In this section, we extend the constant correlation model in Section 2 by modeling the correlation coefficient ρ as a stochastic process. To motivate our model, we first take a look at the data we use in the empirical study. Our data contain daily rates for 10-year constant maturity Treasury bonds and Moody’s Aaa and Baa seasoned bond indices. The data cover the period January 2, 1996 to July 25, 2016, for a total of 5146 observations. In Figure 1, we plot the time series of three yields and it is clear that all of them display mean reversion. We also construct the time series of correlation by the following procedure (similar to [10] [14] and [15] ):

1) At time t , we regress daily changes in the values of X 1 ( t ) and X 2 ( t ) on the values of X 1 ( t ) and X 2 ( t ) a day before for a time window n T :

Δ X 1 ( t ) = X 1 ( t ) − X 1 ( t − Δ t ) = α 1 + β 1 X 1 ( t − Δ t ) + ε 1 ( t ) (15)

and

Δ X 2 ( t ) = X 2 ( t ) − X 2 ( t − Δ t ) = α 2 + β 2 X 2 ( t − Δ t ) + ε 2 ( t ) . (16)

2) The correlation ρ ( t ) is calculated by the correlation coefficient between ε 1 ( t ) and ε 2 ( t ) for a time window n T , that is

ρ ( t ) = ∑ i = 1 n T ( ε 1 ( t − i Δ t ) − ε ¯ 1 ( t ) ) ( ε 2 ( t − i Δ t ) − ε ¯ 2 ( t ) ) ∑ i = 1 n T ( ε 1 ( t − i Δ t ) − ε ¯ 1 ( t ) ) 2 ∑ i = 1 n T ( ε 2 ( t − i Δ t ) − ε ¯ 2 ( t ) ) 2 , (17)

where ε ¯ 1 ( t ) = 1 n T ∑ i = 1 n T     ε 1 ( t − i Δ t ) and ε ¯ 2 ( t ) = 1 n T ∑ i = 1 n T     ε 2 ( t − i Δ t ) .

We then roll over to time t + Δ t and so on to obtain a series of correlations through the time.

In Figure 2 we plot the dynamics of ρ ( t ) with n T = 30 . We can see clearly that the correlation is far from being constant and it varies very abruptly. The correlation is supposed to be bounded between −1 and 1, but in our dataset, it is concentrated on [0, 1]. In addition, the correlation process displays strong mean-reverting property. In Table 1, we provide the summary statistics for ρ ( t ) calculated from the two defaultable bond indices. The average correlations between the yields of Treasury bonds and Aaa and Baa bonds are around

0.87, but the correlations fluctuate between 0.33 and 0.99. The density functions for the correlation series have negative skewness and fat tail.

To satisfy the properties of the correlation series displayed in our data, we model the correlation as a Jacobi process, which takes the value between ρ m and ρ M , i.e.

ρ ( t ) = ρ m + ( ρ M − ρ m ) Y ( t ) , (18)

and

d Y ( t ) = κ ( θ − Y ( t ) ) d t + σ Y ( t ) ( 1 − Y ( t ) ) d B ( t ) , (19)

where κ > 0 , σ > 0 and 0 < θ < 1 . B ( t ) is a standard Brownian motion and independent of B 1 ( t ) and B 2 ( t ) in (1) and (2).

Under the martingale measure Q , we assume the stochastic process of Y ( t ) is

d Y ( t ) = κ Q ( θ Q − Y ( t ) ) d t + σ Y ( t ) ( 1 − Y ( t ) ) d B Q ( t ) , (20)

where κ Q > 0 , κ Q > 0 and 0 < θ Q < 1 . B Q ( t ) is a standard Brownian motion under the measure Q and independent of B 1 Q ( t ) and B 2 Q ( t ) in (4) and (5).

The standard Jacobi process Y takes values between 0 and 1. Following [16] , we also impose the following conditions to ensure that the boundaries 0 and 1 are inaccessible to the process Y ( t ) :

σ 2 2 κ ≤ θ ≤ 1 − σ 2 2 κ ,       σ 2 2 κ Q ≤ θ Q ≤ 1 − σ 2 2 κ Q . (21)

It is well known that there is no closed-form expression for the transition density for the Jacobi process (see e.g. [17] ). However, the transition density of Jacobi process can be represented in terms of eigenfunction expansions.

From (19), we know the infinitesimal generator L of Jacobi process is

L f ( x ) = κ ( θ − x ) f ′ ( x ) + 1 2 x ( 1 − x ) f ″ ( x ) , (22)

where f is transformation function. f ′ and f ″ are first- and second-order derivatives of f , respectively.

