無匯率風險之跨通貨股權交換是新型態的衍生性金融商品,投資人可用此從事跨國投資,而不承擔匯率風險.本文立基於Dravid,Richardson and Sun(1993), Amin and Bodurhta(1995), 及Lin(1997), 首次推導間斷時間,可計算之無匯率風險下跨通貨單向及雙向股權交換之評價模型.在現金流量法之下,股價指數及匯率的隨機過程設定為lognormal process,利率則跟隨HJM模型.文中發現影響無匯率風險之跨通貨股權交換價格的主要變數為股價指數的波動度,外國股價指數與匯率間的相關性,以及兩國利率差距,與匯率的波動度無關.最後,以三個例子說明此新型態金融工具之運作情形. 1. Preliminary....................................................................................................................................................................1 1.1 Background...............................................................................................................................................................1 1.2 Brief Literature Review on the Pricing of Equity Swaps...........................................................................................3 1.3 The Aim of This Study.............................................................................................................................................4 1.4 Chapter Outline and Results......................................................................................................................................5 2.Introduction to Equity Swaps..........................................................................................................................................7 2.1 Basic Equity Swaps...................................................................................................................................................7 2.2 Variants of Baisc Equity Swaps...............................................................................................................................11 2.3 International Investment Environments....................................................................................................................12 2.4 Cross-Currency Equity Swaps.................................................................................................................................14 3. Literature Review...........................................................................................................................................................19 3.1 Marshall,Sorensen, and Tucker(1992)......................................................................................................................19 3.2 Rich(1995)................................................................................................................................................................20 3.3 Jarrow and Turnbull(1996).......................................................................................................................................22 3.4 Lin(1997)..................................................................................................................................................................25 3.5 Chance and Rich(1998).............................................................................................................................................27 4. Valuation of Cross-Currency Two-Way Equity Swaps Without Currency Risks Under Constant Interest Rates.........35 5. Valuation of Cross-Currency Two-Way Equity Swaps Without Currency Risks Under Stochastic Interest Rates.......43 6. Valuation of Cross-Currency One-Way Equity Swaps Without Currency Risks..........................................................55 7. Case Study......................................................................................................................................................................61 7.1 Case One...................................................................................................................................................................61 7.2 Case Two..................................................................................................................................................................63 7.3 Case Three.................................................................................................................................................................64 8. Conclusions...................................................................................................................................................................67 Appendixes........................................................................................................................................................................69 References .........................................................................................................................................................................92 / Based on Dravid,Richardson, and Sun(1993), Amin and Bodurtha(1995), and Lin(1997), this thesis first derives the computable discrete-time pricing formulas for the cross-currency one-way and two-way equity swaps without currency risks, which are exotic financial derivatives used for cross-border investments without the exchange rate exposure. Under the cash flow approach, equity indexes and the exchange rate are modeled by the lognormal processes, and the interest rate processes follow the HJM model. The swap price is shown to depend on the volatilities of equity indexes, the interaction between the foreign equity index and the exchange rate, as well as the interest rate differential of two countries. It does ont depend on the volatility of the exchange rate. Finally, three cases illustrate the usage of these two exotic financial instruments.

近年來在金融市場上出現了新型的金融衍生性商品，這些衍生性商品相當於一般的共同基金，但是額外鑲嵌了喊價選擇權，可以使投資人在市場表現不理想的時候，還能拿回一部份的本金；而在市場表現良好時，可以重置保本的水準來鎖定既得的利潤。因此新型的這類商品更容易吸引投資人。 　　分離基金便是這樣的商品，它募集投資大眾的資金進行投資，它不但提供保本的承諾，同時給予隨時重置保本水準的權利。然而，對發行商而言，它的風險相當地高，因此本篇論文將利用Martingale（平賭過程）的方法，針對這類商品鑲嵌的喊價選擇權來進行理論模型的推導、經濟意涵的分析、數值方法的評價，並且提出重要的避險策略。 / In recent years, there have been a lot of derivatives similar with mutual funds which guarantee that not only should investors receive a minimum benefit after a certain period of time, but also be provided with the reset right . Whenever investors reset, they can change a minimum benefit at the higher guarantee level. 　　Those kinds of funds are known as segregated funds in Canada. This paper develops a pricing model of shout options which are embedded in segregated funds by means of martingale method and it offers a hedge strategy for writers.

論文題目：信用風險下的股酬交換評價 校所組別：國立政治大學金融研究所 畢業日期：中華民國九十一年六月 學位類別：碩士 指導教授：廖四郎、呂桔誠 研究生：廖政芳 論文摘要： 衍生性金融商品在近年來不斷的推陳出新，其申股酬交換亦在國際金融市場上穩定且快速的成長，其所扮演的角色也愈趨重要，文獻上關於股酬交換評價模式的相關探討也陸續出爐。而在股酬交換日益成長的同時，信用風險的考量也漸漸的浮上台面。因此本文將利用Jarrow and Turnbull(1995)對考慮信用風險下金融資產評價的方法，結合股酬交換評價，來探討考量信用風險下股酬交換的評價。 本文所探討的信用風險包含交換資產本身以及交易對手兩種違約風險，透過遠期機率測度的平賭過程分別對普通股酬交換、上限型股酬交換以及雙向股酬交換做評價。在普通股酬交換方面，本文的模型若不考慮信用風險下，其價值只受到當期的利率期間結構的影響，其價值與股價指數無直接關連，而在兩支付點期間其價值也只是受到當時的股價指數與前一個支付時點的比例的影響，其結果與Chance and Rich(1998)利用無套利機會複製法所得到的結論一致。在上限型股酬交換部分，本文將上限型股酬交換各期的現金流量分解為一個普通股酬交換減掉遠期生效選擇權求出其在考慮信用風險下的價值。在雙向股酬交換方面，本文提出在考慮信用風險下，雙向股酬交換利率並不一定為零，而是端看交易資產的信用風險程度。 關鍵字：股酬交換、信用風險、上限型股酬交換、雙向股酬交換、平賭過程

政府為推動台灣成為亞太金融中心，逐漸放寬許多金融管制，因此，規避匯率風險將是台灣落實金融自由化與國際化的重要課題。 過去探討外幣選擇權訂價模式的文獻通常在利率固定的假設下進行研究，本研究將HJM利率模型應用於評價外幣選擇權，考慮國內外利率皆為隨機性，歐式與美式外幣選擇權的訂價。本文運用風險中立評價法，推導出與Grabbe(1983)類似的歐式外幣選擇權封閉解，並採用Amin and Bodurtha(1995)的模型設定，以間斷時間的HJM模型為基礎，運用模擬的方法決定美式外幣買權的價格，進而改變各參數的設定，進行敏感度分析。模擬結果顯示長天期的美式外幣買權對遠期利率波動度的敏感度較短天期大。本文呈現另一種外幣選擇權的評價模式，後續的研究可考慮將本文所採用的方法應用於外匯期貨選擇權、交換選擇權等衍生性金融商品的評價上。 第一章 緒論 第一節 研究背景與動機 1 第二節 研究目的 2 第三節 研究架構 3 第二章 相關文獻探討 第一節 歐式外幣選擇權之固定利率模式 4 第二節 歐式外幣選擇權之隨機利率模式 8 第三節 美式外幣選擇權評價模式 13 第三章 外幣選擇權定價模式 第一節 隨機利率下歐式外幣選擇權訂價理論 16 第二節 隨機利率下美式外幣選擇權訂價模式 26 第四章 模擬結果分析 33 第五章 結論與建議 43 附錄一 45 附錄二 46 附錄三 47 附錄四 49 附錄五、美式外幣選擇權電腦模擬程式 50 參考文獻 53

本研究探討通貨膨脹風險下長期投資人之最適資產配置。由於長年期通貨膨脹之估計誤差於投資決策上容易產生顯著差異，我們延伸 Brennan and Xia (2002)的模型，嘗試以消費者物價指數預估及修正通貨膨脹率，利用貝氏過濾方法預估未來通貨膨脹率。以平賭過程描述基金的限制條件，最適化投資人之效用值求得加入可預測性效果後之最適多期資產組合模型。研究結果顯示，長期投資人之最適策略可表示為固定比例股票指數基金及不同存續期間固定收益基金之組合。以不同存續期間之固定收益債券可以有效建構規避通貨膨脹風險之避險組合。本研究並提供數值計算與分析。 / This paper examines the optimal portfolio selection for a long-term investor. In order to consider the uncertainty of inflation rate, we extend the work in Brennan and Xia (2002) and use the consumer price index (CPI) to estimate and update the inflation rate through the filtering mechanism. The stochastic real interest rate is assumed to follow the Vasicek-type model. The investor’s optimal portfolio selection is solved through the Martingale method. The result is given in a simple closed form solution. We show that the optimal strategy for the fund manager in hedging the inflation uncertainty is to incorporate a dynamic fixed income portfolio with different durations. Numerical illustration is provided to clarify our findings.

The purpose of this thesis is to use the martingale approach to solve dynamic international portfolio problems. This thesis consists of three essays in dynamic international portfolio allocation. In demonstrating that foreign consumption plays an important role in international portfolio allocations, in Chapter 2, we present the first essay where we provide the optimal consumption plan and portfolio allocation for a representative investor with continuoustime and complete market assumptions in a simple two-country setting. Due to the demand for foreign consumption, the optimal portfolio allocation requires suitable foreign bonds to hedge against the changes in the foreign investment opportunity set and the exchange rate. The empirical results not only show that the optimal portfolio allocation with