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A sequential quadratic programming method for volatility estimation in option pricing

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  • Düring, Bertram
  • Jüngel, Ansgar
  • Volkwein, S.

Abstract

Our goal is to identify the volatility function in Dupire's equation from given option prices. Following an optimal control approach in a Lagrangian framework, we propose a globalized sequential quadratic programming (SQP) algorithm with a modified Hessian - to ensure that every SQP step is a descent direction - and implement a line search strategy. In each level of the SQP method a linear-quadratic optimal control problem with box constraints is solved by a primal-dual active set strategy. This guarantees L? constraints for the volatility, in particular assuring its positivity. The proposed algorithm is founded on a thorough first - and second-order optimality analysis. We prove the existence of local optimal solutions and of a Lagrange multiplier associated with the inequality constraints. Furthermore, we prove a sufficient second-order optimality condition and present some numerical results underlining the good properties of the numerical scheme. Dupire equation ; parameter identification ; optimal control ; optimality conditions ; SQP method ; primal-dual active set strategy

Suggested Citation

  • Düring, Bertram & Jüngel, Ansgar & Volkwein, S., 2006. "A sequential quadratic programming method for volatility estimation in option pricing," CoFE Discussion Papers 06/02, University of Konstanz, Center of Finance and Econometrics (CoFE).
    • Handle: RePEc:zbw:cofedp:0602
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    1. Ronald Lagnado & Stanley Osher, "undated". "A Technique for Calibrating Derivative Security Pricing Models: Numerical Solution of an Inverse Problem," Computing in Economics and Finance 1997 101, Society for Computational Economics.
    2. Yves Achdou & Olivier Pironneau, 2002. "Volatility Smile By Multilevel Least Square," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 5(06), pages 619-643.
    3. Marco Avellaneda & Craig Friedman & Richard Holmes & Dominick Samperi, 1997. "Calibrating volatility surfaces via relative-entropy minimization," Applied Mathematical Finance, Taylor & Francis Journals, vol. 4(1), pages 37-64.
    4. Jackwerth, Jens Carsten, 2000. "Recovering Risk Aversion from Option Prices and Realized Returns," Review of Financial Studies, Society for Financial Studies, vol. 13(2), pages 433-451.
    5. Martin Hanke & Elisabeth Rösler, 2005. "Computation Of Local Volatilities From Regularized Dupire Equations," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 8(02), pages 207-221.
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