Let { λ n } be the eigenvalues of − L and { ψ n } be the corresponding eigenfunctions, i.e.

− L ψ n = λ n ψ n . (23)

We know for the Jacobi process Y defined on I = ( 0 , 1 ) , the eigenvalues and eigenfunctions can be written as (see e.g. [16] and [17] )

λ n = σ 2 2 n ( n − 1 + 2 κ σ 2 ) , (24)

and

ψ n ( x ) = N n P n ( α , β ) ( 2 x − 1 ) , (25)

where α = 2 κ ( 1 − θ ) σ 2 − 1 , β = 2 κ θ σ 2 − 1 and

N n 2 = ( α + β + 2 ) n − 1 ( 2 n + α + β + 1 ) n ! ( α + 1 ) n ( β + 1 ) n , (26)

where ( a ) 0 = 1 , ( a ) n = a ( a + 1 ) , ⋯ , ( a + n − 1 ) , n > 0 . P n ( α , β ) ( x ) is the Jacobi polynomials defined as

P n ( α , β ) ( x ) = ( α + 1 ) n n ! F 2 1 ( − n , 1 + α + β + n ; α + 1 ; 1 2 ( 1 − x ) ) , (27)

where F 2 1 ( a , b ; c ; x ) is the hypergeometric function defined by

F 2 1 ( a , b ; c ; x ) = ∑ n = 0 ∞ ( a ) n ( b ) n ( c ) n x n n ! . (28)

The spectral decomposition allows us to find the transition density function of Jacobi process Y as

p Y ( t , y ( 0 ) , y ( t ) ) = π ( y ( t ) ) ∑ n = 0 ∞ exp ( − λ n t ) ψ n ( y ( 0 ) ) ψ n ( y ( t ) ) , (29)

where π ( ⋅ ) is the stationary density function of Jacobi process Y ,

π ( y ) = y β ( 1 − y ) α B ( α + 1 , β + 1 ) , (30)

where B ( a , b ) is the Beta function defined by

B ( a , b ) = Γ ( a ) Γ ( b ) Γ ( a + b ) , (31)

where Γ is the standard Gamma function.

The most important property of Jacobi polynomials is that they satisfy the following orthogonality condition (see e.g. [16] ):

Proposition 2.

∫ 0 1 P n ( α , β ) ( 2 x − 1 ) P m ( α , β ) ( 2 x − 1 ) π ( x ) d x = { 1 N n 2 , n = m 0 , n ≠ m (32)

Using this property, we can immediately have the following results:

Corollary 1. Let k , k 1 and k 2 be nonnegative integers.

・ If k < n , then ∫ − 1 1 ( 1 + x ) β + k ( 1 − x ) α P n ( α , β ) ( x ) d x = 0.

・ If k < n , then ∫ − 1 1 ( 1 + x ) β ( 1 − x ) α + k P n ( α , β ) ( x ) d x = 0.

・ If k 1 + k 2 < n , then ∫ − 1 1 ( 1 + x ) β + k 1 ( 1 − x ) α + k 2 P n ( α , β ) ( x ) d x = 0.

In this paper, we need to compute the following two integrals:

∫ − 1 1 ( 1 − x ) α + k 1 ( 1 + x ) β + k 2 P n ( α , β ) ( x ) d x ,

and

∫ − 1 1 ( 1 − x ) α ( 1 + x ) β + 1 P n ( α , β ) ( x ) P m ( α , β ) ( x ) d x ,

where k 1 and k 2 are two nonnegative integers.

We can prove the following results:

Lemma 1.

(i)

∫ − 1 1 ( 1 − x ) α + k 1 ( 1 + x ) β + k 2 P n ( α , β ) ( x ) d x = { 0 , k 1 + k 2 < n a n ( α , β , k 1 , k 2 ) , k 1 + k 2 ≥ n (33)

where

a n ( α , β , k 1 , k 2 ) = 2 α + k 1 + β + k 2 + 1 Γ ( α + k 1 + 1 ) Γ ( β + k 2 + 1 ) Γ ( n + 1 + α ) n ! Γ ( α + k 1 + β + k 2 + 2 ) Γ ( α + 1 )       ×   F 3 2 ( − n , α + β + n + 1 , α + k 1 + 1 ; α + 1 , α + k 1 + β + k 2 + 2 ; 1 ) , (34)

where F 3 2 ( a , b , c ; d , e ; x ) is the hypergeometric function defined by

F 3 2 ( a , b , c ; d , e ; x ) = ∑ n = 0 ∞ ( a ) n ( b ) n ( c ) n ( d ) n ( e ) n x n n ! . (35)

