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BASIC OF PRICING 2.

Ilya Gikhman

6077 Ivy Woods Court

Mason OH 45040 USA

Ph. 513-573-9348

Email: ilyagikhman@mail.ru

Abstract. In this paper we develop a model of corporate bonds pricing. We begin with default definition

which is similar to one that is used in the standard reduced form of default model. The primary distinction

between our model and reduced form of default model is interpretation of the date-t price of the bond. In

reduced form model date-t corporate bond price is a single number which in practical applications is the

close price of the bond at date t. It could be a reasonable reduction when deviation between maximum and

minimum of the bond prices at date t is sufficiently small. Otherwise the reduction’s error could be

remarkable. In our interpretation date-t bond price is a random variable taking values between minimum -

maximum prices at the date t. In such setting random value of the bond is considered as the present value

of the recovery rate assuming that default occurs at maturity of the bond. Random format of the recovery

rate does not convenient to compare different risky bonds. It makes sense to assume that recovery rate is a

nonrandom constant. This reduction reduced default problem to finding unknown recovery rate and

correspondent default probability. This is the case of two unknowns and one can derive equations for the

first and second moments of the bond price to present a solution of the problem. This approach can be

extend to resolve default problem in more general cases.

JEL : G13. Keywords: no arbitrage, mark-to-market, cash flow, market risk, credit risk, reduced form pricing, credit

risk, interest rate swap.

1. Risk Free Bond Pricing.

Introduction. The price notion is the basis of the finance theory and practice. In standard trading of the

risk free bond investors pay the bond price B ( t , T ) at date t ≥ 0 and receive bond which promises its

face value of $1 at bond’s expiration date T. In [1] we presented a formal definition of the no arbitrage

pricing. The no arbitrage pricing of a financial instrument is defined as pricing which starts with zero

value of the initial investment at initiation date. The zero value at initiation is formed by buying the

instrument and borrowing this amount at risk free interest rate from the bank. Conditioning on zero initial

value of the investor’s position investor should arrive at the 0 value at expiration date T. Such

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interpretation of the no arbitrage pricing is an ideal scheme. It should be adjusted for a stochastic market.

Interest rates at the future dates are unknown which are interpreted as random variables. It can be higher

or lower than market implied forward rates estimated at initiation date. By using market implied forward

rates one can produce market implied forward estimate of the future coupon bond price, which does not

contain arbitrage opportunity with respect to the spot price. No arbitrage price of the risk free coupon

bond does not eliminate market risk. It stems from the fact that coupon payments received by bond buyer

at future dates should be invested at unknown at initiation real forward risk free rates while on no

arbitrage price set at initiation uses market implied forward rates. The difference between real market

forward rates and its market implied forward estimates defines market risk of the risk free bond during

lifetime of the bond prior to its expiration date.

We use cash flow as a formal definition of the financial instrument. Risk free coupon bond from the

buyer perspective can be defined by the cash flow

CF = – B c ( t 0 , T ) χ { t = t 0 } +

n

1j

c χ { t = t j } + 1 χ { t = T } (1)

Here c > 0 is a coupon payment taking place at the dates t j , j = 1, 2, … n and t n = T Function

χ { t = T } denotes indicator of the event { t = T }. One usually interprets the CF’s portion

n

1j

c χ { t = t j } + 1 χ { t = T } (2)

as a portfolio of the risk free bonds with face values c at t j , j < n and 1 at T correspondingly. Portfolio

interpretation of the coupon bond price B c ( t 0 , T ) makes it possible reduce no arbitrage of the coupon

bond to a sum of no arbitrage prices of the zero coupon bonds in the portfolio. The equivalence of the

coupon bond and portfolio of the zero coupon bonds with different maturities takes place in the perfect

market. Indeed, in the perfect market buyer of the bond can go short at initiation with the portfolio of the

zero coupon bonds to compose the value B c ( t 0 , T ). More realistic setting of the bond pricing problem

implies borrowing funds equal to the bond price from the bank at risk free interest at date t 0 and then

return borrowing amount plus interest at the bond maturity T. In other words, we do not assume that

money borrowed from bank can be returned to the bank by parts. Cash flow (1.2) admits two types

representation of the date-t 0 price of the bond. One representation uses the real world market scenarios to

define B ( t j , T , ω ). It will be used to present stochastic market price at the date-t 0 . Other

representation uses market implied forward rates to present the current spot price of the bond. Here

B c ( t j , T ) , j > 0 denote future values of the bond at t j that are unknown at t 0 . One usually apply a

stochastic equation as a theoretical model representing future values of the bond B c ( t j , T , ω ). Using a

model of the B c ( t j , T , ω ) we can construct date-T future value FV of the CF

n

1j

c B – 1 ( t j , T , ω ) + 1

Market risk of the coupon bond investment over [ t 0 , T ] period. Bond buyer loses money if

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B c ( t 0 , T ) > B ( t 0 , T ) [

n

1j

c B – 1 ( t j , T , ω ) + 1 ]

and he gets money otherwise. Nevertheless applying implied market forward estimate to valuation of the

PV of the cash flow (2) shows that B c ( t 0 , T ) is indeed no arbitrage price. Indeed one can easy verify

the equality

B c ( t 0 , T ) = B ( t 0 , T ) [

n

1j

c B – 1 ( t j , T , t 0 ) + 1 ]

where date-t 0 market implied forward discount rate B ( t j , T , t 0 ) is defined as

B ( t 0 , T ) = B ( t 0 , t j ) B ( t j , T , t 0 )

j = 1, 2, … n. This equality confirms the fact that the value B c ( t 0 , T ) borrowed from the bank to buy

coupon bond at t 0 at risk free rate will be covered by the date-T value of the cash flow generated by the

bond.

