FREE ESSAY ON A RISK NEUTRAL FRAMEWORK FOR THE PRICING OF CREDIT DERIVATIVES

A RISK NEUTRAL FRAMEWORK FOR THE PRICING OF CREDIT DERIVATIVES

1. INTRODUCTION
Considerable research effort has gone into Credit Derivatives since the early 1990's. The
roots of credit derivatives can be traced back to the notion that the credit risk of a
firm can be captured by the credit rating ascribed to it. This premise is also the
cornerstone of loan pricing and credit risk management models the world over, including
J.P. Morgan's CreditMetricsTM. 
Empirical research enables the predictability of the event of default as well as the Loss
in the Event of Default (LIED). This information is expressed in terms of a 'transition
matrix' - a matrix that traces out the probabilities the migration of a firm's credit
rating. Rating agencies such as Standard & Poor (S&P) provide transition matrices
computed from periods of data about bonds - default record and post-default behaviour in
the US markets. Lack of adequate data precludes the computation of such matrices in the
Indian context, although it is possible to map ratings of Indian rating agencies such as
CRISIL onto S&P ratings.
2. TYPES OF CREDIT DERIVATIVES
Here is a brief description of some popular types of credit derivatives:
2.1 Credit Default Swaps
A credit default swap provides a hedge against default on some payment, such as a bond.
The counterparty buying credit protection pays the provider a certain amount in return
for a guarantee to make good the loss in the event of default.
2.2 Total Return Swaps
In this contract, the 'payer' gives a 'receiver' the total return on an asset in return
for the returns on a benchmark asset, typically a risk-free asset. The payer has thus
eliminated the risk of default in return for a lower but certain risk-free rate of
return.
2.3 Credit Spread Derivatives
Credit spread derivatives take the form of credit spread options, forwards or swaps. A
credit spread call option, for example, is a call option written on the level of the
spreads for a given bond. The option, thus increases in value as the spread increases, so
that the value of the bond is protected.
3. RISK-NEUTRALITY
Hypothesising the existence of a 'risk-neutral' world is extremely useful in the pricing
of instruments whose value is derived from a stochastic process. In the real world, the
present price is less than the expected net present value of the likely outcomes in
future. Thus, for example, if the price of a commodity can become either Rs. 100 or Rs.
200 in the next period, its value today (ignoring time value of money) will be less than
Rs. 150 (say it is Rs. 140). This is because the uncertainty associated with future
values tends to depress current values. This follows from a risk-averse mindset of the
real world. However, this poses severe problems in the pricing and valuation of such
instruments, because the time-tested expected value criterion fails to hold good.
It therefore becomes necessary to risk-neutral values of the underlying factor that
causes the uncertainty in prices, for instance, interest rate in case of bond prices.
These risk-neutral values can be used in the expected present value context, because the
probabilities of the possible outcomes (Rs. 100 and Rs. 200 in our above example) are
suitably adjusted so as to yield the observed prices. Thus, the risk-neutral
probabilities that the price in the next period would be Rs. 100 and Rs. 200 would be 0.6
and 0.4 respectively so as to yield a current price of Rs. 140.
4. RISK-NEUTRAL PROBABILITIES OF CREDIT RATING MIGRATION
4.1 Derivation of 1-period risk neutral transition matrix
For the purpose of illustration, let us say there are four credit ratings - A, B, C and D
(default). It is also assumed that the empirically observed one-period transition matrix
is as shown in Table -1 and that the amount recovered in case of default is 40% of the
face value of the bond. Further, the zero yields for the first three periods are assumed
to be 15%, 14.35% and 13.58% respectively.
Table-1
Empirical Transition Matrix
Rating after 1 period
Current rating A B C D
A 0.80 0.12 0.06 0.02
B 0.10 0.75 0.10 0.05
C 0.05 0.10 0.75 0.10
D 0.00 0.00 0.00 1.00
In order to value a credit derivative, we need risk-neutral transition probabilities
which may be obtained from the prevailing bond prices (refer Table-2). Let p(M)ij be the
risk neutral probability of transition from rating i to rating j over M periods. We shall
show how p(1)Aj are to be calculated.
If the price of a 1-period zero-coupon bond with a rating A is V(1)A , the following
relationship should hold:
(1)
where r01 = 15% ( the current one period rate)
Table-2
Prices of Zero coupon Bonds
Maturity? Rating? 1 2 3 4
A 85.86 74.10 64.59 55.32
B 84.09 71.32 61.30 51.91
C 80.96 66.76 56.06 46.55
In order to estimate p(1)Aj we assume that the risk-neutral probabilities of transition
from state A to other states are proportional to empirically observed probabilities. 
p(1)AB = p(1)A x 0.12 p(1)AC = p(1)A x 0.06 
p(1)AD = p(1)A x 0.02 p(1)AA = 1 - p(1)A x 0.2 (2)
Equations (1) and (2) are used to obtain p(1)A and hence p(1)Aj
(3) 
From eqn.(3), we get p(1)A = 1.05 and hence the risk neutral probabilities p1Aj are 
p(1)AA = 0.79 p(1)AB = 0.126 p(1)AC = 0.063 p(1)AD = 0.021
The same methodology may be used to obtain p(1)Bj and p(1)Cj and thus the risk neutral
transition matrix for 1 period is obtained (refer Table-3).
