Abstract. Regulations of the market require disclosure of information about the nature and extent of risks arising from the trades of the market instruments. There are several significant drawbacks in fixed income pricing modeling. In this paper we interpret a corporate bond price as a random variable. In this case the spot price does not a complete characteristic of the price. The price should be specified by the spot price as well as its value of market risk. This interpretation is similar to a random variable in Probability Theory where an estimate of the random variable completely defined by its cumulative distribution function. The buyer market risk is the value of the chance that the spot price is higher than it is implied by the market scenarios. First we quantify credit risk of the corporate bonds and then consider marked-to-market pricing adjustment to bond price. In the case when issuer of the corporate bond is the counterparty of the bond buyer counterparty and credit risks are coincide.

1. 1
BOND PRICING AND CVA.
Ilya I. Gikhman
6077 Ivy Woods Court,
Mason, OH 45040, USA
ph. 513-573-9348
email: ilyagikhman@mail.ru
JEL : G12, G13
Key words. Bond, mark-to-market, counterparty risk, CVA.
Abstract. Regulations of the market require disclosure of information about the nature and extent of risks
arising from the trades of the market instruments. There are several significant drawbacks in fixed income
pricing modeling. In this paper we interpret a corporate bond price as a random variable. In this case the
spot price does not a complete characteristic of the price. The price should be specified by the spot price
as well as its value of market risk. This interpretation is similar to a random variable in Probability
Theory where an estimate of the random variable completely defined by its cumulative distribution
function. The buyer market risk is the value of the chance that the spot price is higher than it is implied by
the market scenarios. First we quantify credit risk of the corporate bonds and then consider marked-to-
market pricing adjustment to bond price. In the case when issuer of the corporate bond is the counterparty
of the bond buyer counterparty and credit risks are coincide.
Default at Maturity. Let us briefly recall the reduced form of risky bond valuation. Denote R ( t , T )
and B ( t , T ) prices of the risky and risk free bonds correspondingly at date t with expiration dates at T.
Both bonds promise $1 at the date T. The difference
B ( t , T ) – R ( t , T )  0
is known as credit spread. It is a characteristic of a chance of default of the corporate bond. We illustrate
default pricing by using an example [ me] in which default of the corporate bond may occurred only at
expiration date. Our interpretation differs from the original interpretation [1] where only a single value
was admitted at the default moment. Suppose that risky bond might default only at expiration date T and
R ( T , T ,  ) =  j , where D j = {  : R ( t , T ,  ) =  j } , P { D j } = p j
where 0 ≤  n <  n – 1 < … <  0 = 1, 
n
1j
p j = 1. Default distribution p j and recovery rate δ j are
assumed to be known. Default values imply the formula

2. 2
R ( t , T ; ω ) = B ( t , T ) 
n
1j
 j χ ( D j ) (1)
Formula (1) shows that market price of the risky bond at t is a random variable. The spot price of the bond
R spot ( t , T ) is the price which is used for buying or selling bonds at the market at t. The spot price is
established as a settlement between buyers and sellers by weighting their estimates of possible profits and
losses based on (1). Applying spot price buyers and sellers are subject of the market risk. Buyer market
risk is measured by the probability
P { R ( t , T ; ω ) < R spot ( t , T ) }
This probability represents the value of a chance that spot price is higher than it would be realized by
market scenarios. Average and dispersion of the losses
m b risk = E R ( t , T ; ω ) χ { R ( t , T ; ω ) < R spot ( t , T ) }
D b risk = E R 2
( t , T ; ω ) χ { R ( t , T ; ω ) < R spot ( t , T ) } – m 2
riskb
are primary characteristics of the buyer credit risk at t. Similarly market risk of the bond seller is equal to
P { R ( t , T ; ω ) > R spot ( t , T ) }
and variables
m s risk = E R ( t , T ; ω ) χ { R ( t , T ; ω ) > R spot ( t , T ) }
D s risk = E R 2
( t , T ; ω ) χ { R ( t , T ; ω ) > R spot ( t , T ) } – m 2
risks
define market risk of the bond seller. Note later variables specify market risk as far as buyer of the bond
does not responsible for the bond default. The later formulas specify the chance that seller of the bond
receives less than it is implied by the market. In theory, given market and spot prices we enable to
estimate market risks, expected returns, and possible market deviations. Suppose that spot price is defined
by the formula
R spot ( t , T ) = E R ( t , T ; ω )
Then
R spot ( t , T ) = B ( t , T ) 
n
1j
 j p j , E R 2
( t , T ; ω ) = B 2
( t , T ) 
n
1j
 2
j p 2
j
D R ( t , T ; ω ) = E R 2
( t , T ; ω ) – E [ R spot ( t , T ) ] 2
Randomness of the date-t market price of the corporate bond which is stipulated by a chance of default
represents credit risk of the contract.
Remark. Let us recall the benchmark risk neutral valuation following by [1]. Let A ( t ) denote value of
the money market account (MMA) which is constructed as following.

