本研究主要採用有限差分內隱法來衡量嵌入違約及提前清償選擇權的固定利率貸款價值，並且加以考慮恆定比率(Permanent)、固定利率(Fixed)、罰款遞減(Step-down)以及收益維持(Yield maintenance,或稱之為報酬率維持)此四種不同提前清償罰則對於固定利率貸款價值的影響。藉由數值模擬幫助銀行了解不同提前清償罰則在不同的房價波動度以及貸款比率(Loan-to-value)下，對於銀行所持有的固定利率貸款價值有何影響，使得銀行能夠在考量總體經濟以及金融環境下，擬定適合的貸款契約條款，進而加以減少因為貸款者提前清償以及違約所導致的再投資風險以及違約呆帳風險，藉此穩定銀行的獲利以及營運。
本研究模擬結果顯示，提前清償罰則的設立對於固定利率貸款價值具有顯著且正向的影響，特別是提前清償罰金佔整個契約期間越長且貸款價值比率大者，越是能提高固定利率貸款的價值。除此之外，在利率高時，雖然金融機構可以藉由提前清償罰則的約定提高貸款價值，但是在利率低時，卻會因為提前清償罰則的訂立而使得違約邊界條件提高，進而使得違約以及拖欠付款的機率上升。
另外，在不同貸款比率(Loan-to-value ratio)下，上述提前清償罰金與契約時間及貸款價值比率的關係仍然存在，但是若進一步探討不同提前清償罰則下的選擇權價值，我們將會發現在相同罰則下，隨著貸款比率的提高，選擇權價值增加的幅度呈現遞減，此結果表示雖然較高的貸款比率會使貸款人有較高的聯合選擇權價值，進而使得違約或提前清償的機會增加，但是對於銀行所持有的固定利率貸款價值卻不會線性等比例的減少。
為了讓借款者簽訂提前清償罰則的契約，金融機構降低借款利率雖然會使得貸款的總價值降低，但是其他條件相同時，金融機構卻能因此獲得吸引更多貸款者、獲取更高的利潤。

We use the implicit difference method to measure the influence of four types of penalties on the value of a fixed-rate mortgage (FRM) that has a default and prepayment option and different types of penalties. Bankers realize the influence of prepayment penalties on a mortgage’s value for different house value volatilities and loan-to-value ratios (LTV ratios) through a simulation procedure. Financial institutions can consider the macroeconomic and financial conditions and then design a suitable mortgage contract to avoid borrowers’ prepayment and default behavior. Furthermore, financial institutions can stabilize their profit and operations.
Our simulation results reveal that when the percentage of the prepayment penalty of the whole contract is higher, the influence of this penalty on the mortgage’s value is larger. Financial institutes can use different types of penalties to increase their mortgage loan, but penalties also raise the boundary of default when they are at a low interest rate.
Additionally, the relationship between prepayment penalty and the contract also exists at different LTV ratios. We further addressed the option’s value and found that it did not have a linear relationship with the LTV ratios.

論文審定書...................................................................................................i
致謝............................................................................................................ii
摘要................. ..........................................................................................iii
Abstract......................................................................................................iv
1. Introduction............... ..........................................................................1
2. Literature Review..................................................................................4
3. Model Development and Method............................................................9
3.1 Research Structure..............................................................................9
3.2 Model Assumptions.............................................................................9
3.3 Partial Differential Equation (PDE) Of the Mortgage Value......................10
3.3.1 Fixed-Rate Schedule Payments..........................................................10
3.3.2 The Joint Value of the Prepayment and Default Options.........................12
3.3.3 The Economic Variables Model.......................................................... 13
3.3.4 The Mathematical Model after Variable Transformation.......................... 15
3.4 The Boundary Conditions of the PDE.................................................. 16
3.4.1 When the Interest Rate Is Zero........................................................... 16
3.4.2 When the Log Value of the House Price Is Zero....................................16
3.4.3 Very Large House Price..................................................................... 17
3.4.4 Extremely Large Interest Rates.......................................................... 17
3.4.5 Maturity Date.................................................................................... 17
3.5 Prepayment Penalty Mechanism ........................................................18
4. Simulation Results............................................................................ 20
4.1 Setting the Parameters...................................................................... 20
4.2 What Is the Shape of the Results when the Mortgage Value Does Not Include Penalties?.................................................................................................21
4.3 When will Borrowers Exercise the Default or Prepayment options? ........23
4.4 How does the House Value Return Volatility Affect the Value of FRM?....28
4.5 How do Loan-To-Value Ratios Influence the Mortgage Value?.................31
4.6 How does Profit Change if Lenders Reduce Initial Rate? ........................35
5. Conclusion........................................................................................41
Reference................................................................................................. 45
Appendix.................................................................................................. 48
Appendix A: The derivation process of mortgage value.................................. 48
Appendix B: The derivation process of transformational variable......................50

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