(ii)

∫ − 1 1 ( 1 − x ) α ( 1 + x ) β + 1 P n ( α , β ) ( x ) P m ( α , β ) ( x ) d x = { 0 , | n − m | ≥ 2 b n , m ( α , β ) , | n − m | < 2 (36)

where

b n , m ( α , β ) = ∑ s = 0 n ( n + α s ) ( n + β n − s ) ( − 1 ) n − s 1 2 n a m ( α , β , n − s , s + 1 ) . (37)

Proof. To prove (i), from Corollary 1, we immediately know that for k 1 + k 2 < n , the integral is clearly zero. For k 1 + k 2 ≥ n , we can use the following formula (see e.g. [18] ):

∫ − 1 1 ( 1 − x ) k 1 ( 1 + x ) k 2 P n ( α , β ) ( x ) d x = 2 k 1 + k 2 + 1 Γ ( k 1 + 1 ) Γ ( k 2 + 1 ) Γ ( n + 1 + α ) n ! Γ ( k 1 + k 2 + 2 ) Γ ( 1 + α )       ×   F 3 2 ( − n , α + β + n + 1 , k 1 + 1 ; α + 1 , k 1 + k 2 + 2 ; 1 ) . (38)

To prove (ii), we first use another representation of the Jacobi polynomials,

P n ( α , β ) ( x ) = ∑ s = 0 n ( n + α s ) ( n + β n − s ) ( x − 1 2 ) n − s ( x + 1 2 ) s . (39)

Then, we have

∫ − 1 1 ( 1 − x ) α ( 1 + x ) β + 1 P n ( α , β ) ( x ) P m ( α , β ) ( x ) d x = ∑ s = 0 n ( n + α s ) ( n + β n − s ) ∫ − 1 1 ( x − 1 2 ) n − s ( x + 1 2 ) s ( x + 1 ) β + 1 ( 1 − x ) α P m ( α , β ) ( x ) d x . (40)

Using Corollary 1, we know that for | n − m | ≥ 2 , the above integral will be zero. If | n − m | < 2 , we can compute the integral using the definition of function a n . □

In order to obtain the closed-form solution for the credit spread option prices, we are interested in computing the moments of the following integral

Y ¯ ( T ) = ∫ 0 T Y ( s ) exp ( ( κ 1 Q + κ 2 Q ) s ) d s . (41)

We can prove the following result:

Theorem 1.

E Q [ Y ¯ ( T ) n ] = n ! B n ( α + 1 , β + 1 ) 1 2 n ( α + β + 2 ) ∑ ( k 1 , ⋯ , k n ) ∈ I n ∏ j = 1 n ( N k j ) 2     × P k 1 ( α , β ) ( 2 y ( 0 ) − 1 ) ∏ j = 1 n − 1 b k j , k j + 1 ( α , β ) a k n ( α , β , 0 , 1 )     × ∫ 0 T ∫ s 1 T ⋯ ∫ s n T ∏ j = 1 n exp ( − λ k j ( s j − s j − 1 ) + ( κ 1 Q + κ 2 Q ) s j ) d s n ⋯ d s 1 , (42)

where s 0 : = 0 ,

I n = { ( k 1 ⋯ k n ) ∈ ℤ + n : k n ∈ { 0 , 1 } , | k j − k j + 1 | ≤ 1 , j = 1 , 2 , ⋯ , n − 1 } , (43)

α , β , N k and λ k are assumed to be calculated using the coefficients under the measure Q .

Proof. First, we know

E Q [ Y ¯ ( T ) n ] = E Q [ ( ∫ 0 T Y ( s ) exp ( ( κ 1 Q + κ 2 Q ) s ) d s ) n ] . (44)

We have

E Q [ ( ∫ 0 T Y ( s ) exp ( ( κ 1 Q + κ 2 Q ) s ) d s ) n ] = n ! ∫ 0 T ∫ s 1 T ⋯ ∫ s n T E Q [ ∏ j = 1 n Y ( s j ) exp ( ( κ 1 Q + κ 2 Q ) s j ) ] d s n ⋯ d s 1 . (45)

Using the eigenfunction expansion for the transition density of Jacobi process Y , we have

