基于时间变换分数布朗运动的期权定价：分数方差伽马模型
Option Pricing with Time-Changed Fractional Brownian Motion: A Fractional Variance Gamma Model
AI总结：
本文针对分数布朗运动非半鞅导致传统期权定价方法无法适用的问题，提出时间变换分数布朗运动，开发分数方差伽马模型并采用广义矩估计，对标普500实证得到约0.45的赫斯特指数，为期权定价提供新方法。
AI中文摘要：
分数布朗运动（fBm）具备金融建模所需的诸多优良特性，包括长程依赖性、路径粗糙性与异常扩散性。然而其非半鞅的本质使得传统无套利期权定价方法无法适用。为解决这一局限，本文引入一种时间变换fBm，该模型通过在随机伽马活动时间上评估fBm得到，其中活动时间代表累计已执行交易时间。所得过程既保留了fBm的核心特性，又恢复了半鞅结构。基于此构造，本文开发了分数方差伽马（fVG）模型，并提出用于期权定价的广义矩估计（GMM）方法。对标普500的实证分析显示，其估计的赫斯特指数约为0.45，与收益率矩的轻度亚线性时间标度一致。
英文摘要：
Fractional Brownian motion (fBm) exhibits attractive features for financial modeling, including long-range dependence, path roughness, and anomalous diffusion. However, its non-semimartingale nature precludes the use of conventional no-arbitrage approaches to option pricing. We address this limitation by introducing a time-changed fBm, obtained by evaluating fBm at stochastic gamma activity time, where activity time represents cumulative executed trading time. The resulting process retains the defining properties of fBm while recovering the semimartingale structure. Building on this construction, we develop the fractional Variance Gamma (fVG) model and propose a generalized method of moments (GMM) estimation procedure for option pricing. An empirical analysis of the S\&P 500 yields an estimated Hurst exponent of approximately 0.45, consistent with mildly sublinear temporal scaling of return moments.
