FIXED RATES MODELING.

Ilya Gikhman

6077 Ivy Woods Court

Mason OH 45040 USA

Ph. 513-573-9348

Email: ilyagikhman@yahoo.com

JEL : G12, G13

Key words. Forward rate agreement, Market risk, Stochastic Libor model, Interest Rate Swap, Liquidity of Corporate Bonds, Stochastic default intensity.

Abstract. This paper focuses on the concept of a discount rate. In [1] one expressed some concerns regarding the popular models of the randomization of the discount rates. This paper proposes a new approach to construction of variable deterministic and stochastic interest rates. This approach is based on the randomization of the forward rate consept. In [2] we introduced a synthetically construction price of the contracts associated with LIBOR. From our point of view the standard modeling of the LIBOR are rather empirical than formal.

I. Basic notations and definitions. Denote B ( t , T ) , 0 ≤ t ≤ T the price of zero

coupon default free bond price at date t with expiration at date T and B ( T , T ) = 1. The simple interest i s rate and discount rate i d are defined as

B ( t , T ) = [ 1 + i s ( t , T ) ( T – t ) ] – 1 = 1 - i d ( t , T ) ( T – t ) (1)

Here T – t is expressed in appropriate 365 or 360 day year format. Continuous time model of the bond price is governed by the equation

d B ( t , T ) = r ( t , T ) B ( t , T ) d t (1′)

Here the function r ( t , T ) > 0 in is called annual interest rate. The value of a coupon bond at t which pays coupon c at the moments t 1 < t 2 < … < t n = T is equal to

1

B c ( t , T ) = ∑j = 1

n

c B ( t , t j ) + F B ( t , T )

where F is a face value of the bond. It is useful to define a path dependent financial contract with the help of cash flow. For example the value of the coupon bond B c ( t , T ) can be interpreted as the present value at t of the cash flow

CF = ∑j = 1

n

c χ ( t = t j ) + F χ ( t = T )

where χ ( A ) is the indicator of the event A. It is equal to 1 when A is true and 0 otherwise. Thus,

B c ( t , T ) = PV t { CF }.

II. Forward rate agreement, FRA. Though the FRA pricing is well known we pay more

attention to some details that might occasionally be missed. FRA is a two party OTC contract for a future transaction. The value of the transaction is a notional principal multiplied by the difference between the realized reference rate and its estimate called implied forward rate.

Let t denote initiation date and the fixed and realized and implied forward rates are assigned for a future period [ T , T + H ], H > 0. A FRA contract can be specified as follows. Let

t 0 < t spot < t fixing < t settle < t mature be a set of the time moments. The date t 0 is called trade or deal date. On this date the FRA contract is specified as :

the spot date t spot is usually equal t 0 + 2 is the beginning and t settle = T is the end of the m’s period, i.e.

t settle - t spot = m (months)

Notional principal N,

FRA fixed rate, FRA ( T , T + H ; t 0 )

period specification m × H,

and the FRA time moments.

The date t fixing = T - 2 is usually two business days prior t settle = T is the date at which value of the floating (reference) rate over the period [ T , T + H ] , t mature = T + H. At the settlement date T the settlement (netted) sum

N [ L ( T , T + H ) – FRA ( T , T + H ; t 0 ) ] H

If the latter value is positive then the FRA seller pays it to the FRA buyer (FRA holder). If the value (2) is negative then the FRA buyer pays the value to the FRA seller.

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The FRA pricing problem is a determination of the fixed rate FRA ( T , T + H ; t 0 ). It is common practice to study a simplified scheme of this pricing problem. In such simplification one assumes that

t = t 0 = t spot and T = t fixing = t settle . Let us study the valuation problem in such simplified setting. A FRA contract can be interpreted as a forward loan over [ T , T + H ]. At the date T + H a borrower of the fund should pay the interest specified by the reference rate that usually LIBOR or other similar rate. The rate L ( T , T + H ) is known as T and therefore we can admit that FRA payoff is scheduled on the date T. A variation of the FRA when the payoff is scheduled for T + H. Therefore, the date-T or the date-( T + H ) value of the contract is

V ( t , T , T + H ) = N [ L ( T , T + H ) – FRA ( T , T + H ; t ) ] H (2)

The rate FRA ( T , T + H ; t ) should be determined at t while the rate L ( T , T + H ) is unknown at this date. It is the market practice to approximate unknown value L ( T , T + H ) by date-t implied forward rate l ( t , T + H ; t ) over the period [ T , T + H ]. The implied forward rate l ( t , T + H ; t ) is defined as

l ( T , T + H ; t ) =

1H

[1 − L ( t , T + H ) ( T + H − t )

1 − L ( t , T ) ( T − t )− 1 ]

Here L ( t , T ) denotes date-t spot reference interest rate with expiration at T. Applying implied forward rate l ( T , T + H ; t ) as an approximation of the unknown L ( T , T + H ) we replace the real transaction (2) by its implied approximation

v ( t , T , T + H ) = N [ l ( T , T + H ; t ) – fra ( T , T + H ; t ) ] H (2′)

Here, fra ( T , T + H ; t ) denotes ‘no-arbitrage’ solution of the reduced FRA pricing problem corresponding to еру payoff (2′). As far as the value of the implied form contract at the settlement for either buyer or seller should be equal to zero then

fra ( T , T + H ; t ) = l ( T , T + H ; t )

Then the buyer price at t is the discounted payment of the contract at T. Hence,

N L ( t , T ) l ( T , T + H ; t ) H

If the settlement date is the date T + H then in the latter expression discount factor L ( t , T ) should be replaced by the L ( t , T + H ). Note that no-arbitrage solution of the reduced FRA with the payoff (2′) does not coincide with the real world transaction defined by the actual payoff (2). This inequality implies the market risk of the FRA contract. The value of the risk stipulated by the value

δ l =def

L ( T , T + H ) - l ( T , T + H ; t ) ≠ 0

effective at the settlement date T. The market risk from the buyer perspective is associated with the market scenarios for which δ l < 0 while the scenarios for which δ l > 0 specify the seller risk.

