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MARKET RISK and LIQUIDITY OF THE RISKY BONDS.

Ilya Gikhman

6077 Ivy Woods Court

Mason OH 45040 USA

Ph. 513-573-9348

Email: ilyagikhman@yahoo.com

Key words. Corporate bond, liquidity spread, reduced form approach,

Abstract. In this paper, we present effect of the liquidity on risky bonds pricing. The liquidity effect is

represented by the adjustment to the single price format. We begin with bid-ask pricing format and it

helps to observe effect of the liquidity on each step of pricing. In the first section, we present simplified

scheme when default can occur only at maturity. Next, we present discrete time approximation for default

occurrence.

I. Let B k ( t , T ) and R k ( t , T ) denote k = bid, ask prices of the risk free and a corporate bond prices at

t [ 0 , T ] correspondingly with maturity date T. Consider the reduce form of default setting to study

pricing problem. First, let us briefly recall a single price framework. Next, we extend this approach to

account the liquidity spread. Liquidity spread of the corporate bond is defined as

R ask ( t , T ) – R bid ( t , T )

The difference between the corporate bond price R ( t , T ) and corresponding bid-ask bond prices

represent liquidity premium. For illustration for either single price or bid-ask prices we begin with a

simplified assumption that default can occur at the last moment of the lifetime of the bond. In such

setting, we introduce the main notions used in the reduced form of default. There are two real types of the

market prices. One is the purchase price and other is selling price. A single price is usually denote the

middle price of these two

R ( t , T ) = 2

1[ R ask( t , T ) – R bid( t , T ) ]

In the single pricing format the reduce form the default event is defined as following

2

1 , if no default

R ( T , T ) = {

, if default

< 1. In order to interpret R ( T , T ) as a random variable we need to add probabilities of the no default

and default scenarios. Let d and 0 denote default and no default scenarios. Then R ( T , T ) =

= R ( T , T , ) is a random variable on probability space = { d , 0 }, P ( d ) = p d ,

P ( 0 ) = p 0 , p d + p 0 = 1. For each scenario there exists unique corporate bond price

defined as

R ( t , T , ) = 1 B ( t , T ) ( 0 ) + B ( t , T ) ( d ) = B ( t , T ) [ 1 – ( 1 – ) ( d ) ] (1)

Formula (1) defines the market price of the bond at date t that depends on a market scenario. The other

price is the spot price at t, which one should apply purchasing or selling bonds. Market participants define

the spot price of the bond, which reflects scenario distributions. Let R spot ( t , T ) denote the spot price of

the corporate bond at t. Then the market risk for buyer and seller of the bond are

P { R ( t , T , ) < R spot ( t , T ) } , P { R ( t , T , ) > R spot ( t , T ) } (2)

correspondingly. The first probability represents value of the chance that that a buyer of the corporate

bond paying R spot ( t , T ) is overpaid as far as realized scenario suggests lower price. Similarly, the

second probability is the value of the chance that the seller of the bond sells it cheaper than it implies by

the market scenario. Equalities (2) are the base of the risk management.

Remark 1. Note that we deal with the real probabilities while primary default studies use the risk neutral

distribution. Recall that initially the risk neutral distribution was introduce in order to establish connection

between real stock and the heuristic underlying implied by the Black Scholes formula. It is clear that the

reason to change probability measure is only to help to hide the heuristic underlying in option pricing.

Under the risk neutral measure, one can use initial underlying which is defined on original probability

space that in finance called the real probability space.

In theory, we assume that every parameter of the model is given. Once, there is no additional market

information is available the formula

R spot ( t , T ) = E { R ( t , T , ) | F [ 0 , t ] } = E R ( t , T , ) (3)

can be considered as definition of the spot price. Here, F [ 0 , t ] denotes -algebra generated by the bonds

price values over the time [ 0 , t ]. If other stochastic bonds on the market then -algebra F should include

all this information. Hence, in general the -algebra F [ 0 , t ] might be larger than the -algebra that

generated by the random process R ( t , T , ).

