In this notice we are comment popular approaches to the credit risk modeling.Credit risk; credit derivatives; risk neutral world; risk neutral probability; structural model; reduced form

ilya, gikhman

English

Research Papers in Economics

  1 
Some critical comments on credit risk modeling.  
 
 
 
 
Ilya Gikhman 
Meda Care Transportation, Inc 
Wyoming OH 45215 
Home address: 6077 Ivy Woods Ct 
Mason OH 45040 
Ph. 513 573 9348 
Email: iljogik@yahoo.com 
 
 
Abstract. In this notice we are focus on drawbacks in the basis of the credit risk 
modeling. 
 
Keywords: Credit risk, credit derivatives, risk neutral world, risk neutral probability, 
structural model, reduced form. 
 
JEL Classification: A23, C6, G13, G32. 
 
The credit risk along with the credit derivatives is a modern area of the financial 
business. In recent years this field becomes the most successful innovation, which 
accumulates significant cash flows as well as the highest attention within financial 
community. In this notice we present some general comments concerning mathematical 
techniques used for the credit risk modeling. We will not discuss here specific forms of 
the credit derivatives.  
Unfortunately looking at the progress in this area one might note that it has been 
achieved using somewhat inaccurate modeling techniques. We highlight several 
drawbacks related to the benchmark models that in turn imply a problem of using 
correspondent software.  
Next example illustrates an error of the benchmark plain vanilla option price 
definition using a very simple algebra.  
Let S ( t ), t = 1, 2 be a security price at date t and K a strike price. Assume for 
simplicity that S(1) = K = $2 and there are two hypothetical securities value at the date t 
= 2 which is also an option maturity date 
 
$4   with probabilities 0.99 
S1 ( 2 , ω ) = {  
$1  with probabilities 0.01 
 
$4   with probabilities 0.01 
S2 ( 2 , ω ) = { 
$1  with probabilities 0.99 
   2 
The average return on the first security is equal to 98.5% and   48.5% on the second 
security. The volatility in both cases is the same 0.0891. One can easily to recall that the 
binomial scheme suggests the same call option price C = 2/3 for either security. Thus, for 
investors the binomial scheme suggests the option price that does not take into account 
the real expected return on underlying security. Indeed one can see that the call option on 
the first security in 99 cases out of 100 promises positive payoff and only in 1 case a loss. 
With the second security the situation is opposite. Nevertheless in both scenarios 
binomial scheme suggests the same price. We also note that there is a significant 
difference in expected option returns. The expected rate of return on the call option for 
the first security is about 197% = [ ( 4 – 2 )*0.99 – 2/3 ] / ( 2/3 ) and for the second 
security is   97%  =  [ ( 4 – 2 )*0.01 – 2/3 ] / ( 2/3 ) that explicitly demonstrates failing of 
the option pricing method. Indeed the present value of the future cash flows is different 
but their spot prices are equal. This observation contradicts the understanding of the price 
itself.  
Recall that one of the basic interpretations of the spot price is the present value of 
the future cash flow. In stochastic setting it means in particular that if the volatility of two 
instruments is equal and expected returns are different then the spot prices of the two 
instruments should be different otherwise the price is not correct. Sometimes using the 
arbitrage argument as a necessary condition of the fair pricing could be helpful vehicle to 
identify incorrect pricing. On the other hand an incorrect pricing does not automatically 
leads to the arbitrage opportunity. More details could be found in [2]. 
There is a discrete space time approach used for derivatives valuation. For the 0 
default options it is the binomial scheme. The above example is a simple one of the 
binomial scheme. In a case when the chance of default could not be assumed equal to 0 
the binomial approach was implemented by Jarrow and Turnbull, see for instance [7]. 
The core of the model is the risk neutral probabilities prescribed to the risky bond. Recall 
that risk neutral (martingale) probabilities were first developed for the option valuation in 
discrete space time setting. In the above example we demonstrated that the risk neutral 
option price construction does not have any sense it looks also reasonable to highlight its 
application to default studies. In the equity case presented above the security value S ( 1 ) 
=  2 is known at t = 1. The value of S ( 2 ) is  random with given probability distribution. 
Risk neutralization replaces real world probability distribution by so called martingale 
probabilities. This transformed instrument used then for derivatives valuation only. That 
is if an investor ask about the security price then it is S ( t ) , t = 1,2 supplied by the 
original probability distribution. In risky bond evaluation the mistaken approach was 
extended. The same risk neutralization framework was applied for risky bond valuation. 
Follow [7] let ν ( 1 , 2 ) denote the value at t = 1 a zero coupon bond issued by the firm. 
The only date when the risky bond might default is assumed to be the bond’s maturity  
t = 2. The value of the risky bond at t = 2 is a random variable. This variable admits two 
values 1, if no default and δ ∈ [ 0 , 1 ] if default occurred at t = 2. The value δ is known 
as the recovery rate. Having these two value at t = 2 and deterministic value at t = 1 one 
can easy to construct “risk neutral” probabilities. As it follows from the binomial scheme 
the real world probability distribution of the states { δ , 1 } does not have any effect on 
risk neutral probabilities, i.e. the risk neutral price at t = 1 is the same for either risky 
bonds   
   3 
 
