论文总字数：31257字

摘 要

传统金融工程在完备市场假设下，能够从真实概率测度得到风险中性概率测度，从而求得衍生品在风险中性测度下的价格。然而从风险中性概率恢复出真实概率存在理论难度。虽然已经有方法解决，但是大多是以历史数据来预测未来分布。Ross给出在一定条件下，从风险中性概率测度恢复出真实概率的方法，即从期权信息中恢复出标的股价相关信息，所以Ross恢复是不依赖于历史数据的。

本文的目的是首先回顾传统期权定价理论，然后介绍Ross Recovery定理，包括定理的假设、模型推导过程，并分析定理的局限性。本文的创新点在于运用实证分析，将该理论应用在国内上证50ETF市场，并对实证结果进行分析和反思总结。

实证过程按照Recovery定理的三大步骤进行，先通过离散状态下对执行价格的二次差分实现求出到期日状态价格矩阵S；然后通过Matlab实现有约束的优化计算，计算出状态价格转移矩阵P；最后通过求矩阵P的特征值和特征向量，来恢复出真实转移矩阵F。最后将恢复出的真实概率边际概率与通过Bootstrap产生的边际概率进行了对比，并将恢复出的真实分布与风险中性分布也进行对比。实证结果表明Recovery定理在我国的股票市场上有一定的可行性。

关键词：Ross Recovery定理，测度变化，上证50ETF，Bootstrap

Abstract

Under the assumption of complete market, the traditional financial engineering can obtain the risk-neutral probability measure from the real probability measure, so as to obtain the price of derivative under the risk-neutral measure. However, it is theoretically difficult to recover the real-world probability from the risk neutral probability. Although some solutions have been developed, most of them rely on the historical data to predict the future distribution. Ross proposes a novel method to recover the real probability from the risk neutral probability measure under some hypothesis, that is, to recover the relevant information of the underlying price from the option information, so Ross recovery is not dependent on the historical data.

The purpose of this paper is to introduce Ross recovery theorem after reviewing the traditional option pricing theory, including the hypothesis of the theorem, the derivation process of the model, and analyze the limitations of the theorem. The innovation of this paper lies in the application of empirical analysis in the domestic SSE 50ETF market, and the analysis and discussion of empirical results are summarized.

The empirical process is carried out in accordance with the three steps of the recovery theorem. First, the process of derivation in continuous state can be used to solve the state price matrix by tenors S through difference in discrete state, and then the constrained optimization calculation can be realized through Matlab to calculate the state price transition matrix P. Finally, the natural probability transition matrix F is recovered by finding the eigenvalues and eigenvectors of the matrix P.

Finally, the recovered marginal probability of real probability is compared with the marginal probability generated through bootstrap and the recovered marginal distribution is also compared with the risk-neutral distribution. The empirical results showed that the recovery theorem is feasible in China's stock market

KEY WORDS: Ross recovery theorem, Change of measure, Domestic SSE 50ETF, Bootstrap

目录

摘要 I

Abstract II

第一章 引言 2

1.1 Recovery定理的研究现状 2

1.2 本文研究内容 4

第二章 Ross Recovery定理 5

2.1 传统期权定价理论 5

2.1.1 B-S微分方程 5

2.1.2 风险中性定理 6

2.1.3 测度变换 6

2.2 Ross recovery理论 8

2.2.1 定义 8

2.2.2 假设 9

2.2.3 Recovery定理 9

第三章 Recovery定理在我国市场的实证探究 12

3.1 实证步骤介绍 12

3.2 实证前的数据处理 13

3.3 Recovery定理的实证过程 16

3.4 实证结果 19

3.5 讨论和分析 23

3.5.1 数据及方法的限制 24

3.5.2 Ross Recovery定理的局限性 25

第四章 结论与展望 26

致谢 27

参考文献 28

附录A 29

附录B 30

引言

传统衍生品定价理论是在无套利的市场中，根据资产定价的基本定理，利用风险中性测度来确定证券的价格，风险中性测度通常表示为Q。而真实世界的概率测度是市场客观的实际概率测度，通常表示为P。

也有很多从Q测得估计P测度的方法：历史估计法、股息收益模型、市场投资者情绪指数。然而，这些方法通常需要先假设出模型以及参数，然后利用历史数据计算求出参数值，通过模型再预测未来，这就存在模型的设定会有局限性以及未来并不总依赖于历史等问题。风险中性分布可从期权价格中提取，如果能从理论上恢复出真实世界的概率分布那么就摆脱了以往的用历史数据预测未来的方法，Ross于2015年提出了“Recovery定理”，提供了连接两种测度的纽带。

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