In practice, we observe the prices Rspot ( t , T ), B spot ( t , T ). Our problem is to define the recovery rate

, default distribution, and market price of the corporate bond at any moment t, t T. In practice the

spot price at t is assigned to the close price of the day t. This reduction of the whole date t prices of the

bond to a single price makes sense if the value

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}t{smax

R ( s , T ) – }t{s

min

R ( s , T )

is sufficiently small. Here { t } denote the date t time interval. If this difference does not small the price

reduction of the whole period t to a single number R ( t close , T ) can be not appropriate approximation

which lose too much of the real risk. Let R ( t , T ) and B ( t , T ) denote close prices of the risky and risk

free bonds. Consider a problem of estimation of the recovery rate and probability of default p d given

that default of the risky bond can occur at maturity T. It is clear that observation of the spot prices

R ( s, T ) and B ( s, T ) up to the moment t do not provide us default information. One needs an additional

assumption to establish a connection between observed data and the market price R ( t , T , ) which

helps to estimate default parameters.

Remark 2. The standard reduce form of default assume that the time of default has distribution equal to

the distribution of the first jump of the Poisson process. This assumption contradicts our hypothesis that

default occurs at T only. Indeed, the continuous distribution of the first jump of the Poisson process

prescribes the probability 0 to the first jump at a fixed date. Indeed, let ( t ) = P { > T } we note

that

P { d } = 0u

lim

[ ( T ) – ( T – u ) ] = 0

Therefore, it looks more correctly to say that : the probability of event ‘ the first jump of the Poisson

process occurs until date T ‘ is assigned to probability that default occurs at the date T, and

P { > T } = P { 0 } = 1 – P { d }

Then

( t ) = P { > T } = P { t < t + u } + P { > t + u } =

= P { t < t + u } + ( t + u )

and therefore

P { t < t + u } = ( t ) – ( t + u )

By the definition of the conditional probability, we note that

P { t + u | > t } = )tτ(P

)utτt(P

=

)t(Π

)ut(Π)t(Π

Assume that ( t ) is continuously differentiable function and there exists a continuous function h ( t ) for

which

)t(Π

)t(Πd = h ( t ) d t

Then

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P { t < t + u } = )t(Π

)t(Πd + o ( u )

and ( t ) = exp – t

0

h ( u ) d u . Hence,

P ( 0 ) = exp – T

0

h ( u ) d u = 1 – P ( d ) (4)

The problem is to find estimates for probability of default and recovery rate. If we assume that recovery

rate is an unknown constant then from (1) and (3) it follows that

R ( t , T ) = B ( t , T ) [ 1 – ( 1 – ) P ( d ) ] (5)

The equation (5) has two unknowns , P ( d ).

Remark 3. Let us recall resolution of the similar problem by the standard reduced form approach

following [3, p.561-562]. In the example, they used notations:

t = 0 , T = 1 , R ( t , T ) = ( t , T ) , B ( 0 , 1 ) = A ( 1 ) , p d = λ ( 0 ) , p 0 = 1 – λ ( 0 )

In the modern finance theory, the market risk does not exist in the theory. The spot prices are considered

with respect to whether they admit or do not admit arbitrage. Mo arbitrage prices are interpreted as

‘perfect’ or ‘no-free lunch’. We should point on the fact that theoretical no arbitrage pricing also admits

market risk, i.e. possibility to lose money. The ‘perfect’ are in an estimate of prices in stochastic market

and also implies invisible for Black Scholes theory market risk. In order to present solution of the

equation (5) one usually supposes [3] that the value comes from our credit risk analysts or from the

Moody’s Special Report. This is a quite subjective approach and other value of the recovery in (5)

presents other probability of default.

Note that formula (5) is a particular case that follows from (1). Assuming here that recovery rate is

unknown constant from the formula (1) it follows a formula for the higher order moments of the market

price. In particular

E R 2 ( t , T , ) = B 2 ( t , T ) [ 1 – ( 1 – 2 ) P ( d ) ] (5)

The left hand side of the formula (5) is the spot rate, which is observable variable while the left hand side

of the equation (5) does not observable variable. Thus, one need to provide randomization of the

problem setting and represent a construction of the random process R ( t , T , ). For illustration,

consider a numeric example.