Table-3
Risk neutral probabilities of transition over 1 period
Rating after 1 period
Current Rating A B C D
A 0.790 0.126 0.063 0.021
B 0.110 0.725 0.110 0.055
C 0.058 0.115 0.713 0.115
D 0.000 0.000 0.000 1.000
4.2 Derivation of M-period risk-neutral transition matrix
The M-period risk-neutral transition matrix requires the M-period real-world transition
matrix. If we assume that rating migration in any period is independent of the previous
migrations, the M-period real-world transition matrix is given by 
{T(M)ij } = {T(1)ij }M (4)
Tables 4a-c show the real-world transition matrices for 2, 3 and 4 periods respectively.

Table-4a
Real world probabilities of transition over 2 periods
Rating after 2 periods
Current Rating A B C D
A 0.655 0.192 0.105 0.048
B 0.160 0.585 0.156 0.100
C 0.088 0.156 0.576 0.181
D 0.000 0.000 0.000 1.000
Table-4b
Real world probabilities of transition over 3 periods
Rating after 3 periods
Current Rating A B C D
A 0.548 0.233 0.137 0.081
B 0.194 0.473 0.185 0.148
C 0.114 0.185 0.452 0.248
D 0.000 0.000 0.000 1.000
Table-4c
Real world probabilities of transition over 4 periods
Rating after 4 periods
Current Rating A B C D
A 0.469 0.254 0.159 0.118
B 0.212 0.397 0.198 0.194
C 0.133 0.198 0.365 0.305
D 0.000 0.000 0.000 1.000
In order to obtain the 2-period risk-neutral transition probabilities, p(2)Aj , we
require the price V(2)A of a two-period zero coupon bond currently rated A. The expected
pay-off in the risk neutral framework is p(2)AA x100 + p(2)AB x100 + p(2)AC x100 + p(2)AD
x40. Thus, the following relationship must hold :
V(2)A = (5)
Since the expected pay-off is calculated using risk-neutral probabilities, it is free
from credit risk. Further, since the pay-off is independent of the interest rate, using
the 2-period risk free rate for discounting is justified.
As in the 1-period case, we assume that the risk-neutral probabilities of transition from
state A to other states are proportional to real-world probabilities. The R.H.S of
Eqn.(5) reduces to a function of p(2)A. Thus, the risk neutral transition probabilities
p(2)Aj can be obtained. The same methodology may be used to obtain p(2)Bj and p(2)Cj and
thus the 2-period risk neutral transition matrix is obtained. Tables 5a-c show the 2,3
and 4-period risk neutral transition matrices respectively.
Note that the M-period risk-neutral transition matrix is not the Mth power of the
1-period matrix. 
Table-5a
Risk neutral probabilities of transition over 2 periods
Rating after 2 periods
Current Rating A B C D
A 0.627 0.207 0.113 0.052
B 0.181 0.530 0.176 0.112
C 0.102 0.183 0.503 0.212
D 0.000 0.000 0.000 1.000
Table-5b
Risk neutral probabilities of transition over 3 periods
Rating after 3 periods
Current Rating A B C D
A 0.503 0.256 0.151 0.089
B 0.223 0.394 0.213 0.170
C 0.137 0.222 0.343 0.298
D 0.000 0.000 0.000 1.000
Table-5c
Risk neutral probabilities of transition over 4 periods
Rating after 4 periods
Current Rating A B C D
A 0.405 0.285 0.178 0.132
B 0.248 0.294 0.231 0.226
C 0.163 0.243 0.219 0.375
D 0.000 0.000 0.000 1.000
Now that the risk neutral transition matrices have been calculated, any credit derivative
may be priced.
5. VALUATION OF CREDIT DERIVATIVES
The present model does not consider the credit risk associated with seller of the
derivative product. Further, it does not consider the correlation between interest rate
changes and credit rating migrations. In the following sections, we illustrate the
pricing of two credit derivative products. 
5.1 A simple derivative
Here again, the valuation of the credit derivative requires only the 2-period risk free
zero rate because the pay-off from the derivative is adjusted for credit risk and is
independent of the interest rate. 
5.2 A multi-period derivative
Consider the bond 'XYZ' again. Now the derivative pays Rs. 100 if the credit rating of
the bond changes to C during period 2 or period 3. This derivative can be decomposed into
the simple derivative described above and another derivative that pays Rs. 100 in the
third period if the rating of the bond is C in the third period but not in the second.
For the valuation of this derivative, let us define events Ei and F as under
Ej : The rating of the bond changes to i at the end of 2 periods (where i = A or B)
F : The rating of the bond changes to C at the end of 3 periods
The probability that the derivative yields Rs. 100 in the third period is given by
where i = A or B (6)
The probabilities of transition from period 2 to period 3 are obtained as:
{RN}02 {RN}23 = {RN}03
Where {RN}ij is the risk-neutral transition matrix from period i to period j
Thus, {RN}23 = {RN}-102 {RN}03 
Table -6 shows the risk neutral probabilities of transition from period 2 to period 3.
From this table, it can be seen that P (F / EA ) = 0.074 and P (F / EB ) = 0.176. In
addition, we know that P(EA) = 0.181 and P(EB) = 0.530 (refer Table -5a). Thus, the
risk-neutral probability that Rs. 100 is received in period 3 is 0.074 x 0.181 + 0.176 x
0.530 = 0.107
The value of the derivative is obtained as 
20.76 
Table-6
Risk neutral probabilities of transition from period 2 to period 3
Rating after 3 periods
Rating after 2 periods A B C D
A 0.742 0.169 0.074 0.015
B 0.145 0.625 0.176 0.053
C 0.069 0.180 0.603 0.148
D 0.000 0.000 0.000 1.000
6. VALUATION OF A BOND
6.1 A Plain Bond
The risk-neutral transition matrices developed for the valuation of a credit derivative
can be used to value a bond. Let us consider a 3-period zero-coupon bond rated B.
The pay-offs from the bond are the same as that from the following portfolio:
1. A credit risk free zero-coupon bond with a maturity of 3 periods.
2. A credit derivative which requires payment of Rs. 60 if the rating of a bond rated B
currently touches D anytime during the first three periods. 
The value of the credit-risk free bond is 
100 / (1+ 0.1358)3 = 68.25
The value of the credit derivative calculated as in the previous section turns out to be
Rs. 6.96. This amount is arrived as follows:
RN(3)BD * 60 / (1 + r03 )3 = 6.96
Thus, the value of the bond is obtained as the difference between the credit risk free
bond and the credit derivative, i.e. Rs. 68.25 - Rs. 6.96 = Rs. 61.29. In this case
again, the interest rate tree is not required for the valuation of the bond. This gives a
powerful framework to split up the loan price into its building blocks - a time value of
money (represented by the risk-free rate), and a risk premium (for bearing credit risk).
6.2 Coupon Bond with Prepayment Option
So far, interest rate risk has not been relevant to the valuation of any of the loan or
derivative products. However, in case of a loan with an embedded prepayment option, the
interest rate volatility needs to be considered. It is here that our model departs from
presently available models in giving a framework to price prepayment options embedded in
loans. 
To understand the justification for using the interest rate tree, let us examine the
drivers that lead the borrower to prepay the loan:
? Fall in interest rates: With a fall in interest rates, a borrower can obtain cheaper
funding in the market, and hence would be motivated to prepay the current (fixed rate)
loan.
? Improvement in credit rating: As spreads narrow with enhancement in credit rating,
borrowers with rating upgrades are driven to opt for the prepayment because they too can
obtain refinance at cheaper rates now.
However, the decision to prepay is not independently determined by interest rate decline
and credit rating upgradation. In some situations, the changes in interest rate and in
credit ratings could exert opposite influences on the decision to prepay. Hence, we need
to consider the simultaneous impact of changes in interest rates and credit ratings on
the valuation of the bond. 
We shall value the loan (& the embedded option) using a backward recursive method for
computing the expected present value of the loan in a risk-neutral world. In order to
move into a risk-neutral framework with respect to interest rates, we have used a
recombining binomial interest rate tree (refer Chart-1). 
Chart-1
The Risk-Neutral Interest Rate Tree
Consider a 12% coupon bond rated B with a maturity of 3-periods. Let us suppose the
issuer has the option to prepay starting from the first period. The prepayment amount is
fixed at Rs. 98. The steps in the valuation of the bond and the embedded option are: 
Step-1: List all possible interest rate paths from period 0 to period 3. All these paths
are equally likely.
Period
Path 0-1 1-2 2-3
1 15.0% 16.0% 14.4%
2 15.0% 16.0% 12.0%
3 15.0% 11.5% 12.0%
4 15.0% 11.5% 10.0%
Step-2: For each interest rate path (the following calculation illustrates the valuation
procedure for the second interest rate path, i.e. 15.0%, 16.0% and 12.0%), obtain the
value of the bond as described under:
? Let Vij be the value of the bond in the ith period if its rating is j. At maturity, the
bond's value is Rs. 112 if its rating is A, B or C and Rs. 40 if the rating is D.
? For valuing the bond in period 2, we use the risk-neutral transition matrix from
period-2 to period-3. So, V2A is obtained as 
(7)
Here 12.0% is the interest rate for period 3 in the first interest rate path. 
? Similarly other V2j are obtained. Since there is a prepayment option we compare the
value of the bond with the prepayment amount Rs. 98. If the value of the bond V2j is
greater than the prepayment amount, the issuer is likely to prepay. 
? V2A, V2B and V2C are 99.03, 96.67 and 90.42 respectively. Since V2A is greater than Rs.
98 (the prepayment amount), the issuer will exercise the prepayment option and the value
of the bond will become Rs. 98. Since default is an absorbing state, the risk-neutral
transition matrix indicates that the probability of migrating to any other state from a
state of default is zero. Hence, the value of the bond at the 'default node' is simply
the present value of Rs. 40 received at the end of the original maturity period, the
interest rate tree providing the discount rates.
? At this point, the coupon payment will be received, and the value of the bond will be
incremented by Rs. 12, except in case of default, where no cash flow occurs.
? V2A, V2B V2C and V2D can then be used to compute V1A, V1B V1C and V1D in a similar
fashion, except that the interest rate applicable will be r12 along the chosen interest
rate path.
(8)
? This procedure is repeated till the current period is reached. At this point, the
credit rating applicable is known (B in the present illustration), and thus the bond
value can be ascertained.
A schematic representation of the backward recursion of the bond cash flows is shown
below for the second interest rate path, i.e. r01 = 15%, r12 = 16%, and r23 = 12%,
yielding the bond value (net of the value of the embedded prepayment option) to be Rs.
84.47.
Strike price / Prepayment amount: 98 
Period 0 1 2 3
Rating 
A 104.83 110.00 112
B 84.47 101.47 108.67 112
C 94.10 102.42 112
D 30.79 35.71 40
The value at each node includes the coupon received at that node.
Step-3: Since we are operating in a risk neutral framework and each of the interest rate
paths is equally likely, the value of the bond is the simple average of values obtained
for each of the interest rate paths. 
Step-4: A similar exercise can be carried out to price a plain bond, i.e. where there are
no embedded options. In this case, at each node, there would be no need to compare the
amount arrived at by the backward recursion with any other amount, because there is no
option available to exercise.
Step-5: The difference between the two values (Step-3 and Step-4) represents the value of
the embedded option. Table-7 contains an analysis of the price of the bond with a
prepayment option into the price of a plain bond, and that of a prepayment option.
Table-7
Components of the price of a bond with an embedded prepayment option
r01 r12 r23 Bond value with embedded option Value of plain bond Value of embedded
prepayment option
15.0% 16.0% 14.4% 83.21 83.21 0.00
15.0% 16.0% 12.0% 84.47 84.61 0.14
15.0% 11.5% 12.0% 87.48 87.63 0.15
15.0% 11.5% 10.0% 88.31 88.89 0.58
Price of instrument 85.87 86.09 0.22
7. CONCLUSION
A loan or a bond can be viewed as a portfolio of a risk-free instrument and a credit
derivative that pays an amount equal to the loss in the event of default. This gives a
powerful framework to split up the loan price into its building blocks - a time value of
money (represented by the risk-free rate), and a risk premium (for bearing credit risk).
Ascertaining the risk-neutral transition matrices is critical in this framework, where
the payoff depends on the credit rating of a certain party. These transition matrices can
be obtained from empirically observed or 'real world' transition matrices, observed bond
prices and observed interest rates.
However, when it comes to the pricing of bonds with embedded prepayment options, the
demarcation between credit rating upgradation as a trigger for the exercise of the option
and interest rate decline as the trigger for the exercise of the option is not very
clear. In this scenario, the best way to combine interest rate and credit risk is to use
both, the risk neutral transition matrix, and the risk neutral interest rate tree.
The valuation procedure involves backward recursion of the bond cash flows (including the
embedded option) starting out with the terminal period, separately for each interest rate
path. The price of the bond is the mean of the values arrived at in each path. It is
noteworthy that the mean can be used only in a risk-neutral setting. The option value is
the difference between the bond value (alongwith the embedded option) and the value of a
plain bond.
Bibliography
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Creating Value from Both loan structure and price
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3. Das, Sanjiv R.; An Overview of Credit Derivatives, Harvard Business Review, July 1997
4. Belkin Barry; Dr. Forest, Lawrence Jr. and Dr. Suchower, Stephen, Measures of Credit
Risk and Loan Value in KPMG's LAS, Financial Services Consulting -Risk Solutions KPMG
Peat Marwick LLP.