Systems and methods for modeling credit risks of publicly traded companies
There are provided new structural default models for modeling the likely default of publicly traded companies. In a first embodiment, the invention is straight-forward to implement and allows the capture of some important ingredients of the actual default, including positive short-term CDSs. In a second embodiment them model is somewhat more versatile and complex. Provided is a very efficient method for dealing with the timing of a default boundary, that is, jumps in the company's value, etc. Further provided is a process using Fast Fourier Transform matrix processing for processing the structural default models in a computationally efficient manner.
1. A method for calculating the financial status of a company, comprising the steps of:
calculating, using a computer, a value of the company in accordance with a formula
dV
V
=
(
r
+
λ
)
dt
+
σ
dW
-
dN
wherein
V is the value of the company,
r is an interest rate,
λ is an intensity of a jump arrival,
σ is a volatility of the company,
N is a standard Poisson process,
W is a standard Wiener process,
and t is a time between 0 and T, T being a maturity of a debt; and
using the value of the company to calculate, using the computer, a financial metric for the company.
2. The method of claim 1 , wherein the financial metric is an equity value of the company calculated in accordance with a formula
S=VN ( d + )− e −(r+λ)T DN ( d − )− V ( LD/V ) 2(r+λ)/σ 2 +1 N (ƒ + )+ e −(r+λ)T D ( LD/V ) 2(r+λ)σ 2 −1 N (ƒ − )
wherein like variables to those defined above define the same values, and
wherein
d ± =[ln( S/D )+( r+λ±σ 2 /2) T]/σ√{square root over (T)}
ƒ ± =[ln( L 2 D/V )+( r+λ±σ 2 /2) T]
D is a value of the debt,
T is the maturity of the debt, and
L is a relative level at which the company defaults.
3. The method of claim 1 , wherein the financial metric is a debt value of the company in accordance with a formula
δ= VN ( d − )+ e −(r+λ)T DN ( d − )+ V ( LD/V ) 2(r+λ)/σ 2 +1 N (θ + )− e −(r+λ)T D ( LD/V ) 2(r+λ)/σ 2 −1 N (ƒ − )
wherein like variables to those defined above define the same values, and
wherein
d ± =[ln( S/D )+( r+λ±σ 2 /2) T ]/σ√{square root over (T)}
ƒ ± =[ln( L 2 D/V )+( r+λ±σ 2 /2) T]/σ√{square root over (T)}
D is the value of the debt,
T is the maturity of the debt, and
L is a relative level at which the company defaults.
4. The method of claim 1 , wherein the financial metric is a survival probability of the company in accordance with a formula
Q ( t )= e −λT └N ( g + )−( LD/V ) 2r/σ 2 N ( g − )┘, t<T
wherein like variables to those defined above define the same values, and
wherein
g ± =[±ln( V/LD )+( r+λ−σ 2 /2) t]/σ√{square root over (t)}
D is a value of the debt,
T is the maturity of debt,
L is a relative level at which the company defaults.
5. The method of claim 1 , further comprising the step of calculating a credit default spread (CDS(t)) for very short maturities of the company in accordance with a formula
CDS( t )→λ(1− R ) when t→0,
wherein R is a recovery level for a selected debt seniority.
6. A system for calculating the financial status of a company, comprising:
a processor; and
a memory connected to the processor and storing instructions for controlling the operation of the processor,
the processor operative with the instructions in the memory to perform the steps of:
calculating a value of the company in accordance with a formula
dV
V
=
(
r
+
λ
)
dt
+
σ
dW
-
dN
wherein
V is the value of the company,
r is an interest rate,
λ is an intensity of a jump arrival,
σ is a volatility of the company,
N is a standard Poisson process,
W is a standard Wiener process,
and t is a time between 0 and T, T being a maturity of a debt; and
using the value of the company to calculate a financial metric for the company.
7. The system of claim 6 , wherein the financial metric is an equity value of the company in accordance with a formula
S=VN ( d + )− e −(r+λ)T DN ( d − )− V ( LD/V ) 2(r+λ)/σ 2 +1 N (ƒ + )+ e −(r+λ)T D ( LD/V ) 2(r+λ)/σ 2 −1 N (ƒ − )
wherein like variables to those defined above define the same values, and
wherein
d ± =[ln( S/D )+( r+λ±σ 2 /2) T]/σ√{square root over (T)}
ƒ ± =[ln( L 2 D/V )+( r+λ±σ 2 /2) T]/σ√{square root over (T)}
D is a value of the debt,
T is the maturity of the debt, and
L is a relative level at which the company defaults.