In stochastic setting no arbitrage pricing does not eliminate market risk. In finance theory market risk

does not formally defined. Informally it can be associated with profit-loss diagram which represents profit

and loss of an investment based on a market scenario. Market scenario is associated with the future price

at a particular date or at a series of dates. The modern finance theory deals with modeling of the spot

price. Spot price of an instrument is interpreted as present value, PV of the future cash flow associated

with CF. For example, for the coupon bond defined by cash flow (0.1) we can write the formula

B c ( t 0 , T ) = PV CF

Market risk of the bond buyer or seller is associated with lower return than it is expected at the initial

moment. If risk free bond is used for financing of a business than going short and buying back bond

during lifetime of the bond is a market risky deal. On the other hand buying risk free bond and holding it

until expiration does not risky purchase.

Corporate bond pricing. Let { Ω , F , P }be a complete probability space. Elements of the set Ω

represent market scenarios which represent stochastic prices of the debt instruments, F is the sigma

algebra generated by the observed market scenarios, and P is a complete probabilistic measure. Consider

credit risk effect on bonds pricing. Credit risk is associated with bond default. Other risk that also effects

on pricing is counterparty risk. Counterparty risk is a risk when one of the participants could not fulfill its

obligations. In such a case, underlying financial instrument does not default. In case of a corporate bond

the credit risk is associated with the issuer of the bond that could not pay face value of the bond to bond

buyer at maturity. In this case counterparty risk coincides with the credit risk. For more complex financial

instruments credit and counterparty risks are different.

A corporate bond is a risky instrument in which only buyer of the bond is exposed to risk. This is credit

risk which implies that bond’s seller could not accomplish its obligations to pay notional value of the

bond at its expiration. In standard ’cash-and-carry’ trading buyer if the bond pays the spot price R ( t , T )

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and gets the bond which promises say $1 at T. If default occurs prior to maturity buyer of the bond

receives bond’s recovery rate, RR = δ < $1.

Reduced form of default is probably the most popular model of the corporate bonds pricing theory. There

are two primary parameters such as recovery rate and probability of default that effect on corporate bond

price. Benchmark reduced form of default theory does not present simultaneously calculations of the

probability of default and recovery rate values. In practice, agencies heuristically assign 40% or other

popular percentage to the bond’s face value of the recovery rate. This assumption simplifies the reduced

form default model and makes it possible calculation of the probability of default that corresponds to a

chosen recovery rate. In contrast to the benchmark approach, we do not use known recovery rate

assumption. We considered more general approach to default problem in [2,3]. For example based on

pricing data at date t , t < T and assuming that default takes place only at maturity we present closed

form formulas for recovery rate and probability of default. Here we present more detailed study regarding

default.

Let briefly recall basic corporate coupon bond valuation formulas. Let corporate bond admits default

only at maturity. Then at maturity T the value of the bond can be defined by the formula

R ( T , T , ω ) = χ { > T } + δ χ { = T }

where the constant δ [ 0 , 1 ) is assumed to be known depending on t. Functions χ { > T } and

χ { = T } denote indicators of the no default and default at T scenarios correspondingly. Probabilities

of the default and no default events are also depend on t. Value δ and ( 1 – δ ) are referred to as recovery

rate and loss given default . Stochastic value of the corporate bond at maturity T implies stochastic price

of the bond at date t

R ( t , T , ω ) = B ( t , T ) [ χ { > T } + δ χ { = T } ] (3)

= B ( t , T ) [ 1 – ( 1 – δ ) χ { = T } ]

Formula (3) defines date-t ‘fair’ price for each market scenario ω. Denote R spot ( t , T ) the spot price of

the bond at date t. It is a statistics of the observed data during a period associated with date t. One can

interpret spot price of the day t as open, close, middle, or expected value of the random market price (1.3).

For given spot price probabilities

P { R ( t , T , ω ) > R spot ( t , T ) } , P { R ( t , T , ω ) < R spot ( t , T ) }

define seller’s and buyer’s market values of risk correspondingly. First probability represents the chance

that bond is underpriced at t while the second probability represents the chance that bond at t is overpriced

by the market. In practice the value of recovery rate is unknown and should be estimated by historical

data. Note that our interpretation of the notion price significantly broader than it is used in modern finance

theory. For example in the standard reduced form model of default the only spot price is defined, which is

expectation of the random price (3).

Formula (3) presents date-t bond price given value of the bond at maturity. There are two values of the

bond ( δ , 1 ) at T with known probability distribution. To each value of the bond corresponds a unique

value of the bond ( δ B ( t , T ) , B ( t , T ) ) at t and probability distribution does not changed. This

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construction of the stochastic bond price does not be changed if we assume that value of the bond at T is a

discrete or continuous distributed random variable taking values on interval [ 0 , 1 ]. In practice we deal

with inverse problem. We observe distribution of the bond prices during a date t and our goal is to make a

conclusion regarding default distribution. Note that given price R ( t , T ; ω ) distribution at t we can

define market implied forward value δ ( ω ) = B – 1 ( t , T ) R ( t , T ; ω ) to value of the bond at maturity

which represents market implied stochastic recovery rate at T assuming that default occurs at maturity. It

is not convenient to justify which bond is more reliable dealing with continuous distribution of recovery

rates.