E Q [ ∏ j = 1 n Y ( s j ) exp ( ( κ 1 Q + κ 2 Q ) s j ) ] = ∫ 0 1 ⋯ ∫ 0 1 ∏ j = 1 n y ( s j ) exp ( ( κ 1 Q + κ 2 Q ) s j )         × p Y ( s j − 1 , s j , y ( s j − 1 ) , y ( s j ) ) d y ( s 1 ) ⋯ d y ( s n ) = ∑ k 1 = 0 ∞ ⋯ ∑ k n = 0 ∞ ∏ j = 1 n exp ( − λ k j ( s j − s j − 1 ) + ( κ 1 Q + κ 2 Q ) s j ) ψ k 1 ( y ( 0 ) )         × ∏ j = 1 n − 1 ∫ 0 1 x π ( x ) ψ k j ( x ) ψ k j + 1 ( x ) d x ∫ 0 1 x π ( x ) ψ k n ( x ) d x . (46)

For the first integral in the above equation,

∫ 0 1 x π ( x ) ψ k j ( x ) ψ k j + 1 ( x ) d x = N k j N k j + 1 B ( α + 1 , β + 1 ) 1 2 α + β + 2 ∫ − 1 1 ( 1 − x ) α ( 1 + x ) β + 1 P k j ( α , β ) ( x ) P k j + 1 ( α , β ) ( x ) d x . (47)

For the second integral,

∫ 0 1 x π ( x ) ψ k n ( x ) d x = N k n B ( α + 1 , β + 1 ) 1 2 α + β + 2 ∫ − 1 1 ( 1 − x ) α ( 1 + x ) β + 1 P k n ( α , β ) ( x ) d x . (48)

Using Lemma 1, we can see that E Q [ ∏ j = 1 n Y ( s j ) exp ( ( κ 1 Q + κ 2 Q ) s j ) ] = 0 if ( k 1 ⋯ k n ) ∉ I n . Otherwise, we have

E Q [ ∏ j = 1 n Y ( s j ) exp ( ( κ 1 Q + κ 2 Q ) s j ) ] = 1 B n ( α + 1 , β + 1 ) 1 2 n ( α + β + 2 ) ∑ ( k 1 ⋯ k n ) ∈ I n ψ k 1 ( y ( 0 ) )         × ∏ j = 1 n − 1 N k j N k j + 1 b k j , k j + 1 ( α , β ) N k n a k n ( α , β , 0 , 1 )         × ∏ j = 1 n exp ( − λ k j ( s j − s j − 1 ) + ( κ 1 Q + κ 2 Q ) s j ) . (49)

Finally, using the definition of ψ k , we obtain the final result. □

We are also interested in calculating the moments of the following quantity

ρ ¯ ( T ) = 1 ∫ 0 T exp ( − ( κ 1 Q + κ 2 Q ) ( T − s ) ) d s     × ∫ 0 T ρ ( s ) exp ( − ( κ 1 Q + κ 2 Q ) ( T − s ) ) d s = ρ m + ( ρ M − ρ m ) ( κ 1 Q + κ 2 Q ) exp ( − ( κ 1 Q + κ 2 Q ) T ) 1 − exp ( − ( κ 1 Q + κ 2 Q ) T )     × ∫ 0 T Y ( s ) exp ( ( κ 1 Q + κ 2 Q ) s ) d s . (50)

Using binomial expansion, we immediately have

E Q [ ρ ¯ ( T ) n ] = ∑ j = 0 n ( n j ) ρ m n − j ( ( ρ M − ρ m ) ( κ 1 Q + κ 2 Q ) exp ( − ( κ 1 Q + κ 2 Q ) T ) 1 − exp ( − ( κ 1 Q + κ 2 Q ) T ) ) j E Q [ Y ¯ ( T ) j ] . (51)

It is also easy to obtain the centered moments of ρ ¯ ( T ) from the uncentered moments. The general equation for converting the n^th uncentered moment to the centered moment is

E Q [ ( ρ ¯ ( T ) − E Q [ ρ ¯ ( T ) ] ) n ] = ∑ j = 0 n ( n j ) ( − 1 ) n − j E Q [ ρ ¯ ( T ) j ] ( E Q [ ρ ¯ ( T ) ] ) n − j . (52)

We provide the first two centered moments for ρ ¯ ( T ) as follows:

Corollary 2.

