# 臺灣博碩士論文加值系統

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 美式選擇權需要使用樹狀法才能求出較精確的價格，但是在大於一個狀態變數時，其計算會變的相當費時且困難。本論文提供了在隨機利率、隨機波動性及存在股價跳躍環境下美式買權上界的解析解（封閉解），它提供了一個有效的指導方針可以讓我們知道美式選擇權理論價格的最高界限。我們亦修正了Chen 與 Yeh 兩位學者在2002年文章中的數值與公式錯誤，並且將他們文章中所提出的定理1做更一般化的修改來得到一個更接近美式買權理論價格的上界。在我們更一般化的觀念下，我們可以得到定理二，並且將其應用到當利率小於股利下，美式買權上界的例子。最後，我們應用定理2至一個美式互換選擇權的例子作為結尾。
 American options require lattice method to provide accurate price estimates. But it will be very time-consuming and difficult when more than one state variable is involved. In this paper, we develop analytical (closed form) upper bounds for American call options under stochastic interest rates, stochastic volatility, and jumps environment. It will provide a useful guideline for how high American values can reach. Also, we correct the numerical errors and formula typo in Chen and Yeh (2002) and generalize their theorem1 to derive a tighter upper bound of American calls when interest rate is greater than dividend yield. With our generalized concept, we can derive theorem 2 and apply the result to the call option case where interest rate is less than dividend yield. Finally, we will demonstrate another case about the exchange option using our theorem 2 as ending.
 Abstract……………………………………………………………………………..…I1. Introduction……………………………………………………………………..…12. Literature Review………………………………………………………………….33. General Analysis……………………………………………………………….……..53.1 Generalized Theorem 1 of Chen and Yeh (2002)………………………… …...53.1.1 American Calls on Dividend Paying Stocks (when r > d)………….…..73.2 Theorem 2 and its applications…………………………………………...……73.2.1 American Call Options on Dividend Paying Stocks (when r < d)…….83.2.2 American Exchange Option on Dividend Paying Stocks……………...94. Modeling……………………………………………………………………………..114.1 American Calls on Dividend Paying Stocks when r > d………………………114.2 American Calls on Dividend Paying Stocks when r < d………………………134.3 Jumps……………………………………………………………..………...144.4 American Exchange Options on Dividend Paying Stocks…………………….155. Numerical Result……………………………………………………………………..185.1 Correct errors in Chen and Yeh (2002)………………………………………...185.2 American Calls on Dividend Paying Stocks (when r > d)……………………..215.3 American Calls on Dividend Paying Stocks (when r < d)……………………..215.4 American Exchange Options…………………………………………………..226. Conclusion……………………………………………………………………………24Appendix A. Correct formula typo of Chen and Yeh (2002)…………… … ………….. 25A1. Derivation of the Futures Price……………………………………..……… …25A2. Derivation of the Characteristic Functions of Calls………………………....…26A3. Derivation of the Characteristic Functions of Puts……………………………..28A4. Derivation of the Characteristic Functions of Futures Options………………...29Appendix B. Derivation of and the Characteristic Functions ofTighter Upper Bound of Calls When r>d………………………………….31Appendix C. Derivation of the Characteristic Functions of UpperBound of Calls When r
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