In order to present valuation in more explicit form it is necessary to make a discrete reduction of the

recovery rates. We begin with theoretical model where values of the bond are assumed to be known at

maturity. Given that default can be only at maturity T we introduce a total set of scenarios Ω = m

1j

ω j ,

where ω j = { ω : δ j – 1 ≤ R ( T , T ; ω ) < δ j } . Here 0 = δ 0 ≤ ≤ δ j < δ j + 1 , j = 1, 2, … m – 1

and δ m = 1. Thus a continuously distributed recovery rate δ ( ω ) can be approximated by a discrete

random variable δ λ ( ω ) =

m

0j

δ j χ ( ω j ) where

P ( ω j ) = p j = P { R ( T , T ; ω ) [ δ j , δ j + 1 ) }

Let χ D ( ) denote indicator of the default. We can present price of the corporate bond as

R ( t , T ; ω ) = B ( t , T ) [ 1 – ( 1 – ( ω ) ) χ D ( ) ]

This equation defines the price for each market scenario ω. We can approximate latter equality

1 – λ ( t , T ; ) = ( 1 – λ ) χ ( ) =

m

1j

( 1 – δ j ) χ ( ω j )

where

λ ( t , T ; ) = )T,t(B

)ω;T,t(R

λ

λ

Therefore

E kλ ( t , T ; ) =

m

1j

( 1 – δ j ) k p j

k = 1, 2, … m. This is a linear system of the m order with respect to unknown p j . It could be solved by

the standard methods of the Linear Algebra. The probability of the no default is equal to

p 0 = 1 – p 1 – p 2 – … – p m – 1

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The solution of the system represents the approximation of the default distribution corresponding to the

stochastic recovery rate.

Corporate bond pricing.

Let us assume that recovery rate δ = δ ( ω ) is a continuously distributed random variables on [ 0 , 1 ).

Introduce a discrete approximation of the recovery rate given that default occurs only at maturity date T.

Denote

p j = P { R ( T , T , ω ) [ δ j – 1 , δ j ) }

(4)

ω j = { ω : R ( T , T , ω ) [ δ j – 1 , δ j ) } , Ω = m

1j

ω j

where 0 ≤ δ j < δ j + 1 , j = 1, 2, … m – 1, δ 0 = 0 and δ m = 1 . In theory we suppose that distribution

of the random variable R ( T , T ; ω ) is known and therefore probabilities p j are known too. From (4) it

follows that market price of the bond that admits default at maturity with known recovery rate δ j is

defined as

R mkt ( t 0 , T ; Ω j ) = δ j B ( t 0 , T )

Hence a discrete approximation of the bond can be written as

R mkt ( t 0 , T , ω ) =

m

1j

δ j B ( t 0 , T ) χ Ω j ( ω )

We have introduced theoretical valuation formulas of corporate bond. In practice we have only historical

data available. Recovery rate is unknown and should be estimated based on observed data. Hence the

problem is how using observed data present stochastic market price and recovery rate estimate.

Let us first introduce randomization of the bond’s price. Recall that market risk of the no arbitrage price

of the default free coupon bond is stipulated by the unknown at t 0 future values of the bond

B ( t j , T , ω ) [1] which are estimated by market implied forward rates. On the other hand later model of

the corporate zero coupon bond given possibility of default at maturity moment the date-t 0 uncertainty of

the PV reduction of the corporate bond comes with unknown recovery rate. In [1] we presented a

theoretical solution of the problem.

Remark. Benchmark reduced form model of default begins with the similar definition of default. Next

they introduce date-t spot price ignoring stochastic price at maturity of the bond. Recall that the price of

the risky bond is a random variable which takes two values 1 and δ with known probabilities. Difference

of these two approaches relates to the ways how one interprets date-t price of the bond. In theory asset

price is interpreted as a continuous time random process. In practice we usually use close price of asset as

the price at date-t. In such reduction of the price we can consider future price at T as close price at T. It

fallows from the fact that all data we use represent only close historical prices. In other interpretation of

the asset price close price of the date t can be interpreted as a good approximation of the whole trading

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period which we associate with date-t. Good approximation can be justified by a low volatility of the

date-t asset prices. If volatility cannot by considered as sufficiently small then use deterministic statistics

like close, open admits additional risk leading to possible losses. Standard reduced form model deals with

one equation and two unknowns. It could not present a unique solution of the default problem in simplest

setting when default might occur only at maturity. That is why the primary financial institutions and

rating agencies need to make additional assumption that recovery rate is a known such as for example

30%, 40% or other pre-specified value of recovery rate. Using a heuristic recovery rate helps to produce

heuristic probability of default. It is clear that such simplification of the problem leads to distortion of the

estimates of the recovery rate as well as probability of default which are the primary quantitative rating

parameters.

