結論與建議

本文首先將 Lévy 過程應用到信用風險中的結構式模型，求算負債價值與違約 機率。Lévy 過程具有獨立、定態增量與無限分割的性質，可以藉由調整相關參數來 符合實證上資產報酬的高峰、厚尾、偏態及波動叢聚等發現，有助於更正確地描繪 資產過程。

本文除了假設跳躍過程服從普瓦松過程，更進一步推廣至馬可夫調整普瓦松過 程。多數研究假設跳躍過程服從普瓦松過程，跳躍頻率固定不變。然而，實際觀察 市場現象，企業處在經濟穩定或是經濟相對不穩定的兩種狀態之下，公司價值發生 跳躍的頻率並不一樣，因此假設跳躍過程服從馬可夫調整普瓦松過程，將更具一般 性。

由數值分析的結果發現，在跳躍頻率服從普瓦松過程之下，隨著負債到期日的 增加，跳躍幅度的平均值、變異數、跳躍頻率與負債價值呈現負相關，與違約機率 呈現正相關。在跳躍頻率服從馬可夫調整普瓦松過程之下，越容易停留狀態之跳躍 頻率，對於負債價值的影響越大。

未來研究，將利用本文推導模型應用至信用價差選擇權及信用價差交換等衍生 性商品，並進一步將跳躍過程服從馬可夫調整普瓦松過程，推廣至雙重隨機普瓦松 過程，放寬普瓦松過程的移轉速度與移轉機率成為隨機。

參考文獻

1. Akgiray, V. and Booth, G. (1988), "Mixed diffusion-jump process modeling of exchange rate movements," Review of Economics and Statistics, Vol.70, pp.631-637 2. Altman, E.I. and Rijken, H.A., (2004), "How rating agencies achieve rating

stability," Journal of Banking and Finance, Vol.28, pp.2641-2677

3. Altman, E.I., (1968), "Financial ratios, discriminant analysis and the predication of corporate bankruptcy," Journal of Finance, Vol.23, pp.589-609

4. Altman, E.I., Haldeman,R. G. and Narayanan, P. (1977), "Zeta T M analysis: A new model to identify bankruptcy risk of corporations," Journal of Banking and Finance, Vol.1, pp.29-54

5. Ashbaugh-Skaife, H., Collins, D. and LaFond, R. (2006), "The effects of corporate governance on firms’ credit ratings," Journal of Accounting and Economics, Vol.42, pp203-243

6. Ball, A.C. and Torous, W.N. (1993), "A simplified jump process for common stock returns," Journal of Financial and Quantitative Analysis, Vol.16, pp.53-65

7. Ball, A.C. and Torous, W.N. (1995a), "On jumps in stock returns," Journal of Financial ans Quantitative Analysis, Vol.10, pp.337-351

8. Ball, A.C. and Torous, W.N. (1995b), "On jump in common stock prices and their impact on call option pricing," Journal of Finance, Vol.40, pp.155-173

9. Ballotta, L. (2005), "A Lévy process-based framework for the fair valuation of participating life insurance contracts," Insurance: Mathematics and Economics, Vol.

37, pp.173-96

10. Barndorff-Nielsen, O. E. (1995), “Normal inverse Gaussian distributions and the modeling of stock returns,” Research Report No.300, Department of Theoretical Statistics, Aarhus University.

11. Baxter, M. (2006), “Lévy simple structural models,” Nomura Fixed Income Quantitative, (Technical report), London: Nomura International

12. Benth, F.E. and Saltyte-Benth, J. (2005), "The volatility of temperature and pricing of weather derivatives," Quantitative Finance, Vol.7, pp.553-561

13. Bhojraj, S. and Sengupta, P. (2003), "Effect of corporate governance on bond ratings and yields: The role of institutional investors and outside directors," The Journal of Business, Vol.76, pp.455-476.

