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Term Structures Of Implied Volatilities: Absence Of Arbitrage And Existence Results

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  • Martin Schweizer
  • Johannes Wissel

Abstract

This paper studies modeling and existence issues for market models of stochastic implied volatility in a continuous‐time framework with one stock, one bank account, and a family of European options for all maturities with a fixed payoff function h. We first characterize absence of arbitrage in terms of drift conditions for the forward implied volatilities corresponding to a general convex h. For the resulting infinite system of SDEs for the stock and all the forward implied volatilities, we then study the question of solvability and provide sufficient conditions for existence and uniqueness of a solution. We do this for two examples of h, namely, calls with a fixed strike and a fixed power of the terminal stock price, and we give explicit examples of volatility coefficients satisfying the required assumptions.

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  • Martin Schweizer & Johannes Wissel, 2008. "Term Structures Of Implied Volatilities: Absence Of Arbitrage And Existence Results," Mathematical Finance, Wiley Blackwell, vol. 18(1), pages 77-114, January.
    • Handle: RePEc:bla:mathfi:v:18:y:2008:i:1:p:77-114
    DOI: 10.1111/j.1467-9965.2007.00323.x
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    References listed on IDEAS

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