不確定環境下之選擇權評價模型

Abstract

選擇權評價模型(option pricing model)係由Black及Scholes於1973年聯合提出，而過去在處理選擇權之定價問題時都是假設變數是明確的(crisp)，所以大多數之研究集中在如何放鬆B-S 模型之假設，因為無風險利率和股價波動性被認為是隨機過程而非常數，在放鬆這些假設後，就能建立一個新的模型。本文即應用模糊理論(Fuzzy Theory)的觀念，針對Black-Scholes (1973)選擇權評價模式，採用模糊決策理論的主張，在實際選擇權評價分析中，輔以模糊狀態、模糊樣本訊息、模糊行動及評價函數之四項維度所構成的決策空間，來描述投資者於模糊環境下之決策，推論出若不考慮模糊因素則: (1)將高估風險利率之期望值，造成買權價格偏離的現象; (2)將高估股價變動性之期望值，高估投資人之獲利空間，易造成市場之不穩定性; (3)除非投資者已掌握完全訊息或客觀環境已確定情況下，否則投資者估計B-S選擇權評價模式中相關變數時，在價內 及價平 時，將高估買權價格期望值，而在價外 時，將低估買權價格期望值，此舉將導致訂價決策之錯誤，造成投資人之損失。 為了驗證以上模型之推導，本研究更進一步地應用模糊集合理論於二項式選擇權評價模型中，亦即CRR (Cox, Ross and Rubinstein, 1979)模型，據以建立模糊二項式選擇權評價模型。此模型能提供合理之選擇權歸屬函數和價格區間，使許多投資者可藉此套利或避險，故我們若能預測合理之選擇權價格區間，不同風險偏好之投資者就能從中獲利。然CRR模型或Black-Scholes模型亦僅能提供一個理論之參考值，在以上模型假設中，不論是無風險利率或股價報酬波動性皆為估計值，亦即隱含不確定性，故我們可以將模糊集合理論應用於CRR模型中，求算出三角模糊數，則投資者就可根據三角模糊數中極左值或極右值之選擇權價格區間來修正其投資組合策略，就不同風險偏好之投資者而言，亦能據此作出正確之投資判斷。 針對B-S模型、CRR模型及模糊二項式模型之比較，茲以S&P500股價指數選擇權為例，於價外 時，其市價皆比B-S模型與CRR模型所求算之理論價還高，權證之價值有被低估之現象。大致而言，B-S模型所求出之理論價格比CRR模型及模糊二項式模型所求算之理論價格還低，且誤差較大。而模糊二項式模型所求算之理論價格在敏感度較高之情況下，比一般CRR模型更接近實際值，且也較其收斂，同時又可創造出一個買權價格區間，提供不同風險偏好者做投資選擇。

The Black-Scholes Option Pricing Model (OPM) developed in 1973 has always been taken as the cornerstone of option pricing model. The generic applications of such a model are always restricted by its nature of not being suitable for fuzzy environment since the decision-making problems occurring in the area of option pricing are always with a feature of uncertainty. When an investor faces an option-pricing problem, the outcomes of the primary variables depend on the investor’s estimation. It means that a person's deduction and thinking process uses a non-binary logic with fuzziness. Unfortunately, the traditional probabilistic B-S model does not consider fuzziness to deal with the aforementioned problems. The purpose of this study is to adopt the fuzzy decision theory and Bayes' rule as a base for measuring fuzziness in the practice of option analysis. This study also employs “Fuzzy Decision Space” consisting of four dimensions, i.e. fuzzy state, fuzzy sample information, fuzzy action, and evaluation function, to describe the decision of investors, which is used to derive a fuzzy B-S OPM and to determine an optimal pricing for option under fuzzy environment. Finally, this study finds that the over-estimation exists in the expected value of risk interest rate, the expected value of variation stock price, and the expected value of the call price of in the money and at the money, but under-estimation exists in the expected value of the call price of out of the money without a consideration of the fuzziness. To prove above conclusions, we apply fuzzy set theory to the Cox, Ross and Rubinstein (CRR) model to set up the fuzzy binomial OPM. The model can provide reasonable ranges of option prices, which many investors can use it for arbitrage or hedge. Because of the CRR model can provide only theoretical reference values, it is expected that the fuzzy volatility and riskless interest rate replace the crisp values, which were used in generalized CRR model. In the fuzzy binomial OPM, investors can correct their portfolio strategy according to the right and left value of triangular fuzzy number and they can interpret the optimal difference, according to their individual risk preferences. Finally, this study compares with B-S model, CRR model and fuzzy binomial OPM. We use an empirical analysis of S&P 500 index options to find that the call prices of three models are lower than the market price, the call price of B-S model is lower than that of CRR model and fuzzy binomial model, and that of the fuzzy binomial OPM is much closer to the reality and more convergent than that of the generalized CRR model when the sensentivity is large.