Our approach to default is based on stochastic market price was introduced in [1]. Stochastic price at t is

defined for each market scenario which associated with a particular value of the bond during date-t

period. With the help of stochastic price one can present independent equations for higher moments of the

market price. The system of two equations for the first and the second moments is sufficient to calculate

nonrandom recovery rate along with correspondent default probability. Following [1] let us briefly recall

this construction. From equality (3) it follows that

1 – )T,t(B

)ω,T,t(R = ( 1 – δ ) χ { = T } (5)

Then recovery rate can be written in the form

)T,t(B

)ω,T,t(R , for ω { ( ω ) = T }

δ = δ ( t , T , ω ) = {

1 , for ω { ( ω ) > T }

The distribution of the random variable δ ( ω ) is the distribution of the stochastic recovery rate. One choice of the statistical estimate Δ of the random recovery rate δ ( ω ) is its expectation Δ = E δ ( ω ). Stochastic interpretation of the recovery rate implies that reduction Δ of the rate δ ( ω ) implies market risk. This risk for the bond buyer is the higher value of the Δ than recovery rate which is specified by a scenario at the default moment. Buyer and seller risks are measured by probabilities of undervalue or overvalue of the recovery rate

P { δ ( ω ) < Δ } , P { δ ( ω ) > Δ } correspondingly. Consider for example the estimate δ ( ω ) χ ( ω D ), where D = { ω : δ ( ω ) < 1 }. This estimate specifies credit risk when bond seller could not pay initially promised amount of $1 at expiration date. At the same time this risk is appealing bond sellers. Define normalized spread function θ ( t , T ; ω ) by the formula

)T,t(B

)ω,T,t(R1)ω,T,t(θ

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Note that this random function associated with normalized estimate of the LGD. The random function θ ( t , T ; ω ) is an observable function. Stochastic recovery rate is complete credit information of the bond. Nevertheless it is difficult to compare two distribution functions. To present credit characteristics in more comparable form we assume that observations on bond’s prices at t correspond to unknown deterministic recovery rate in case of default. We use function θ ( t , T ; ω ) to calculate a nonrandom market implied recovery rate < Δ > and its correspondent probability of default. From (5) it follows that

E θ ( t , T , ω ) = ( 1 – < Δ > ) P ( D )

E θ ² ( t , T , ω ) = ( 1 – < Δ > ) ² P ( D )

Solving the system for < Δ > and P ( D ) we arrive at the solution

< Δ > = 1 – )ω , T,t(θE

)ω,T,t(θE 2

, P ( t , T , D ) = )ω,T,t(θE

])ω , T,t(θE[2

2

(6)

Value < Δ > is market implied estimate and it does not equal to expected value of the stochastic recovery

rate

δ ( t , T , ω ) = )T,t(B

)ω,T,t(R

Value < Δ > depends on distribution of the random process R ( t , T ; ω ) which is an assumption of the

model. Note that recovery rate < Δ > and correspondent probability of default represents estimates of the

real credit risk. In general

E θ p ( t , T , ω ) ≠ ( 1 – < Δ > ) p P ( D )

p = 3, 4, … . Hence the approach is good for price distribution which completely defined by its first and

second moments. More crude but more explicit estimate can be presented as following. Assume for

example that we fix a particular recovery rate < Δ > . Applying first moment equation we arrive at the

formula

P ( D ) = ( 1 – < Δ > ) – 1 E θ ( t , T , ω )

Thus one can compare probabilities of default for different bonds given the same value of the recovery

rate. On the other hand one can fix probability of default and consider values of recovery rate of different

bonds. For example one can fix 45% recovery rate and determine bonds which probabilities of default

less than 0.2. Then fixing probability of default 0.3 one can find the bond which recovery rate is maximal

or a set of bonds which recovery rates exceed 80%.

Let us consider an implementation of randomization of the bond price. Let { t 0 } denote trading time

interval of the date t 0 . Value of { t 0 } can be either a day, week , or other appropriate period. Define

minimum and maximum values of the bond prices over the day period { t 0 }. Denote

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D min ( t 0 , T ) = }t{u 0

min

D spot ( u , T ) , D max ( t 0 , T ) = }t{u 0

max

D spot ( u , T )

Symbol D spot is interpreted as spot price of the corporate bond. We interpret date-t 0 bond price as a

random variable taking values on the interval [ D min ( t 0 , T ) , D max ( t 0 , T ) ] . There are different ways

to assign distribution to random bond price D mkt ( t 0 , T , ω ). A simple distribution that can be used for

D mkt ( t 0 , T , ω ) is uniform distribution. This distribution actually does not have any advantages or

drawbacks with respect to other types of probability distributions that can be applied for randomization.

Discrete approximation of the uniform distribution can be introduced as following. Fix a number m and

denote δ j = j / m , j = 0, 1, … m Defined numbers k and q are defined by inequalities

δ k ≤ R min ( t 0 , T ) < δ k + 1 and δ k + q – 1 ≤ R max ( t 0 , T ) < δ k + q

Then putting Ω j = { u { t 0 } : δ j ≤ R spot ( u , T ) < δ j + 1 } we put P { Ω j } = q – 1 ,

j = k + 1, k + 2, … k + q – 1. Note that numbers k and q depends on t 0 . This is a heuristic distribution

that approximates real prices of the bond observed over the date t 0 .

Other more practical way of randomization of the date-t 0 pricing data is an assumption that distribution

P ( Ω j ) is proportional to the time when spot price

P ( Ω j ) ~ measure { u { t 0 } : R spot ( u , T ) [ δ j , δ j + 1 ] }

We also can use other distributions that approximated prices of the bond. These randomizations are

assigned to a particular date. The benchmark pricing uses one close price to represent a date t 0 bond

price. Such approximation can be good if the value D max ( t 0 , T ) – D min ( t 0 , T ) is sufficiently small.

