The implied volatility surface is a fundamental building block in modern financial markets, as it underpins the pricing of both vanilla and exotic instruments and supports key risk-management functions such as hedging and scenario analysis. It has been modeled extensively in traditional finance; in crypto, however, few studies exist. Given the volatile nature of the crypto market, it is important to examine this area.
Reference [1] proposes a numerical method for reconstructing the implied vol surface of major cryptocurrencies: Bitcoin, Ethereum, Solana, and Ripple. The main steps are as follows:
- Apply the Black-Scholes-Merton (BSM) equation,
- Convert it into a discretized framework using the Finite Difference Method, and
- Fit the resulting implied volatilities into bivariate polynomials.
The authors pointed out,
In this study, we developed and implemented a method for reconstructing smooth local volatility surfaces for cryptocurrency options by extending the generalized BS model with a bivariate polynomial volatility function. The proposed approach combines a finite difference method with an optimization routine to calibrate volatility surfaces from observed option prices. The resulting local volatility functions are smooth, flexible, and differentiable with respect to both the underlying asset price and time, which enhances their analytical tractability and practical usability…
Through extensive computational tests using real option data from BTC, ETH, SOL, and XRP, we confirmed that the reconstructed local volatility surfaces successfully reproduce observed market prices across different maturities and strike ranges. In particular, the method captures the market phenomenon that volatility tends to increase as the underlying asset deviates from the spot level. Numerical comparisons showed that the model-generated prices closely matched the actual market prices, which highlights the effectiveness of the proposed calibration procedure. Our algorithm provides a tractable and robust methodology for approximating volatility surfaces in highly volatile crypto markets…
In short, the authors successfully develop a numerical procedure that accurately reconstructs the implied volatility surface for major cryptocurrencies.
This paper carries important practical relevance. However, we note the confusing terminology: the authors refer to their volatility surface as “local volatility,” which may be misleading, as it can be mistaken for the classical local volatility surface introduced by Derman, Kani, and Dupire. These two concepts are distinct.
Let us know what you think in the comments below or in the discussion forum.
References
[1] Yunjae Nam, Youngjin Hwang & Junseok Kim, Reconstructing Smooth Local Volatility Surfaces for Cryptocurrency Options, Int. J. Appl. Comput. Math (2025) 11:242
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