Asymptotic Analysis of Option Pricing Functions

  • Archil Gulisashvili
Part of the Springer Finance book series (FINANCE)


This chapter introduces call and put pricing functions in general asset price models. The chapter first takes up the question of what are the conditions under which a function of two variables (strike price and maturity) is a call pricing function. The answer to this question is given in the form of a characterization theorem for general call pricing functions. The best known call pricing function is without doubt the pricing function in the Black-Scholes model. This celebrated model is discussed in the present chapter and an analytical proof of the Black-Scholes formula is given. Moreover, sharp asymptotic formulas are obtained for call pricing functions in the Hull-White, Stein-Stein, and Heston models.


Asset Price Option Price Price Function Stochastic Volatility Model Scholes Model 
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  1. [BS73]
    Black, F., Scholes, M., The pricing of options and corporate liabilities, Journal of Political Economy 81 (1973), pp. 635–654. CrossRefGoogle Scholar
  2. [BH95]
    Bleistein, N., Handelsman, R. A., Asymptotic Expansions of Integrals, Holt, Rinehart and Winston, New York, 1995. Google Scholar
  3. [Bue06]
    Buehler, H., Expensive martingales, Quantitative Finance 6 (2006), pp. 207–218. MathSciNetzbMATHCrossRefGoogle Scholar
  4. [CN09]
    Carmona, R., Nadtochiy, S., Local volatility dynamic models, Finance and Stochastics 13 (2009), pp. 1–48. MathSciNetzbMATHCrossRefGoogle Scholar
  5. [CM05]
    Carr, P., Madan, D. B., A note on sufficient conditions for no arbitrage, Finance Research Letters 2 (2005), pp. 125–130. CrossRefGoogle Scholar
  6. [Cou07]
    Cousot, L., Conditions on option prices for absence of arbitrage and exact calibration, Journal of Banking and Finance 31 (2007), pp. 3377–3397. CrossRefGoogle Scholar
  7. [DH07]
    Davis, M. H. A., Hobson, D. G., The range of traded option prices, Mathematical Finance 17 (2007), pp. 1–14. MathSciNetzbMATHCrossRefGoogle Scholar
  8. [Gul10]
    Gulisashvili, A., Asymptotic formulas with error estimates for call pricing functions and the implied volatility at extreme strikes, SIAM Journal on Financial Mathematics 1 (2010), pp. 609–641. MathSciNetzbMATHCrossRefGoogle Scholar
  9. [HPRY11]
    Hirsh, F., Profeta, C., Roynette, B., Yor, M., Peacocks and Associated Martingales, with Explicit Constructions, Springer, Italia, 2011. CrossRefGoogle Scholar
  10. [Kel72]
    Kellerer, H. G., Markov-Komposition und eine Anwendung auf Martingale, Mathematische Annalen 198 (1972), pp. 99–122. MathSciNetzbMATHCrossRefGoogle Scholar
  11. [RY04]
    Revuz, D., Yor, M., Continuous Martingales and Brownian Motion, Springer, Berlin, 2004. Google Scholar
  12. [Rop10]
    Roper, M., Arbitrage free implied volatility surfaces, preprint, 2010. Google Scholar
  13. [Str65]
    Strassen, V., The existence of probability measures with given marginals, The Annals of Mathematical Statistics 36 (1965), pp. 423–439. MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Archil Gulisashvili
    • 1
  1. 1.Department of MathematicsOhio UniversityAthensUSA

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