3. 3
By definition we put A ( 0 ) = $1 and next by induction define A ( t + 1 ) = A ( t ) R , t = 0, 1, …
where R denotes risk free interest rate over a single period. R – 1
is used as risk-free discount factor in
binomial scheme option valuation. In a single period economy t = 0, T = 1 we assume that underlying
security admits two values S d < S u at maturity t + 1 and
S d < R S ( t ) < S u (2)
Condition (2) guarantees no arbitrage opportunity. For given R there exists unique weights ( 1 –  ) , 
which depends on t such that
E π
S ( t + 1 )
def
 ( 1 –  ) S d +  S u = R S ( t ) =
)t(A
)1t(A 
S ( t )
Then
E 
{
)1t(A
)1t(S


| F t } =
)t(A
)t(S
(3)
Here F t is the - algebra generated by values of the stock up to the moment t and weights ( 1 –  ) , 
are called unique equivalent martingale probabilities [1]. This equality shows that random process
S ( t ) A – 1
( t ) is a martingale with respect to risk neutral measure { ( 1 -  ) ,  }.
Equality (3) can be also represented in the form
E 
)t(S
)t(S)1t(S 
=
)t(A
)t(A)1t(A 
It is proved that equality (3) follows from (2) for the case when S ( T ) takes two values. If S ( T ) takes
arbitrary finite number of states the existence of the unique equivalent martingale probabilities did not be
proved. If uniqueness of the risk neutral probabilities does not exist then the two state stock problem can
be considered as degenerating case.
In the latter construction investor buying a stochastic stock is effected by the market risk. This risk is
represented by the lower price S d at maturity in no arbitrage price setting. Therefore no arbitrage price
does not eliminate the market risk. The construction of the unique equivalent martingale probabilities can
be briefly formulated as following.
Statement. Given stock price with two states at T which is defined on original probability space there
exists a new measure π such that the stock expected rate of return is equal to risk free interest rate.
Actually risk neutral distribution construction does not represent a new pricing model of pricing. It is used
for Black-Scholes option pricing to reveal a connection between risk neutral Black Scholes pricing
underlying and real security used for the option pricing.
The binomial scheme can be applied for a corporate bond. Assume that risky bond at maturity is
defined by equality (1). Let n = 0, 1 and bond takes only two values1 and δ  [ 0 , 1 ). Let us for
simplicity default may occur only at maturity. The spot price of the risky bond is assumed to be defined
by unique equivalent martingale probabilities (3). Then
E 
{
)T(A
)ω;T,T(R
| F t } =
)t(A
)ω,T,t(R
(3)