14. Bielecki, T.R. and Rutkowski, M. (2002), "Credit risk: modeling, valuation and hedging," Springer-Verlag, Berlin Heidelberg New York, pp51-55

15. Black, F. and Cox, J.C. (1976), "Valuing corporate securities: Some effects of bond indenture provisions," Journal of Finance, Vol.31, pp.351-367

16. Black, F. and Scholes, M.S. (1973), "The pricing of options and corporate liabilities," Journal of Political Economy, Vol.81, pp.637-654

17. Carr, P.P., Geman, H., Madan, D.B. and Yor, M. (2002), "The fine structure of asset returns: An empirical investigation," Journal of Business, Vol.75, pp.305-332

18. Chan, T. (1999), “Pricing contingent claims on stocks driven by Lévy process,” The annals of applied probability, Vol.9, pp.504-528

19. Collins, R.A. and Green, R.D. (1982), "Statistic methods for bankruptcy forecasting," Journal of Economics and Business, Vol.34, pp.349-354

20. Cont, R. (2001), "Empirical properties of asset returns: stylized facts and statistical issues," Quantitative Finance Vol.1, p.223-236

21. Cont, R. and Tankov, P. (2004), “Financial modeling with jump processes,” U.K., Chapman & Hall/CRC

22. Corcuera, J. M., Nualart, D. and Schoutens, W. (2005a), “Moment derivatives and Lévy-type market completion,” In: Kyprianou, W. Schoutens, and P. Wilmott, editors, Exotic Option Pricing and Advanced Lévy Models, pp.169-193. Wiley, 23. Corcuera, J. M., Nualart, D. and Schoutens, W. (2005b), “Completion of a Lévy

market by power-jump assets,” Finance and Stochastics, Vol.9, pp.109-127

24. Cox, J.C., Ingersoll, J.E. and Ross, S.A. (1985). "A Theory of the Term Structure of Interest Rates," Econometrica, Vol.53, pp.385-407

25. Das, S.R. and Tufano, P. (1996), "Pricing Credit-Sensitive Debt When Interest Rates, Credit Ratings, and Credit Spreads are Stochastic," Journal of Financial Engineering, Vol.5, pp.161-198

26. Eberlein, E. and Jacod, J. (1997), “On the range of options prices,” Finance and Stochastics, Vol.1, pp.131-140

27. Eberlein, E. and Keller, U. (1995), "Hyperbolic distributions in finance," Bernoulli, Vol.1, pp.281-299.

28. Eberlein, E. and Keller, U. (1995), “Hyperbolic distributions in finance,” Bernoulli, Vol.1, pp.281-299

29. Esscher, F. (1932), "On the probability function in the collective theory of risk,"

Skandinavisk Aktuarietidskrift, Vol.15, pp.175-195

30. Fama, E.F. (1965), "The behavior of stock market price", Journal of business, Vol.38, pp.34-105

31. Feinstone, L. (1985), "Minute by minute: Efficiency, normally and randomness in intradaily asset prices." Unpublished manuscript.

32. Gerber, H.U. and Shiu, E.S.W. (1994), "Option pricing by Esscher transforms,"

Transactions of the Society of Actuaries, Vol.46, pp.99-140

33. Geske, R. (1977), "The Valuation of Corporate Liabilities as Compound Options,"

Journal of Financial and Quantitative Analysis, Vol.12, pp.541-552

34. Geske, R. (1979), "The Valuation of Compound Options," Journal of Financial Economics, Vol.7, pp.63-81

35. Hull, J. and White, A. (1998), "Value at risk when daily changes in market variables are not normally distributed," Journal of Derivatives, Vol.5, pp.9-27

36. Jaimungal , S. and Young, V.R. (2005), “Pricing equity-linked pure endowments with risky assets that follow L´evy processes,” Insurance: Mathematics and Economics, Vol.36, pp.329-346

37. Jamshidian, F. (1996), "Bond and option evaluation in the Gaussian interest model,"

Research in Finance, Vol.9, pp.131-170

38. Jarrow, R.A. and Rosenfeld, E. (1984), "Jump risks and the intertemporal capital asset pricing model," Journal of Business,Vol.57, pp.337-351

39. Jarrow, R.A. and Turnbull, S.M. (1995), "Pricing Derivatives on Financial Securities Subject to Credit Risk," Journal of Finance, Vol.50, pp.53-86

40. Jarrow, R.A., Lando, D. and Turnbull, S.M. (1997), "A Markov model for the term structure of credit risk spread," The Review of Financial Studies, Vol.10, pp.481-523 41. Jasen, D.W. and De Vries, C.G. (1991), "On the Frequency of Large Stock Return:

Putting Booms and Busts into Perspective," Review of Economics and Statistics, Vol.73, pp.18-24

42. Kearns, P. and Pagan, A. (1997), "Estimating the Density Tail Index for Financial Time Series," The Review of Economics and Statistics, Vol.79, pp.171-175.