E Q [ ρ ¯ ( T ) ] = ρ m + ( ρ M − ρ m ) ( κ 1 Q + κ 2 Q ) exp ( − ( κ 1 Q + κ 2 Q ) T ) 1 − exp ( − ( κ 1 Q + κ 2 Q ) T )     × 1 B ( α + 1 , β + 1 ) 1 2 α + β + 2 × ∑ k = 0 1     N k 2 P k ( α , β ) ( 2 y ( 0 ) − 1 ) a k ( α , β , 0 , 1 )     × exp ( ( − λ k + κ 1 Q + κ 2 Q ) T ) − 1 − λ k + κ 1 Q + κ 2 Q . (53)

V a r Q [ ρ ¯ ( T ) ] = ( ( ρ M − ρ m ) ( κ 1 Q + κ 2 Q ) exp ( − ( κ 1 Q + κ 2 Q ) T ) 1 − exp ( − ( κ 1 Q + κ 2 Q ) T ) ) 2 2 B 2 ( α + 1 , β + 1 ) 1 2 2 ( α + β + 2 )       × ( U 1 + U 2 + U 3 + U 4 + U 5 ) − ( E Q [ ρ ¯ ( T ) ] − ρ m ) 2 , (54)

where

U 1 = N 0 4 P 0 ( α , β ) ( 2 y ( 0 ) − 1 ) b 0 , 0 ( α , β ) a 0 ( α , β , 0 , 1 ) 1 κ 1 Q + κ 2 Q     × [ exp ( 2 ( κ 1 Q + κ 2 Q ) T ) − exp ( ( κ 1 Q + κ 2 Q ) T ) κ 1 Q + κ 2 Q − exp ( 2 ( κ 1 Q + κ 2 Q ) T ) − 1 2 ( κ 1 Q + κ 2 Q ) ] , (55)

U 2 = N 0 2 N 1 2 P 1 ( α , β ) ( 2 y ( 0 ) − 1 ) b 1 , 0 ( α , β ) a 0 ( α , β , 0 , 1 ) 1 κ 1 Q + κ 2 Q     × [ exp ( ( − λ 1 + 2 ( κ 1 Q + κ 2 Q ) ) T ) − exp ( ( κ 1 Q + κ 2 Q ) T ) − λ 1 + κ 1 Q + κ 2 Q     − exp ( ( − λ 1 + 2 ( κ 1 Q + κ 2 Q ) ) T ) − 1 − λ 1 + 2 ( κ 1 Q + κ 2 Q ) ] , (56)

U 3 = N 0 2 N 1 2 P 0 ( α , β ) ( 2 y ( 0 ) − 1 ) b 0 , 1 ( α , β ) a 1 ( α , β , 0 , 1 ) 1 − λ 1 + κ 1 Q + κ 2 Q     × [ exp ( 2 ( κ 1 Q + κ 2 Q ) T ) − exp ( ( − λ 1 + κ 1 Q + κ 2 Q ) T ) λ 1 + κ 1 Q + κ 2 Q     − exp ( 2 ( κ 1 Q + κ 2 Q ) T ) − 1 2 ( κ 1 Q + κ 2 Q ) ] , (57)

U 4 = N 1 4 P 1 ( α , β ) ( 2 y ( 0 ) − 1 ) b 1 , 1 ( α , β ) a 1 ( α , β , 0 , 1 ) 1 − λ 1 + κ 1 Q + κ 2 Q     × [ exp ( ( − λ 1 + 2 ( κ 1 Q + κ 2 Q ) ) T ) − exp ( ( − λ 1 + κ 1 Q + κ 2 Q ) T ) κ 1 Q + κ 2 Q     − exp ( ( − λ 1 + 2 ( κ 1 Q + κ 2 Q ) ) T ) − 1 − λ 1 + 2 ( κ 1 Q + κ 2 Q ) ] , (58)

U 5 = N 1 2 N 2 2 P 2 ( α , β ) ( 2 y ( 0 ) − 1 ) b 2 , 1 ( α , β ) a 1 ( α , β , 0 , 1 ) 1 − λ 1 + κ 1 Q + κ 2 Q     × [ exp ( ( − λ 2 + 2 ( κ 1 Q + κ 2 Q ) ) T ) − exp ( ( − λ 1 + κ 1 Q + κ 2 Q ) T ) λ 1 − λ 2 + κ 1 Q + κ 2 Q     − exp ( ( − λ 2 + 2 ( κ 1 Q + κ 2 Q ) ) T ) − 1 − λ 2 + 2 ( κ 1 Q + κ 2 Q ) ] . (59)

If ρ ( t ) is not a constant, but deterministic, the credit spread X 1 ( T ) − X 2 ( T ) defined in Section 2 will still be a Gaussian process with same mean μ 1 − μ 2 but different variance

η ¯ s 2 = η 1 2 + η 2 2 − 2 ρ ¯ ( T ) η 12 , (60)

where ρ ¯ ( T ) is defined in (50).