Example. Let the face value of the risky and risk free bonds are 100 and R ( t , T ) = R close ( t , T ) =

= 94.5, B ( t , T ) = B ( t , T ) = 98, and risky spot price R ( t , T ) is defined by formula (3). An

example of admissible distribution of the random variable R ( t , T , ) is the uniform distribution on the

interval

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I ( t , T ) = [ R min ( t , T ) , R max ( t , T ) ]

where the end points of the interval are minimum and maximum of the corporate bond prices during the

day t. Let R min ( t , T ) = 94, R max ( t , T ) = 96. The density of the uniform distribution on [ 94, 96 ] is

0.5. Consider a model in which is interpreted as an unknown constant. Then

R ( t , T ) = E R ( t , T , ) = 96

94

0.5 x d x = 95 , E R 2 ( t , T , ) = 96

94

0.5 x 2 d x = 9025

The system (5), (5) can be rewritten in the form

0.969 = 1 – ( 1 – ) p d , 0.94 = 1 – ( 1 – 2 ) p d

The solution of the system is the values = 0.945 , p d = 0.564. Correspondent risk management is

represented by equalities (2). Then the buyer and seller risks are equal correspondingly to

P { R ( t , T , ) < 95 } = 95

94

0.5 d x = 0.5

P { R ( t , T , ) > 95 } = 96

95

0.5 d x = 0.5

It is easy to calculate other risk characteristics. The average profit-loss exposure of the buyer is defined as

E R ( t , T , ) { R ( t , T , ) > R spot ( t , T ) } , E R ( t , T , ) { R ( t , T , ) < R spot ( t , T ) }

Similarly to uniform distribution one can use Gaussian distribution with mean equal the middle point of

the interval I ( t , T ) and 3 = R max ( t , T ) – R mid ( t , T ). Here the low index ‘mid’ stands for the

middle point of the interval I ( t , T ). One can also present the risk adjustment to close prices used as the

spot price for the date t. At the end of the trading day one can make a conclusion regarding implied

recovery rate and probability of default. Consider general case that does not reduce the recovery rate to a

constant. Given the data of the example, define the market scenarios set by putting

= ( x ) , x I = [ 94 , 96 ]

For each value x I define the one-to-one preset value (PV) correspondence x ↔ B – 1 ( t , T ) x. Then

the PV image of the interval I will be the range of the recovery rate

I PV = [ B – 1 ( t , T ) R min ( t , T ) , B – 1 ( t , T ) R max ( t , T ) ] = [ 95.92 , 97.96 ]

Uniform distribution defined on [ 94 , 96 ] with probability density 0.5 will be converted to the uniform

distribution on [95.92 , 97.96 ] with the constant probability density equal to

( x ) = [ 97.96 – 95.92 ] – 1 ≈ 0.4902

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The interval I PV supplied by the probabilistic density ( x ) are complete reduced form of default

information related to the risky bond. Any particular spot price of the bond R spot ( t , T ) = x can be

converted into correspondent value of the recovery rate = B – 1 ( t , T ) x. For example, the credit risk

of the buyer is equal to

P { R ( t , T , ) < x } = x

94

0.5 d x = 98

92.95

x

0.4902 d

Consider liquidity adjustment of the reduced form pricing of risky bonds presented above in the single

price format. We discussed liquidity adjustment in [5]. The traded liquidity is represented by the

difference between bid and ask prices and a single price that is usually interpreted as the middle price.

The difference also called liquidity premium. Given that default can occur only at maturity the liquidity

adjustment can be easily calculated. In the bid-ask format default of the corporate bond can be defined as

R bid ( T, T , d ) = R ask ( T, T , d ) = < 1

R bid ( T, T , 0 ) = R ask ( T, T , 0 ) = 1

Here the recovery rate is a known constant. Nevertheless, an implementation of the model in stochastic

market suggests that each scenario ω realization R k implies at least in theory a unique value of the

recovery rate δ that promises the rate of return equal to risk free rate, i.e.