8. The system of claim 6 , wherein the financial metric is a debt value of the company in accordance with a formula
δ= VN ( d − )+ e −(r+λ)T DN ( d − )+ V ( LD/V ) 2(r+λ)/σ 2 +1 N (ƒ + )− e −(r+λ)T D ( LD/V ) 2(r+λ)/σ 2 −1 N (ƒ − )
wherein like variables to those defined above define the same values, and
wherein
d ± =[ln( S/D )+( r+λ±σ 2 /2) T]/σ√{square root over (T)}
ƒ ± =[ln( L 2 D/V )+( r+λ±σ 2 /2) T]/σ√{square root over (T)}
D is a value of the debt,
T is the maturity of the debt, and
L is a relative level at which the company defaults.
9. The system of claim 6 , wherein the financial metric is a survival probability of the company in accordance with a formula
Q ( t )= e −λT └N ( g + )−( LD/V ) 2r/σ 2 N ( g − )┘, t<T
wherein like variables to those defined above define the same values, and
wherein
g ± =[±ln( V/LD )+( r+λ−σ 2 /2) t]/σ√{square root over (t)}
D is a value of the debt,
T is the maturity of the debt,
L is a relative level at which the company defaults.
10. The system of claim 6 , wherein the processor is further operative with the instructions in the memory to perform the step of calculating a credit default spread (CDS(t)) for very short maturities of the company in accordance with a formula
CDS( t )→λ(1− R ) when t→0,
wherein R is a recovery level for a selected debt seniority.
11. A system for calculating a financial status of a company, comprising:
means for calculating a value of the company in accordance with a formula
dV
V
=
(
r
+
λ
)
dt
+
σ
dW
-
dN
wherein
V is the value of the company,
r is an interest rate,
λ is an intensity of jump arrivals,
σ is a volatility of the company,
N is a standard Poisson process,
W is a standard Wiener process,
and t is a time between 0 and T, T being a maturity of a debt; and
means for using the value of the company to calculate a financial metric for the company.
12. The system of claim 11 , wherein the financial metric is an equity value of the company in accordance with the formula
S=VN ( d + )− e −(r+λ)T DN ( d − )− V ( LD/V ) 2(r+λ)/σ 2 +1 N (ƒ + )+ e −(r+λ)T D ( LD/V ) 2(r+λ)/σ 2 −1 N (ƒ − )
wherein like variables to those defined above define the same values, and
wherein
d ± =[ln( S/D )+( r+λ±σ 2 /2) T]/σ√{square root over (T)}
ƒ ± =[ln( L 2 D/V )+( r+λ±σ 2 /2) T]/σ√{square root over (T)}
D is a value of the debt,
T is the maturity of the debt, and
L is a relative level at which the company defaults.
13. The system of claim 11 , wherein the financial metric is a debt value of the company in accordance with the formula
δ= VN ( d − )+ e −(r+λ)T DN ( d − )+ V ( LD/V ) 2(r+λ)/σ 2 +1 N (ƒ + )− e −(r+λ)T D ( LD/V ) 2(r+λ)/σ 2 −1 N (ƒ − )
wherein like variables to those defined above define the same values, and
wherein
d ± =[ln( S/D )+( r+λ±σ 2 /2) T]/σ√{square root over (T)}
ƒ ± =[ln( L 2 D/V )+( r+λ±σ 2 /2) T]/σ√{square root over (T)}
D is a value of the debt,
T is the maturity of the debt, and
L is a relative level at which the company defaults.
14. The system of claim 11 , wherein the financial metric is a survival probability of the company in accordance with the formula
Q ( t )= e −λT └N ( g + )−( LD/V ) 2r/σ 2 N ( g − )┘, t<T
wherein like variables to those defined above define the same values, and
wherein
g ± =[±ln( V/LD )+( r+λ−σ 2 /2) t]/σ√{square root over (t)}
D is a value of the debt,
T is the maturity of the debt,
L is a relative level at which the company defaults.
15. The system of claim 11 , further comprising means for calculating a credit default spread (CDS(t)) for very short maturities of the company in accordance with the formula
CDS( t )→λ(1− R ) when t→0,
wherein R is a recovery level for a selected debt seniority.