基于粗糙伯格米（rBergomi）模型的比特币反向期权定价与校准
Pricing and Calibration of Bitcoin Inverse Options via the Rough Bergomi Model
中文总结 AI 辅助
本文基于粗糙伯格米模型构建比特币反向期权的定价校准框架，对比三条计算管线后发现混合方案与混合估计管线最优，证实比特币波动率的粗糙特性及校准误差与平值隐含波动率的线性关系。
中文摘要 AI 辅助
比特币反向期权在Deribit交易所交易，以标的加密货币而非法定货币结算，结合了极端且真正的粗糙波动率动态，以及非线性、依赖货币的收益结构。本文开发并实证验证了基于Bayer、Friz和Gatheral（2016）提出的粗糙伯格米（rough Bergomi，rBergomi）模型的此类工具的定价与校准框架。我们将rBergomi动态适配到反向收益max(S_T - K, 0)/S_T，并实现并比较了三条计算管线，它们在驱动分数布朗运动的模拟方案（粗网格乔列斯基法 vs. Bennedsen等人2017年提出的混合方案）和蒙特卡洛定价估计器（普通对数欧拉法 vs. McCrickerd和Pakkanen 2018年提出的混合估计器）上存在差异。该模型针对2022年5月至2025年3月期间从Deribit交易数据中提取的30条隐含波动率曲面进行了校准，涵盖7次主要市场压力事件和9种按波动率水平分层的基准 regime。混合方案与混合估计管线同时具有最高的准确性（平均未加权均方根误差为22.83个百分点，而乔列斯基法与欧拉法基准为41.76个百分点）和最快的速度（每个快照17秒，速度提升20倍）。校准得到的赫斯特指数始终接近搜索空间的下界（在大多数 regime 中H约为0.01至0.06），证实比特币的波动率确实是粗糙的，且校准误差与平值隐含波动率水平呈近似线性比例关系（皮尔逊相关系数r=0.89）。
英文摘要
Bitcoin inverse options, traded on the Deribit exchange and settled in the underlying cryptocurrency rather than in fiat currency, combine extreme and genuinely rough volatility dynamics with a non-linear, currency-dependent payoff structure. This paper develops and empirically validates a pricing and calibration framework for these instruments based on the rough Bergomi (rBergomi) model of Bayer, Friz and Gatheral (2016). We adapt the rBergomi dynamics to the inverse payoff max(S_T - K, 0)/S_T, and implement and compare three computational pipelines that differ in the simulation scheme for the driving fractional Brownian motion (coarse-grid Cholesky vs. the Hybrid Scheme of Bennedsen et al., 2017) and in the Monte Carlo pricing estimator (plain log-Euler vs. the Mixed Estimator of McCrickerd and Pakkanen, 2018). The model is calibrated to thirty implied volatility surfaces extracted from Deribit trade data between May 2022 and March 2025, spanning seven major market-stress events and nine baseline regimes stratified by volatility level. The Hybrid and Mixed pipeline is simultaneously the most accurate (mean unweighted RMSE 22.83 percentage points, versus 41.76 pp for the Cholesky and Euler benchmark) and the fastest (17 seconds per snapshot, a 20-fold speed-up). The calibrated Hurst exponent is consistently close to the lower bound of the search space (H approximately equal to 0.01--0.06 in most regimes), confirming that Bitcoin's volatility is genuinely rough, and calibration error scales approximately linearly with the level of at-the-money implied volatility (Pearson r = 0.89).
