無套利市場、完全市場與風險中立機率的唯一存在 (An Arbitrage-Free and Complete Market and the Unique Existence of Risk-Neutral Probabilities)
15 Pages Posted: 17 Oct 2019
Date Written: October 8, 2019
Chinese Abstract: 衍生性金融商品的基本訂價理論，是根植於不存在套利機會與完全市場假設，它的對等命題是存在唯一的風險中立測度。闡述以上定理的直覺、精簡模型是單期、二資產且二狀態的二項樹模型；因此，原文的衍生性金融商品教科書，都將二項樹模型安排在布雷克修斯連續時間模型之前。然而，大部分的中文教科書卻將二項樹模型安排在連續模型之後，而且普遍忽略風險中立訂價的必要條件：無套利機會假設，以及忽略風險中立機率唯一存在的必要條件：完全市場。本文利用單期、三狀態與三獨立基礎資產假設，呈現三維度報酬率的圖形表達、闡述不存在套利機會市場、完全市場與平賭測度機率的直覺意涵。最後，本文列表比較十數冊教科書對於本文討論議題的表達方式，並且提出評論。本文作者相信，也試圖為期貨與選擇權教科書貢獻清楚、嚴謹的學理意涵，也啟發對無套利市場、完全市場與平賭測度的解讀。
English Abstract: The fundamental pricing theory of derivatives is based on the assumptions of arbitrage-free and complete markets, which is equivalent to the unique existence of a risk-neutral measure. A single-period binomial-tree model with two assets and two states provides an intuitive and concise way to interpret this theory. As a result, English textbooks on derivatives present binomial-tree models before the Black-Scholes continuous-time model. However, most Chinese textbooks on derivatives present them in reverse order. In particular, they tend to neglect that the assumption of arbitrage-free markets is a necessary condition for the legitimacy of risk-neutral pricing, and that a condition of complete markets is for the uniqueness of a risk-neutral measure. The objective of this paper is to provide an intuitive interpretation of arbitrage-free and complete markets, as well as martingale measures, in three-dimensional figures. It is done by presenting discrete single-period models with three states, accompanied by three underlying assets. Finally, this paper compares and critiques the contents of more than ten textbooks. This paper intends to contribute to a clearer theoretical illustration of arbitrage-free, complete markets, and martingale measures.
Note: Downloadable document is in Chinese.
Keywords: Risk-neutral pricing, Futures and Options, complete market, arbitrage-free pricing, equivalent measure
JEL Classification: G12, G13
Suggested Citation: Suggested Citation