臺灣博碩士論文加值系統

本論文第一章將 LIBOR 市場模型加入股價動態，並求出其風險中立過程下的動態模型，並利用此模型評價股籌交換契約。第二章將 LIBOR 市場模型擴展成兩國的市場模型，加入兩國股價動態，並求出風險中立過程下的動態模型，並利用此模型評價跨國股籌交換契約。本論文第二部份說明如何實際使用此模型，並使用蒙地卡羅模擬檢驗此評價模型的正確性。

This thesis includes two main chapters. Chapter 2 is entiled as "Equity Swaps in a LIBOR Market Model" and Chapter 3 is entitled as "Cross-Currency Equity Swaps in a LIBOR Market in a Model". The conclusions of this thesis are made in Chapter 4.

In Chapter 2, we extends the BGM (Brace, Gatarek and Musiela (1997))interest rate model (the LIBOR market model) by incorporating the stock price dynamics under the martingale measure. As compared with traditional interest rate models, the extended BGM model is easy to calibrate the model parameters and appropriate for pricing equity
swaps. The general framework for pricing equity swaps is proposed and applied to the pricing of floating-for-equity swaps with either constant or variable notional principals. The calibration procedure and the practical implementation are also discussed.

In Chapter 3, under the arbitrage-free framework of HJM, we
simultaneously extends the BGM model (the LIBOR market model) from a single-currency economy to a cross-currency case and incorporates the stock price dynamics under the martingale measure. The resulting model is very general for pricing almost every kind of (cross-currency) equity swaps traded in OTC markets. The calibration procedure and the hedging strategies are also provided in this paper for practical operation. The pricing formulas of the equity swaps with either a constant or a variable notional principal and with hedged or un-hedged exchange rate risk are derived and discussed as examples.

Contents

1. Introduction --------------------------------------------------------------- 1

1.1. Traditional Interest Rate Models ---------------------------------- 2

1.1.1. Vasicek model (1977) ---------------------------------------- 3

1.1.2. Hull and White (1990, HW) --------------------------------- 3

1.1.3. Heath, Jarrow and Morton (1992, HJM) ------------------- 4

1.2. Market Models ------------------------------------------------------ 6

1.2.1. The LIBOR Market Model (LMM) ------------------------- 6

1.2.2. The Swap Market Model (SMM) --------------------------- 7

1.3. The Thesis ------------------------------------------------------------ 8

2. Equity Swaps in a LIBOR Market Model --------------------------- 10

2.1. Introduction -------------------------------------------------------- 10

2.2. The Model ---------------------------------------------------------- 12

2.2.1. Step I : Arbitrage-free Extended HJM Model ------------ 13

2.2.2. Step II : The Arbitrage-Free Extended BGM Model ---- 18

2.3 Pricing Equity Swaps ---------------------------------------------- 22

2.3.1. Pricing Floating-for-Equity Swaps with a Constant Notional Principal ------------------------------------------------------- 22


2.3.2. Pricing Floating-for-Equity Swaps with Variable Notional Principal ------------------------------------------------------- 24

2.4. Calibration and Numerical Examples -------------------------- 26

Appendix A: Proof of Theorem 2.1 ---------------------------------- 30

Appendix B: Proof of Theorem 2.2 ---------------------------------- 32

Appendix C: Examining the Accuracy of the Approximation in Equation (B.4) -------------------------------------------- 37

3. Cross-currency Equity Swaps with LIBOR Market Model ------- 40

3.1. Introduction -------------------------------------------------------- 40

3.2. Arbitrage-free Extended HJM Model --------------------------- 43

3.3. Arbitrage-Free Extended BGM Model ------------------------- 51

3.4. Pricing Cross-Currency Equity Swaps -------------------------- 55

3.4.1 Pricing Hedged Cross-Currency Equity Swaps with a Constant Notional Principal (HCESC) ------------------------------- 55

3.4.2 Pricing Hedged Cross-Currency Equity Swaps with a Variable Notional Principal (HCESV) ------------------------------- 58

3.4.3 Pricing Unhedged Cross-Currency Equity Swaps with a Variable Notional Principal (UHCESV) ------------------ 60

3.5. Calibration Procedure --------------------------------------------- 62

Appendix D: Proof of Theorem 3.1 ---------------------------------- 66

Appendix E: Proof of Theorem 3.2 ----------------------------------- 71

Appendix F: Proof of Theorem 3.3 ----------------------------------- 75

4 Conclusions -------------------------------------------------------------- 80

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