Otherwise a lot of real pricing information will be lost. Pricing in stochastic environment implies market

risk regardless the chance of default. For example buyer’s market risk is measured by the probability that

buyer pays a higher price for the bond than it is implied by the market. If spot price at t 0 is identified as

the close price of the date t 0 then market risk of the buyer is the probability

P { R close ( t 0 , T ) > R ( t 0 , T , ω ) } = ) T , t ( R - ) T , t ( R

) T , t ( R - ) T , t ( R

0min0max

0close0max

Let us define recovery rate and default distribution implied by the stochastic price of the corporate bond

R ( t 0 , T ; ω ). Assuming for example uniform distribution of the recovery rate at maturity we arrive at

the formula

R ( t 0 , T , ω ) = δ j B ( t 0 , T ) , ω Ω j

P ( Ω j ) = q – 1, j = k + 1, … k + q – 1

Hence given the date-t 0 pricing data of the risky bond δ j = δ j ( t 0 ) , k = k ( t 0 ) , q = q ( t 0 ) we can

calculate credit risk based on prior assumed uniform distribution.

Remark. Randomization of the date t spot price of the bond is a primary assumption of our approach to

credit risk valuation. In modern finance it is common rule to use historical data time series as independent

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observation. It is not quite accurate interpretation of the statistical sample. Indeed consider for example

rate of return on an asset S

i ( t , T ) = )t(S

)t(S)T(S

and let t j < t j + 1 , j = 0 , 1, … n – 1 be a partition of the interval [ t , T ]. Then historical data

represented by time series i ( t j , t j + 1 ) is usually interpreted as independent observation over real rate of

return. Given historical data one should first to complete test of independence. Second unknown historical

data such as rate of return is assumed to present independent observation of the random variable. Hence

unknown parameters mean and variance are assumed to be constant. If not one should expect a large

deviation between model and real data. Also if parameters of the model could not be assume to be

constant one can observe other effects such as ‘fat’ tails. It can occur because variance depends on time

and fuzzifies data.

Consider corporate zero coupon bonds that admit default at any moment during its lifetime. Recall

approach that leads us to exponential distribution of the default moment. Introduce no default probability

distribution function P ( t ) = P { > t } and let t 0 < t 1 <…< t n = T be a partition of the interval

[ t 0 , T ]. Function P ( t ) is monotonic decreasing function in variable t. Our problem is: given no default

up to the moment t j to calculate probability that there is no default up to the future moment t k , k > j. A

solution of the problem can be represented by the conditional probability

P ( t k , t j ) = P { > t k | > t j }

Bearing in mind that default time depends on initiation date t 0 all functions defined bellow are also

depend on t 0 . Bearing in mind that

P { > t k > t j } = P { > t k } = P ( t k )

we note that

P ( t k , t j ) = }tτ{P

}tτ{P

j

k

=

)t(P

)t(P

j

k

The probability of default on ( t j , t k ] is then equal to

Q ( t k , t j ) = 1 – P ( t k , t j ) = 1 – )t(P

)t(P

j

k = )t(P

)t(P-)t(P

j

kj

Putting t j = t and t k = t + Δ t we arrive at the equality

Q ( t + Δ t , t ) = – )t(P

)t('P Δ t + o ( Δ t )

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Denote λ ( t ) = )t(P

)t('P . Then

P ( t ) = exp – t

t 0

λ ( s ) d s

There are other types of distributions that can be applied for default time modeling. These types are

Weibull distribution, Lognormal, Power, Gamma distributions. These distributions have multiple

parameters that can be used for better than exponential approximations of the default time. Function

λ ( s ) in exponential distribution is known as a hazard rate.

A discrete time approximation of the continuously distributed default time can be represented in the form

λ ( ω ) =

n

1k

t k χ { ( ω ) ( t k – 1 , t k ] } + T χ { ( ω ) > T }

For simplicity we assumes that zero coupon bond admits default at the dates t j . There are a few pricing

settlements of the corporate bond at the time of default. Fractional recovery of the market value

δ j B ( t j , T ) that is paid at t j . Other types of default settlements are the contingent claim of value

defined as δ j B – 1 ( t j , T ) paid at T and the fractional recovery of the Treasury value δ j at T. Given a

particular choice of the default settlement a discrete time approximation of the cash flows of the corporate

bond can be represented in one of the forms

CF A 1 ( ω ) =

n

1j

χ ( λ = t j ) δ j B ( t j , T ) χ ( t = t j ) + χ ( λ > T ) χ ( t = T )

CF A 2 ( ω ) =

n

1j

χ ( λ = t j ) δ j B – 1 ( t j , T ) χ ( t = T ) + χ ( λ > T ) χ ( t = T )

CF A 3 ( ω ) =

n

1j

χ ( λ = t j ) δ j χ ( t = t j ) + χ ( t > T ) χ ( t = T )

Here χ ( t = t j ) is indicator function in t which specifies value of transaction between buyer and seller

which takes place at t j . Cash flow CF A 1 defines recovery rates δ j of the corporate bond with respect to

the value of the risk free bond with equal maturity. Also note that as far as the bid and ask prices are

assumed in theory to be equal for a fixed maturity date T the bond price at a future date t j , t j < T does

not depend on time when bond was issued. In other words on the run risky or risk free bond issued at t j

and the similar bond issued prior to the date t j have the same price at the date t j . Note that theoretically

cash flows CF A k , k = 2, 3 can be rewritten in the CF A 1 form. Hence we assume that bond is

represented by the formula CF A 1 . The stochastic market price of the bond at t 0 can be represented as

the PV of the cash flow CF A 1 . Thus

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PV { CF ( ω ) } = R ( t 0 , T , ω ) =

n

1j

χ { = t j } R ( t 0 , T , ω ) + χ { > T } B ( t 0 , T ) =

=

n

1j

χ { = t j } B ( t 0 , t j ) δ j B ( t j , T ) + χ { > T } B ( t 0 , T ) = (7)

=

n

1j

χ { = t j } R ( t 0 , t j , ω ) B ( t j , T ) + χ { > T } B ( t 0 , T )

Next for writing simplicity we omit index λ and low index A 1 which specifies cash flow. For the fixed t 0

ratio δ j on the right hand side (7) depends on t j and does not depend on T . The recovery rate of the bond

at the date of default δ j B ( t j , T ) depends on T .

