Continuous Financial Models

  • Peter Carr
  • Qiji Jim Zhu
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


We turn to discuss continuous financial models. These models in general involve infinite dimensional spaces and are more complex. Our focus here is to use relatively simple models to illustrate the convex duality between the price of a contingent claim and the process of cash borrowed in delta hedging. This reveals the root of the convexity in contingent claims. Interestingly, when hedging with a contingent claim instead of the underlying, a similar duality in the sense of generalized Fenchel conjugate holds. Correspondingly, this generalized duality leads to the generalized convexity of the contingent claims with many interesting applications. Much of the material presented in this chapter appear here for the first time.


  1. 3.
    Bachelier, L.: Théorie de la spéculation. Ann. Sci. Éc. Norm. Supér. 3(17), 21–86 (1900)CrossRefGoogle Scholar
  2. 6.
    Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–645 (1973)MathSciNetCrossRefGoogle Scholar
  3. 11.
    Carr, P., Madan, D.: Toward a theory of volatility trading. In: Jarrow, R. (ed.) Volatility Estimation Techniques for Pricing Derivatives, pp. 417–427. Risk Books, London (1998)Google Scholar
  4. 31.
    Leland, H.: Corporate debt value, bond covenants, and optimal capital structure. J. Financ. 49(4), 1213–1252 (1994)CrossRefGoogle Scholar
  5. 42.
    Oksendal, B.: Stochastic Differential Equations, 6th edn. Springer, New York (2003)CrossRefGoogle Scholar
  6. 54.
    Steele, J.M.: Stochastic Calculus and Financial Applications. Springer, New York (2001)CrossRefGoogle Scholar

Copyright information

© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Peter Carr
    • 1
  • Qiji Jim Zhu
    • 2
  1. 1.Department of Finance and Risk EngineeringTandon School of Engineering, New York UniversityNew YorkUSA
  2. 2.Department of MathematicsWestern Michigan UniversityKalamazooUSA

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