4. 4
Here randomness of the R ( t , T , ω ) is stipulated by the randomness of the face value at maturity
T = t + 1. Thus date-t price of the risky bond is always a random variable. If buyer purchases bond for a
spot price R spot ( t , T ) then
δ ≤ A R spot ( t , T ) ≤ 1
If we calculate risk neutral probabilities we arrive at risk neutral distribution  and 1 –  .
Assume that the random pricing process R ( t , T ; ω ) is constructed for any t , t  [ 0 , T ]. Let
us estimate recovery rate (RR) and correspondent default probability (DP). By definition it follows that
)T,t(B
)ω,T,t(R
1  = ( 1 –  ) χ ( ω , D ) (4)
Hence, the relative value of the corporate bond with respect to its risk free counterpart at t can be
presented as
1 , for ω ≠ D
δ = δ ( ω ) = {
)T,t(B
)ω,T,t(R
, for ω  D
Formulas
P { δ ( ω ) < u } = P {
)T,t(B
)ω,T,t(R
< u } , u  [ 0 , 1 )
(5)
P ( D ) = 1 – P {
)T,t(B
)ω,T,t(R
= 1 }
represent date-t stochastic recovery rate δ ( ω ). For zero default scenario price of the corporate bond
coincides with the risk free bond price for any t ≤ T. The distribution of the random variable
δ ( ω ) = δ ( t , T ; ω ) defined by the formula (5) is the distribution of the recovery rate which depends
on parameters t and T. If the function B ( t , T ) is a function of the time to maturity T – t then risky bond
price would be represented by a random function of the variable T – t.
Remark. It is popular in applications to assume a reduction of the RR in which δ is a given number equal
to for example 30%, 40% or other known fraction. Such assumption is extended on a set of corporate
bonds. From formula (4) one can easy find that given δ the probability of default PD is equal
P ( D ) = ( 1 -  ) – 1
E [
)T,t(B
)ω,T,t(R
1  ]
The known δ helps to overcome the problem of solving one equation with two unknowns RR and DP. By
taking expectation from both sides of the equation (1) we arrive at the primary equation of the reduced
form of default theory. The assumption of known recovery rate looks over simplified and correspondent
value of the default probability does not represent a sufficiently reliable estimate.
A popular statistical pointwise estimate of a random variable is the expected value. The pointwise
estimate should be replaced by a confidence interval when one takes into account deviation or volatility of
the random variable does not very small. Default at maturity T is by definition associated with the event

5. 5
D T = { ω : R ( T , T ,  ) < 1 }
Introduce a random function
θ ( t , T ; ω ) = 1 –
)T,t(B
)ω,T,t(R
(6)
This function represents estimate of the loss given default, LGD. Using function θ ( t , T ; ω ) we note that
P { δ < x } = P {
)T,t(B
)ω,T,t(R
< x } (6′)
Cumulative distribution function (6′) performs complete information about RR and PD. The use of a
nonrandom constant δ as a point estimate of the RR implies market risk. From buyer perspective it is
measured by probability that the realized at default recovery rate would be bellow than the chosen
constant δ. This risk represents additional losses for the buyer. At the same time buyer’s risk implies
profit for the bond seller. Assuming that recovery rate δ is an unknown nonrandom constant < δ > it
follows from equality (4) that
E θ ( t , T ; ω ) = ( 1 – < δ > ) P ( D )
E θ ² ( t , T ; ω ) = ( 1 – < δ > ) ² P ( D )
This is a system of two equations for two unknowns < δ >, P ( D ). Solving system for RR and PD we
arrive at the values
)ω;T,t(θE
])T,t(θE[
)D(P
)7(
)ω;T,t(θE
)ω;T,t(θE
1δ
2
2
2


Here RR and PD depend on parameters t , T.
Randomization. Our goal is a construction of the random price of the corporate bond. This
randomization is stipulated by the credit risk of the bond. Recall that only historical data R spot ( t , T ) at
each moment t is available. One can use historical data to construct stochastic prices function
R ( t , T ,  ). It is common practice to use close prices as a date-t asset price. Such reduction can be
accurate when the difference max - min of the date-t asset prices is sufficiently small. Formulas (4) – (7)
present a solution of the default problem when distribution R ( T , T ,  ) is known and default time
 = T. In practice, distributions of the default time and prices of the bond at default are unknown.
Suppose for simplicity that default may occur only at maturity date T and distribution R ( T , T ,  ) is
unknown. Formula (6) uses stochastic price R ( t , T ; ω ) to derive RR and PD. Thus randomization
problem is a construction random variable R ( t , T ; ω ) by using historical data. It is clear that a single