Risk factor that affects bond price (7) associated with default time distribution is referred to as credit risk.

Values B ( t j , T ) which are unknown at initiation date t 0 specify market risk of the recovery rate of the

corporate bond. Indeed replacing values B ( t j , T ) in formula (7) by market implied forward discounting

rate B ( t j , T , t 0 ) we use date-t 0 estimate of the real rate which arrives at the date of default. Future

value of the CF ( ω ) is equal to

FV { CF ( ω ) } =

n

1j

χ ( λ = t j ) δ j B ( t j , T ) B – 1 ( t j , T ) + χ ( λ > T ) =

=

n

1j

χ ( = t j ) δ j ( ω ) + χ ( > T )

No arbitrage pricing on [ t 0 , T ] we defined in [1] by equality

B – 1 ( t 0 , T ) PV { CF ( ω ) } = FV { CF ( ω ) }

where cash flow CF was associated with risk free bond. Spot price of the risk free bond is assumed here

to be a constant while risky bond spot price is a random variable. Therefore

B – 1 ( t 0 , T ) PV { CF ( ω ) } = B – 1 ( t 0 , T ) R ( t 0 , T , ω ) =

= B – 1 ( t 0 , T ) [

n

1j

χ ( = t j ) B ( t 0 , t j ) δ j B ( t j , T ) + χ ( > T ) B ( t 0 , T ) ] =

=

n

1j

χ ( = t j ) B – 1 ( t j , T , t 0 ) δ j B ( t j , T ) + χ ( > T ) =

=

n

1j

χ ( = t j ) δ j ( ω ) + χ ( > T ) = FV { CF ( ω ) }

13

Therefore PV of the stochastic cash flow represents no arbitrage (stochastic) market price of the bond.

There is no market risk on [ t 0 , T ] as no arbitrage pricing excludes market risk. Nevertheless early

selling of the bond might represent market risk to counterparty. Indeed let us define market risk of the

corporate bond. We defined spot price as a random variable. By definition scenario ω is equal to a

particular price of the bond during trade time of the date t 0 . Note that prices of the bond at different

moments can be equal to each other. Investor buying bond at date t 0 does not know either exact moment

of default or recovery rate at this moment. Such uncertainty implies that counterparty could loss or makes

a profit based on realized market scenario. On the other hand price defined by (7) does not eliminate

credit risk which can be specified by a choice of default time distribution.

Remark. Recall that standard approaches to corporate bond pricing deal with expected value of the

market implied PV of the CF. This approach replace future rate by its market implied estimate. Such

reduction incorporates market risk. This risk is specified by the difference between real future rate and its

market implied estimate. For a small standard deviation it might be reasonable approximation of the

stochastic price R ( t 0 , T , ω ) to a single number R spot ( t 0 , T ). It can be open, close price, or for

example

R spot ( t 0 , T ) = E R ( t 0 , T , ω )

For each future date t j market risk is implied by the fact that B ( t j , T , t 0 ) ≠ B ( t j , T ). Applying a

stochastic model of the future rates [4] one can estimate a chance P { B ( t j , T , t 0 ) > B ( t j , T ) } or

P { B ( t j , T , t 0 ) < B ( t j , T ) } at the date t 0 . Such estimates can justify reduction of the stochastic

price to single number as a representation of the date-t 0 bond price.

Now let us consider a generalization of the method used for calculation recovery rate and probability of

default (6) to cover general discrete time pricing model. In formula (7) dates t j , j = 1,2 … n represent

possible dates of default. Suppose (in theory) that bonds with expiration dates at t j , j = 1, 2, … n are

available on the market. Applying (3) we arrive at the formula

( 1 – δ 1 ) χ { = t 1 } = 1 – )t,t(B

)ω,t,t(R

10

10

From this equality it follows that

( 1 – δ 1 ) P { = t 1 } = E [ 1 – )t,t(B

)ω,t,t(R

10

10 ]

( 1 – δ 1 ) 2 P { = t 1 } = E [ 1 –

)t,t(B

)ω,t,t(R

10

10 ] 2

Solving the system for recovery rate and default probability we arrive at the values

14

δ 1 = 1 –

])t,t(B

)ω,t,t(R1[E

])t,t(B

)ω,t,t(R1[E

10

10

2

10

10

,

P { = t 1 } = ( 1 – δ 1 ) – 1 E [ 1 –

)t,t(B

)ω,t,t(R

10

10 ] = (8)

= 2

1010

21010

])ω,t,t(R)t,t(B[E

}])ω,t,t(R)t,t(B[E{

,

P { > t 1 } = 1 – P { = t 1 }

Bearing in mind Jensen’s inequality it is easy to verify inequality

P { = t 1 } = 2

1010

21010

])ω,t,t(R)t,t(B[E

}])ω,t,t(R)t,t(B[E{

≤ 1

Formulas (8) represent first term on the right hand side (7) which corresponds to a set of scenarios

D 1 = { ω : = t 1 }. Next we note that value of probabilities P ( D j ) = P { ω : = t j } and

correspondent recovery rates could be adjusted by taking into account that { > t j – 1 }, j = 2, 3, … n .

