考慮最小交易批量之選擇權避險投資組合

Abstract

選擇權的主要功能在於為股票等資產提供一種降低投資風險的工具。建構選擇權避險投資組合的方法之一是利用隨機規劃(stochastic programming)模式以及情境產生(scenario generation)，而情境產生是一具高度挑戰性的工作。另一個方法是透過投資組合中和Black–Scholes評價模式所定義的各種風險(Greek risks)。這些風險分別代表選擇權價格對股價、履約價、波動度、無風險利率、以及到期日的導數。根據這些風險進行避險毋須依賴情境產生。 目前已有線性規劃模式可用於選擇權的無風險套利。這些模式在中和Greek risks，極大化無風險報酬的同時，也存在一些問題。首先，可完全避險的可行解可能不存在。其次，唯一一個考慮多個資產的選擇權的投資組合模式其實並不正確。第三，這些模式因未考慮最小交易批量而缺乏實用性，而考慮最小交易批量可能進一步使模式的求解更為困難。這些存在於選擇權投資組合建構模式的問題都必須加以解決。 此外，現有的選擇權的投資組合模式大多屬於套利模式，少有為避險所建立的投資組合模式。避險與套利的不同，在於套利必須是零風險，而避險則只是盡量降低風險，並不一定要完全零風險。唯一的選擇權避險模式基本上也是套利模式，不同之處在於風險可以有上下限，然而這也正是其缺點所在。由於選擇權的風險因子眾多，而且數值大小不一，如何各別設定上下限卻不會導致某些風險因子主導投資組合的選擇，成為一個棘手的問題。 本研究首先為多資產選擇權投資組合問題提出一個修正模式，接著結合模糊目標規劃法進一步延伸此模式，使其可以考慮最小交易批量的要求。實證結果發現，當考慮最小交易批量時，現有的模式無可行解，而本研究的模糊目標規劃模式可有效求解此問題。本研究接著為選擇權避險投資組合選擇問題提出一個新的模式。此模式以歸一化的絕對風險為基礎，以期望報酬為限制條件，極小化絕對風險的總和。實證結果顯示，此模式可以直接應用於具最小交易批量的選擇權投資組合問題，並發現選擇權投資組合具有類似股票投資組合的效率前緣，不同的是某些效率前緣片段呈直線而非曲線。 針對目前為止的選擇權投資組合最佳化模式，本研究除了解決基本的風險聚合(risk aggregation)問題外，另提出一個通用於避險與套利的選擇權投資組合最佳化模式，並且考慮最小交易批量的需求。本研究的成果除了具有實用性外，更期望能為未來關於選擇權投資組合的相關研究奠下基礎。

Options are designed to hedge against risks to their underlying assets. One method of forming option-hedging portfolios is using stochastic programming models that depend heavily on scenario generation, a challenging task. Another method is neutralizing the Greek risks derived from the Black–Scholes formula, which expresses the option price as a function of the stock price, strike price, volatility, risk-free interest rate, and time to maturity. Greek risks are the derivatives of the option price with respect to these variables. Hedging Greek risks requires no scenario generation. Linear programming models were proposed for constructing option portfolios with neutralized risks and maximized profit. However, problems with these models exist. First, feasible solutions might not exist to neutralize Greek risks perfectly. Second, models that involve multiple assets and their derivatives were incorrectly formulated. Third, these models lack practicability because they consider no minimum transaction lots. Minimum transaction lots can exacerbate the infeasibility problem. These problems must be resolved before option-hedging models can be further applied. Furthermore, current models for option portfolio selection are mainly for arbitrage; models for hedging are scarce. Hedging differs from arbitrage in that arbitrage requires zero-risk, whereas hedging aims to reduce risks as much as possible. Therefore, zero-risk is not mandatory to hedging. The only model for hedging stems from an arbitrage model. The difference lies in that each Greek risk can have an upper and a lower limit, but this is exactly where its shortcomings. Because options have multiple risk factors covering different value ranges, how to set the respective upper and lower limits without having certain risks dominating the portfolio selection process has become a thorny issue. This study proposed a revised model for option portfolios with multiple underlying assets, and then extended the model by incorporating it with a fuzzy goal programming approach to consider the requirement of minimum transaction lots. Numerical examples show that current models failed to obtain feasible solutions when minimum transaction lots were considered. By contrast, the proposed models solved the problems efficiently. Afterwards, this study proposed a new model for the option-hedging portfolio selection problem. This model minimizes the sum of normalized absolute risks with respect to a desirable profit. Empirical results show that this model can be applied directly to problems with minimum transaction lots. Moreover, it was found that option portfolios have efficient frontiers similar to those of stock portfolios, except that some segments of the efficient frontier appeared to be lines than curves. In addition to having resolved the risk aggregation problem existing in current option portfolio optimization models, this study has proposed a new option portfolio selection model that can be used for hedging as well as arbitrage. Minimum transaction lots can be easily took into consideration in this model. In addition to having applicability, hopefully the results of this study can lay the foundation for future studies on option portfolios.