δ

)ω,T,t(R k = )T,T(B

)T,t(B

k

k

Hence, δ should be interpreted as a random variable. Assume for a while that recovery rate is unknown

constant. Then

R k ( t , T , ) = [ 1 – ( 1 – ) ( ) ] B k ( t , T ) (6)

k = bid, ask , = { d , 0 }, and t [ 0 , T ]. The random R k ( t , T , ) is market price of the

risky bond at t. For each market scenario formula (6) presents a unique value of the risky bond. It is

common practice to use close or open prices as date t asset price. In this case, this price is implicitly

considered as an approximation of the prices price for the whole date t. Such approximation makes sense

if the values

R k max ( t , T ) – R k min ( t , T)

k = bid, ask is sufficiently small. Here R k max ( t , T ) , R k min ( t , T ) are maximum and minimum bid an

ask values of the bond at date t. For each R k ( t , T , ) one can apply methods used earlier for the single

price R ( t , T , ). Along with the market price R k ( t , T , ) there exist the spot prices of the bond,

which are non random constants R k spot ( t , T ) , k = bid, ask. Formulas (2) in bid-ask format can be

rewritten as

P { R ask ( t , T , ) < R ask spot ( t , T ) }

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is the value of the chance that buyer overpaid for the bond while the probability

P { R bid ( t , T , ) > R bid spot ( t , T ) }

represents the value of the chance that seller of the bond sells it for the lower price than it implies by

market scenarios.

Assume that is an unknown constant in the system (5), (5). Solving the system for bid prices we arrive

at the solution

= 1 –

])T,t(B

)ω,T,t(R1[E

])T,t(B

)ω,T,t(R1[E

bid

bid

2

bid

bid

, P ( d ) = 2

bid

bid

2

bid

bid

])T,t(B

)ω,T,t(R1[E

}])T,t(B

)ω,T,t(R1[E{

(7)

From (6) it follows that

)T,t(B

)ω,T,t(R

)T,t(B

)ω,T,t(R

ask

ask

bid

bid (8)

This equality makes sense under the simplified assumption that default time ( ) = T. The solution (7)

for the ask prices brings the same values for recovery rate and default probability. In case, when

observations show inequality of the right and left hand sides of the latter equality one can conclude that

the simplified assumption does not look realistic. Nevertheless, given the assumption that ( ) = T let

us note that probability of default as well as recovery rate are calculated similar to the single price format

by using either bid or ask prices. Note that from equality (8) does not equal to the middle bid-ask price.

This observation might suggest that making adjustment for liquidity starting from single pricing model

we need to be accurate.

We can present a liquidity solution of the risky bond pricing in case when recovery rate is a random

variable and default can be observed at maturity. Let 0 = 0 < 1 < … < n = 1 be a partition of the

interval [ 0 , 1 ]. Introduce a discrete approximation of the recovery rate

=

1n

0j

j χ { δ ( ω ) [ j , j + 1 ) }

Denote p j = P { δ ( ) [ j , j + 1 ) }. Then

E [ 1 – )T,t(B

)ω,T,t(R

bid

bid ] n =

1n

0j

( 1 – j ) n p j

Solving this linear algebraic system with respect to p j we arrive at the discrete approximation of the

continuous density of the random variable δ ( ). Let us estimate the value of the liquidity spread. Note

[ R ask ( t , T , ) – R bid ( t , T , ) ] – [ B ask ( t , T ) – B bid ( t , T ) ] =

8

= [ B ask ( t , T ) – B bid ( t , T ) ] ( 1 – δ ( ) ) χ ( d )

Therefore

)T,t(B-)T,t(B

)ω,T,t(R-)ω,T,t(R

bidask

bidask – 1 = ( 1 – δ ( ) ) χ ( d )

In the case, when recovery rate is interpreted as unknown constant we arrive at the simple formula

E )T,t(B-)T,t(B

)ω,T,t(R-)ω,T,t(R

bidask

bidask – 1 = ( 1 – δ ) P ( d )

This formula represents relative value of the risky liquidity spread with respect risk free liquidity spread

given that default can occur at maturity.

