordère PH (2005) A General Asymptotic Implied Volatility for Stochastic Volatility Models. Working Paper available at ArXiv:cond-mat/0504317
  14. Labordère PH (2008) Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing. Chapman and Hall/CRC Financial Mathematics SeriesGoogle Scholar
  15. Lesniewski A (2002) WKB Method for Swaption Smile http://lesniewski.us/papers/presentations/Courant020702.pdf
  16. Medvedev AN (2004) Asymptotic Methods for Computing Implied Volatilities Under Stochastic Volatility. Working Paper available at http://ssrn.com/abstract=667281
  17. Merton R (1973) Theory of rational option pricing. Bell J Econ Manag Sci (The RAND Corporation) 4(1):141–183MathSciNetCrossRefzbMATHGoogle Scholar
  18. Pagliarani S, Pascucci A (2012) Analytical approximation of the transition density in a local volatility model cent. Eur J Math 10(1):250–270MathSciNetCrossRefzbMATHGoogle Scholar
  19. Paulot L (2007) Asymptotic Implied Volatility at the Second Order with Application to the SABR Model. Working paper available at http://ssrn.com/abstract=1413649
  20. Zühlsdorff C (2002) Extended Libor Market Models with Affine and Quadratic Volatility. Bonn Econ Discussion PapersGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Stephen Taylor
    • 1
    Email author
  • Scott Glasgow
    • 2
  • James Taylor
    • 2
  • Jan Vecer
    • 3
    • 4
  1. 1.Hutchin Hill CapitalNew YorkUSA
  2. 2.BYU Department of MathematicsProvoUSA
  3. 3.Vysoka skola aplikovanecho pravaPrague 4Czech Republic
  4. 4.Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic

Personalised recommendations