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" used CME BTC/ETH futures options data to compare 6 pricing models, and 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 handle jumps crush models that cannot handle jumps. 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 long-dated options). The reason is that the implied volatility of crypto options is approximately 4–6 times that of the S&P 500, and the return distribution exhibits fat tails and skewness, completely deviating from BS's normal distribution assumption.
**Model Selection Recommendations:**
- For cross-asset selection: Merton's jump-diffusion model (4 parameters, ranking top on both assets)
- For asset-specific optimization: Kou for BTC, Bates for ETH (MAPE only 1.9%, best overall)
**The paper uses three metrics to measure the difference between model pricing and market prices:**
- MAE (Mean Absolute Error) is most intuitive—take the absolute value of pricing deviation for each option and calculate the average. Kou's MAE on BTC is 258, meaning each option deviates by an average of $258.
- RMSE (Root Mean Square Error) squares first then takes the square 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 similar, but RMSE will skyrocket. It reflects how bad the worst case can be.
- MAPE (Mean Absolute Percentage Error) divides the deviation by market price and converts to percentage. This neutralizes the effect of price magnitude, making pricing deviations comparable across different assets (e.g., BTC and ETH).
**Other Interesting Findings:**
- BTC and ETH have different price jump characteristics: MJD calibration shows ETH's price jump frequency is approximately twice that of BTC. This may explain why ETH requires the more complex Bates model (to handle both high-frequency jumps and stochastic volatility simultaneously), while BTC suffices with the relatively simple Kou model.
- BTC and ETH have completely different term structures: The ν parameter of the VG model shows that BTC increases monotonically with time to maturity—the market believes extreme events are more likely further out. ETH's extreme volatility concentrates in the mid-term, trending toward stability over longer periods.
**Paper 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)
- No discussion of calibration stability, e.g., using parameters from March 11 to predict prices on March 12
- Data comes from CME, but CME and Deribit differ in liquidity, participant structure, and margin mechanisms—model rankings may differ on Deribit
- No cost comparison analysis: live trading is latency-sensitive. BS has an analytical solution with instant results, while Bates requires numerical integration. The paper completely omits computation time, but this could be a decisive factor in high-frequency scenarios.
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" used CME BTC/ETH futures options data to compare 6 pricing models, and 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 handle jumps crush models that cannot handle jumps. 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 long-dated options). The reason is that the implied volatility of crypto options is approximately 4–6 times that of the S&P 500, and the return distribution exhibits fat tails and skewness, completely deviating from BS's normal distribution assumption.
**Model Selection Recommendations:**
- For cross-asset selection: Merton's jump-diffusion model (4 parameters, ranking top on both assets)
- For asset-specific optimization: Kou for BTC, Bates for ETH (MAPE only 1.9%, best overall)
**The paper uses three metrics to measure the difference between model pricing and market prices:**
- MAE (Mean Absolute Error) is most intuitive—take the absolute value of pricing deviation for each option and calculate the average. Kou's MAE on BTC is 258, meaning each option deviates by an average of $258.
- RMSE (Root Mean Square Error) squares first then takes the square 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 similar, but RMSE will skyrocket. It reflects how bad the worst case can be.
- MAPE (Mean Absolute Percentage Error) divides the deviation by market price and converts to percentage. This neutralizes the effect of price magnitude, making pricing deviations comparable across different assets (e.g., BTC and ETH).
**Other Interesting Findings:**
- BTC and ETH have different price jump characteristics: MJD calibration shows ETH's price jump frequency is approximately twice that of BTC. This may explain why ETH requires the more complex Bates model (to handle both high-frequency jumps and stochastic volatility simultaneously), while BTC suffices with the relatively simple Kou model.
- BTC and ETH have completely different term structures: The ν parameter of the VG model shows that BTC increases monotonically with time to maturity—the market believes extreme events are more likely further out. ETH's extreme volatility concentrates in the mid-term, trending toward stability over longer periods.
**Paper 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)
- No discussion of calibration stability, e.g., using parameters from March 11 to predict prices on March 12
- Data comes from CME, but CME and Deribit differ in liquidity, participant structure, and margin mechanisms—model rankings may differ on Deribit
- No cost comparison analysis: live trading is latency-sensitive. BS has an analytical solution with instant results, while Bates requires numerical integration. The paper completely omits computation time, but this could be a decisive factor in high-frequency scenarios.