1. VIP 2.0 Next Generation Volatility InterPolation

2. VIP 2.0 - Load Page See, what input data is available for a given underlying See input data in detail Click two buttons to start game VIP 2.0 is also available with real-time data connection

3. VIP 2.0 - On-the-Fly Calculation Type of surface and cleansing parameters Choose underlying Get option grid filled + your extra grid-points… Hit Calc to create a single surface Detailed output file

4. VIP 2.0 - Batch Calculation File-based Inputs Holds parameters for batch run Holds the extra grid for all underlyings adjust settings for surface Hit Calc to run batch Find detailed output here global options for batch run

5. Underlying: BASF Option Type: Put Spot: 50.33 € Time Stamp: 11/10/2011 16:27 BASF Put Surface Black dots mark the option quote input Interpolated grid- points (also green) Stochastic model recognizes a “smile” in quotes (black) and extrapolates accordingly (green) Stochastic model recognizes a “smile” in input data (black) and extrapolates accordingly (green) VIP 2.0 uses extrapolation based on a stochastic volatility model. It is able to fit a large number of input patterns and fails over to special splines for the remainder

6. E-Stoxx Call Surface Underlying: EURO STOXX Option Type: Call Spot: 2504.49 Time Stamp: 2/6/2012 16:25 Input curvature fits a ”smile”, here … … but rather a (moderate) “skew”, here … and even a (moderate) “frown”, here The stochastic vol model is able to recognize different patterns in the input of a single surface.

7. E-Stoxx X-over Surface (Put and Call) Underlying: EURO STOXX Option Type: X-over (Put and Call) Spot: 2254.01 Time Stamp: 11/10/2011 16:25 B&C X-over produces a single arbitrage-free vol surface from put and call options simultaneously. Out- of-the-money options are weighted heavier than in-the-money options using the option delta as driving parameter

8. VIP 2.0 handles arbitrage afflicted input Deutsche Bank Option Type: Call Spot: 38.66€ Time Stamp: 03/16/2012 16:27 Quoted option vols contain arbitrage. The VIP surface is calculated on an optimal arbitrage- free trade-off. Hence, it is obliged to leave some points manifestly off the surface.

9. Arbitrage Correction (II) Calendar arbitrage yields riskless gain of 17.51€ - 16.98€ = 0.53€ Quoted Vol arbitrage relative to VIP Vol surface: 17.51€ - 17.291€ = 0.219€ 17.217€ - 16.98€ = 0.237€ VIP vol does not allow for a riskless gain: 17.217€ - 17.291€ = - 0.074€ VIP eliminates arbitrage of Quoted Option Prices VIP 2.0 output comes always with an arbitrage measure, which is the maximum riskless gain obtainable from the 5 trading strategies that enforce the 5 arbitrage conditions. Positive values indicate arbitrage opportunities. This example of calendar arbitrage is particularly easy, since the arbitrage measures coincides with price differences.

10. Simply, because we claim we have the next generation answer to the four long standing problems of volatility surface calculation (see next slide). In total, VIP 2.0 delivers features in a numerically accessible way that have been reach-able (if at all?) only at tremendous numerical effort We suggest a scientific and practicable way to extrapolation that brings about a dramatic numeri- cal advantage enabling us to enforce all strike and calendar no-arbitrage conditions in one go. How come B&C dares to talk about “Next Generation”?

11. The „Big 4“ of Vola Surface Calculation I.Only options for some strike maturity pairs are quoted – often not the ones needed. A volatility surface is always a Sparse Grid Problem that leads to very badly conditioned matrices when an optimization shall be carried out. II.Even less options are really liquid. A “simultaneous” option quotes snapshot leads to rather Polluted Input Data. III.There is no commonly agreed way, how to extrapolate volatility in (a) scientific way, (b) at acceptable numerical costs and (c) for any given input dataset and. IV.Options prices can contain arbitrage, which is difficult to spot or avoid numerically in volatility space. There are 4 arbitrage conditions in strike plus one calendar arbitrage condition in time dimension.

12. How we tackle Polluted Input Data (II): Benoist & Company is a Market Data Company. We have tremendous experience in setting up market data solutions and satisfying all sorts of data cleansing needs. Thus, VIP 2.0 comes equipped with sophisticated and well-tested data cleansing mechanisms based on our large in-house expertise. The cleansing criteria can be adjusted by the user to obtain best data quality from given option quote inputs. Default parameters for cleansing have been calibrated on a very large number of input quote sets.

13. How we tackle the Sparse Grid (I): In principle it is easy to fill a surface, but if no-arbitrage conditions have to be met by a non-trivial optimization the resulting matrix is very poorly conditioned. This forced prior solutions (Fengler, etc) to do the optimization tenor- wise and put up with non unique solutions and failing optimizations. Our trick is to do the extrapolation of a so called “target” first, create a well conditioned optimization matrix. We are then able to enforce all arbitrage conditions in strike and calendar direction (2D) at a time for the whole grid. Thus, VIP 2.0 generates plausible vola surfaces for any user-supplied grid - reproducing “good” option quotes with high accuracy (<1bp).

14. How we tackle extrapolation (III) Early extrapolation was seen as pure data fitting problem which lead to “nice” surfaces but lacked “scientific” foundation on the basis of any financial model. Stochastic vol models published over the last decade were successful in explaining volatility patterns but often not practicable for near-time calculations and – even worse – there is no model that can fit to all observable inputs. VIP 2.0 uses a carefully designed extrapolation technique employing a stochastic vol model (Heston) to provide extrapolations based on financial mathematics. We chose a parameterization that is numerically efficient and reproduces a large number of input patterns For the remainder of situations we found an ultra-robust extrapolation based on special surface splines that avoids problems (like wiggles or implausible behavior) present in previously suggested approaches.

15. How we tackle arbitrage (IV) Algorithms that enforce arbitrage in price space have to transform as last step prices into BS-Vol. This introduced minimal arbitrage in the order of magnitude of the numerical error, but made results worthless for model calibrations. So practitioners calculated in vola space with tremendous numerical effort. We are able to align the numerical effort and distance from the arbitrage boundary. Together with the well conditioned optimization matrix (see (I) VIP 2.0 is the first tool to produce a unique, 100% arbitrage free solution, quicker than 1 sec (on average on a single processor)

16. Further Benefits VIP 2.0 is designed to handle on the fly calculations, file-based bulk processing, as well as continuous near-time processing in connection with real-time data provision Comes as C++ standalone with its GUI. Algorithm is encapsulated and can be incorporated in many other systems. It allows for 3D visualization and comparison with external vol surfaces It is optimal for arbitrage detection and allows to trace back to the option input Extrapolation of volatilities to extremely short and extremely long expiries Reproduces market implied volatilities very well (up to ca. 1 bp) Pricing of any strike/maturity combination And the best: VIP comes at a true one-to-many price

17. Contact Contact: Benoist & Company GmbH Seefelder Str. 15 82229 Seefeld Germany Mail: contact@benoist-company.com Web: www. benoist-company.com Tel.: +49 8152 99 81 82 Fax.: +49 8152 99 81 83