6. 6
number the close price R close ( t , T ) at the date t does not sufficient to present distribution of the bond
price at t and construction recovery rate at maturity.
Usually one chooses a particular period and historical close prices over this period. Following
mathematical statistics historical prices are interpreted as independent observations in equal market
conditions. Such observations in statistics are known as a random sample taking from the total population.
A random sample is interpreted then as a set of independent equally distributed random variables. If
observations over a random variable cannot be drawn in equal conditions then sample is not a random and
it cannot adequately represent general population and statistical theory fails to make a good statistical
conclusion. Taking into account latter remark we note that historical data which is represented by the set
of close prices over extended period of time may not be interpreted as a set of equally distributed random
variables as far as market conditions change over the time. Hence, historical close price data may not
represent independent equally distributed observations and therefore the estimated parameters of a pricing
model can be bad. On the other hand if max - min spread of the historical prices over a single date does
not small enough stochastic effect of the price over a single date can not be ignored.
In order to present stochastic price R ( t , T ,  ) at t we first should specify the meaning of the date-t price
notion. Dealing with close prices historical data our statistical forecast of the date-T future price would
relate to close price at T. If deviation of the prices during a date can not be ignored then a single number
reduction (close price) can be oversimplified.
Let us interpret the price of the bond at t as a random variable having for example the uniform distribution
on interval [ d – , d + ] where
d – = R min ( t ) =
}t{u
min

R ( u , T ) , d + = R max ( t ) =
}t{u
max

R ( u , T )
Here { t } denotes the trading period associated with the date t. Date t can be a trading day, or a week or
other convenient period. Assumption regarding uniform distribution is taking for simplicity. One can also
apply other type of distribution though it does not guarantee a higher accuracy. Consider two possible
scenarios
( α ) 0 ≤ R min ( t ) ≤ R max ( t ) ≤ B ( t , T ) ≤ 1
( β ) R min ( t ) ≤ B ( t , T ) ≤ R max ( t ) ≤ 1
Case ( α )
Assume that occurs at maturity, i.e. P {  = T } = 1 and that the random variable δ = δ ( ω ) is
uniformly distributed on the interval
[ B – 1
( t , T ) d –_( t ) , B – 1
( t , T ) d + ( t ) ]
Then the first two moments of the random variable which is defined in (6) are equal to
E θ ( t , T ; ω ) =
  dd
1



d
d
[ 1 –
)T,t(B
u
] d u =
=
)dd(2
1
 
{ [ 1 –
)T,t(B
d 
] 2
– [ 1 –
)T,t(B
d 
] 2
}
E θ ² ( t , T ; ω ) =
  dd
1



d
d
[ 1 –
)T,t(B
u
] 2
d u =

7. 7
=
)dd(3
1
 
{ [ 1 –
)T,t(B
d 
] 3
– [ 1 –
)T,t(B
d 
] 3
}
Then from (7) it follows that point estimate < δ > of the recovery rate and correspondent probability of
default, PD at t can be represented by the formulas
< δ > = 1 –
3
2
22
33
]
)T,t(B
d
1[]
)T,t(B
d
1[
]
)T,t(B
d
1[]
)T,t(B
d
1[




PD =
)dd(4
3
  33
222
]
)T,t(B
d
1[]
)T,t(B
d
1[
}]
)T,t(B
d
1[]
)T,t(B
d
1[{




Case ( β )
In this case we supposed that d –_( t ) ≤ B ( t , T ) ≤ d + ( t ) ≤ 1 and P {  = T } = 1. The date-t
market implied estimate of the recovery rate is a random variable
1 , with probability P { R ( t , T ,  )  [ B ( t , T ) , d + ( t ) ] }
δ ( ω ) = {
u , with density P { δ ( ω )  [ u , u + ∆ u ] } =
  dd
uΔ
where u < 1. Then the values of the first and second moments of the θ ( t , T ; ω ) can be represented by
formulas
E θ ( t , T ; ω ) =
  dd
1


