摘要
證券借貸是一種貸款，就是讓擁有一檔股票的人可以拿此檔股票當作抵押品
向金融機構貸出現金。然而，證券借貸的評價是類似於一個無到期日的美式買權，然而美式買權並無封閉解，因為證券借貸為無到期日的美式買權，因此我們必須去找出最適贖回時間，並且找出最適的贖回價格。因此自從 Xia and Zhou (2007) 證券借貸的評價問題吸引各界極大興趣，引發後來各種相關探討。本文主要研究目標在利率為隨機過程下，證券借貸的評價問題，並且找出最適的贖回價格，以供市場上借貸者 (Lender) 和貸款者 (Borrower) 做為參考。因此我們從貸款者的角度出發，關注在股價與利率為動態隨機過程 (Stochastic process)，在考量股價報酬率為負之下，我們利用在力學上解最適化問題的變分不等式 (Variational inequality)，找出最適的贖回價格。

Abstract
A stock loan or securities lending is a loan which borrowers have a share of stock can use it as collateral. However, closed form solution is not available for perpetual America options, the valuation of stock loan is an optimal stopping problem related to a perpetual American option. As Xia and Zhou (2007) published their work about stock loans. The stock loan pricing problem has thus attracted a great deal of attention since their work.
Our purpose is that under interest rate is stochastic to find an optimal exercise boundary. As a result, we focus on the stochastic process about stock and interest rate as well as the correlation between both. Finally, we use Variational Inequality to figure out the optimal stopping time which indicates
the optimal exercise price.

目錄
1 緒論 3
2 定價模型與假設 6
2.1 均值回歸隨機利率模型 6
2.2 證券借貸 6
2.3 複製 7
3變分不等式 9
4 實證研究 12
5 結論 16
A 附錄 17
A.1 考慮股價用貸款利率折現推導 17
A.2 證券借貸現值推導 17
A.3 考慮證券借貸現值推導 21
參考文獻 22

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