domestic and foreign consumption is different from that without consumption or with domestic consumption only, but also demonstrate the need for the foreign bonds to hedge against the change in the exchange rate risk. We present the second essay in which we extend the research of the investor's portfolio allocation problem into a continuous dynamical international market where the investment barrier of international portfolio exists. With deterministic market prices of risks, CRRA utility function and the existence of a simple investment barrier, the investor optimally hedges against the investment opportunity by allocating funds into three portfolios which are constructed by unconstrained bank accounts, equities and bonds. The first portfolio is the so called mean-variance portfolio, the second is the hedge portfolio, and the third is the synthetic portfolio which mimics the expected excess return of the constrained security in foreign country. This issue displays in Chapter 3. The third essay is presented in Chapter 4. Here we develop a continuous-time intertemporal portfolio allocation model in an international mildly segmented market. With deterministic market prices of risks and CRRA utility function, the domestic investor in the segmented market optimally hedges against the stochastic interest rates by allocating funds into two portfolios. The restricted mean-variance portfolio is derived from the traditional mean-variance portfolio without foreign constrained securities. The hedge portfolio is comprised of domestic bonds with a specific horizon for hedging against the change in the domestic interest rate. The numerical results indicate that when the volatility of the stochastic discount factor increases due to the less diversification caused by market segmentation, the less risk-averse investor benefits accordingly. Chapter 5 summarizes the main findings of the three studies and concludes the thesis by suggesting some future research venues related the current subject. / The purpose of this thesis is to use the martingale approach to solve dynamic international portfolio problems. This thesis consists of three essays in dynamic international portfolio allocation. In demonstrating that foreign consumption plays an important role in international portfolio allocations, in Chapter 2, we present the first essay where we provide the optimal consumption plan and portfolio allocation for a representative investor with continuoustime and complete market assumptions in a simple two-country setting. Due to the demand for foreign consumption, the optimal portfolio allocation requires suitable foreign bonds to hedge against the changes in the foreign investment opportunity set and the exchange rate. The empirical results not only show that the optimal portfolio allocation with domestic and foreign consumption is different from that without consumption or with domestic consumption only, but also demonstrate the need for the foreign bonds to hedge against the change in the exchange rate risk. We present the second essay in which we extend the