)T,t(B
d
[ 1 –
)T,t(B
u
] d u =
)dd(2
]
)T,t(B
d
1[ 2-
 

E θ ² ( t , T ; ω ) =
  dd
1


)T,t(B
d
[ 1 –
)T,t(B
u
] 2
d u =
)dd(3
]
)T,t(B
d
1[ 3-
 

Therefore
< δ > = 1 –
3
2
2
3
]
)T,t(B
d
1[
]
)T,t(B
d
1[




, PD =
)dd(4
3
  3
4
]
)T,t(B
d
1[
]
)T,t(B
d
1[





8. 8
Default in Continuous Time. In continuous time corporate bond admits default at any future moment
prior to expiration date. Introduce a discrete time approximation of the continuous default moment
 λ ( ω ) = 
n
1k
t k χ {  ( ω )  ( t k – 1 , t k ] } + T χ {  > T }
where 0 = t 0 < … < t n = T is a λ-partition of the interval [ 0 , T ] , where λ = t k – t k – 1 does not
depend on k. Continuous real time cash flow CF A ( ω ) associated with zero coupon corporate bond from
bond buyer perspective can be then approximated then by a discrete time cash flow
CF A ( λ ) = – R ( t 0 , T ) χ { t = t 0 } + 
n
1k
R (  , T ,  ) χ {   ( t k – 1 , t k ] } χ { t = t k } +
+ χ {  > T } χ { t = T } = – R ( t 0 , T ) χ {  λ > 0 } χ { t = t 0 } + (8)
+ 
n
1k
R ( t k , T ,  ) χ {  λ = t k } χ { t = t k } + χ {  λ > T } χ { t = T }
Cash flow CF A ( λ ) corresponds to transactions:
*) A pays R ( t 0 , T ) to B at t 0 ,
**) A receives R (  , T ) from B at t k if  ( ω )  ( t k – 1 , t k ] , k = 1, 2, … n ,
***) A receives $1 at T for the no default scenario {  > T }.
For each market scenario ω only one term on the right hand side (8) does not equal to zero. On the right
hand side of the formula (8) the value of the bond at the moment  λ = t k is defined as
R ( t k , T ) = δ k ( ω ) B ( t k , T ) , k = 1, 2, … n
Assume that corporate bonds with different maturities default simultaneously with equal recovery rate
with respect to risk free bonds with the same maturities, i.e.
δ k ( ω ) = δ k ( t 0 ,  ) ,   {  λ = t k }
and δ k ( ω ) does not depend on T. Rates δ k ( ω ) , k = 1, 2, … n do not known at initiation. Thus
R ( t 0 , T , ω ) = 
n
1k
B ( t 0 , t k ) δ k B ( t k , T ) χ {  λ = t k } + B ( t 0 , T ) χ {  λ > T } (9)
We make additional assumptions which will help us to reduce discrete in time valuation problem to the
set of default at maturity problems. Suppose that
*) if a corporate bond with maturity t k , t k < T does not exist then we will use market implied bond
**) random variables δ k ( ω ) = δ ( ω ) does not depend on k.
One can interpret later assumption as an assignment of a particular fixed credit rating of the bond. Bearing
in mind market implied forward discount rate we define the market implied bond value at t k
B ( t k , T ; t 0 ) by the equality
B ( t 0 , T ) = B ( t k , T ; t 0 ) B ( t 0 , t k )
Then (9) can be rewritten as

