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Jim Bodurtha Georgetown University

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Jim Bodurtha Georgetown University

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  1. Estimating a Local (Time- and State-Dependent) Volatility Surface(A Linearization-Based Solution to the Ill-Posed Local Volatility Estimation Problem,and Non-Parametric Estimation of an Implied Volatility Surface with Martin Jermakyanfromhttp://www.gsb.georgetown.edu/dept/facserv/faculty/bodurthj/research/research.html) • Jim Bodurtha • Georgetown University • Chicago Risk Management ConferenceMay 7, 1998 - Chicago

  2. Term Volatility Surfaces - • Shimko, Dumas, Fleming and Whaley ... • Ait-Sahalia-Lo, Longstaff, Elliot-Madan, Derman-Kani-Zhou… • Rubinstein, Rubinstein-Jackwerth ... Local Volatility Surfaces - • Avellanede • Bodurtha-Jermakyan • Dupire, Derman-Kani (interpolate artifical prices) • Lagnado-Oscher

  3. Table 2: Example European FX Call Options, Pricing and Vol “Smile” Implied Vol across options (s0) = 20%

  4. Binomial Tree Valuation Set upSee also Kamrad-Ritchken (1991)

  5. Trinomial Lattice Algebra

  6. Trinomial Lattice European Call Option Values and Volatilities

  7. Redefine Variance and European Call Option Values

  8. Estimation Procedure with Regularization

  9. Linear (Smoothed) Volatility Surface Estimator

  10. Risk Management - Valuing a European Option Book

  11. Augment discrepancy function for Gamma estimation error:

  12. Figure 2: PHLX DM European Options Local Volatility Surface - 11/25/91(2/9/95-4/14/97 average bid-ask spread 0.64% and standard deviation 0.52%)

  13. Other issues • Computation-estimation and benchmarking Restricted non-linear models vs. linearizations Higher-order regularizers (large GSVD problem) • Volatility surface dynamics • Exotics • American options • Interest rates • Convergence

  14. “Non-Parametric Estimation …” Problem statement:

  15. 5. A Finite Difference-Based Numerical Implementation - Change state variable (moving frame of coordinates), discretize in state and time, first-order forward difference for time derivatives, second-order central difference for state derivatives, and introduce artificial state boundary conditions for L,

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