# Everyone may have heard that the BS model is unsuitable for crypto options pricing, but the extent of its unsuitability may lack quantitative understanding. A 2025 paper by Kończal titled "Options Pricing Based on Crypto Futures Contracts" compared 6 pricing models using CME BTC/ETH futures options data, finding that the BS model's error is 3.5-5.5 times that of the optimal model.

## Core findings of the paper:

- For crypto options, models that can handle jumps overwhelmingly outperform models that cannot. Sudden price jumps are the defining characteristic of crypto markets, so capturing sudden price movements is more important than precisely modeling continuous volatility changes.

- The BS model's error far exceeds other models and is nearly unusable for practical pricing(especially for long-dated options), because crypto options' implied volatility is approximately 4–6 times that of the S&P 500, and return distributions have fat tails and skewness, completely departing from the normal distribution assumption of BS.

## Model selection recommendations:

- Cross-currency: Choose the Merton jump-diffusion model (4 parameters, ranks highly for both currencies)

- Currency-specific optimization: Kou for BTC, Bates for ETH (MAPE only 1.9%, optimal overall)

## The paper uses three metrics to measure model pricing vs. market price differences:

- **MAE (Mean Absolute Error)** is most intuitive: take the absolute value of pricing deviation for each option, then average. Kou's MAE on BTC is 258, meaning each option deviates by an average of $258.

- **RMSE (Root Mean Square Error)** squares before taking the root, so large deviations are amplified. If a model deviates by only $10 on 99 options but $5,000 on 1 option, MAE might look relatively small, but RMSE will spike. It reflects how bad the worst case is.

- **MAPE (Mean Absolute Percentage Error)** divides the deviation by market price and converts to percentage. This eliminates the impact of price magnitude, enabling horizontal comparison of pricing deviations across different currencies (e.g., BTC and ETH).

## Other interesting findings:

- BTC and ETH have different price jump characteristics: MJD calibration shows ETH's jump frequency is roughly twice that of BTC, which may explain why ETH requires the more complex Bates model (simultaneously handling high-frequency jumps and stochastic volatility), while BTC works fine with the relatively simpler Kou model.

- BTC and ETH have drastically different term structures: The ν parameter of the VG model shows BTC increases monotonically with time-to-expiration, with the market believing extreme events more likely at longer horizons. ETH's extreme volatility concentrates in the mid-term, while the long term is relatively stable.

## Limitations of the paper:

- All conclusions are based on data from a single day, March 11, 2024 (when BTC broke through the previous cycle high, representing extreme market conditions)

- No discussion of calibration stability, such as using March 11 parameters to forecast March 12 prices

- Data sourced from CME; CME and Deribit differ in liquidity, market participant structure, and margin mechanisms, so model rankings on Deribit may differ.

- No cost comparison: real-world trading is latency-sensitive. BS has an analytical solution outputting instantly, while Bates requires numerical integration; the paper completely omits computational time, which could be the decisive factor in high-frequency scenarios.