9. 9
R ( t 0 , T , ω ) = δ ( ω ) 
n
1k
B ( t 0 , t k ) B ( t k , T ; t 0 ) χ {  λ = t k } + B ( t 0 , T ) χ {  λ > T } =
(10)
= δ ( ω ) B ( t 0 , T ) χ {  λ ≤ T } + B ( t 0 , T ) χ {  λ > T }
The market implied forward discount rate B ( t k , T ; t 0 ) is a market estimate of the unknown at t 0
date- t k forward starting bond value B ( t k , T ). Unknown forward discount rate is assumed to be random
variable and therefore the replacement of the real random forward rate by its estimate implies market risk
for both counterparties. Dividing both sides of the latter equation by B ( t 0 , T ) we arrive at the equation
)T,t(B
)ω,T,t(R
1
0
0
 = ( 1 –  ) χ {  λ ≤ T } (11)
Equation (11) is similar to (4). Consider case when recovery rate is random δ ( ω ). Similar to (5) we
conclude that
1 , for ω  {  λ > T }
δ = δ ( ω ) = {
)T,t(B
)ω,T,t(R
0
0
, for ω  {  λ ≤ T }
P { δ ( ω ) < u } = P {
)T,t(B
)ω,T,t(R
0
0
< u } , u  [ 0 , 1 )
P (  λ ≤ T ) = 1 – P {
)T,t(B
)ω,T,t(R
0
0
= 1 }
Then taking into account formula (7) we arrive at formulas
< δ > = 1 –
]
)T,t(B
)ω,T,t(R
-1[E
]
)T,t(B
)ω,T,t(R
-1[E
0
0
2
0
0
, P (  λ ≤ T ) =
2
0
0
2
0
0
]
)T,t(B
)ω,T,t(R
-1[E
}]
)T,t(B
)ω,T,t(R
-1[E{
(12)
Formulas (12) represent deterministic market estimates of the recovery rate and correspondent default
probability over [ t 0 , T ]. These formulas do not depend on a particular distribution P (  = t k ).
Thus the initial continuous time default distribution we approximated by a discrete time default
distribution which produces nonrandom recovery rate.
The assumption δ k ( ω ) = δ ( ω ) can be too strong for a long term bonds. Let us consider more accurate
construction. Using market implied estimate of the forward discount rate it follows from (9) that
R ( t 0 , T , ω ) = B ( t 0 , t k ) δ k B ( t k , T )  B ( t 0 , t k ) δ k B ( t k , T , t 0 )
for ω  {  λ = t k }. Then

10. 10
B – 1
( t k , T , t 0 ) R ( t 0 , T , ω ) χ {  λ = t k } = B ( t 0 , t k ) δ k χ {  λ = t k } =
= R ( t 0 , t k , ω ) χ {  λ = t k }
k = 1, 2, … n. Right hand side of the later equality represents market implied estimate of the risky bond
price with expiration date which might not exist on the market. This estimate is constructed similar to
market implied forward rate which is an important estimate of the trading fixed income forward type
contracts. Market implied corporate bond R ( t 0 , t k , ω ) with expiration date t k admits defaults at any of
the dates t j = 1, 2, … k. Then we note that
R ( t 0 , t k , ω ) χ {  λ = t k } = B ( t 0 , t k – 1 ) R ( t k – 1 , t k , t 0 , ω ) χ {  λ = t k }
where forward market implied corporate bond price is defined by equality
R ( t k – 1 , t k , t 0 , ω ) = B – 1
( t 0 , t k – 1 ) R ( t 0 , t k , ω )
Latter equality takes place for the scenarios ω  {  > t k – 1 }. Therefore
B ( t k – 1 , t k , t 0 ) , for {  λ > t k }
R ( t k – 1 , t k , t 0 , ω ) = {
B – 1
( t 0 , t k – 1 ) R ( t 0 , t k , ω ) , for {  λ = t k }
The latter equality can be rewritten in the form
θ k = B ( t k – 1 , t k , t 0 ) –
)t,t(B
)ω,t,t(R
1-k0
k0
= ( 1 –  k ) χ (  λ = t k ) (13)
Hence, conditional probability of default and conditional distribution of the recovery rate are defined by
the formulas
P (  λ = t k |  > t k – 1 ) = P {
)t,t(B
)ω,t,t(R
1-k0
k0
< B ( t k – 1 , t k , t 0 ) |  > t k – 1 }
(14)
P { δ k ( ω ) < u |  > t k – 1 } = P {
)t,t(B
)ω,t,t(R
1-k0
k0
< u |  > t k – 1 } , u  [ 0 , B ( t k – 1 , t k , t 0 ))
Let us reduce the default problem. Let recovery rate δ k be a nonrandom constant. Taking conditional
expectation E k – 1 { ∙ } = E { ∙ |  > t k – 1 } from both sides (13) we arrive at the system
E k – 1 θ k = ( 1 – < δ k > ) P {  = t k |  > t k – 1 }
E k – 1 θ 2
k = ( 1 – < δ k > ) ² P {  = t k |  > t k – 1 }
Solving the system for RR and PD we arrive at the values

