Arbitrage-free Volatility Surface Interpolation. Author: Dr. Kay Moritzen (B&C) Dr. Ulrich Leiner (B&C)

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1 Arbitrage-free Volatility Surface Interpolation Author: Dr. Kay Moritzen (B&C) Dr. Ulrich Leiner (B&C)

2 Challenges we see Market prices of options reflect both, a modelinduced shape of the volatility surface and preference-induced deformations which give rise to arbitrage Scattered data only gives an incomplete impression of the real surface Option prices for very long maturities are barely exchange quoted and available as OTC quotes Prices need to be verified and compared quickly - 2 -

3 Questions we want to address Is it possible to cleanse market data from arbitrage effects? Is it possible to compute a reliable volatility surface based on sparse data? Is it possible to extrapolate this data for long maturities or extreme moneyness? - 3 -

4 led us to develop VIP the Volatility InterPolator a tool that retrieves Option prices from the data vendor (e.g. Reuters) cleanses the input data interpolates and extrapolates arbitrage-free vola and option price surfaces handles dividends, repos and yield curve information optimally delivers data ín arbitrary strike/maturity and tenor/moneyness grids - 4 -

5 Example Option Price Data Stock: Deutsch Bank AG Underlying Price: Timestamp: (16:53) Source: Reuters RIC: /DBKGn.DE - 5 -

6 Inconsistent Price Data Inconsistent price Stock: Deutsch Bank AG Underlying Price: Timestamp: (16:53) X } Source: Reuters RIC: /DBKGn.DE - 6 -

7 Inconsistent Price Data Observed option (call) prices Stock: Deutsch Bank AG Maturity Strike Mid Bid Ask Tenor Underlying Price: Timestamp: (16:53) Source: Reuters RIC: /DBKGn.DE - 7 -

8 How to get the volatility surface (ToDo list) Taking Snapshot of option and underlying prices for quotation date and time t (synchronicity issue) Choosing Call/Put and Ask, Bid or Mid option prices and additional parameters (dividends, interest rate, ) Unifying for each observed tenor option prices to a family (set) Cleansing of price data by performing a separate spline approximation along the strike axis for each family of option prices Least Squares Problem (Optimization Problem) under market relevant constraints Retrieving the volatility surface from cleaned option prices!core problem! - 8 -

9 Optimization Algorithm Overview Spline approximation for a single tenor (family of option price with same fixed maturity) Least Squares Approach by Quadratic Programming (QP) subject to Inequality constraints for convexity along strike Inequality constraints along tenor for arbitrage elimination (monotony in the total variance) - 9 -

10 Optimization Setup Objective Function: Formulating an approximization function (objective function) being fitted against observed market data Least squares approach with curvature penalty

11 Optimization Setup Inequality Constraints: Formulating Inequality constraints for convexity and arbitrage elimination Formulating Inequality constraints for monotony in the total variance Literature: M. Fengler, Arbitrage-Free Smoothing of the Implied Volatility Surface,

12 Optimization Setup All together in one optimization problem Least Squares Problem: Fitting design function against observed market data with linear constraints Spline condition Arbitrage-freeness and Convexity condition

13 System Overview Das Bild kann nicht angezeigt werden. Dieser Computer verfügt möglicherweise über zu wenig Arbeitsspeicher, um das Bild zu öffnen, oder das Bild ist beschädigt. Starten Sie den Computer neu, und öffnen Sie dann erneut die Datei. Wenn weiterhin das rote x angezeigt wird, müssen Sie das Bild möglicherweise löschen und dann erneut einfügen. Internal Data Base User MS-EXCEL Reuters PowerPlus Pro Data Query (RICs) (API: RtChain, RtUpdate) VIP Excel-Add-In Data Selection and Storage Yields / Dividends / Own Positions Controlling Approximation and Interpolation Controlling Storage MATLAB ExcelLink (Excel-Add-In) VIP MatlabKernel Interpolation and Approximation Routines 3D Visualization of V I P Volatility / Quote Surfaces and Active Positions

14 Retrieval of Option Prices Retrieving option peice data via Reuters 3000Xtra (MS Excel-Add-In) Das Bild kann nicht angezeigt werden. Dieser Computer verfügt möglicherweise über zu wenig Arbeitsspeicher, um das Bild zu öffnen, oder das Bild ist beschädigt. Starten Sie den Computer neu, und öffnen Sie dann erneut die Datei. Wenn weiterhin das rote x angezeigt wird, müssen Sie das Bild möglicherweise löschen und dann erneut einfügen

15 Volatility InterPolator Choosing set of option prices, interest rate curve and estimated dividends Starting VIP

16 Volatility InterPolator Visualizing Volatility Surface with VIP

17 Cons Standalone application Connectivity to non-standard systems cause additional implementation efforts

18 Pros Direct rescaling (moneyness/strike) of interpolation results Direct 3D analysis of volatility and option price surface Direct analysis of own positions Automatic data cleansing Extrapolating surfaces Easy installation and connection to Reuters or other standard market data vendors

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