43. Kijima, M. and Komoribayashi, K. (1998), "A Markov Chain Model for Valuing Credit Risk Derivatives," The Journal of Derivatives, Vol.6, pp.97-108

44. Kim, I.J., Ramaswamy, K. and Sundaresan, S.M. (1993), "Does default risk in coupons affect the valuation of corporate bonds? : A contingent claims model,"

Financial Management, Vol. 22, pp.117-131

45. Kim, M. J., Oh, Y. H. and Brooks, R. (1994), "Are jumps in stock returns diversifiable? Evidence and implication for option pricing," Journal of Financial and Quantitative Analysis, Vol.29, pp.409-631

46. Kyprianou, A.E. (2006), "Introductory lectures on fluctuations of Lévy processes with applications," Germany, Springer

47. Lando, D. (1998), "On Cox processes and credit risky securities," Review of Derivatives Research, Vol.2, pp.99-120

48. Longstaff, F. A. and Schwartz, E. S. (1995), "A Simple Approach to Valuing Risky Fixed and Floating Rate Debt," Journal of Finance, Vol.3, pp.789-819

49. Loretan, M. and Phillips, P.C.B. (1994), "Testing the covariance stationarity of heavy-tailed time series: An overview of the theory with applications to several financial datasets,", Journal of Empirical Finance, Vol.1, pp.211-48.

50. Madan, D. B. and Seneta, E. (1987), “Chebyshev polynomial approximations and characteristic function estimation,” Journal of the Royal Statistical Society, Vol.49, pp.163-169

51. Madan, D. B. and Seneta, E. (1990), “The VG model for share market returns,”

Journal of Business, Vol.63, pp.511-524

52. Madan, D.B. and Seneta, E. (1990), "The variance gamma (VG) model for share market returns," Journal of Business, Vol.63, pp.511-524

53. Madan, D.B., Carr, P. and Chang, E. (1998), "The variance gamma process and option pricing," European Finance Review, Vol.2, pp.79-105

54. Mandelbrot, B. (1963), "The variation of certain speculative prices," Journal of Business, Vol.36, pp.394-419

55. Martin, D. (1977), "Earning warning of bank failure: A logit regression approach,"

Journal of Banking and Finance, Vol.1, pp249-276

56. Mason, S. and Bhattacharya, S. (1981), "Risky debt: Jump processes and safety covenants," Journal of Financial Economics, Vol.9, pp.281-307

57. Merton, R.C. (1974), "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates," Journal of Finance, Vol.28, pp.449-470

58. Merton, R.C. (1976), "Option Pricing When Underlying Stock Returns Are Discontinuous," Journal of Financial Economics, Vol.3, pp.125-144

59. Modigliani, F. and Miller, M. (1958), "The Cost of Capital, Corporation Finance and the Theory of Investment," American Economic Review, Vol.48, pp.261-297 60. Schoutens, W. (2003), "Lévy processes in finance: pricing financial derivatives,"

British, Wiley

61. Selivanov, A.V. (2005), “On the martingale measures in exponential Lévy models,”

Theory of Probability and its Applications, Vol.49, pp.261-274

62. Shimko, D. C., Tejima, N. and Van Deventer, D.R. (1993), "The Pricing of risky debt when interest rate are stochastic," Journal of Fixed Income, Vol.3 pp.58-66 63. Vasicek, O. (1977), "An Equilibrium characterization of the Term Structure,"

Journal of Financial Economics, Vol.5, pp.177-188

64. Zhou, C. (1997), "A Jump Diffusion Approach to Modeling Credit Risk and Valuing Defaultable Securities," Working Paper

65. Zhou, C. (2001), "The term structure of credit spreads with jump risk," Journal of Banking and Finance, Vol.25, pp.2015-2040