11. 11
(RR) k = < δ k > = 1 –
)ω;t,t(θE
)ω;t,t(θE
k0k1-k
k0
2
k1-k
(15)
(PD) k = P {  = t k |  > t k – 1 } =
)ω;t,t(θE
])ω;t,t(θE[
k0
2
k1-k
2
k0k1-k
As far as P (  > t 0 ) = 1 then
(PD) 1 = P {  = t 1 |  > t 0 } = P (  = t 1 )
Next for k = 2 we note that
(PD) 2 = P {  = t 2 |  > t 1 } =
)tτ(P
)tτtτ(P
1
12


=
)tτ(P
)tτ(P
1
2


where P (  > t 1 ) = 1 – P (  = t 1 ). Hence
P (  = t 2 ) = P {  = t 2 |  > t 1 } [ 1 – P (  = t 1 ) ]
Applying induction method let us assume that P (  = t k – 1 ), k = 2, 3, … is known. Then similar to the
case k = 2 we note that
P {  = t k |  > t k – 1 } =
)tτ(P1
)tτ(P
1-k
k


and therefore
P (  = t k ) = P {  = t k |  > t k – 1 } [ 1 – P (  = t k – 1 ) ] (16)
Formulas (16) and (15) define probability of default and deterministic recovery rate at date t k ,
k = 1, 2, … n which represent complete information regarding date-t corporate bond price.
Implementation of this pricing model interprets date-t as a day of trade rather than the price at the say
t = 10 am, 3.30 pm, or close price at day t. Therefore the date-t price can be thought as a random variable
R ( t , T , ω ). If an investor pays a particular deterministic price R ( t , T ) for the bond during trading
time at t then there exists market risk the price R ( t , T ) is higher than it is implied by the market. Such
market risk is valued by P { R ( t , T ) > R ( t , T , ω ) } and average of the overpayment is equal to
E [ R ( t , T ) – R ( t , T , ω ) ] χ { R ( t , T ) > R ( t , T , ω ) }
Mark-to-Market (MtM) bond pricing.
Bonds are single party risky instruments. Buyer, A pays bond price at initiation of the deal and expects to
receive the face value at expiration date. If the bond price remarkably falls at some point of time the seller
of the bond may declare default in obligation to deliver face value at the expiration date. In this case
issuer of the bond pays a settlement price of the bond. Note that the bond itself might not default at this
moment. By using MtM account bond buyer is guaranteed to receive higher price at the default settlement
than in the standard plain trading.

12. 12
Assume that counterparties of the bond trading agree to use MtM account to hedge the risk of default.
They also need to establish a benchmark risk rate to keep MtM account until expiration date. For US
corporate bonds such risk free benchmark can be represented by T-bond rate.
Let t 0 and t j + 1 = t j + 1, j = 0, 1, … n, t n = T denote initiation date and MtM adjustment moments
correspondingly. Let us present a sketch of the MtM account transactions. At initiation date t 0 bond
buyer pays R ( t 0 , T ), R ( t 0 , T ) < B ( t 0 , T ) and receives corporate bond that promises $1 at T.
At the date t 0 the value of the MtM account is set to be equal
MtM ( t 0 ) = R ( t 0 , T ) + ad ( t 0 ) = B ( t 0 , T )
where R ( t 0 , T ) is paid by A and the value of adjustment at t 0 is
ad ( t 0 ) = B ( t 0 , T ) – R ( t 0 , T )
is paid by B. The MtM account is under bond seller supervision. Hence initial payment made by bond
buyer does not go directly to bond seller. It may or may not go to bond seller, B later at the settlement
moment. Counterparty B as owner of MtM account pays interest i MtM ( t 0 , t 1 ) for keeping money during
[ t 0 , t 1 ) period. Hence just before the new prices comes to the market at the date t 1 the value of the MtM
account is equal to
MtM ( t 1 – 0 ) = B ( t 0 , T ) ( 1 + i MtM ( t 0 , t 1 )  )
where  = t j + 1 – t j does not depend on j and interest rate i MtM ( t 0 , t 1 ) may or may not coincide with
the risk free bond rate. At thee date t 1 the value of the MtM account is equal to
MtM ( t 1 ) = B ( t 1 , T ) = MtM ( t 1 – 0 ) + ad ( t 1 )
where
ad ( t 1 ) = B ( t 1 , T ) – MtM ( t 1 – 0 )
is date-t 1 adjustment of the MtM account. If ad ( t 1 ) > 0 then B deposits ad ( t 1 ) and if ad ( t 1 ) < 0 then
B withdraws this sum from MtM account. In general if MtM ( t j ) is known then
MtM ( t j + 1 – 0 ) = B ( t j , T ) ( 1 + i MtM ( t j , t j + 1 )  ) ,
MtM ( t j + 1 ) = B ( t j + 1 , T ) = MtM ( t j + 1 – 0 ) + ad ( t j + 1 ) ,
ad ( t j + 1 ) = B ( t j + 1 , T ) – MtM ( t j + 1 – 0 )
j = 0, 1, … n. The value of ad ( t j + 1 ) is added by B to MtM account when it is positive and it is
withdrawn from MtM account if it is negative. Define bond seller default time . For writing simplicity
low index  bellow will be omitted. The cash flows to MtM account can be represented by the formula
CF MtM = MtM ( t 0 ) χ ( t = t 0 ) + 
n
1k
χ {  = t k } { 
1-k
1j
ad ( t j ) χ ( t = t j ) +
+ δ k B ( t k , T ) χ ( t = t k ) } + χ {  > T } [ – 1 + MtM ( T – 0 ) ] χ ( t = T )