The pricing formula in (14) will be changed to

C ( T , K , ρ ¯ ) = exp ( − r T ) [ ( μ 1 − μ 2 − K ) N ( d ¯ ) + η ¯ s 2 π exp ( − d ¯ 2 2 ) ] , (61)

where d ¯ = μ 1 − μ 2 − K η ¯ s .

If the correlation coefficient ρ ( t ) is a stochastic process, but independent of underlying processes for the two yields X 1 ( t ) and X 2 ( t ) , we can employ Taylor series expansion method to obtain the analytical formula for the spread credit spread option prices (see e.g. [11] [12] [13] )

C ¯ ( T , K , ρ ¯ ) = E Q [ C ( T , K , ρ ¯ ) ] = E Q [ ∑ k = 0 ∞ C ( k ) ( T , K , E Q [ ρ ¯ ] ) k ! ( ρ ¯ − E Q [ ρ ¯ ] ) k ] = ∑ k = 0 ∞ C ( k ) ( T , K , E Q [ ρ ¯ ] ) k ! E Q [ ( ρ ¯ − E Q [ ρ ¯ ] ) k ] , (62)

where C ( k ) ( T , K , E Q [ ρ ¯ ] ) denotes the k^th derivative of C ( T , K , ρ ) with respect to ρ evaluated at the point ρ = E Q [ ρ ¯ ] .

In practice, we need to approximate the series using the first n terms to obtain the option prices

C ^ n ( T , K , ρ ¯ ) = ∑ k = 0 n C ( k ) ( T , K , E Q [ ρ ¯ ] ) k ! E Q [ ( ρ ¯ − E Q [ ρ ¯ ] ) k ] . (63)

To estimate the truncation error, we use the remainder of Taylor’s formula. We know there exists ρ ¯ * such that

C ( T , K , ρ ¯ ) = ∑ k = 0 n C ( k ) ( T , K , E Q [ ρ ¯ ] ) k ! ( ρ ¯ − E Q [ ρ ¯ ] ) k     + C ( n + 1 ) ( T , K , ρ ¯ * ) ( n + 1 ) ! ( ρ ¯ − E Q [ ρ ¯ ] ) n + 1 . (64)

Then, the bound for the error of C ^ n ( T , K , ρ ¯ ) is

| C ¯ ( T , K , ρ ¯ ) − C ^ n ( T , K , ρ ¯ ) | = | E Q [ C ( n + 1 ) ( T , K , ρ ¯ * ) ( n + 1 ) ! ( ρ ¯ − E Q [ ρ ¯ ] ) n + 1 ] | ≤ sup 0 < ρ ¯ * < 1 | C ( n + 1 ) ( T , K , ρ ¯ * ) | ( n + 1 ) ! | E Q [ ( ρ ¯ − E Q [ ρ ¯ ] ) n + 1 ] | . (65)

For example, the second-order approximation to the credit spread option price is

C ^ 2 ( T , K , ρ ¯ ) = C ( T , K , E Q [ ρ ¯ ] ) + 1 2 C ″ ( T , K , E Q [ ρ ¯ ] ) V a r Q ( ρ ¯ ) , (66)

where E Q [ ρ ¯ ] and V a r Q [ ρ ¯ ] can be found from Corollary 2. The second- order derivative of C is

C ″ ( T , K , ρ ) = exp ( − r T ) ( σ 1 σ 2 κ 1 Q + κ 2 Q ) 2 ( 1 − exp ( − ( κ 1 Q + κ 2 Q ) T ) ) 2 exp ( − d 2 2 ) 2 π         × { − d ( μ 1 − μ 2 − K ) [ ( μ 1 − μ 2 − K ) 2 η 12 − 3 − η 12 − 2 − d ( μ 1 − μ 2 − K ) η 12 − 5 2 ]         + [ 2 ( μ 1 − μ 2 − K ) 2 η 12 − 5 2 − η 12 − 3 2 − 2 d ( μ 1 − μ 2 − K ) η 12 − 2 ] } . (67)

A Lévy subordinator T ( t ) is a non-decreasing Lévy process with jumps and non-negative drift (see e.g. [26] ). The Laplace transform of a Lévy subordinator can be written as

E [ exp ( − λ T ( t ) ) ] = exp ( − ϕ ( λ ) t ) , (68)

where function ϕ ( λ ) is known as the Lévy exponent of the subordinator T ( t ) and is given by the Lévy-Khintchine formula (see e.g. [26] )

ϕ ( λ ) = γ λ + ∫ 0 ∞ ( 1 − exp ( − λ s ) ) ν ( d s ) , (69)

where γ ≥ 0 is the drift term of T ( t ) . ν is the Lévy measure which must satisfy

