- The paper proposes a novel Hawkes process model for tick-by-tick implied volatility surface dynamics.
- It establishes kernel constraints to ensure no calendar spread or butterfly arbitrage, reducing computational complexity.
- The study demonstrates convergence towards rough volatility processes and informs market impact and option market-making strategies.
High-frequency Dynamics of the Implied Volatility Surface
Introduction
The paper "High-frequency dynamics of the implied volatility surface" (2012.10875) presents a novel approach to modeling the high-frequency dynamics of the volatility surface using Hawkes processes. The authors emphasize the importance of directly modeling the implied volatility surface instead of solely relying on the dynamics of the underlying asset. This approach is driven by the non-constant behavior observed in options markets, such as skew, smile, and term structure, which traditional models based on underlying asset dynamics often fail to capture accurately. The objective is to generate a volatility surface that adheres to no-arbitrage conditions while reducing computational complexity and parameter estimation.
Microscopic Volatility Surface Modeling
The microscopic model proposed in the paper leverages Hawkes processes to articulate the tick-by-tick dynamics of the implied volatility surface. It considers a matrix process where each element corresponds to the high-frequency dynamics of options characterized by their strike and maturity. The Hawkes process framework accounts for self-exciting and cross-exciting mechanisms between different options, using kernel coefficients to influence the skew and convexity of the volatility surface. This approach captures nuanced interactions within the market, reflecting both endogenous and exogenous factors.
No-Arbitrage Conditions
The paper meticulously outlines conditions under which the volatility surface is free from arbitrage opportunities, focusing on calendar spread and butterfly arbitrage. These are articulated in terms of implied variance and kernel parametrization. Specifically, the authors provide sufficient conditions in the form of kernel constraints to ensure no calendar spread and butterfly arbitrage. This allows the volatility surface to remain statistically sound and reduces the dimensionality of the parameter space, facilitating more efficient model calibration.
Parametrization Techniques
The authors illustrate the parametrization of the volatility surface through three-point and five-point models. These models are specifically designed to control skew and convexity by adjusting the Hawkes kernel parameters. For instance, the three-point model involves crucial strikes, such as at-the-money (ATM) and out-of-the-money (OTM), allowing researchers to capture market dynamics more precisely.
Skew and Convexity Control
Critical to the methodology is the ability to manipulate kernel parameters to obtain desired skew and convexity levels in the volatility surface. The paper provides conditions for right and left skewness and at-the-money convexity, offering clarity on how variations in kernel coefficients impact these characteristics.
Macroscopic Behavior of the Volatility Surface
At larger time scales, the model exhibits convergence towards rough volatility processes, typified by fractional Brownian motion dynamics. By employing limit theorems, the authors demonstrate that the volatility surface can be characterized by a sum of orthogonal factors with rough volatility processes. This is pivotal for understanding the emergent properties of the surface at macroscopic scales, such as "level-skew-convexity" interactions observed traditionally in financial markets.
Practical Applications
Market-making Strategies
The model serves as a framework for backtesting trading strategies, particularly in option market-making. By simulating the impact of trades on the high-frequency dynamics, researchers can paper the implications of market-making strategies under both Poisson and Hawkes assumptions. This allows for enhanced testing of strategies, reflecting real-market conditions more appropriately.
Market Impact Assessment
The authors introduce methodologies for measuring market impact, where trades alter the implied volatility surface. They provide models to quantify the cumulative and localized impact of trading activities, facilitating a deeper understanding of trading strategies' influence on market dynamics.
Conclusion
The paper presents a comprehensive method for modeling high-frequency dynamics of the implied volatility surface using Hawkes processes, including robust no-arbitrage conditions and efficient parametrizations. The detailed exploration of microscopic modeling and scaling limits extends the understanding of volatility surfaces in dynamic markets. Future extensions could incorporate spot-volatility interactions and varying Hurst parameters, further enriching no-arbitrage conditions and enhancing model robustness. The implications for trading strategies and market impact demonstrate practical applications of this theoretical model in financial markets.