research of the investor's portfolio allocation problem into a continuous dynamical international market where the investment barrier of international portfolio exists. With deterministic market prices of risks, CRRA utility function and the existence of a simple investment barrier, the investor optimally hedges against the investment opportunity by allocating funds into three portfolios which are constructed by unconstrained bank accounts, equities and bonds. The first portfolio is the so called mean-variance portfolio, the second is the hedge portfolio, and the third is the synthetic portfolio which mimics the expected excess return of the constrained security in foreign country. This issue displays in Chapter 3. The third essay is presented in Chapter 4. Here we develop a continuous-time intertemporal portfolio allocation model in an international mildly segmented market. With deterministic market prices of risks and CRRA utility function, the domestic investor in the segmented market optimally hedges against the stochastic interest rates by allocating funds into two portfolios. The restricted mean-variance portfolio is derived from the traditional mean-variance portfolio without foreign constrained securities. The hedge portfolio is comprised of domestic bonds with a specific horizon for hedging against the change in the domestic interest rate. The numerical results indicate that when the volatility of the stochastic discount factor increases due to the less diversification caused by market segmentation, the less risk-averse investor benefits accordingly. Chapter 5 summarizes the main findings of the three studies and concludes the thesis by suggesting some future research venues related the current subject.

雖然跨通貨股酬交換在國際投資市場扮演著重要的角色，但文獻上關於股酬交換評價模式的相關探討並不多，且多集中於國內市場或以本國貨幣做為支付幣別的股酬交換。對於跨通貨股酬交換而言，其評價模式較國內股酬交換之評價模式複雜許多，如何將影響其價值之股價指數、匯率與利率此三個主要因子間的交互相關性同時加入考量，即是此產品之評價過程的重點。 本文在完全市場的假設下，同時放寬傳統評價方法之各變數之相關係數為固定值的假設，提出一新的股酬交換評價方法，即以『兩階段兩步驟』之較具經濟含意的複製方式，推導出股酬交換的一般化評價公式。透過此複製方法，可更清楚得知股酬交換於存續期間的價值變動，更可進一步求得其避險方式，以提供股酬交換交易商在面臨不對稱風險(mismatch risk)時的避險方法。而本文的第二個貢獻在於，將本文所提出之『兩階段兩步驟』的複製方法應用於利率交換的評價上，推導出跨通貨利率交換的一般化評價模式，以進一步比較股酬交換與利率交換此兩種商品的差異性，並試圖釐清市場上對於跨通貨股酬交換評價上的誤解。 與傳統評價公式最大的差異在於：本文評價公式額外考慮了一修正項，複製投資組合可藉由此修正項，對未來各參數間的變動隨時做出調整，以使投資組合能完全複製跨通貨股酬交換的價值。 本文發現，對於國內投資人支付固定利率，以交換B市場的股價指數報酬，且以C國的貨幣做為支付幣別的跨通貨股酬交換而言，其價值除了受到當期利率期間結構的影響外，在期初或每期交換後，其價值與股價指數無直接關聯，但在兩支付間，其價值則會受到當時股價指數與前期股價指數之相對比例的影響。同時，C國對本國的未來匯率並未直接影響跨通貨股酬交換的價值。且若假設各國遠期利率的波動度為零下，則當B國股價指數與C國對本國的匯率呈現正關係或當B國股價指數與B國對本國的匯率呈現負關係時，跨通貨股酬交換的價值愈大。另外，市場上投資人通常誤認股酬交換的價值等於利率交換價值，對於股酬交換與利率交換的比較，本文發現在大多數的情況下，股酬交換的價值與利率交換的價值並不相等。

本論文含蓋三篇文章，分別從評價、避險和風險控管三方面，分析上限型股權連結保本票券。 第一篇文章為上限型股權連結保本票券之設計、評價和比較。本文考量投資人保守的投資行為與設限型股權連結票券所存在的delta跳躍(delta jump)現象，延伸Brennan and Schwartz (1976)模型，提出一個能在股價波動之際，使發行的避險部位delta呈現平滑變動且兼具保本(protected principal)功用的一般化模型(general form)。相較於一般的設限型股權連結保本模型，本模型具有以下特色。第一，加入股價成長率的調整因子(adjustable factor)，當景氣低靡，股價不停下跌時，正的調整因子可減緩股價下滑之勢，進而增加投資人在票券到期日時獲取更多資本利得(capital gain)的機會。同時，調整因子縮小了當期股價成長率與股價上限成長率(capped stock growth rate)之間的差距，繼而減緩delta 跳躍的幅度，降低發行者的避險成本。並且在HJM利率模型下，delta隨股價與股價波動度的變化更顯平滑(smooth)。