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Title Page

Abstract

Contents

1. Introduction 10

2. A First-Passage-Time Model under, Regime-Switching Market Environment 13

2.1. The model 15

2.2. Individual and joint default probability 17

2.3. Default correlation dynamics 20

2.3.1. Asset correlation 20

2.3.2. Asset-to-liability ratio 20

2.3.3. Market condition 22

2.3.4. Asset return volatility 23

2.3.5. Regime-switching intensity 24

2.4. Applications to credit derivatives 25

2.4.1. Credit default swap with counterparty default risk 25

2.4.2. Basket default swap 27

2.4.3. Implications 27

2.5. Conclusion 34

2.6. Appendix 35

3. Optimal Policies with a Loss Rise and a Higher Borrowing Rate 37

3.1. The model 37

3.2. The optimal consumption, investment and life insurance purchase 39

3.2.1. Investment in the risky assets more than the wealth level: 41

3.2.2. Invest all the wealth in the risky assets: 42

3.2.3. Investment in the risky assets less than the wealth level: 42

3.3. CRRA utility function case 43

3.3.1. Investment in the risky assets more than the wealth level 44

3.3.2. Invest all the wealth in the risky assets 45

3.3.3. Investment in the risky assets less than the wealth level 47

3.4. Implications from CRRA case 49

3.5. Conclusion 52

요약문 53

References 55

감사의 글 59

이력서 61

List of Figures

Figure 2.1: The relation between default correlation and maturity for given asset return correlations. Here, p represents the asset return correlation between firms where X₁/K₁＝X₂/K₂ ＝4, σ1B ＝ 0.13, σ1b ＝ 0.26, σ2B ＝ 0.16, σ2b ＝ 0.33, λB ＝ 0.23,...(이미지참조) 21

Figure 2.2: The relation between default correlation and maturity for various asset-to-liability ratios. Here, p ＝ 0.4, σ1B ＝ 0.13, σ1b ＝ 0.26, σ2B ＝ 0.16, σ2b ＝ 0.33, λB ＝ 0.23, λb ＝ 1.73, rB ＝ rb ＝0.05, and η₁ ＝ η₂ ＝ 0.05. Current market is bullish.(이미지참조) 22

Figure 2.3: The relation between default correlation and maturity for various asset-to-liability ratios (with different volatility change). Here, p ＝ 0.4, λB ＝ 0.23, λb ＝ 1.73, rB ＝ rb ＝ 0.05, and η₁ ＝ η₂ ＝ 0.05. Current market is bullish and we...(이미지참조) 23

Figure 2.4: The relation between default correlation and maturity in two regimes : bull and bear. Here, X₁/K₁ ＝ X₂/K₂ ＝ 4 and other parameters are p ＝ 0.4, σ1B ＝ 0.13, σ1b ＝ 0.26, σ2B ＝ 0.16, σ2b ＝ 0.33, λB ＝ 0.23, λb ＝ 1.73, rB ＝ rb ＝ 0.05, and...(이미지참조) 24

Figure 2.5: The relation between default correlation and volatility for different market conditions. For both cases, the average volatilities are assumed to be σ₁ ＝ 0.145 and σ₂ ＝ 0.18. Other parameters are X₁/K₁ ＝ X₂/K₂ ＝ 4, p ＝ 0.4, λB ＝ 0.23,...(이미지참조) 25

Figure 2.6: The relation between default correlation and intensity for different market conditions. Here, the parameters are X₁/K₁ ＝ X₂/K₂ ＝ 4, p ＝ 0.4, σ1B ＝ 0.13, σ1b ＝ 0.26, σ2B ＝ 0.16, σ2b ＝ 0.33, T ＝ 20, rB ＝ rb ＝ 0.05, and η₁ ＝ η₂ ＝0.05.(이미지참조) 26

Figure 2.7: The relation between default swap rates and maturity for different market conditions. Here, the default parameters are X₁/K₁ ＝ X₂/K₂ ＝ 3, p ＝ 0.4, R ＝ 0.4, σ2B ＝ 0.3, σ2b ＝ 0.4, λB ＝ 0.23, λb ＝ 1.73, rB ＝ rb ＝ 0.05, and η₁ ＝ η₂ ＝ 0.05. Furhermore,...(이미지참조) 29

Figure 2.8: The relation between default swap rate and maturity for various stabilities. (a) X₁/K₁ ＝ X₂/K₂ ＝3, p ＝ 0.4, R ＝ 0.4, σ2B ＝ 0.2, σ2b ＝ 0.5, λB ＝ 0.23, λb ＝ 1.73, rB ＝ rb ＝ 0.05, and η₁ ＝ η₂ ＝ 0.05. (b) p ＝ 0.4, R ＝ 0.4, λB ＝ 0.23, λb ＝ 1.73,...(이미지참조) 30

Figure 2.9: The relation between default swap rate and maturity for various regime-switching intensities λB. Here, the parameters are X₁/K₁ ＝ X₂/K₂ ＝ 3, p ＝ 0.4, R ＝ 0.4, σ1B ＝ 0.2, σ1b ＝ 0.5, σ2B ＝ 0.3, σ2b ＝ 0.4, λb ＝ 1.73, rB ＝ rb ＝ 0.05, and η₁ ＝ η₂ ＝ 0.05....(이미지참조) 32

Figure 2.10: The relation between default swap rate and maturity for various legime-switching intensities λb. Here, thc parameters are X₁/K₁ ＝ X₂/K₂ ＝ 3, p ＝ 0.4, R ＝ 0.4, σ1B ＝ 0.2, σ1b ＝ 0.5, σ2B ＝ 0.3, σ2b ＝ 0.4, λB ＝ 0.23, rB ＝ rb ＝ 0.05, and...(이미지참조) 33

Figure 3.1: The relation between the asset correlation and the optimal investment ratio. γ ＝ 1.5, R ＝ 0.07, r ＝ 0.03, β ＝ 0.03, f ＝ 0.15, μ ＝ 0.2, σ ＝ 0.3, v ＝ 0.4. 49

Figure 3.2: The relation between σ and the optimal investment ratio. γ ＝ 1.5, R ＝ 0.07, r ＝ 0.03, β ＝ 0.03, f ＝ 0.15, μ ＝ 0.2, v ＝ 0.4. p ＝ -0.2 for Figure 3.2(a) and p ＝ 0.2 for Figure 3.2(b). 50

Figure 3.3: The optimal life insurance purchase for various risk aversion coefficients. R ＝ 0.07, r ＝ 0.03, β ＝ 0.03, p ＝ 0.03, f ＝ 0.15, μ ＝ 0.2, σ ＝ 0.3, v ＝ 0.4, T ＝ 40. 51

Figure 3.4: The optimal life insurance purchase for uarious forces of mortality. γ ＝ 1.5, R ＝ 0.07, r ＝ 0.03, β ＝ 0.03, p ＝ 0.03, f ＝ 0.15, μ ＝ 0.2, σ ＝ 0.3, v ＝ 0.4, T ＝ 40. 51

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