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Local Volatility Calibration using the “ Most Likely Path ”. 19 December 2006 for Computational Methods in Finance Prof Ali Hirsa/ Paris Pender. Option Data Extraction. Use “ Option Metrics ” from the WRDS (Wharton Data Research Services)
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Local Volatility Calibrationusing the “Most Likely Path” 19 December 2006 for Computational Methods in Finance Prof Ali Hirsa/ Paris Pender
Option Data Extraction • Use “Option Metrics” from the WRDS (Wharton Data Research Services) • Option Metrics is a comprehensive source of historical price and implied volatility data for the US equity and index options markets. • Volatility Surface contains the interpolated volatility surface for each security on each day, using a methodology based on kernel smoothing algorithm.
Data Fields • We download the following fields from the database: • Days to Expiration. • Interpolated Implied Volatility • Implied Strike Price • Implied Premium. • Spot price.
Mechanism • A standard option is only included if there exists enough option price data on that date to accurately interpolate the required values. • We have designed a data processing module in Matlab that pulls this data in Matlab vectors and then fed into out local volatility processing engine. • The Matlab vectors contain implied volatility data only for OTM calls and puts.
Calibration to SPX • Given a finite set of implied volatility ( )
Calibration to SPX • Given a finite set of implied volatility ( ) • We interpolate onto a “calibration grid” using Matlab’s gridfit function
Calibration to SPX • Given a finite set of implied volatility ( ) • We interpolate onto a “calibration grid” using Matlab’s gridfit function • This is the “market” implied volatility surface that use to calibrate on
Table of Call Prices Results Table of Put Prices
Overview of scheme Take market impled volatility surface as first guess of Local Vol Local volatility surface Converged. Stop!
Two Key Concepts • Most Likely Path: • Implied Volatility Proxy
Two Key Concepts • Most Likely Path \ • Definition: • Difficult to compute directly from the original local volatility dynamics: • Under simpler dynamics, however, we have a closed form solution: where
Two Key Concepts Recall: 1) Compute by our iterative algorithm 2) Compute by Monte-Carlo
Two Key ConceptsComparison of the most likely path • Using iterative algorithm (black) • Using Monte Carlo Simulation (blue) • They are very similar!
Two Key Concepts • Implied volatility proxy • This states that the BS implied volatility of an option with strike K and expiration T is given approximately by the path-integral from valuation date (t=0) to the expiration date (t = T) of the local volatility along the “‘most likely path”
How does our method work?? (2/5) Based on a fixed-point iteration scheme: • Initialize • Repeat the following until convergence under
How does our method work?? (3/5) For each (K,T) on the calibration grid: • Get: a. initialize b. set c. set d. repeat (b-c), until converges in • Set
How does our method work?? (5/5) Conclusion: The method is robust and calibration takes around 3 minutes
Overview of scheme Take market impled volatility surface as first guess of Local Vol Local volatility surface Converged. Stop!
Questions/ Comments Presentation by: Kwasi Danquah, Saurav Kasera, Brian Lee, Sonky Ung