附錄

附錄 1. 跳躍風險可分散下，跳躍頻率服從普瓦松過程，Lévy 過程中隨機變數測度 轉換P→Q

(1)布朗運動的測度轉換

G t t W G

W

W e

dP Q

d = ^{− ()−}₂²

2 2

2 2

] ) ( [ 2

1 2

) ( ) 2

2 ( ) (

2 1 2

1 ^W _t^t _t^{W t Gt}

G t t W G W G t t W G

W e

e t e t

dP e

Q

d = ^{− −} = ^{− − −} = ^{− +}

⇒

π π

所以布朗運動項的測度轉換為W₁^Q(t)=W₁^P(t)+Gt (2)跳躍幅度的測度轉換

=1

X X

dP dQ

( )²

2 2

1

2

1 _x ^{X x}

x X

X dP e

dQ ^{σ μ}

π σ

−

= −

=

⇒

所以在風險中立Q 測度下x的分配仍為常態X^Q ~N(

μ

_x,

σ

_x²)。 (3)跳躍頻率的測度轉換

=1

N N

dP Q d

! ) ( m

t dP e

Q d

m t N N

λ

λ

= −

=

⇒

所以可知跳躍次數在風險中立Q 測度或真實測度皆相同，N^Q(t)~Poisson(

λ

t)。

附錄 2. 將零息債券與資產的布朗運動項由風險中立 Q 測度轉到遠期測度Q ^F 根據 Jamshidian (1996)處理測度轉換的方式，將零息債券與資產的布朗運動項 測度由風險中立Q 測度轉成遠期測度，首先介紹零息債券的轉換方式。

據 Girsanov Theorem，當等號右邊第二項等於一時，即可找到測度轉換的關係：

)

⎭⎬

根據 Cont and Tankov (2004)命題 8.18，我們將(1)(2)兩式組合成新的衍生性金融 商品：

⎭⎬

− ⎫

− +

−

+

∫

^{B s t dW s B s t ds}

∑

₌ ^x

∫ ∫

^t R ^{e dxds} t x

N

n n

t

r F

r 0

) (

0

σ

( , )( 2 ( )

σ

( , ) ) 0 ( 1)

υ

( )

⎩⎨

⎧− + + + +

=

⇒^{Vp t VP}

∫

0^{t rB2 s t rB s t ds}

∫

0^{t dW}1^{F s}

∫

0^{t rB sT dW}2^{F s} 2

2 ( , ) 2 ( , ) ) ( ) ( , ) ( )

2 ( exp 1 ) 0 ( )

(

σ ρσσ σ σ σ

⎭⎬

− ⎫

− +

∑ ∫ ∫

=

ds dx e

x ^t

R t x

N

n n 0

) (

0

) ( ) 1 (

υ

其中 ( , )

) ) (

( P t t t t V

V_P = ，

) , 0 (

) 0 ) (

0

( P t V_P = V

附錄 3. 簡化飄移項與布朗運動項的積分

[ ]

⎭⎬

⎫

⎩⎨

⎧ + + − −

−

= (0, )

) 2 , 0 2 (

2

1 ²

2 2

2 B t

t k B k t

t kρσσ^r σ^r σ^r σ

) , 0 2 (

1

σ

2 t

−

≡

因為資產在遠期測度之下為平賭，所以A 可以組成 Radon-Nikodym derivative：

⎭⎬

⎫

⎩⎨

⎧− (0,t)+ (0,t)W^F 2

exp 1

σ

²

σ

，其中布朗運動項W^F ~ N(0,1)。

根據上述對飄移向與布朗運動項的整理，我們得到資產動態過程可以表示為：

⎭⎬

⎫

⎩⎨

⎧− + + − −

=

∑ ∫ ∫

=

ds dx e

x W

t t

V t

V ^t

R t x

N

n n

F P

P 0

) (

0

2(0, ) (0, ) ( 1) ( )

2 exp 1 ) 0 ( )

(

σ σ υ

其中

[ ]