Such adjustment implies on credit risk exposure and it can be realized by applying conditional probability

P { λ = t j | > t j – 1 }. Taking into account inclusion

{ > t j } { > t j – 1 }

we can conclude that

P { λ = t j + 1 | > t j , … > t 1 } = P { λ = t j + 1 | > t j }

Given that { ( ω ) > t j – 1 }, j = 2, 3, … n we use estimate of the price of the bond at the moment t j – 1

provided by market implied forward value. Definition of the market implied forward risky discount rate

which in contrast to the risk free discount rate should take into account default time distribution. It

follows from (7) that

R ( t 0 , T , ω ) χ { = t j } = R ( t 0 , t j , ω ) B ( t j , T ) χ { = t j } (9)

j = 1, 2, … n. Here B ( t j , T ) = B ( t j , T , ω ) is future rate known only at t j and it is unknown at t 0.

Let us apply market implied forward rate as an estimate of the real forward rate. Then we arrive at the

reduction which decompose bond price

R ( t 0 , T , ω ) χ { = t j } = [ δ j B ( t 0 , t j ) ] B ( t j , T , t 0 ) χ { = t j } =

= R ( t 0 , t j , ω ) B ( t j , T , t 0 ) χ { = t j }

Solving latter equation for R ( t 0 , t j , ω ) brings us the formula

15

R ( t 0 , t j , ω ) χ { = t j } = B – 1 ( t j , T , t 0 ) R ( t 0 , T , ω ) χ { = t j } (9)

Note that transition from (9) to (9) implies market risk and the fact that bonds R ( t 0 , t j , ω ) , j = 1, 2,

… n – 1 might not exist on the market. On the other hand taking into account equality

R ( t 0 , t j – 1 , ω ) χ { = t j } = B ( t 0 , t j – 1 ) χ { = t j }

we define market implied forward risky discount rate R ( t j – 1 , t j ; t 0 , ω ) , ω { = t j } by equality

R ( t 0 , t j , ω ) χ { = t j } = R ( t 0 , t j – 1 , ω ) R ( t j – 1 , t j , t 0 , ω ) χ { = t j } =

= B ( t 0 , t j – 1 ) R ( t j – 1 , t j , t 0 , ω ) χ { = t j }

Bearing in mind that

R ( t 0 , t j – 1 , ω ) χ { > t j – 1 } = B ( t 0 , t j – 1 ) χ { > t j – 1 }

we conclude that

R ( t j – 1 , t j , t 0 , ω ) χ { = t j } = B – 1 ( t 0 , t j – 1 ) R ( t 0 , t j , ω ) χ { = t j } = (10)

= B – 1 ( t 0 , t j – 1 ) R ( t 0 , T , ω ) B – 1 ( t j , T , t 0 ) χ { = t j }

Market implied forward discount rate R ( t j – 1 , t j , t 0 , ω ) is represented by forward starting bond that

admits default only at its maturity t j . We can now apply estimates represented in formula (8).

Applying formulas (8) to the function R ( t j – 1 , t j , t 0 , ω ) defined by (10) we arrive at the formulas

δ j ( t 0 ) = 1 –

])t,t,t(B

)t,T,t(B)ω,T,t(R)t,t(B1[E

])t,t,t(B

)t,T,t(B)ω,T,t(R)t,t(B1[ E

0j1-j

0j1

01-j01

R

2

0j1-j

0j1

01-j01

R

(11)

with correspondent conditional probabilities of default

P { = t j | > t j – 1 } = ( 1 – δ j ( t 0 ) ) – 1 [ 1 – E

)t,t,t(B

)t,T,t(B)ω,T,t(R)t,t(B

0j1-j

0j1

01-j01

]

No default conditional probability is then equal to

P { > t j | > t j – 1 } = 1 – P { = t j | > t j – 1 }

Unconditional probability of default at the date t j and no default over lifetime of the bond could be

defined by the formula

P { = t j } = P ( = t j | > t j – 1 )

1j

0k

[ 1 – P ( = t k | > t k – 1 ) ]

16

(11)

P { > T } = 1 –

n

1j

P ( = t j | > t j – 1 )

1j

0k

[ 1 – P ( = t k | > t k – 1 ) ]

Formulas (1.11), (1.11) represent recovery rate and probability of default values of the zero coupon

corporate bond in discrete time setting.

Remark. Let us add some remarks on randomization. We generalized spot price notion of the bond.

