Asia-Pacific Financial Markets

, Volume 17, Issue 4, pp 391–436 | Cite as

The Instantaneous Volatility and the Implied Volatility Surface for a Generalized Black–Scholes Model

  • Koichiro Takaoka
  • Hidenori Futami


Takaoka (Asia–Pacific Financial Markets 11:431–444, 2004) proposed a generalization of the Black–Scholes stock price model by taking a weighted average of geometric Brownian motions of different variance parameters. The model can be classified as a local volatility model, though its local volatility function is not explicitly given. In the present paper, we prove some properties concerning the instantaneous volatility process, the implied volatility curve, and the local volatility function of the generalized model. Some numerical computations are also carried out to confirm our results.


Black–Scholes model Option pricing Local volatility model Implied volatility 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Black F., Scholes M. (1973) The pricing of options and corporate liabilities. Journal of Political Economy 81(3): 637–659CrossRefGoogle Scholar
  2. Brigo D., Mercurio F. (2001) Displaced and mixture diffusions for analytically tractable smile models. In: Geman H. et al (eds) Mathematical finance—bachelier congress 2000. Springer, New YorkGoogle Scholar
  3. Brigo D., Mercurio F. (2003) Analytical pricing of the smile in a forward LIBOR market model. Quantitative Finance 1: 15–27CrossRefGoogle Scholar
  4. Cox J. C., Ross S. A. (1976) The valuation of options for alternative stochastic processes. Journal of Financial Economics 3: 145–166CrossRefGoogle Scholar
  5. Dupire B. (1994) Pricing with a smile. Risk 7: 18–20Google Scholar
  6. Feller W. (1970) An introduction to probability and its applications. Wiley, New YorkGoogle Scholar
  7. Gatheral J. (2006) The volatility surface: A practitioner’s guide. Wiley, New YorkGoogle Scholar
  8. Heston S. L. (1993) A closed form solution for options with stochastic volatility with applications to bond currency options. Review of Financial Studies 6: 327–342CrossRefGoogle Scholar
  9. Hull J. C. (2006) Options, futures, and other derivatives. Prentice-Hall, New JerseyGoogle Scholar
  10. Ishimura N., Sakaguchi T. (2004) Exact solutions of a model for asset prices by K. Takaoka. Asia-Pacific Financial Markets 11: 445–451CrossRefGoogle Scholar
  11. Karatzas I., Shreve S. E. (1991) Brownian motion and stochastic calculus. Springer, New YorkGoogle Scholar
  12. Long J. B. (1990) The numeraire portfolio. Journal of Financial Economics 26: 29–69CrossRefGoogle Scholar
  13. Madan D. B., Miline F. (1991) Option pricing with V.G. (variance gamma) martingale components. Mathematical Finance 1: 39–55CrossRefGoogle Scholar
  14. Merton R. C. (1976) Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3: 125–144CrossRefGoogle Scholar
  15. Numazawa, Y. (2007). A generalized Black–Scholes model and its no arbitrage condition (in Japanese). Master Thesis, Graduate School of Science, Tohoku University.Google Scholar
  16. Takaoka, K. (2000). An equilibrium model of the short-term stock price behavior, Working Paper 49, Faculty of Commerce, Hitotsubashi University.Google Scholar
  17. Takaoka K. (2004) A complete-market generalization of the Black-Scholes model. Asia-Pacific Financial Markets 11: 431–444CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  1. 1.Graduate School of Commerce and ManagementHitotsubashi UniversityKunitachi-City, TokyoJapan
  2. 2.Tokio Marine & Nichido Fire Insurance Co., Ltd.Chiyoda-ku, TokyoJapan

Personalised recommendations