 
        1    with probability  0.001 
ν 1 ( 1 , 2 )   =     { 
      δ    with probability  0.999 
 
1    with probability  0.999 
ν 2 ( 1 , 2 )   =     { 
      δ    with probability  0.001 
 
that in turn contradict the company ratings. It would be good if the rating companies have 
not used risk neutralization in their business. In this comment we somewhat simplified 
original reduced form approach by separating two different issues. The first one is the 
interpretation of the default in the simplest discrete time case and the second one is the 
interpretation of the default with the help of Poisson random process. In this case the 
probability [ 1   λ ( 1 )   ] is prescribed to the risk neutral probability of  the  
0 default scenario and  λ ( 1 )   if default occurred at t = 2. Here the time interval  
  = 2 – 1 = 1. These transformations make it possible to interpret the time of default as 
the time of the first jump of the Poisson process. Starting from this point the well 
developed mathematical techniques have been adjusted for the reduced form approach to 
the credit derivatives valuations.  
Now let us take a look at a credit derivatives structural model. The primary results 
were introduced in [4 6]. In these papers a new canonical distribution was used to 
describe losses of the large debt portfolio. This distribution plays now a key role in the 
structural models as well as for CreditMark product of the Moody’s KMV. In the New 
Basel Accord the regulatory capital of a Bank should be calculated using Internal Rating 
Based method. Under this method the canonical distribution must be applied for 
regulatory capital calculations. 
Follow [4] we assume that Wiener Z i  processes are equally correlated and 
 
E [   Z i  ( t ) ] ²  =   t  ,    E    Z i  ( t )   Z j  ( t )  =  ρ   t  ,      i  ≠  j 
 
where   Z i  ( t )  = Z i  ( t  +   t )     Z i  ( t ), i = 1,2,…n.  
  
Statement 1. [4]. The Wiener processes Z i ( t ) admit representation 
 
 
where x  ( t ) and ε i  ( t ), i = 1,2,…n are mutually independent Wiener processes. The 
idea of the proof is presented bellow follows Vasicek’s private communication.  
Let U ( t ) be a Wiener process independent on the given Wiener processes  
Z i ( t ), i = 1,2,…n.  Putting 
 
) 1 ( t ε ρ   1     )  t  (  x   ρ         )  t  (    i   Z i  ) (     + =
) ρ ) ( ) ( (
ρ 1
1
) ( ε , ) ( ) ( ) ( i
1
t x t Z t t U b t Z a t x i
n
i
i −
−
= + = ∑
=  4 
 
where 
 
 
one can check that x ( t ) and ε i ( t ), i = 1,2,…n are independent Wiener processes. 
There are several pitfalls here that we wish highlight bellow.  
Remark. It is clear that the proof of the statement 2 should be refined. Indeed the 
decomposition used in the statement 2 is correct when and only when the assumption 
regarding the Wiener process U ( t ) is true. If the Wiener process U ( t ) did not exist 
then stated decomposition for the given system { Z i ( t ) } fails. In [6] it is stated that (1) 
is not an assumption but a property of the equicorrelated normal distributed system. This 
statement does not correct and does not correspond to the real situation. Also note that all 
terms in (1) are depend on the number n by the construction.  
Thus the main result [4,5] regarding existence of the limit distribution of the large loan 
portfolio is subject to the assumption that for any system of Wiener processes  
Z  = { Z i ( t ), i = 1,2,…n }( n is an arbitrary number) there exist a Wiener process  
U( t ) independent on the system Z. Here the Wiener process U( t ) might depend or 
independent on n.  
A new approach that at some degree is close to the structural model was 
introduced in [3]. In this paper the presentation (1) is used indirectly in order to describe 
joint defaults on underlying loans. Thus the problem that related to the construction of the 
process U ( t ) does not exist in such model.  
We comment that construction. Let M and Z i , i = 1, 2, …, n be independent 
random variables with mean 0 and variance 1. Define random variables X i  , i = 1,2,…,n 
with a help of equality  
 
where the constants a i satisfy a condition: | a i |  <  1. Let t i be the time of default of a i th 
obligator and Q i is the cumulative distribution function ( cdf ) of the random time t i  and 
H i ( x ) is the cdf of Z i . Then 
 
 
Let F i ( x ) denotes the cdf of the random variable X i . Define mappings 
 
  
ρ ) 1 ( 1
ρ 1
and
ρ ) 1 ( 1
ρ
− +
−
=
− +
=
n
b
n
a
i
2
i i i Z a 1 M a X − + =
)
a 1
M a x
( H | }
a 1
m a x
Z { P
) 2 (
} M | x Z a 1 M a { P } M | x X { P
2
i
i
i M m 2
i
i
i
i
2
i i i
−
−
=
−
−
< =
= < − + = <
=
) 3 ( ) ) x ( F ( Q t , ) ) t ( Q ( F x
1
i i
1
i
− − = =  5 
Hence x = x ( t ) and t = t ( x ). Then from (3) it follows   
Conclusion [3]. Conditionals on M defaults are independent. Indeed 
 
