Detecting and Repairing Arbitrage in Traded Option Prices

ABSTRACT

Option price data are used as inputs for model calibration, risk-neutral density estimation and many other financial applications. The presence of arbitrage in option price data can lead to poor performance or even failure of these tasks, making pre-processing of the data to eliminate arbitrage necessary. Most attention in the relevant literature has been devoted to arbitrage-free smoothing and filtering (i.e., removing) of data. In contrast to smoothing, which typically changes nearly all data, or filtering, which truncates data, we propose to repair data by only necessary and minimal changes. We formulate the data repair as a linear programming (LP) problem, where the no-arbitrage relations are constraints, and the objective is to minimize prices’ changes within their bid and ask price bounds. Through empirical studies, we show that the proposed arbitrage repair method gives sparse perturbations on data, and is fast when applied to real-world large-scale problems due to the LP formulation. In addition, we show that removing arbitrage from prices data by our repair method can improve model calibration with enhanced robustness and reduced calibration error.

KEYWORDS:

• Static arbitrage
• data cleansing
• option prices

Acknowledgments

This publication is based on work supported by the EPSRC Centre for Doctoral Training in Industrially Focused Mathematical Modelling (EP/L015803/1) in collaboration with CME Group. We thank Florian Huchede, Director of Quantitative Risk Management, and other colleagues at CME Group for providing valuable data access, suggestions from the business perspective, and continued support. Samuel Cohen and Christoph Reisinger acknowledge the support of the Oxford–Man Institute for Quantitative Finance, and Samuel Cohen also acknowledges the support of the Alan Turing Institute under the Engineering and Physical Sciences Research Council grant EP/N510129/1.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1. Our implementation of this algorithm in Python is available in the repository https://github.com/vicaws/arbitragerepair.

2. We focus on European style vanilla options in this study. Specifically, we only consider call options, since the static arbitrage constraint between call and put options is the put-call parity, which can be easily incorporated in our approach. The framework of our arbitrage repair method is applicable to a mixture of a wider range of options, as long as their arbitrage constraints can be defined by feasible linear inequalities of prices.

3. When applying our method to other asset classes, dividends of stock shares are comparable to foreign currency interest rates for FX rates, or convenience yields for commodities.

4. Note that the ${\mathrm{\ell }}^0$-norm is not actually a ‘norm’ as it violates the homogeneity and triangle inequality properties that a vector norm must satisfy.

5. Given the instrument bid-ask spreads for ATM, RR and BF, one cannot uniquely determine the corresponding vanilla spreads without specifying some rule. For example, in practice, trading desks may estimate vanilla spreads only using ATM spreads, which makes the spread of each option at the same expiry equal, see Section 4.2.1 of Wystup (2017). Since vanilla IVs are linear transformations of instrument IVs, we conservatively assume that vanilla spreads are weighted sums of instrument spreads. This does not take into account that delta-symmetric vanilla spreads are dependent on each other, and generates the widest possible bid-ask spreads for vanilla IVs.

6. Note that we must apply exactly the same numerical procedure for these two separate calibrations, i.e., the same optimization algorithm, terminal criteria, lower and upper bounds, and initial values.

7. Heston model parameters are chosen as those that reproduce a typical call price surface for USDBRL options. Noise simulation parameters $\lambda$ and ${\sigma }_{\xi }$ are chosen to mimic severe but not extreme arbitrage scenarios (measured by the fraction of perturbed prices by the repair method) observed in real-world data.

Additional information

Funding

This work was supported by the Engineering and Physical Sciences Research Council [EP/L015803/1, EP/N510129/1].