序列複合選擇權之評價、分析、計算與應用

Abstract

在專案鑑價方法的需求之下，本文提出序列複合選擇權(Sequential Compound options，SCOs)、它們的一般化評價公式以及敏感度分析。傳統專案鑑價的評價方法忽略了複雜專案的內在本質，例如內部高度交互作用或是多層堆疊，使得這些方法不適用，進而誤導策略制定。基於專案的特質，本研究提出序列複合選擇權，以提升專案鑑價的效能。 文獻中大部分的複合選擇權，大多是參數固定的簡單兩層選擇權。在多層複合選擇權的現有研究，也只侷限在序列複合買權(Sequential Compound CALL options，SCCs)。本研究提出多層的序列複合選擇權(SCOs)，定義為以(複合)選擇權為標的的選擇權，而它們每一層的買權(call)或賣權(put)性質是可以任意指定。此外，隨機利率與隨時間改變之資產價格波動度讓模型更加彈性。評價公式是由risk-neutral方法與change of numéraire方法分別推導而得到。一個多維度常態積分的偏微分關係，可以被視為萊布尼茲法則(Leibnitz’s Rule)的推廣，也在本研究裡推導而得，並且被用來推導序列複合選擇權(SCOs)的敏感度分析。 序列複合選擇權(SCOs)的計算，比起其他傳統的選擇權還要複雜許多。傳統歐式選擇權與(兩層或更多層)複合選擇權在演算上的差異，在於約當資產價格(Equivalent Asset Prices，EAPs)的槽套迴圈計算以及常態積分的維度。本研究克服這些困難，提出序列複合選擇權(SCOs)的演算法與三層複合選擇權的數值例。 序列複合選擇權(SCOs)可以強化並增廣複合選擇權理論在專案鑑價、風險管理與財務衍生性商品定價領域的應用。對於里程碑專案(例如新藥開發)而言，里程碑專案的達成代表擁有選擇進入下一個階段與否的權利，因此這類專案可以用序列複合選擇權(SCOs)來評價。擁有擴張、縮小規模、中止、放棄、轉換或成長選擇權在裡面交互作用的複雜專案，也可以運用序列複合選擇權(SCOs)來評價。序列複合選擇權(SCOs)的優點，包括較便宜的權利金、允許決策後延、費用分期支付、較高的彈性，可以提高風險控管的效果。一些金融機構所關心的最重要議題，例如波動度風險、抵押貸款提前還款風險與天氣風險，也可以透過序列複合選擇權(SCOs)而得到良好的控管。此外，序列複合選擇權(SCOs)也可以被運用於財務衍生性商品的定價，例如新奇美式選擇權。 本文提出序列複合選擇權(SCOs)的數值範例，包括政府營收保證評估與外匯避險運用。另外，以序列複合選擇權(SCOs)為核心的資訊系統也被提出，以作為專案與衍生性商品的評價。

This paper proposes the sequential compound options (SCOs), their generalized pricing formula and sensitivity analysis under the necessity from project valuation. Traditional methods for project valuation ignoring complicated projects' intrinsic properties, such as highly internal interacting or multiple-fold stacks, are far beyond the adequacy and will cause misleading for strategy-making. Based on project's characteristics, this study propose SCOs in order to have better effectiveness for project valuation. Most compound options described in literatures are simple 2-fold options whose parameters are constant over time. Existing research on multi-fold compound options has been limited to sequential compound CALL options (SCCs). The multi-fold sequential compound options (SCOs) proposed in this study are defined as compound options on (compound) options where the call/put property of each fold can be arbitrarily assigned. Besides, the random interest rate and time-dependent variance of asset price make the model more flexible. The pricing formula is derived by risk-neutral method and change of numéraire method. The partial derivative of a multivariate normal integration, a extension case of Leibnitz’s Rule, is derived in this study and used to derive the SCOs sensitivities. Evaluations of SCOs are more complicated than those of conventional options. The computation differences between European options and compound options (2-fold or more) lie in the equivalent asset prices (EAPs) evaluation with nested loops and the dimension of normal integrals. This study overcomes these difficulties and proposes the computing algorithm for SCOs and the numerical illustration of 3-fold SCOs. SCOs can enhance and broaden the use of compound option theory in the study of project valuation, risk management and financial derivatives valuation. For milestone projects (e.g., the new drug development), the milestone completion has the choice to enter the next stage or not, and hence the projects can be pricing by SCOs. Complex projects, within which expansion, contraction, shutting down, abandon, switch and or growth option interacting, can also be evaluated by the SCOs. Several most important issues, such as volatility risk, prepayment risk of mortgage and weather risk, concerned by the finance institutions can be well controlled through SCOs. The advantages of SCOs, including the cheaper premium, permission of decision postponement, split-fee and better flexibility, can enhance the risk management effectiveness. In addition, the SCOs can also be applied for the pricing of financial derivatives, e.g. exotic American options. The numerical examples of SCOs are proposed, including evaluation of government revenue guarantee and currency hedging. In addition, the information management system with SCOs as its core module is also proposed in order to evaluating projects and financial derivatives.