∫ 0 ∞ ( 1 ∧ s ) ν ( d s ) < ∞ . (70)

An important sub-class of Lévy subordinators are the tempered stable subordinators. For such subordinators, the Lévy measure ν ( s ) is given by

ν ( s ) = δ s − 1 − p exp ( − η s ) , (71)

where δ > 0 and η > 0 . Important special cases are the Gamma subordinator

with p = 0 , the IG subordinator with p = 1 2 and the compound Poisson sub-

ordinator with p = − 1 and η > 0 . For such subordinators, the Lévy exponent is given by

ϕ ( λ ) = { γ λ − δ Γ ( − p ) [ ( λ + η ) p − η p ] , p ≠ 0 γ λ + δ log ( 1 + λ η ) , p = 0 . (72)

We model the correlation as the TC-Jacobi process, i.e.

ρ ( t ) = ρ m + ( ρ M − ρ m ) Y ϕ ( t ) , (73)

Y ϕ ( t ) = Y ( T ( t ) ) , (74)

d Y ( t ) = κ ( θ − Y ( t ) ) d t + σ Y ( t ) ( 1 − Y ( t ) ) d B ( t ) , (75)

where T ( t ) is a Lévy subordinator and Y ϕ ( t ) is a time-changed Jacobi process. Under the martingale measure Q , we assume the stochastic process of Y ( t ) is

d Y ( t ) = κ Q ( θ Q − Y ( t ) ) d t + σ Y ( t ) ( 1 − Y ( t ) ) d B Q ( t ) . (76)

Therefore, the original Jacobi process ρ ( t ) is time changed by Lévy subordinator ϕ to generate jumps. With these specifications, the new correlation pro- cess ρ ( t ) is guaranteed to lie between ρ m and ρ M .

An important advantage of modeling the correlation as Jacobi process time changed by Lévy subordinator is that we can express the transition density of Y ϕ via eigenfunction expansion (see e.g. [24] and [25] ), i.e.

p Y ϕ ( t , y ϕ ( 0 ) , y ϕ ( t ) ) = π ( y ϕ ( t ) ) ∑ n = 0 ∞ exp ( − ϕ ( λ n ) t ) ψ n ( y ϕ ( 0 ) ) ψ n ( y ϕ ( t ) ) , (77)

where λ n , ψ n and π ( y ) are the eigenvalues, eigenfunctions and steady-state density of Jacobi process, respectively, which can be found in (24), (25) and (30). Thus, the eigenfunction expansion of Y ϕ remains the same form as Y , but with exp ( − λ n t ) replaced by exp ( − ϕ ( λ n ) t ) .

In order to obtain the closed-form solution for the credit spread option prices under the TC-Jacobi process, we need to calculate the moments of the following integral

Y ¯ ϕ ( T ) = ∫ 0 T Y ϕ ( s ) exp ( ( κ 1 Q + κ 2 Q ) s ) d s . (78)

Using the eigenfunction expansion of the transition density of Y ϕ , we see that

the moments of E Q [ ( Y ¯ ϕ ( T ) ) n ] can be computed exactly same as E Q [ ( Y ¯ ( T ) ) n ]

by replacing λ j with ϕ ( λ j ) . We immediately obtain the following result:

Theorem 2.

E Q [ Y ¯ ϕ ( T ) n ] = n ! B n ( α + 1 , β + 1 ) 1 2 n ( α + β + 2 ) ∑ ( k 1 , ⋯ , k n ) ∈ I n ∏ j = 1 n ( N k j ) 2         × P k 1 ( α , β ) ( 2 y ( 0 ) − 1 ) ∏ j = 1 n − 1 b k j , k j + 1 ( α , β ) a k n ( α , β , 0 , 1 )         × ∫ 0 T ∫ s 1 T ⋯ ∫ s n T ∏ j = 1 n exp ( − ϕ ( λ k j ) ( s j − s j − 1 ) + ( κ 1 Q + κ 2 Q ) s j ) d s n ⋯ d s 1 , (79)

where s 0 : = 0

I n = { ( k 1 ⋯ k n ) ∈ ℤ + n : k n ∈ { 0 , 1 } , | k j − k j + 1 | ≤ 1 , j = 1 , 2 , ⋯ , n − 1 } . (80)

The credit spread options prices under the TC-Jacobi process can then be computed by employing exactly the same technique as Jacobi process.