^{(0, )}

) 2 , 0 2 (

) , 0 (

2 2

2 2

2 B t

t k B k t

t k

t

σ ρσσ

^r

σ

^r

σ

^r

σ

+ − −

+

=

∫

∫

⁺

= ^{t t} _r ^F

o F

F dW s B s t dW s

W

t) 1 ( ) 0 ( , ) 2 ( ) ,

0

(

σ σ

σ

附錄 4. 風險可分散下，跳躍頻率服從普瓦松過程歐式賣權

(2)跳躍強度的測度轉換

P Q Q^F R

其中 2 2

附錄 5. 跳躍風險可分散下，跳躍頻率服從馬可夫調整普瓦松過程，Lévy 過程中隨 機變數測度轉換P→Q

(1)布朗運動的測度轉換

G t t W G

W

W e

dP Q

d = ^{− ()−}₂²

2 2

2 2

] ) ( 2[

1 2

) ( ) 2

2 ( ) (

2 1 2

1 ^W _t^t _t^{Wt Gt}

G t t W G W Gt t W G

W e

e t e t

dP e

Q

d = ^{− −} = ^{− − −} = ^{− +}

⇒

π π

所以布朗運動項的測度轉換為W₁^Q(t)=W₁^P(t)+Gt (2)跳躍幅度的測度轉換

=1

X X

dP dQ

( )²

2 2

1

2

1 _x ^{X x}

x X

X dP e

dQ ^{σ μ}

π σ

−

= −

=

⇒

所以在風險中立Q 測度下x的分配仍為常態X^Q ~ N(

μ

_x,

σ

_x²)。 (3)跳躍頻率的測度轉換

1

) (

=

= Φt m prob

prob

dP Q d

prob

prob dP

dQ =

⇒

令真實測度之下的移轉機率為P(m,t)，風險中立Q 測度之下為Q(m,t)

t z m

n m

m P m t z P m t e

z t m Q t

m

Q ^{* [ (1 ) ]}

0 0

*( , ) ( , ) ( , ) ( , ) ^{Ψ− − Λ}

∞

=

∞

=

=

=

=

=

∑ ∑

在真實測度之下的移轉機率為P(m,t)，其移轉速度為Ψ ，跳躍頻率 Λ 為I× 對I 角矩陣，對角線上元素

λ

_i。在測度轉換到風險中立Q 測度之後，移轉機率Q(m,t)仍 為移轉速度Ψ ，跳躍頻率 Λ 之對角矩陣，對角線上元素

λ

_i。所以在風險中立Q 測度 下移轉機率並沒有改變：Q(m,t)=P(m,t)。

附錄 6. 風險可分散下，跳躍頻率服從馬可夫調整普瓦松過程之歐式賣權

⎥⎥

P Q Q^F R

⋅

⎟⎟

附錄 7. 跳躍風險不可分散下，跳躍頻率服從普瓦松過程，Lévy 過程中隨機變數測

，所以 Randon-Nikodym derivative 也可以 用比較簡易的方式表達：

]

附錄 8. 風險不可分散下，跳躍頻率服從普瓦松過程之歐式賣權

2

1

( )

( )

3.結合J與K ，可得賣權

根據上述對於J與K 的整理，我們可以得到歐式賣權為：

∑

^∞

=

+

− − − −

= ⁺

0

, 1 ,

2 1 *

0 ¹( ) (0, )[ ( ) (0) ( )]

m

m R P m F m

h t

d N V d

N B t m! P

t P e ^λμh λμ

附錄 9. 跳躍風險不可分散下，跳躍頻率服從馬可夫調整普瓦松過程，Lévy 過程中

其中 _h ^m

m

R hx m

hx m

t x

h E e e f x dx

e E

t

n n

) ( )

( ]

)

) [(

(

) (

0 ⎟ =

μ

⎠⎞

⎜⎝

=⎛

⎟=

⎟

⎠

⎞

⎜⎜

⎝

⎛ ^{∑ Φ =}

∫

Φ

=

∑

^∞

=

∗ =

⇒

0

) , ( )

, (

m

m h

h m t Q m t z

Q

∑

^∞

=

∗ = −Λ −

⇒

0

) , ( ] ) 1 ( exp[

) ( ) , (

m

m h

m h

h m t t P m t z

Q

μ μ

∑

^∞

=

∗ = −Λ −

⇒

0

) , ( ] ) 1 ( exp[

) ( ) , (

m h

m h

h m t z t P m t

Q

μ μ

) ](

) 1 ( [ ) )(

1 ( ) ](

) 1 (

) [

,

( ^{z t t z t}

h m t e h e h e h

Q ^∗ = ^{Ψ− −μ Λ −Λ μ−} = ^{Ψ− − Λμ}

⇒

在真實測度之下，移轉機率P(m,t)代表時間由0到t ，狀況由i 移動到 j ，發生m次 跳躍的機率，其中包含移轉速度Ψ 與對角元素為

λ

_i，i=1,2,3...I之跳躍頻率矩陣Λ ； 在風險中立 Esscher 測度之下，移轉機率Q^h( tn, )的移轉速度仍為Ψ ，但跳躍頻率矩 陣為Λ ，其對角元素為

μ

_h

λ

_i

μ

_h，i=1,2,3...I。

附錄 10 風險不可分散下，跳躍頻率服從馬可夫調整普瓦松過程之歐式賣權

2

t

⎥⎦

⎟⎟⎠

在文檔中 隨機利率下Lvy 過程在信用風險之應用：結構式模型 (頁 53-89)