Usually date-t price of the bond is associated with a particular price such as close price of the bond at the

date t. Our approach interprets bond price at date t as a random variable R ( t , T , ω ) taking values in the

interval

[ }t{u

min

R ( u , T ) , }t{u

max

R ( u , T ) ]

Therefore market scenario associated with a particular value of the bond, i.e. ω = R ( u , T ) for a some

moment u { t }. The use of a fixed price such as close price at date t should be interpreted as an

approximation of the random variable implies market risk. If we consider random price at a future

moment t + Δ t then the probability space

[}tt{u

min

R ( u , T ) , }tt{u

max

R ( u , T ) ]

does not coincide with the one that is defined at t. Therefore dealing with dynamic market we should

introduce a unique probability space. Next we associate probability space Ω R with the set of measurable

functions ω ( t ) such that

}t{umin

R ( u , T ) ≤ ω ( t ) ≤ }t{u

max

R ( u , T ) }

t [ t 0 , T ]. Prior to expiration date values of the corporate bond does not represent default event. Hence

time of default of the bond can be thought as an internal company factor and it can be interpreted as a

random variable. As far as values of the bond do not completely define time of default distribution the

probability space Ω ( R ) should not interpret default time ( ω ) ( t 0 , + ) , ω Ω . The term ‘

should not interpret default time’ suggests that in general ( ω ) does not measurable function on Ω ( R ).

We introduce measurable space { Ω , B } where Ω = Ω ( R ) Ω , B = B ( R ) B . Here

symbol B denote σ-algebra of Borel sets on correspondent space. Probability measure P on measurable

space { Ω , B } is defined by equality

P ( A Δ ) = P { R ( t , T , ω ) A , ( ω ) Δ }

A B ( R ), Δ B . For a random variable ξ ( ω ) on { Ω , B , P } denote

expectation with respect to market scenarios generated by the values of the bond while default time

remains stochastic. Such construction implies that values of the bond and default time are mutually

17

independent random variables and symbol E R denotes conditional expectation with respect to -algebra

generated by the default time

E R ξ ( ω ) = E { ξ ( ω ) | B }

Note that it is quite a strong assumption and it is very popular in credit risk valuations.

Corporate coupon bearing bond pricing. Zero coupon corporate bonds pricing scheme can be applied

to coupon bearing bond. Suppose for simplicity that default dates coincide with coupon payment dates

T 1 < T 2 < … < T m = T. Then cash flow of the coupon bearing bond from bond to bond buyer

perspective can be written in the form

CF ( ω ) =

m

1j

χ ( = T j ) {

1-j

1k

c χ ( t = T k ) + δ j [ c

m

ji

B ( T j , T i ) +

+ B ( T j , T m ) ] χ ( t = T j ) + χ ( > T ) [

m

1j

c χ ( t = T k ) + χ ( t = T m ) ]

In this formula recovery rate is assigned to outstanding debt value defined at default moment. The PV

reduction at t of the cash flow CF ( ω ) defines market price of the corporate bond depending on market

scenario which incorporate unknown values of the risk free bonds B ( T j , T i ) at future moments T j , T i

and time of default. Assume that corporation simultaneously issued zero coupon and coupon bonds. In

this case investors can expect that loss-ratio due to default of the zero and nonzero coupon bonds are

equal. Otherwise there exists hypothetical arbitrage opportunity. The notion ‘equal loss-ratio’ in latter

statement should be refined. We interpret bond default as default of the corporation and therefore default

of the corporation affects on all corporate bonds issued by this corporation. One should assume that

probability distribution of default time for corporate zero and nonzero coupon bonds are the same. Recall

that recovery rate of the zero coupon bond conditional on default at T j can be represented in the form

c δ j ( t , ω ) B ( T j , T , ω ), which is unknown at t and can be estimated by deterministic value

δ j ( t ) B ( T j , T , t ) . Hence recovery rate of the zero coupon corporate bond is interpreted as a portion

δ j ( t ) of the date-T j PV of the face value of the bond. Following this idea we define recovery value of

the coupon bond at T j as a portion δcj ( t ) of the outstanding balance of the coupon bond on [ T j , T ].

Hence, recovery rate of the coupon bond can be defined as

δ cj [ 1 + c

m

1 ji

B – 1 ( T i , T ) + c B – 1 ( T j , T ) ]

The value δ cj we call recovery ratio. For the zero coupon bond, c = 0 recovery rate notion coincides with

recovery ratio. Total losses due to default are equal to

( 1 – δcj ) [ 1 + c

m

1 ji

B – 1 ( T i , T ) + c B – 1 ( T j , T ) ]

18

Theoretically the value of the losses at T of the zero and nonzero coupon bonds of the corporation that

defaults at T j should be equal for each scenario. In this case we eliminate arbitrage opportunity between

zero and non zero coupon bonds for each scenario. Hence

1

δ1

)T,T(Bc1

])T,T(Bc)T,T(Bc1[)δ1(j

i1

m

1ji

j1

i1

m

1ji

cj

Solving latter equation for the recovery rate of the coupon bond rate we arrive at the formula

δ cj ( t , ω ) = 1 –

)T,T(B)T,T(Bc1

])T,T(Bc1[))ω,t(δ1(

j1

i1

m

1ji

i1

m

1jij

Market implied estimate of the above stochastic recovery rate can be represented as

δ cj = 1 –

)t,T,T(B)T,T(Bc1

])t,T,T(Bc1[)δ1(

j1

i1

m

1ji

i1

m

1jij

Recall that using market implied forward estimate δ cj implies market risk as far as δ c

j ( t , ω ) ≠ δ cj .

19

References.

1. I. Gikhman, BASIC OF PRICING 1, http://www.slideshare.net/list2do/basic-pricing ,

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2428024.

2. I. Gikhman, Corporate Debt Pricing, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1163195 3. Multiple Risky Securities Valuation I,

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1944171

4. Fixed Rates Modeling. 2013 p. 25,

http://www.slideshare.net/list2do/fixed-rates-modeling

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2287165