We estimate the Jacobi and TC-Jacobi processes using daily correlation series constructed as in Section 3. We choose maximum likelihood estimation (MLE) method for both models and the estimation is carried out by using R software. For a sample of size N of { ρ t } t = 1 N the log conditional likelihood function is given by

log ( L ( Θ ) ) = ∑ t = 1 N log ( p ( t , t + Δ t , ρ ( t ) , ρ ( t + Δ t ) ; Θ ) ) , (81)

where Θ is the set of parameters to be estimated. p ( t , t + Δ t , ρ ( t ) , ρ ( t + Δ t ) ; Θ ) is the transition density function of ρ . The transition density functions are

p ( t , t + Δ t , ρ ( t ) , ρ ( t + Δ t ) ) = 1 ρ M − ρ m π ( ρ ( t + Δ t ) − ρ m ρ M − ρ m )         × ∑ n = 0 ∞ exp ( − λ n Δ t ) ψ n ( ρ ( t ) − ρ m ρ M − ρ m ) ψ n ( ρ ( t + Δ t ) − ρ m ρ M − ρ m ) , (82)

for Jacobi process and

p ( t , t + Δ t , ρ ( t ) , ρ ( t + Δ t ) ) = 1 ρ M − ρ m π ( ρ ( t + Δ t ) − ρ m ρ M − ρ m )         × ∑ n = 0 ∞ exp ( − ϕ ( λ n ) Δ t ) ψ n ( ρ ( t ) − ρ m ρ M − ρ m ) ψ n ( ρ ( t + Δ t ) − ρ m ρ M − ρ m ) , (83)

for TC-Jacobi process.

The MLE approach based on eigenfunctions for Jacobi process has been investigated in [17] . They also compare the performance of MLE estimator with other estimators proposed in the literature, such as the estimator in [27] , which is a method of moments based on an approximation of score function or with a generalized method of moments (GMM) estimator. The MLE estimator is further compared with computer-intensive simulation-based estimator, such as the simulated method of moments (SMM). They show that MLE outperforms the other estimators relative to bias and variance, while being easy to implement.

In the estimation, we set ρ m = 0 and ρ M = 1 so that the correlation is bounded by 0 and 1. The estimation results are presented in Table 2. We notice that all the parameters for both Jacobi and TC-Jacobi processes are highly significant. For Jacobi process, the mean parameters are very close for the correlations calculated from Aaa and Baa bonds. The correlations for Aaa bonds have higher mean reversion and smaller variance compared to Baa bonds. Whence Lévy subordinator is included, we clearly see that mean reversion and variance parameters drop significantly. This is as expected since the time-change process introduces the mean-reverting jumps into the correlation process, which is contrary to Jacobi process where the only source to generate mean reversion is through diffusion process.

We also report the log-likelihood function values, Akaike Information Criterion (AIC) and the Bayes Information Criterion (BIC) in Table 2. It is clear for the correlation calculated from both Aaa and Baa bonds, the TC-Jacobi model produces much larger likelihood values than Jacobi model, which

The standard errors are in the parentheses.

indicates that it is necessary to include the jumps into the Jacobi process to improve its performance. We also employ AIC and BIC to compare the relative performance of Jacobi and TC-Jacobi processes. Again, it is clear that TC-Jacobi process captures the dynamics of the correlation much better than the Jacobi process.

We compute the credit spread option prices using second-order approximation that is provided in sections 3 and 4. The higher-order approximation yields similar results.

First, we test if stochastic correlation leads to different credit spread option prices compared to the constant correlation case. For the Jacobi process, we take the parameters from those reported in Table 2 based on the Aaa bonds. From Figure 3, we observe when the correlation process starts from the value lower/higher than the long-run mean, the prices under constant correlation

will be higher/lower than those under the stochastic correlation. When starting value is close to the long-run mean, there is no decisive difference between the two prices. These are due to the fact that under the Jacobi process, the correlation will converge to the long-run mean due to the strong mean-reversion. When the correlation is higher, the variance of credit spread will be lower, which will lead to the lower option prices and vice versa.

Second, we test if there is significant difference in the prices from the Jacobi and TC-Jacobi processes. We use the parameters from those in Table 2 for Aaa bonds for both correlation processes. From Figure 4 it is clear the prices from the TC-Jacobi process are higher than those from Jacobi process. This can be explained by the fact that the long-run mean from the TC-Jacobi process is lower than the Jacobi process.

In Figure 5, we also illustrate the role of different parameters of TC-Jacobi process played in the pricing of credit spread options. We vary each parameter while keeping others at values reported in Table 2. It seems that higher mean reversion parameter κ will lead to higher prices. As expected, increase in long-run mean will decrease the option prices. We also note there is no

significant effect on the prices for different volatility parameters. Finally, we find the higher δ , the lower η in the Lévy process, the higher the price is.