II. Let us construct implied distribution of the continuous distributed default time of a risky bond.

Consider a case when default can be observed during discrete moments t 1 < t 1 < … < t N = T during

the lifetime of the risky bond. Then given the observations over risky and risk free bonds prices the

problem is to draw implied estimates for the recovery rates δ ( t j ) given that default will occur at t j as

well as the of default P { τ j = t j } and j = 1, 2, … N. Introduce a discrete approximation τ λ of the

continuous default time. Denote

τ λ =

n

1j

t j { τ ( t j – 1 , t j ] }

First, we briefly recall pricing in single price format. Market price at t of the corporate bond can be

written as

R λ ( t , T , ) =

N

1j

R ( t , t j , ) χ ( τ λ = t j ) + B ( t , T ) χ ( τ λ > T ) (8)

Next for writing simplicity, we omit index λ. Here R ( t , t j , ) denote market price of the same

corporate bond with expiration at t j . Bearing in mind results of the previous section equality (8) can be

rewritten in the form

R λ ( t , T , ) =

N

1j

δ ( t , t j , ) B ( t , t j ) χ ( τ λ = t j ) + B ( t , T ) χ ( τ λ > T ) (8′)

Remark 4. Equality (8) implies that there is no seniority of the corporate bond with respect to maturity of

the corporate bonds, i.e. the bonds with maturity t k , k = j , j + 1, … N default at t j with equal recovery

rates δ ( t , t j , ). This assumption can be formally represented by the formula

R ( t , t k , ) χ { ( t , t k ) = t j } = R ( t , t j , ) χ { ( t , t j ) = t j } = R ( t , , ) χ ( = t j )

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for all k ≥ j. When the latter equality does not hold the basic formula (8) should be adjusted in order to

present relationship between χ { ( t , t k ) = t j } for k ≥ j.

When default occurs during the lifetime of the bond one can observe that

R ( t , T , ) =

N

1j

δ ( t , t j , ) B ( t , t j ) χ ( = t j ) + B ( t , T ) χ ( > T ) (8′)

Now let us look at liquidity adjustments for the corporate bond. Given that corporate bond can default

prior to maturity we can usually observe inequality

B bid ( t , T ) – R bid ( t , T , ) ≠ B ask ( t , T ) – R ask ( t , T , )

that suggests that probability of default and recovery rate implies ask and bid prices are different in

reduced form of default pricing. This observation suggests different values of the basic risk parameters for

ask and bid prices of the corporate bond. Suppose that right hand side of the latter inequality is smaller

than the right hand side. It can then interpreted as liquidity of the bid prices is better than liquidity of the

ask prices. It might also suggest that in future corporate bond prices will move to the direction that will

promise the equality of the bid-ask liquidities. The constructions developed for the single price

framework can be applied for the bid and ask curves separately. It will lead to the spot liquidity spread

which value is risky. Risk of the long liquidity is

P { R ask ( t , T ) – R bid ( t , T ) > R ask ( t , T , ) – R bid ( t , T , ) }

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References.

1. I. Gikhman. FX Basic Notions and Randomization.

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1964307.

2. I. Gikhman. A Comment On No-Arbitrage Pricing.

3. J. Hull, Options, Futures and other Derivatives. Pearson Education International, 7ed. p. 814

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2195310.

4. I. Gikhman. STOCHASTIC DIFFERENTIAL EQUATIONS AND ITS APPPLICATIONS: Stochastic

analysis of the dynamic systems. ISBN-10:3845407913, LAP LAMBERT Academic Publishing, 2011, p. 252.

5. I. Gikhman. Market Risk of the Fixed Rates Contracts. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2270349