1.
1
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.
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
3.
3
}t{s
max
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
4.
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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
5.
5
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
6.
6
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 ) }
7.
7
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(R
1[E
]
)T,t(B
)ω,T,t(R
1[E
bid
bid
2
bid
bid
, P ( d ) =
2
bid
bid
2
bid
bid
]
)T,t(B
)ω,T,t(R
1[E
}]
)T,t(B
)ω,T,t(R
1[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.
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 )
9.
9
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 , ) }
10.
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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
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