1. Localvol.ppt - 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 Jermakyan from http://www.gsb.georgetown.edu/dept/facserv/faculty/bodurthj/research/research.html ) Jim Bodurtha Georgetown University Chicago Risk Management Conference May 7, 1998 - Chicago

2. Localvol.ppt - 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. Localvol.ppt - 3 Table 2: Example European FX Call Options, Pricing and Vol “Smile” Implied Vol across options (  0 ) = 20%

4. Localvol.ppt - 4 Binomial Tree Valuation Set up See also Kamrad-Ritchken (1991)

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7. Localvol.ppt - 7 Trinomial Lattice Algebra

8. Localvol.ppt - 8 Trinomial Lattice European Call Option Values and Volatilities

9. Localvol.ppt - 9 Redefine Variance and European Call Option Values

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14. Localvol.ppt - 14 Estimation Procedure with Regularization

15. Localvol.ppt - 15 Linear (Smoothed) Volatility Surface Estimator

16. Localvol.ppt - 16 Risk Management - Valuing a European Option Book

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19. Localvol.ppt - 19 Augment discrepancy function for Gamma estimation error:

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21. Localvol.ppt - 21 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%)

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25. Localvol.ppt - 25 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

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27. Localvol.ppt - 27 “Non-Parametric Estimation …” Problem statement:

28. Localvol.ppt - 28 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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