Let us now consider randomization of the FRA pricing problem. There are two functions L ( T , T + H ) and l ( T , T + H ; t ) that are available for randomization. Suppose first that

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L ( T , T + H , ) = L ( t , t + H ) + ∫t

T

( s ) L ( s , s + H ) d s +

(3)

+ ∫t

T

( s ) L ( s , s + H ) d w ( s )

where coefficients , are deterministic or random functions, which satisfy the standard conditions that ensure the existence and the uniqueness of the solution of the Ito equation (3). The solution depends on the parameter H. Taking the limit when H tends to 0 we arrive at the instantaneous forward rate

L ( T , ) = L ( T , T + 0 , ) = P.lim

H ↓ 0 L ( T , T + H , )

which is the solution of the equation

L ( T , ) = L ( t ) + ∫t

T

( s ) L ( s , ) d s + ∫t

T

( s ) L ( s , ) d w ( s )

There exists another way to present a model for the future rate L ( T , T + H , ). It is based on the implied forward rate. At the fixed date t the function l ( T , T + H ; t ) is known. For the fixed t , T , and H let us consider a random function l ( T , T + H ; u ) of the variable u , which represents the value of the implied forward rate over the period [ T , T + H ] at a future date u [ t , T ]. Suppose that

l ( T , T + H ; u ) = l ( T , T + H ; t ) + ∫t

u

( v ) l ( T , T + H ; v ) d v +

(3′)

+ ∫t

u

λ ( v ) l ( T , T + H ; v ) d w ( v )

In (3′) put u = T and bearing in mind that l ( T , T + H ; T ) = L ( T , T + H , ) we arrive at the equation that defines unknown rate L ( T , T + H , ). Given randomization of the future rate L ( T , T + H ) in the form (3) we enable to present market risk in the form P { δ l > 0 } , P { δ l < 0 } for the FRA seller and buyer correspondingly. One can define the cumulative distribution to calculate primary market risk characteristics such as average profit and losses as well as its standard deviations.

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Remark 1. Let us first highlight the difference between the benchmark and our approaches to FRA valuation. Let us recall the benchmark FRA pricing following [3, p.87].

“ FRA can be valued if we:

1. Calculate the payoff on the assumption that forward rates are realized ( that is, on the assumption that R M = R F ).

2. Discount this payoff at the risk-free rate. “

Applying our notations we get the following R F = l ( T 1 , T 2 ; t ) , R M = L ( T 1 , T 2 ). Benchmark approach to FRA valuation makes sense in deterministic problem setting and it is ignoring the fact that R M ≠ R F in stochastic case. In stochastic setting we interpret the date-t known rate R F = l ( T 1 , T 2 ; t ) as an approximation of the unknown at t rate R M = L ( T 1 , T 2 ) . Let { A } denote indicator of the event A. Then the difference

[ l ( T 1 , T 2 ; t ) - L ( T 1 , T 2 ) ] { l ( T 1 , T 2 ; t ) > L ( T 1 , T 2 ) }

is the market risk of the FRA buyer while

[ l ( T 1 , T 2 ; t ) - L ( T 1 , T 2 ) ] { l ( T 1 , T 2 ; t ) < L ( T 1 , T 2 ) }

specifies market risk of the FRA seller. Note that

limt → T 1 l ( T 1 , T 2 ; t ) = L ( T 1 , T 2 )

Nevertheless, it is easy to verify in stochastic setting by using real data that implied forward rate does not equal to the value of the correspondent future rate, i.e. l ( T 1 , T 2 ; t ) ≠ L ( T 1 , T 2 ). The rate L ( T 1 , T 2 ) is the real rate known at T 1 while the rate l ( T 1 , T 2 ; t ) is an approximation or a statistical estimate of this rate. In stochastic setting the first step of the valuation is incorrect as far as R F = l ( T 1 , T 2 ; t ) is a known constant at t while the rate R M = L ( T 1 , T 2 ) is unknown at t and L ( T 1 , T 2 ) is assumed to be a random variable. Therefore, in general P { R M = R F } = 0. It is also important to note that in stochastic theory the probability distribution of the random variable L ( T 1 , T 2 ; ) is assumed to be known at t. This assumption is crucial for stochastic pricing. It leads to new comprehension of the price. The revision of the price notion comes from the fact that a continuous distribution prescribes probability 0 to any particular number regardless whether this number is ‘arbitrage free’ or other meaningful spot price. Complete price of the contract consists from two components. These are a specific spot price l ( T 1 , T 2 ; t ) along with the probability

P { L ( T 1 , T 2 ; ) < l ( T 1 , T 2 ; t ) }

The last probability represents a buyer's market risk while the probability of the inverse inequality represents seller’s risk. Primary risk characteristic from the buyer perspective is mean of the losses

M b = E L ( T 1 , T 2 ; ) χ { L ( T 1 , T 2 ; ) < l ( T 1 , T 2 ; t ) } =

5

= ∫0

l ( T 1 , T 2 ; t )

y P{ L ( T 1 , T 2 ; ) d y }

It is easy to present formula for the corresponding standard deviation of the losses

E [ L ( T 1 , T 2 ; ) χ { L ( T 1 , T 2 ; ) < l ( T 1 , T 2 ; t ) } - M b ] 2

The latter remark on pricing in stochastic setting is usually ignored as far as market risk is always omitted from derivatives pricing concept.

Remark 2. Randomization problem was studied also in the HJM and LMM models. In [1] we discussed the randomization problem of the instantaneous forward interest rates, which is by definition the rate

f ( t , T ) = lim

H ↓ 0 l ( t ; T , T + H ; 0 ). The HJM model is the basis of the LIBOR market model presented a randomization of the instantaneous implied forward rate in the form

f ( t , T , ) = f ( 0 , T ) + ∫0

t

α ( u , T ) d u + ∫0

t

σ ( u , T ) d w ( u )

where α ( u , T ) , σ ( u , T ) are known functions. Putting in the latter equation T ↓ t we arrive at the money market rate r ( t ) = f ( t , t ), which satisfies the equation

r ( t ) = f ( 0 , t ) + ∫0

t

α ( u , t ) d u + ∫0

t

σ ( u , t ) d w ( u )

Note that this formula does not consistent with the money market rate definition. Indeed, the bond prices are only observable data that define r ( t ). Applying latter expression for r ( t ) to the bond definition it follows that

B ( t , T ) = exp – ∫t

T

r ( u ) d u =

= exp – ∫t

T

{ f ( 0 , u ) + ∫0

u

α ( v , u ) d v + ∫0

u

σ ( v , u ) d w ( v ) } du

This formula suggests that the value of the bond at any moment t depends on all observations of the bond prices over [ 0 , T ]. This model assumption does not look accurate. Indeed, from the bond definition we know that

B ( t , T ) = 1 – ∫t

T

r ( u ) B ( u , T ) d u

6

r ( t ) =

dB ( t , T )B ( t )

Therefore, the value of the bond and the instantaneous interest rate at some date t depend on the values of the bond on ( t , T ]. More details on instantaneous implied forward rate randomization are given in [1]. In particular, it was suggested that the use of the backward Ito equations for randomization problem in does not lead to the pointed above contradictions.

III. Interest Rate Swap (IRS) valuation. In this section we investigate market risk which comes

up with a construction of the implied market price of the swap. An IRS is a cash exchange contract in which a floating rate payments are exchanged for the fixed one at a predetermined set of dates. The current swap price is also known as ‘spread’ is the value of the fixed rate which makes the present value of the future fixed payments be equal to the expectation of the present value of the floating rate payments. For calculation of the expectation of the present value of the floating rate payments the real future rates interpreted in a theory as random variables are replaced by its expected by the market estimates. These replacements imply market risk similar to discussed above FRA contracts.

Generic IRS is a financial contract to exchange a fixed rate q for a floating rate L based on a notional principal. Let t = t 0 < t 1 < t 2 < … < t n = T be a known sequence of dates and N notional principal. The buyer of a swap pays fixed rate payments N q to a swap seller and receives the amount N L ( t j – 1 , t j ) at the reset dates t j , j = 1, 2, … , n. As far as fixed and floating transactions are scheduled on the same reset dates the only netted values of transactions take place. The floating rate L is one of the basic market rates like a Treasury rate, a LIBOR or other similar rates. Rates L ( t j – 1 , t j ) are known at the dates t j – 1 , j = 1, 2, … n and therefore the only rate that is known at t is the rate L ( t 0 , t 1 ). Real world cash flow from the swap buyer perspective A to the swap seller B can be represented as

CF A → B ( t , T ) = ∑j = 1

n

[ L ( t j – 1 , t j ) - c ] χ ( t = t j ) (4)

Positive terms on the right hand side in (4) correspond to payments B to A while negative terms in (4) signify payments makes by A to B. The swap valuation problem is determination a fixed rate q that promises to counterparties A and B equality of their positions in the deal. The equality of the A and B positions in the deal should be specified. With the deterministic setting we are dealing with known or implied data the market risk is ignored and the problem has a unique solution. The fixed rate c is a solution of the equality present values of the two legs of the swap . In stochastic setting we need to take into account the market risk characteristics. There are a few types of the dollar denominated basic interest rates. Two primary rates are Treasury and LIBOR rates. They are intensively used by the markets. Treasury rates are specified by the T-bonds. It is a common practice to use one of these rates as discount factor. In this case we can interpret bond price B ( t , T ) at t as a discount factor from date T to the date t. The LIBOR rate L ( t , T ) can be used to form the discount factor D ( t , T )

D ( t , T ) = [ 1 + L ( t , T ) ( T – t ) ] – 1

7

which can also be interpreted as a value of a virtual bond price at t that promises $1 at T. The forward rate implied by the discount factor D can be defined as

l ( t j – 1 , t j ; t ) =

1Δ t j

[D ( t , t j − 1 )

D ( t , t j )− 1 ]

from which it follows that

D ( t , t j – 1 ) - D ( t , t j ) = D ( t , t j ) l ( t j – 1 , t j ; t ) Δ t j

Hence, the present value of the floating leg is equal to

N ∑j = 1

n

[ D ( t , t j – 1 ) - D ( t , t j ) ] = N [ 1 - D ( t , T ) ] = N ∑j = 1

n

D ( t , t j ) l ( t j – 1 , t j ; t ) Δ t j

It is common practice to interpret equality of fixed and floating rate cash flows as equality its present values, PVs. The PV of the fixed rate cash flow is

∑j = 1

n

c D ( t , t j ) Δ t j

In order to present PV of the floating leg we need first replace unknown at t rates L ( t j – 1 , t j ) on its date-t estimates l ( t j – 1 , t j ; t ). Then the PV of the ‘implied’ floating rate cash flow is equal to

N

∑j = 1

n

D ( t , t j ) l ( t j – 1 , t j ; t ) Δ t j = N [ 1 - D ( t , T ) ]

Therefore, the date-t value V ( t , T ) of the swap that is by definition the PV of the difference between floating and fixed legs is

V ( t , T ) = N [ 1 - D ( t , T ) - ∑j = 1

n

c D ( t , t j ) Δ t j ] (5)

Swap rate c is the solution of the equation V ( t , T ) = 0. Hence,

c =

1 − D ( t , T )

∑j = 1

n

D ( t , t j ) Δ t j (6)

The value c = c ( t , T , n ) is called the swap spread. Equality (6) we can resolve with respect to the discount rate D ( t , T ) = D ( t , t n ). Indeed, from (5) it follows that

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1 - D ( t , t n ) - ∑j = 1

n

c D ( t , t j ) Δ t j = 0

Solving this equation for D ( t , t n ) we arrive at recursive formula

D ( t , t n ) =

1 − ∑j = 1

n − 1

c D ( t , t j ) Δ t j

1 + c Δ t n

Remark 3. Note that the formula (6) presents implied swap rate, which is known at t number. Similarly to FRA valuation the market risk of the IRS stipulated by the stochastic rates L ( t j – 1 , t j ) can be represented by the cash flow

∑j = 1

n

N [ L ( t j – 1 , t j , ) - l ( t j – 1 , t j ; t ) ] Δ t j { t = t j }

Expressions in brackets can be either signs positive or negative.

Assume for example, that rate L ( T , T + H , ) is governed by the equation (3). The solution of the equation can be written in the form

L ( T , T + H , ) = L ( t , t + H ) ρ ( t , T )

where

ρ ( t , T , ) = exp {∫t

T

[ ( s ) -

σ 2 ( s )2 d s ] +

∫t

T

( s ) d w ( s ) }

Assume that Δ t j = Δ t . Then PV of the real floating leg cash flow is

N ∑j = 1

n

D ( t , t j ) L ( t j – 1 , t j ) Δ t j = N L ( t , t + Δ t ) ∑j = 1

n

D ( t , t j ) ρ ( t , t j – 1 , ) Δ t

The real world value of the swap can be written as

V ( t , T , ) = N [ L ( t , t + Δ t )∑j = 1

n

D ( t , t j ) ρ ( t , t j – 1 , ) Δ t - ∑j = 1

n

Q D ( t , t j ) Δ t ] (5′)

Therefore, the market realized swap spread depends on a market scenario and equal to

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Q ( ) =

1 − ∑j = 1

n

D ( t , t j ) ρ ( t , t j − 1 ) Δ t

∑j = 1

n

D ( t , t j ) Δ t (6′)

The market risk of the swap from the buyer perspective is associated with the negative terms in (4). Hence, swap buyer's market risk is

P { Q ( ) < c } = P { ∑j = 1

n D ( t , t j )D ( t , T ) ρ ( t , t j – 1 , ) Δ t > 1 }

It represents the value of the chance that the buyer of the swap pays the higher rate than it is implied by the market. Note that right hand side of the latter equality can be approximated by a probability, i.e.

P { ∑j = 1

n D ( t , t j )D ( t , T ) ρ ( t , t j – 1 , ) Δ t > 1 } P {

∫0

TD ( t , u )D ( t , T ) ρ ( t , u , ) d u > 1 }

Remark 4. Note that presented constructions have been dealt with original probability space while it is common practice to use so-called risk-neutral space. The probability measure on risk-neutral space is chosen such that it should replace real return of a security on risk free rate. This construction takes its origin from Black-Scholes’ concept of the option price. It might make sense to bring a short critical remark on Black Scholes (BS) price definition. In more details this concept is discussed in [3,4].

Let C ( t , S ( t ) ) denote BS’s call option price of a European call option with strike price K and maturity at T. It was assumed that underlying stock price is governed by stochastic differential equation (SDE)

d S ( t ) = S ( t ) d t + σ S ( t ) d w ( t )

with constant coefficients , σ. Black and Schloes proposed a portfolio П with one short call option and a number of the underlying stock shares. The value of the portfolio at t is equal to

П ( t , S ( t ) ) = - C ( t , S ( t ) ) +

∂ C ( t , S ( t ) )∂ S S ( t ) (П1)

They also supposed that infinitesimal change in value of the portfolio is given by the formula

d П ( t , S ( t ) ) = - d C ( t , S ( t ) ) +

∂ C ( t , S ( t ) )∂ S d S ( t ) (П2)

Assuming that C ( t , S ) is a smooth function and applying Ito formula to the option price we arrive at the formula

d C ( t , S ( t ) ) = [

∂ C ( t , S ( t ) )∂ t +

σ 2 S ( t )2

∂ 2 C ( t , S ( t ) )∂ S 2

] d t +

∂ C ( t , S ( t ) )∂ S d S ( t )

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It is not difficult to note that

d П ( t , S ( t ) ) = [

∂ C ( t , S ( t ) )∂ t +

σ 2 S ( t )2

∂ 2 C ( t , S ( t ) )∂ S 2

] d t (П3)

The right hand side of the formula (П3) does not contain risk term associated with the ‘white’ noise factor d w ( t ). If a portfolio’s infinitesimal change in value is risk free then the only way to avoid arbitrage opportunity is that its rate of return should be equal to risk free rate r. Therefore

d П ( t , S ( t ) ) = r П ( t , S ( t ) ) d t

Bearing in mind formula (П3) and (П1) we arrive at the Black Scholes equation

∂ C ( t , S ( t ) )∂ t +

σ 2 S ( t )2

∂ 2 C ( t , S ( t ) )∂ S 2

= r [ - C ( t , S ( t ) ) +

∂ C ( t , S ( t ) )∂ S S ( t ) ] (BSE)

Equation is defined in the domain ( t , S ) [ 0 , T ) ( 0 , + ∞ ) with boundary condition

C ( T , S ) = max { S - K , 0 }

Construction and correspondent assumption on portfolio’s dynamics of the portfolio in the form (П1), (П2) is theory fundamental for derivatives pricing in no arbitrage setting.

The easiest way to convince one oneself about the Black Scholes’ error is to use explicit form of the solution of the (BSE) and construct portfolio as it was presented in the formula (П1). Then with help of simple calculus one can easy verify that the assumption that the infinitesimal change in value of the

portfolio follows (П2) is incorrect. The nonzero term S ( t ) d

∂ C ( t , S ( t ) )∂ S was lost on the right hand

side (П2). As an implication of the error follows the fact that portfolio in the form (П1) does not provide ‘perfect’ dynamic hedge of the option price which also serves as another version of the option price definition. Also probabilistic interpretation of the (BSE) has lead derivatives pricing theory in replacing real word, which is defined on original probability space by so-called risk-neutral world. The latter used Girsanov change measure technique to replace real security return on risk free counterpart. In case when real and risk free return are close in average over [ 0 , T ] the numeric calculations can bring close results. Nevertheless, even in this case the replacing the real return on the risk free is formally incorrect.

The error of the Black Scholes pricing approach in stochastic setting is that they did not pay attention to the market risk of the spot price. The market risk is an attribute of any particular model of the spot price of a derivative instrument.

IV. Liquidity. The liquidity problem in a simple interpretation comes up as a premium problem

which one should to add to a single price model associated with the perfect liquidity price. Latter is associated with a single price asset models, which implies equality, bid and ask prices. In theory the liquidity premium exists when pricing uses the bid-ask format. For the last years the liquidity problem attracted theoretical and practical attention . A measure of the liquidity premium is its bid-ask spread.

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Hence, talking about assets pricing in a single pricing format one inexplicitly deals with the perfect liquidity case. The perfect liquidity is interpreted as a trade with no delay, which follows from the equality of the bid and ask prices at any moment time t. Indeed, in such a model a buyer who wishes to buy an asset at t pays at this moment its price while a seller receives exactly the same amount at this moment. There is no loss at any time for selling and buying transactions. Thus, the liquidity problem can be interpreted as an adjustment to a single price representing by the perfect liquidity model in order to take into account asset illiquidity. It is clear that illiquidity stems from the fact that ask price always higher than its bid price. It is a popular practice to use the middle price of the bid-ask spread to present premium in perfect liquidity approximation. In this case one should make an adjustment adding or subtracting the half of the bid-ask spread in final formulas. In such interpretation the effect of illiquidity is a half of the bid-ask spread for the buyer and seller adjustments for the price. This adjustment specifies the liquidity premium.

There are two primary types of liquidity. These are trading and funding liquidity. Trading liquidity shows the ease of the asset trading while funding liquidity is a characteristic of access to funding. Consider the trading liquidity problem. Let us define the value of the bond illiquidity at a moment t by the spread

λ 1 , B ( t , T ) = B ask ( t , T ) - B bid ( t , T )

Indicator λ 1 , B ( t , T ) represents liquidity at t of the bond that expired at T. Obviously that similar instruments with fixed t and different expiration dates or with different moments t with equal T would have different liquidity. Broadly speaking the value of illiquidity represent losses on immediate buy-sell transactions. If { t } represents a trading day period then liquidity can be interpreted as a random variable taking values for the interval [ min λ 1 , B ( u , T ) , max λ 1 , B ( u , T ) ] , u { t }. For numerical calculations one can use for example uniform or other simple distribution. We can also use other definition of the liquidity. Define the profitability of the bond d B ( s , t ; T ) that represent profit or loss ratio purchasing bond at s and selling it at a next moment t

d B ( s , t ; T ) =

B bid ( s , T )B ask ( t , T )

and define value of liquidity as

λ B ( t , T ) = lims → t d B ( s , t ; T ) (7)

The perfect liquidity and complete illiquidity correspond to the value λ B ( t , T ) = 1 and λ B ( t , T ) = 0 correspondingly. Next remark does not depend on liquidity definition. We call asset A more liquid at a moment t than asset G if λ G ( t , T ) < λ A ( t , T ). If the latter inequality takes place on [ 0 , T ] then A is more liquid than G on the interval [ 0 , T ]. Assume that A is more liquid than G on a subinterval of the interval [ 0 , T ] and G more liquid than A on another subinterval then the definition of liquidity should be refined. One way to adjust the definition is to consider an average bid-ask spread. For example asset A is more liquid than asset G on [ 0 , T ] in sense of the average if

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1T

∫0

T

λ G ( t , T ) d t ≤

1T

∫0

T

λ A ( t , T ) d t (7′)

where λ A , λ G denote liquidity of asset A and G correspondingly. This definition of liquidity does not comprise variety of the real market components associated with liquidity. Broadly speaking, one can consider a liquidity adjustment including volume of trades during a particular period. Indeed, let us imagine that assets A and G look similar in terms of (7), (7′) while the number of trades G is visibly larger than A over [ 0 , T ] then it might makes sense to think that G is more liquid than A on this interval.

The risk free bond price at t with expiration at T is defined by its bid-ask prices

0 < B bid ( t , T ) < B ask ( t , T ) ≤ 1 , t [ 0 , T )

B bid ( T , T ) = B ask ( T , T ) = 1

Note that in the single price format defined as

B ( t , T ) =

12 [ B bid ( t , T ) + B ask ( t , T ) ] (8)

is a heuristic price and it does not represent perfect liquidity either for buyer or seller of the bond. The perfect liquidity for buyer is the bid price while the seller’s perfect liquidity is the ask price. Taking into account this remark we note that perfect liquidity is defined for buyer and seller separately. The only real world prices B bid ( t , T ) , B ask ( t , T ) represent liquidity or illiquidity of the bond. Therefore, liquidity of the risk free bond which price is defined by (8) does not perfect for a counterparty. The risky bond liquidity is more complex issue. We will discuss it later. Liquidity problem implies discovery of the adjustment to the formula (8). On the other hand we can deal with bid-ask prices directly.

Formal randomization of liquidity problem for risk free bond can be studied by considering two random processes for bid and ask prices. By definition we put

B k ( t , T ; ) = exp - ∫t

T

r k ( u , ) d u (9)

k = bid , ask. Here the risk free interest rate is defined as

r k ( u , ) = lim

Δ t → 0

1 − B k ( t , t + Δ t )Δ t =

limΔ t → 0

B k ( t + Δ t, t + Δ t ) − B k ( t , t + Δ t )Δ t

From (9) it follows that

B k ( t , T ) = 1 + ∫t

T

r k ( u , ) B k ( u , T ; ) d u (9′)

13

and

r k ( u , ) =

1B k ( t , T )

∂ B k ( t , T )∂ t

From (9′) it follows that bond price and interest rate at moment t depend on values of bond on the interval ( t , T ]. It is quite common in performing randomization of the bond price (8) that the price is governed by Linear Ito equation. In this case we can guarantee the condition B ( t , T ) > 0 while we cannot guarantee the condition that B ( t , T ) ≤ 1. In other words we arrive at the fact that with a positive probability for each t B ( t , T ) > 1 which does not make sense. To state that theoretical formulas

represent good approximation of the historical data makes sense if the probability P { sup

0 ≤ t ≤ T B ( t , T ) > 1 } is sufficiently small. Following [1] consider randomization of the bond price in single price format ,i.e. ignoring liquidity aspect. Assume that r ( t , ) is govern by backward Ito equation

r ( t , ) = r T ( ) – ∫t

T

λ ( u ) r ( u , ) d u – ∫t

T

θ ( u ) r ( u , ) dw⃗ ( u ) (10)

where stochastic integral on the right hand side is the backward Ito integral [5]. The solution of the

equation (10) is interpreted as a random F←

( t , T ] = σ { w ( s ) - w ( T ) , t < s } measurable stochastic process for which sup E | r ( t , ) | 2 < ∞. The backward initial condition r T ( ) = r ( T + , ) is

assumed to be F←

T = F←

( T , + ∞) measurable and thus it independent on σ - algebraF←

( t , T ]

r T = lim

Δ ↓ 0 ( Δ ) – 1 r ( T , T + Δ ) (11)

It represents instantaneous rate of return (1′) over infinitesimal interval [ T , T + dot ) and it will be known at the date T. The choice of the backward Ito equation (10) is stipulated by the connection of the price and interest rate presented by formulas (9) , (9′). Indeed, from (9) it follows that values B ( t , T ; ) and r ( t , ) are completely determined by observations over bond prices over [ t , T ]. Modern interest rates stochastic models assume that interest rate at t depends on r ( u , ) prior to t which implicitly implies its dependence on B ( u , T ; ) , u ≤ t. This construction inconsistent with original definition of the interest rate. Thus, providing randomization of the interest rate we need verify consistency random interest rate with random bond price.

Assume that r ( t , T ) is a known function for all 0 ≤ t ≤ T < + ∞. According to market practice we replace random r T ( ) by its market implied estimate r ( T , T + H ; t ) defined by the equation

1 + r ( t , T + H ) ( T + H - t ) = [ 1 + r ( t , T ) ( T - t ) ] [ 1 + r ( T , T + H ; t ) H ]

Solving this equation for the implied market forward rate r ( T , T + H ; t ) we arrive at

r ( T , T + H ; t ) =

1H

[r ( t , T + H ) ( T + H − t ) - r ( t , T ) ( T − t )

1 + r ( t , T ) ( T − t )]

14

Taking into account latter formula we can present implied estimate of the future rate r T ( )

r ( T , T + 0 ; t ) = lim

H ↓ 0 r ( T , T + H ; t )

Nonrandom variable r T ( t ) = r ( T , T + 0 ; t ) is known at t and the replacement of the random variable r T ( ) on nonrandom value r T ( t ) implies market risk stipulated by the fact that in general market scenarios reveal the fact that r T ( t ) r T ( ). Denote r ( t ; T , r T ) solution of the equation (10). Then r ( t ; T , r T ( t )) denote solution of the equation (10) with initial value given at T r ( T ; T , r T ( t )) = r T ( T ). Bond buyer market risk is that his price r T ( t ) = r ( T , T + 0 ; t ) at date t is higher price than the realized price which corresponds to scenarios

{ : r ( t ; T , r T ( )) < r ( t ; T , r T ( t )) }

Suppose that random function r k ( t , ) , k = bid , ask is the solution of the backward SDE

r k ( t , ) = r T , k ( ) – ∫t

T

λ k ( u , r k ( u , )) d u – ∫t

T

θ ( u . r k ( u , )) dw⃗ ( u ) (10′)

where λ ask < λ bid and with probability 1 r T , ask ( ) < r T , bid ( ). These conditions guarantee correspondence between bid and ask prices of the bond. Exchange random variables r T , k ( ) on nonrandom values r T , k ( t ) = r k ( T , T + 0 ; t ) defines the solution of the equation (10′) r k ( t , ) = r k ( t ; T , r T , k ( t ) ) with boundary condition r T , k ( t ) = r k ( T , T + 0 ; t ), k = bid , ask. One can assume that spot prices in bid-ask format at date t can be approximated by expectation conditional over the observations over bond prices up to the moment t

r spot , k ( t ) = E { r k ( t , ) | F→

t } (12′)

where F→

t = F→

[ 0 , t ) = { B ( u , T ) , u < t }. On the other hand we can use implied market estimate techniques to present other estimate of the spot prices. Replace values r k ( u , ) on the right hand side (10′) by its implied non random estimate r u ( t ) = r ( u , t ). Then market implied estimate of spot interest rate admits representation

r k( im )

( t , ) = r T , k ( t ) – ∫t

T

λ k ( u , r u ( t )) d u – ∫t

T

θ ( u . r u ( t )) dw⃗ ( u )

k = bid , ask. In accordance to (12) one can assume that

r spot , k( im )

( t ) = E { r k( im )

( t , ) | F→

t } (12′′)

The random function r k( im )

( t , ) is a Gaussian process with mean and volatility equal to

15

r T , k ( t ) – ∫t

T

λ k ( u , r u ( t )) d u , ∫t

T

E θ 2 ( u . r u ( t )) d u

correspondingly. Formulas (12′), (12′′) are the basic formulas for default free bond pricing that follow (9). There are two types of the risks exist in our model. One is the model risk which stipulated by the assumption presented by the formulas (10), (12′), (12′′). Other risk type is approximations of the market. This risk is represented by the replacements by the real market unknown random future values by the market implied forward rates know at the spot moment of time.

In our construction we specify buyer and seller liquidities separately. The buyer illiquidity is stipulated by the fact that the ask price of a bond A is larger than its bid price. Similarly, the seller liquidity is defined by the fact that the bid prices of the bond are lower than the ask price. Given distinctive bid and ask prices one can define the date-t implied forward bid-ask spread at date T for the future [ T , T + H ] period. Indeed, implied forward bid and ask rates f k ( T , T + H ; t ), k = ask, bid can be defined as

f k ( T , T + H ; t ) =

1H

[1 − D k ( t , T + H ) ( T + H − t )

1 − D k ( t , T ) ( T − t )− 1 ]

k = ask , bid. Note that we can use discount rates also for the interest rates that do not associate with a bond. Dealing with LIBOR rate we put in the latter formula D k = L k and f k = l k. Hence,

fra k ( T , T + H ; t ) = l k ( T , T + H ; t )

and FRA bid-ask spread of the FRA contract by definition is a difference

λ l ( t , fra ( T , T + H ; t ) ) = l ask ( T , T + H ; t ) - l bid ( T , T + H ; t )

This formula enables to compare liquidity at t for different T , H. Recall the inverse relationship between prices and corresponding interest rates. As it is already pointed out the implied rates represent statistical estimates in stochastic market setting. A future period real rate depends on a market scenario. Therefore, the use of its corresponding implied rate is subject to market risk. This risk represents the fact that the date-t implied forward rates l bid or l ask do not equal to its real values at the future moment T. In other words the liquidity market risk of the FRA contract implied by the bond is stipulated by the inequality

f ask ( T , T + H ; t ) - f bid ( T , T + H ; t ) ≠ B ask ( T , T + H ) - B bid ( T , T + H )

and the probability of the inequality

P { f ask ( T , T + H ; t ) - f bid ( T , T + H ; t ) > B ask ( T , T + H ) - B bid ( T , T + H ) }

is a measure of the illiquidity risk of the FRA contract. Here the FRA contract has been used to illustrate stochastic effect on implied forward liquidity. The date-t implied liquidity spread λ f = f ask - f bid represents a statistical estimate which can be larger or smaller than its date-T real market rate

λ = f ask ( T , T + H ) - f bid ( T , T + H )

16

Higher future liquidity spread corresponds to a higher liquidity risk and future buyer’s additional expenses while lower future illiquidity suggests more liquidity in the future market. Bearing in mind liquidity problem one can extend a single rate bid = ask model of the FRA valuation. Given risk free curves B bid ( t , T ) , B ask ( t , T ) one can present default effect for a corporate bond pricing in bid-ask format.

V. Liquidity of Corporate Bonds. Let B bid ( t , T ), B ask ( t , T ) and R bid ( t , T ), R ask ( t , T )

denote bid and ask prices of the risk free and risky bonds. Suppose that default can occur only at maturity date T. Following reduced form model of default one can define default prices.

R bid ( T , T , ) = R ask ( T , T , ) = Δ < 1 , if D

R bid ( T , T , ) = R ask ( T , T , ) = Δ < 1 , if D

Here D denotes default scenarios set. The standard reduce form theory deals with the single price format [8]. Assuming that constant Δ is known we note that

R k ( t , T , ) = [ 1 - ( 1 - Δ ) χ ( D ) ] B k ( t , T ) (13)

k = ask , bid and t ≤ T. For other credit events more complex than bankruptcy the theory would probably suggest that recovery rate Δ would depend on k too. For numeric calculations one usually uses either close or open prices of the date t. Let date t is associated with a whole trading day t. In this case the close or open prices implicitly are considered as a market approximation of the day-t prices. Such approximation makes sense if the value

R k , max ( t , T ) - R k , min ( t , T )

is sufficiently small and k = bid, ask. Here, the latter differences denote the largest date t value of the ask and bid spreads. Otherwise, the reduction of the daily values to the one price can be too crude estimate. In this case, recovery rates (13) can be interpreted as random variables having an appropriate distribution. For example, one can assume that R k ( t , T , ) has a uniform distribution on interval

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

or a Gaussian distribution with mean equal to middle point of the I k ( t , T ) and standard deviation

σ =

13 [ R k , max ( t , T ) - R k , min ( t , T ) ]

If default occurs at maturity T with known recovery rates Δ k than bid and ask prices оf the bond prior to the moment T are expressed by (8). Equality (8) does not suffice to state that the default should occur prior to maturity T, i.e. reduced form of default model expressed by (8) is a necessary condition of the default. In practice data R k ( t , T ) , t < T is observable. Hence, given R k ( t , T ) one needs to present an additional assumption regarding default time τ distribution and based on this assumption presents estimates of the recovery rates Δ k , k = bid, ask. Thus, given credit spread B k ( t , T ) - R k ( t , T ), k = bid, ask at a moment t < T we can state that corporate bond is traded at t similar to a risky bond that might default at maturity with a particular recovery rate Δ k . The assumption that default occurs at

17

[ t j , t j + ε ) , ε > 0 will effect on an estimate of the value of the recovery rate. Default problem is resolved conditioning on the assumption that default can occurred at maturity T.

Assume now that default time ( 0 , T ]. Then recovery rates implied by bid and ask bond prices in general are different. Given corporate bond prices one need to establish an assumption regarding distribution of the default. This distribution does not directly follows from price observations. Based on this assumption we should present estimates of the values of recovery rates Δ k . Thus, given credit spreads B k ( t , T ) - R k ( t , T ) , k = bid, ask at t , t < T we can state that

1. corporate bonds with unknown default time distributions are traded at t similar to a risky bonds that might default only at maturity with a particular recovery rate Δ k ,

2. an assumption that default of a corporate bond occurs at [ t j , t j + ε ) , ε > 0 will effect on estimate of the recovery rate.

In general theory it is a common to assume that value of the recovery rate and default time are independent random variables. This is a technical assumption and it does not look close to real world events.

In the reduced form framework equality (13) represent necessary conditions of the default. One needs a randomization of the observations in order to prescribe probability density to default at a moment t. For example, if a portion p [ 0 , 1 ] of time during a day t the price of the corporate bond is approximately equal to Δ B k ( t , T ) then it makes sense to state that assuming that default can be observed only at expiration date T then with probability approximately equal to p recovery rate of the bond should be equal Δ . Note that in this case values p and Δ are functions of t. Following this way one can present a construction of a discrete approximation of the continuous distribution of the default at T. In bid-ask format we observe data

B k ( t , T ) - R k ( t , T ) = B k ( t , T ) [ 1 - Δ k ]

k = bid, ask , and t < T. Then assuming that default might occur at maturity T we can state that recovery rates implied by bid and ask prices will be equal to Δ k . Note, that

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

and therefore Δ ask ≠ Δ bid . Admitting that default might occur at t prior to expiration date T we should assume that recovery rate is a function of t.

Remark. In practice, it is common to fixe recovery rate Δ. For example, one assumes that Δ = 40% and default time distribution coincides with the distribution of the first jump of the Poisson process. The intensity of the Poisson process is assumed to be consistent with B ( t , T ) - R ( t , T ).

In general theory in order to present numeric calculations it is usually supposed that moment of default and recovery rate Δ are independent random variables. From our point of view this assumption does not look realistic and it oversimplified stochastic setting. Recovery rate does depend on the default moment.

Recall that prior to default a holder of the bond can sell it for the bid price while a buyer of the bond should pay ask price to purchase bond. On the default time regulatory rules, which would direct default procedure.

18

We illustrate default in bid-ask format assuming that default can occur only at maturity date T. This assumption simplifies the problem ignoring the random default time issue. We make some remarks to reduce form of default bearing in mind liquidity setting of the problem. In single price format default scenarios can be presented as

D = { ω : R ( T , T ; ω ) = Δ < 1 }

and put D = Ω \ D. Then

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

for t ≤ T. In bid-ask format default at T can be defined as following

D = { R bid ( T , T ; ω ) = Δ bid < 1 }

Here 0 ≤ Δ bid < 1. Then

R k ( t , T ; ω ) = B k ( t , T ) [ 1 ( D ) + Δ k ( D ) ] = B k ( t , T ) [ 1 – ( 1 – Δ k ) ( D ) ] (14)

for k = bid, ask. Here is supposed that nonrandom function B k ( t , T ) and constants Δ k are known at t. In this case equality (14) represents market price R k ( t , T ; ω ) of the corporate bond. The market price is defines uniquely for each market scenario . Bearing in mind available information market participants at

form a spot price at t . Given B k ( t , T ) , Δ k , R k ( t , T ; ω ) and R kspot

( t , T ) we enable to estimate market risk of a bond seller with the help of formula

P { R bidspot

( t , T ) < R bid ( t , T ; ω ) }

This formula represent value of the chance that bond’s buyer pays less than it implies by the market. In

reality the only values B k ( t , T ) и R kspot

( t , T ) are observable while values Δ k are unknown. Randomization subtends the random processes R k ( t , T ; ω ) construction based on observations

R kspot

( t , T ). It is common practice to identify value R kspot

( t , T ) with close price at date t. In the case the forecast for the next date price is also relates to the close moment. We can think that close price approximates the prices during the date t. This assumption can fail if the variation of the prices does not sufficiently small. Randomization cal be realized as following. Suppose that R k ( t , T ; ω ) is a random

variable taking values from the interval [ R k minspot

( t , T ) , R k maxspot

( t , T ) ] where the end points of the interval are minimum and maximum spot values during the date {t}. Date {t} can be also denote other that day period. As model distribution of the R k ( t , T ; ω ) can be chosen uniform, triangle, or appropriate Gaussian distributions. Given a model distribution and assuming that default possible only at maturity one can calculate recovery rates Δ k and probability of default Р ( D ). Taking expectation in (14) we get the equation

E R k ( t , T ; ω ) = B k ( t , T ) [ Р (D ) + Δ k Р ( D ) ] = B k ( t , T ) [ 1 – ( 1 – Δ k ) Р ( D ) ] (14′)

19

In general case when recovery rate is interpreted as a random variable on [ 0 , 1 ] equation (14′) can be rewritten as

E R k ( t , T ; ω ) = B k ( t , T ) [ 1 – Р k ( D ) + ∫0

1

v d Р k ( v ) ] (14′′)

where Р k ( v ) = Р { Δ k < v }. The probability Р k ( v ) on the right hand side (14′′) can be interpreted as the portion of time during the date {t} for which

R k ( t , T ; ω )B k ( t , T ) < v

From equality (14′) it follows that

E R bid ( t , T ; ω )B bid ( t , T )

=E R ask ( t , T ; ω )

B ask ( t , T ) (15)

Assuming that default can occur only at maturity we note that equalities (14′) for k = bid, ask are dependent. Violation of the equality (15) can mean that at least one of the basic assumptions does not correct. For example it might be incorrect to admit that default occurs at only expiration date. Let Δ k be unknown constants. Then from (14) follows that for any n

Е [ 1 –

R k ( t , T ; ω )B k ( t , T ) ] n = ( 1 – Δ k ) n P ( D ) (16)

Solving algebraic system (16) when n = 2 we arrive at the solution

Δ k = 1 –

E ( 1 −R k ( t , T ; ω )

B k ( t , T )) 2

E ( 1 −R k ( t , T ; ω )

B k ( t , T ))

, P ( D ) =

[ E ( 1 −R k ( t , T ; ω )

B k ( t , T )) ] 2

E ( 1 −R k ( t , T ; ω )

B k ( t , T )) 2

(16′)

k = bid, ask. In general case we assume that the bond price immediately after default is a random variable Δ = Δ ( t , T ; ω ) with a continuous distribution on [ 0 , 1 ]. Let us approximate it by a discrete random variable

Δ β k ( ω ) = ∑j = 1

N − 1

β j { Δ k ( ω ) [ β j , β j + 1 ) }

where = β 0 < β 1 < … < β N = 1 and Р j k = Р { Δ k ( ω ) [ β j , β j + 1 ) }. Here, Р j k are function of the variable t. It follows from (14) that

20

Е [ 1 –

R k ( t , T ; ω )B k ( t , T ) ] n =

∑j = 1

N − 1

( 1 – β j ) n Р j k (16′′)

n = 1, 2, … N и k = bid, ask. For each fixed k = bid, ask there is a linear algebraic system for unknown variables Р j k , j = 1, 2, … N. Numeric calculations unknowns can be simplified if we note that determinant of the system is of Vandermonde’s type. Values Р j k can depend on time t. For simplicity we can put β n = N – 1 n. Hence, the liquidity problem can be solved assuming that default occurs only at maturity date. Let us estimate liquidity spread between the risk free and the corporate bonds. Note that from (14) it follows that

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

= [ B bid ( t , T ) ( 1 – Δ bid ) – B ask ( t , T ) ( 1 – Δ ask ) ] ( D )

and therefore

E [ R ask ( t , T ; ω ) − R bid ( t , T ; ω ) ] − [ B ask ( t , T ) − B bid ( t , T ) ]B ask ( t , T ) − B bid ( t , T ) =

= – Р ( D )

B ask( t , T ) Δ ask − B bid( t , T )Δ bid

B ask ( t , T ) − B bid ( t , T )

This formula represents the relative change of the value corporate liquidity with respect to risk free liquidity.

Now let us consider the case when corporate bond admits default at some fixed moment of the period [ 0 , T ]. Let τ = τ ( ω ) denote the default time of the bond. The market price of the bond defined for each market scenario can be written in the form

R ( t , T ; ω ) = ∑

i = 1

n − 1

R ( t , t i ; ω ) χ { τ = t i } + B ( t , T ) χ { τ > T } (17)

Here, R ( t , t i ; ω ) denotes the market price of the bond with expiration at t i conditioning that default can be occurred at expiration date. For notation simplicity in (17) the index k was omitted. Assume now that recovery rates and probability rates

Δ = Δ ( t , t i ) , P ( D ) = P ( t , t i )

are unknown. From equality (16′) it follows that market price (17) can be presented as

R ( t , T ; ω ) = ∑

i = 1

n − 1

Δ k ( t , t i ) В ( t , t i ; ω ) χ { τ = t i } + B ( t , T ) χ { τ > T } (17′)

In case when market is such that expected market loss is approximately equal to expected gain of the bond mathematical expectation can be considered as a good estimate of the spot price at t. Then

21

R k spot ( t , T ) Е R k ( t , T ; ω ) =

= ∑

i = 1

n − 1

В k ( t , t i ; ω ) { 1 –

E ( 1 −R k ( t , t i ; ω )

B k ( t , t i )) 2

E ( 1 −R k ( t , t i ; ω )

B k ( t , t i ))

}

[ E ( 1 −R k ( t , t i ; ω )

B k ( t , t i )) ] 2

E ( 1 −R k ( t , t i ; ω )

B k ( t , t i )) 2

+ (17′′)

+ B k ( t , T ) { 1 –

[ E ( 1 −R k ( t , T ; ω )

B k ( t , T )) ] 2

E ( 1 −R k ( t , T ; ω )

B k ( t , T )) 2

}

k = bid, ask. To present liquidity effect we need to present the adjustment to the single format pricing which is interpreted as middle price of bid and ask prices presented by (17′′).

The above formulas for recovery rate and probability of default were obtained under assumption that default can be observed only at maturity of the bond. In order to lessen the assumption and admit that bond can default prior to its maturity we need to make an assumption regarding distribution of the time of default. Different hypotheses regarding distributions of the default time lead to different statistical characteristics of the price and liquidity of the bond. Let us briefly remind reduced form model of default. It was assumed that distribution of the default time could be approximated be the distribution of the first jump of a Poisson process. Let П ( t ) denote probability of the event that default does not occur up to the moment t. Then the value П ( t ) – П ( t + h ) represents probability of event that default would first observed during the interval [ t , t + h ]. Indeed, bearing in mind equalities

П ( t ) = P { τ > t } = P { t < τ ≤ t + h } + P { τ > t + h } = P { t < τ ≤ t + h } + П ( t + h )

it follows that unconditional probability that default occurs at [ t , t + h ] will be equal to

П ( t ) – П ( t + h ) = P { t < τ ≤ t + h }

Next

Р { τ ≤ t + h | τ > t } = Р { τ ≤ t + h ∩ τ > t } [ P { τ > t } ] – 1 =

Π ( t ) − Π ( t + h )Π ( t )

Suppose that function П ( t ) is continuously differentiable and there exists a continuous function h ( t ) for which

1Π ( t )

d Π ( t )d t = h ( t )

Then

22

П ( t ) = exp – ∫0

t

h ( u ) d u (18)

Let us represent default intensity function in the form h ( u ) = r ( u ) + λ ( u ), where r ( u ) is the risk free interest rate. The (18) can be rewritten in the form

П ( t ) = exp – ∫0

t

[ r ( u ) + λ ( u ) ] d u (18′)

This representation takes place on original probability space under original probability measure P. In (18′) function λ ( u ) can be interpreted as a spread of the default intensity.

Remark. It is known a generalization of the representation (18). One assumes that the intensity of default is a random function λ ( t , ω ) = h ( t , X ( t , ω )), where X ( t , ω ) is a random process. The random processes of this type are called Cox processes. In paper [7] the default was presented by using Cox processes. We wish to highlight a complexity of the consistency formal mathematic and financial essence of the problem. It was supposed that conditional probability of the event that default does not occur before date t can be presented by the formula

P { τ > t | X ( u ) , 0 ≤ u ≤ t } = exp – ∫0

t

h ( u , X ( u , ω )) d u = Secu19()(19)

Following [7] define filtration of the σ-algebras :

G t = σ { X ( u , ω ) , 0 ≤ u ≤ t }

H t = σ { χ ( τ ≤ u ) , 0 ≤ u ≤ t }

F t = G t H t

In applications of Cox processes to default problem

*) the random process X is interpreted [7] as the state variable process X , for which the spot rate r t = R ( X t ) at time t. In other words X characterizes the company that issued bond;

**) events from H t shows whether default observed until the moment t or nor;

***) events from F t described simultaneously default and company state until the moment t

Note that formula (19) shows that default distribution is also depend on the process X as far as one inexplicitly assumed in (19) that τ = τ ( X ( )). Nevertheless the formula (19) can be interpreted as following. For each constant value X ( ) = v there uniquely defined default time τ = τ ( v ) of the firm in the state v for which

23

P { τ ( v ) > t } = exp – ∫0

t

h ( u , v ) d u

Then if the state of the firm is a random process X ( t , ) that independent on τ ( v ) then the formula (19) holds. Next, one should explicitly define the random process X ( t , ). The formula

r t = R ( X ( t , ) )

can be used as a definition of the process X if the function R ( x ) is formally defined. These facts were probably missed in [7]. This is a way that stochastic intensity can be studied. Nevertheless it is not clear whether function R can be formally defined and whether or not X will be independent on τ ( v ).

At the end let us highlight the fact that introduced here risky bond pricing approach differs from the reduced form standard though both approaches relates to the reduced form interpretation of the default. In simplified case when default can be observed only at maturity the main distinction is that based on prices observations we calculate probability of default along with recovery rate of the bond. The standard reduce form approach assumes that recovery rate of the corporate bond to be known. It is stipulated by the fact that in the single price format we deal with expected value of the equation of the type (14). Therefore it is impossible to derive two unknown Δ , Р ( D ). Our approach separates market price that depends on market scenario and spot price. This approach makes it possible to present equations for higher moments of the market price (16) that helps to present two unknown in the form (16′). It is clear that different assumptions regarding distributions default time will bring other values for recovery rate and probability of default.

24

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