 
This conclusion would follow from the fact that H i are cumulative distribution functions 
that independent on variables Z i . 
One could see that the link between (3) and (4) does not mathematically accurate. Indeed 
the equality (3) deals with unconditional probabilities and left hand side (4) relates to the 
conditional probabilities. It is not difficult to present an example that shows that from 
equality unconditional expected values of two random processes does not follows the 
equality their conditional expected values. Indeed the Wiener processes w ( t ) and the 
constant 0 have the same expectations equal to the 0 nevertheless the unconditional 
expectation of and E { w ( t ) | Ft
w } = w ( t ) ≠ 0 where Ft
w = σ { w ( s ) , s ≤ t }. Note 
also that by the definition the random variables t i and M do not have any relationship 
between them. Therefore the derivation [3] leading to the equality (4) is correct for either 
when the random variables X i and t i are independent or when they are dependent. Hence 
it follows that the construction introduced in [3] rather incorrect.  
 
  There is another approach to the credit risk modeling referred to as to the reduced 
form of the credit risk [1]. This approach deals with the statistical data regarding the 
default events ignoring structural parameters such as firm’s equity and debt costs.  
Statistical data is used to help providing analysts by probabilities of credit events. In a 
simple setting the default time is associated here with the first jump of the counting 
process N ( t ). The Poisson process or its generalization can be considered as a model 
example of the Process N ( t ). If τ  =  τ ( ω ) is a random time associated with a default 
event then reduced form interprets it as  
 
τ  =  min { t ≥ 0 :  N ( t ) = 1 } 
  
The hazard rate λ ( t ) is defined with a help of conditional probability 
 
P { τ  ≤  t  +    t | τ  > t }  =  λ ( t )   t  + o (  t )      (5)  
 
With the reduced form models we can make two comments regarding to approach 
rather than to the problems solution. The first comment is related to the notorious risk 
neutral world and risk neutral probabilities. As it was shown above the Black Schloes 
option price definition is incorrect and as a conclusion the risk neutral interpretation is 
irrelevant for the derivative pricing. Therefore the reduced form approach interprets the 
credit risk in the risk neutral environment and therefore is inappropriate. In addition recall 
that the risk neutral setting has been used for the modeling of the stochastic short interest 
rates.   
The second comment relates to the drawback of the construction of the doubly 
stochastic hazard rates that is the essence of the reduced form models. In these models the 
stochastic intensity λ ( t )  is assumed to be in the form 
) 4 ( }
1
) ) ( (
{ } | { ) | (
2
1
i
i i i
i i i
a
M a t Q F
H M t t P M t Q
−
−
= < =
−  6 
 
 
λ ( t )  =  Λ ( X ( t ) ) , 
(6) 
dX ( t )  =    ( X ( t ) ) dt  +  σ ( X ( t ) ) dw ( t )       
 
where Λ ( x ) is a nonrandom appropriate function. Our remark relates to the informal 
substitution of the random process X ( t ) at the right hand side (5). Indeed it is clear that 
it makes sense to consider first the case when the function X( t ) is equal to the constant 
scalar or vector x. It implies in turn that the default time τ ( ω ) is also a function on x, 
Thus on the first step of the modeling one should introduce the parameter x and explain 
the way how it effects on default time. On the next step we need to justify the substitution 
x = X ( t ) into both sides of the equality (5). On the right hand side of (5) this 
substitution is correct when for example the function λ ( x ) is a Borel measurable 
function with respect to x (i.e. B  measurable). On the left hand side of the equality (5) 
the corresponding substitution would be correct if the conditional expectation on the left 
hand side of (5) is measurable with respect to σ algebra σ { X ( t ) } x B. This comment 
shows that setting of the model should be done more accurate. Besides that it looks 
possible that some additional assumptions on the counting process N( t ) might be taking.   
 
 
 
 
 
 
 
 
 
References. 
 
1.  Duffie, D. Singleton, K. (2002). Credit Risk: Pricing, Measurement, and 
Management. NJ, Princeton University Press. 
2.   Gikhman, Il.I.(2006). http://papers.ssrn.com/sol3/papers.cfm?abstract_id=900111. 
3.   Hull, J. , White, A. (2005). The Perfect Copula.  
4.   Vasicek, O.A. (1987). Probability of loss on loan portfolio, Working Paper, KMV 
     Corporation. 
5.   Vasicek, O.A. (1991). Limiting Loan Loss Distribution, Working Paper, KMV 
     Corporation. 
6    Vasicek, O.A. (2002). The distribution of loan portfolio value. Risk. 
7.   Jarrow, R. , Turnbull, S. Derivatives Securities, 2nd ed, South Western College 
      Publishing, 2000. 
 

Some critical comments on credit risk modeling.

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics