Everyone has probably heard that the BS model is unsuitable for crypto option pricing, but there may be a lack of quantitative understanding of just how unsuitable it is. Kończal's 2025 paper "Option Pricing Based on Crypto Futures Contracts" compared 6 pricing models using CME BTC/ETH futures option data, and found that the BS model's error was 3.5-5.5 times that of the optimal model.

Core findings of the paper:

- For crypto options, models that can handle jumps crush those that cannot. Sudden price jumps are the core 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 almost unusable for actual pricing(especially for far-dated options), because the implied volatility of crypto options is approximately 4-6 times that of the S&P 500, and return distributions have fat tails and skewness, completely deviating from the normal distribution assumption of BS.

Model selection recommendations:

- For cross-currency use: Merton jump-diffusion model (4 parameters, ranked top for both currencies)

- Currency-specific optimization: Use Kou for BTC, Bates for ETH (MAPE only 1.9%, optimal across the board)

The paper uses three metrics to measure the difference between model pricing and market prices:

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

- RMSE (Root Mean Square Error) squares first then takes the square root, so large deviations are amplified. If a model only deviates by $10 on 99 options but deviates by $5000 on 1 option, MAE might look like the difference is not significant, but RMSE will skyrocket. It reflects how bad the worst case can be.

- MAPE (Mean Absolute Percentage Error) divides the deviation by market price then takes the percentage. This eliminates the impact of price magnitude, making it possible to cross-compare pricing deviations between different currencies (such as BTC and ETH).

Other interesting findings:

- BTC and ETH have different price jump characteristics: MJD calibration shows that ETH's price jump frequency is approximately twice that of BTC, which may explain why ETH requires the more complex Bates model (to handle both high-frequency jumps and stochastic volatility), while BTC can get by with the relatively simple Kou model.

- BTC and ETH have drastically different term structures: The ν parameter of the VG model shows that BTC increases monotonically to expiration, suggesting the market believes extreme events are more likely the farther out you go. ETH's extreme volatility is concentrated in the medium term, while the long term tends toward stability.

Paper's limitations:

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

- Does not discuss calibration stability, such as using March 11 parameters to forecast March 12 prices

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

- Does not calculate cost comparisons: actual trading is sensitive to latency. BS has an analytical solution computable in milliseconds, while Bates requires numerical integration; the paper completely omits computation time, but this could be a decisive factor in high-frequency scenarios