- The paper presents an arbitrage-free SVI parameterization that eliminates static arbitrage with a closed-form calibration method.
- It establishes necessary conditions to avoid calendar spread and butterfly arbitrage, validated with empirical SPX options data.
- The framework enhances real-time trading and risk management, paving the way for dynamic volatility modeling and derivative pricing research.
Arbitrage-free SVI Volatility Surfaces: An Examination
This paper presents an investigation and solution for calibrating the Stochastic Volatility Inspired (SVI) parameterization of the implied volatility smile, ensuring the absence of static arbitrage. The proposed framework introduces a class of arbitrage-free SVI volatility surfaces that feature a straightforward closed-form representation, allowing for efficient and robust calibration to market data, as exemplified with the SPX options data.
Background and Motivation
Since its introduction, the SVI parameterization has gained traction among practitioners due to its ease of use in fitting listed option prices while maintaining compatibility with no-arbitrage bounds at extreme strikes. However, ensuring that SVI smiles are non-arbitrageable remains challenging. Previous work has mainly addressed parametric forms to avoid dynamic arbitrage, yet they often fall short of addressing static arbitrage in a closed form. This paper fills this gap by presenting a large class of SVI volatility surfaces that are guaranteed to be free of static arbitrage, utilizing a combination of theoretical derivation and empirical validation.
Arbitrage-free Conditions for SVI
The SVI parameterizations are revisited to identify the necessary and sufficient conditions for the absence of static arbitrage, which includes both calendar spread and butterfly arbitrage. For calendar spread arbitrage, the condition requires that the total variance be non-decreasing with maturity. For butterfly arbitrage, the paper outlines a detailed criterion linked to the positivity of the probability density derived from option prices, ensuring convexity and monotonicity where needed.
The authors introduce a Surface SVI (SSVI) parameterization aimed at systematically eliminating static arbitrage opportunities. This formulation extends the classical SVI parameterization by incorporating constraints that conditionally modulate its surface structure, leveraging properties from historical mathematical insights.
Key Numerical Results
The paper provides a computational demonstration using recent SPX options data, illustrating the effectiveness of the proposed method. Through the exemplification of particular parameter sets exhibiting arbitrage (like the Axel Vogt example), the authors showcase their approach to rectifying these inconsistencies, achieving a fit devoid of butterfly arbitrage through an optimized adjustment of SVI-JW parameters.
Implications and Future Directions
By determining a class of arbitrage-free surfaces, the presented methodology significantly contributes to the theoretical and practical landscape of financial modeling. It offers a ready-to-implement framework that helps practitioners preserve arbitrage-free constraints across varying market conditions. Furthermore, the efficiency of the calibration algorithm supports practical applications in real-time trading systems and enhances the risk management processes of financial institutions.
For future research, the authors suggest expanding this approach to dynamic scenarios where the evolution of the volatility surface under various economic conditions could be studied. This work also opens avenues for exploring the parameterization's influence on derivative pricing and hedging, potentially offering richer insights into market behaviors and risk premiums.
Overall, this paper provides a comprehensive solution to one of the key challenges in financial modeling, reinforcing the stability and reliability of volatility surfaces used in options pricing and risk management tasks. The detailed mathematical underpinnings and substantial empirical results aim to guide future enhancements in the modelling of implied volatility surfaces.