第二，在保本率(protection rate)和參與率 (participation rate)不變之下，本模型的期初合理價格(fair price)較低，投資人能以較低的成本取得同等的投資保障。第三，若將本票券的名目面額(notional principal)視作共同基金(mutual fund)的淨值(net value)，而該淨值與股價連動，則本模型即成為股權連結的保本型基金(principal-protected fund)。 第二篇文章是路徑依賴之上限型股權連結保本模型之評價和風險測量。該文是擴展Brennan and Schwartz (1976)模型發展一個路徑依賴之上限型股權連結保本模型，並且提出一個比二元數模型更精確的封閉解。此外，也對七個時間序列進行股價波動度之精確檢定，得知AR-ARCH(1)模型對上限型股權連結保本票券而言，較其它時間序列模型，更能有效估計股價之波動度。 第二篇文章是外國資產的風險管理。目前在國內金融市場上，國外金融商品很多，大都以外幣計價，因此匯率風險是投資人不可忽視的因子。本文拓展Kupiec(1999)模型，將匯率風險加入模型中，使投資人更有效進行風險管理。 / This thesis studies valuation, hedge and risk management of capped equity-linked and principal-protected notes by means of the following essays: (1) Design, Valuation and Comparison of Capped Equity-linked and Principal-protected Notes (2) Valuation and Risk Measurement of Capped Equity-linked and Principal-protected Notes with Path Dependence (3) Risk Management of Foreign Assets Capped equity-linked and principal-protected notes are similar with barrier options. There exists delta jump as stock price or growth rate reaches the barrier. But previous studies about equity-linked and principal-protected notes with a restricted growth rate of stock price never explicitly discussed how the delta jump could be solved. In my first essay, I present a new design for capped equity-linked and principal-protected notes and add an adjustable factor to growth rate of stock price in such a way that the adjustable factor narrows the gap between the current stock growth rate and the capped stock growth rate and thus really reduces the magnitude of the delta jump and hence lowers the hedging cost for brokers. Recently, the focus of previous studies about principal-protected notes has been on either the restriction on the rate of stock return or the path dependence on the underlying asset, but not both in the same context. In my second essay, I develop a model on the capped, equity-linked and principal-protected notes with path dependence. There are two issues in this article. The first issue is valuation on the capped, equity-linked and principal-protected notes with path dependence. I find a closed-form approximation using the 2nd-order Taylor approximation and the method of Vorst (1992) that has higher accuracy than binomial tree model as maturity time or volatility becomes large. The second issue is risk measurement. I use VaR model to evaluate market risk of the principal-protected notes, and employ seven univariate time series models to forecast volatility and examine the accuracy. Additionally, investors may well encounter potential loss as the prices of financial products are reduced in the secondary market. The VaR is mainly concerned with the downside risk and becomes a standard measure of financial market risk that is increasingly used by investors. But if we want to apply 〝textbook〞formulation to risk management of foreign assets, there leaves exchange rate risk out of consideration. Therefore, I extend the work by Kupiec (1999) to present VaR formula with exchange rate risk for foreign assets and then to manage market risk usefully.