AI 中文总结
针对相关对数正态动态下一篮子期权定价难题，提出四矩概率框架，通过测度变换结合移位对数正态近似，提升定价精度，适用于多类期权及重复估值场景。
AI 中文摘要
在相关对数正态动态下，一篮子期权难以估值，因为对数正态变量的加权和与差不存在可处理的分布。本文提出一种基于概率的四矩框架，将精确定价表示与分布近似相分离。首先通过测度变换将一篮子期权价格写为概率的线性组合；对于权重全为正的标准一篮子期权，这些概率变为正相关对数正态和的累积分布函数（CDF）值，每个和通过匹配其一阶到四阶矩的移位对数正态方差混合模型近似。对于权重符号不限的混合一篮子期权，带符号的移位对数正态代理给出解析看涨期权定价公式。本文给出可容许条件，提供实用的根选择规则，确立直接代理的主要基于行权价的金融性质，并推导基于累积分布函数（CDF）偏差的精确定价误差恒等式。数值分析结合标准一篮子基准与对由RBOB汽油、ULSD（超低硫柴油）或取暖油及WTI期货构建的标准化3:2:1裂解价差的实证应用，结果表明，该概率重构与四矩条件提升了分布拟合度与定价精度，尤其在到期期限延长和尾部不对称性增强时效果显著。该框架仍保持解析性、透明性，适用于不同行权价与到期期限的重复估值。
英文摘要
Basket options are difficult to value under correlated lognormal dynamics because weighted sums and differences of lognormal variables have no tractable distribution. This paper develops a probability-based four-moment framework that separates the exact pricing representation from the distributional approximation. A change of measure first writes a basket price as a linear combination of probabilities. For a standard basket with one positive weight, these probabilities become CDF values of positive correlated lognormal sums. Each sum is approximated by a shifted lognormal variance mixture matched to its first four moments. For an unrestricted mixed-sign basket, a signed shifted lognormal proxy gives an analytical call-price formula. We state admissibility conditions, provide a practical root-selection rule, establish the main strike-based financial properties of the direct proxy, and derive exact pricing-error identities in terms of cumulative distribution function (CDF) discrepancies. The numerical analysis combines standard-basket benchmarks with an empirical application to a normalized $3{:}2{:}1$ crack spread constructed from RBOB gasoline, ULSD or heating oil, and WTI futures. The results show that the probability reformulation and the fourth-moment condition improve the distributional fit and pricing accuracy, particularly when maturity and tail asymmetry increase. The framework remains analytical, transparent, and suitable for repeated valuation across strikes and maturities.