13. 13
From bond buyer and seller perspective the values of MtM transactions are different. They can be
represented by the following formulas correspondingly
CF A = – R ( t 0 , T ) χ ( t = t 0 ) + 
n
1k
χ {  = t k } [ MtM ( t k – 0 ) χ ( t = t k ) + δ k B ( t k , T ) ] +
+ χ {  > T } 1 χ ( t = T ) = – R ( t 0 , T ) χ ( t = t 0 ) + 
n
1k
χ (  = t k ) [ B ( t k – 1 , T ) 
 [ 1 + i MtM ( t k – 1 , t k )  ] + δ k B ( t k , T ) ] χ ( t = t k ) + χ (  > T ) χ ( t = T ) ,
CF B = – ad ( t 0 ) χ ( t = t 0 ) +
+ 
n
1k
χ (  = t k ) { 
1-k
1j
ad ( t j ) χ ( t = t j ) – MtM ( t k – 0 ) χ ( t = t k ) } +
+ χ (  > T ) [ – 1 + B ( t n – 1 , T ) ( 1 + i MtM ( t n – 1 , t n )  ) ] χ ( t = T) =
= χ (  > T ) { B ( t 0 , T ) χ ( t = t 0 ) + B ( t n – 1 , T ) ( 1 + i MtM ( t n – 1 , t n )  ] χ ( t = T ) }
Note that MtM account applies interest rate i MtM which may be lower than interest rate implied by the
risk free bond prior to counterparty default moment or bond expiration date which one comes first.
Forward interest rate i MtM and forward risk free discount rate B ( t j , T ) are unknown at initiation date.
We assume that these forward rates are random processes. It is market rule to use market implied forward
estimates for pricing of these contracts. The use of market estimates versus stochastic rates implies market
risk. Buyer and seller expected value of the spot prices can be represented by the EPV of the
correspondent cash flows. Applying formulas (12) we receive formulas
EPV CFA ( ω ) = – R ( t 0 , T ) + 
n
1k
P (  = t k ) { B ( t k – 1 , T , t 0 ) [ 1 + i MtM ( t k – 1 , t k , t 0 )  ] +
+ < δ k > B ( t k , T , t 0 ) } B ( t k , T , t 0 ) + P (  > T ) B ( t 0 , T )
EPV CFB = P (  > T ) { B ( t 0 , T ) + B ( t n – 1 , T , t 0 ) [ 1 + i MtM ( t n – 1 , t n , t 0 )  ] B ( t 0 , T ) }
where default distribution P (  = t k ) and recovery rates < δ k > are defined in (15) and (16). E PV CF A
and E PV CF B are associated with bid and ask bond prices in MtM trading format. The gap between
bid and ask prices is a measure of illiquidity of the corporate bonds on MtM bond trading.

14. 14
References.
1. R. Jarrow, S. Turnbull Derivative Securities, 2d ed. South-Western College Publishing, p.684.