1

BASIC OF PRICING 1.

Ilya Gikhman

6077 Ivy Woods Court

Mason OH 45040 USA

Ph. 513-573-9348

Email: ilyagikhman@mail.ru

Abstract. In this paper we begin with details of the no arbitrage pricing scheme. It is common to call the

pricing approach no arbitrage if it is impossible to receive a positive profit on a contract starting from zero

investment at initiation date. We specify no arbitrage pricing by

a) initiation and expiration dates

b) zero values of the contract at initiation and expiration dates

Clause b) presents more precise definition as far as it is a common practice to define the no arbitrage

pricing as the price that assigns zero value to the expected present value of all future transactions. Such

value does not guarantee the zero value of all future transactions at expiration date. Such interpretation of

the price leads us to the important attribute of the pricing known as market risk which does not exist in the

benchmark finance theory. Omitting face value in the pricing theory leads to oversimplified concept of

price which also ignoring market risk factor in price construction. Refined concept of the no arbitrage

pricing is applied for the valuations of the risk free bond and interest rate swap. Next we present a formal

mark-to market valuation of the simple model example of the risky bond. The main goal of this

illustrative example is an explicit effect of the mark-to market valuation format.

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

risk, interest rate swap.

0. Introduction. In this paper we begin with cash flow notion and use it as a formal definition of a

financial instrument. Cash flow is defined by the all future transactions between counterparties specified

by the instrument. Present value is usually used to determine no arbitrage price of an instrument. On the

other hand the present value of the cash flow is equivalent to single initial payment that is sufficient to

cover all transactions as well as upfront and future payments. The premium payment that provides zero

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value of the present value of the cash flow is usually interpreted as no arbitrage price. The intuition of this

interpretation is the fact that with zero present value one can generates future transactions to counterparty.

Nevertheless one can easy single out that original zero value of the cash flow does not always provide

zero value of the cash flow at expiration date. We illustrate our point bellow by using coupon bond

valuation. General valuation approach considers risk free coupon bond as sum of risk free bonds with

different maturities. Risk free bond no arbitrage price is uniquely defined as present value of the final

payment. Therefore the sum of no arbitrage prices should present no arbitrage pricing too. Nevertheless

the present value is equivalent to the cash flow but it is not equivalent to the zero value at expiration date.

Hence, if a bond buyer borrows funds from the bank at risk free interest and receiving coupon payments

immediately pays back coupon to the bank we enable to call the price defined by present value as no

arbitrage price. On the other hand and perhaps it is more popular to hold all coupon payments until

expiration date. In such case no arbitrage should imply value of the coupon bond at expiration date.

We use letters A and B to denote buyer and seller of a financial instrument. Recall definition of the cash

flow. Risk free zero coupon bond from A perspective is formally can be defined by its cash flow

CF A ( B ) = – B ( t 0 , T ) χ { t = t 0 } + B ( T , T ) χ { t = T } (0.1)

Here χ { t = s } is indicator function of the variable t [ t 0 , T ]. It is equal to 1 when t = s and equal to

0 otherwise. We put here that B ( T , T ) = 1. Negative term on the right hand side (0.1) corresponds to

A’s payment to B while positive term is the payment paid by B to A. No arbitrage pricing on [ t 0 , T ] is

defined by equalities of the present value (PV) and face value (FV) to zero, i.e.

PV 0 CF A = FV CF A = 0 (0.2)

These equalities signify that starting from zero value of all transactions at initiation of a contract we

should arrive at zero value all transactions at the expiration date. By definition of the PV and FV it

follows from (0.1) that B ( t 0 , T ) is risk free date- t 0 discount value of 1 at T and therefore

PV 0 CF A = – B ( t 0 , T ) + 1 B ( t 0 , T ) = 0

FV T CF A = – B ( t 0 , T ) B – 1 ( t 0 , T ) + 1 = 0

Equality (0.2) represents a formal definition of the no arbitrage pricing. Starting from zero value of

investment we should arrive at its zero value at expiration date. In other words no arbitrage on a particular

time interval implies that if the initial value of the all transactions of a financial instrument is equal to

zero then the value of all transactions at expiration date should be zero too. If time interval is

infinitesimally small then one can talk about no arbitrage at this moment. Next we will deal with

transactions values from bond buyer perspective and low index A in cash flows will be omitted.

Consider no arbitrage pricing of a coupon bearing risk-free bond. Cash flow of the coupon bond from

bond buyer perspective can be written as

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

n

1j

c χ { t = t j } + 1 χ { t = T }

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where c is a known value of the coupon. It is common rule that solution of the equation PV CF = 0 is

referred to as to ‘fair value’ in accounting documents or no arbitrage price in finance. Hence

PV CF = – B c ( t 0 , T ) +

n

1j

c B ( t 0 , t j ) + B ( t 0 , T ) = 0 (0.3)

and therefore no arbitrage price is defined by the formula

B(PV)

c ( t 0 , T ) =

n

1j

c B ( t 0 , t j ) + B ( t 0 , T ) (0.3)

Underlying argument to call B (PV)c ( t 0 , T ) no arbitrage price is that this sum at date t 0 exactly covers

cash flow

n

1j

c χ { t = t j } + 1 χ { t = T }

paid by B to A.

Remark. One price formulas are approximation of the real bid-ask pricing format. This approximation

might be a good one when value of the bid-ask spread is small. For illiquid or credit risky bonds one price

format perhaps might be not a good approximation.

Buyer of zero coupon risk free bond pays upper ask price and receives $1 at maturity. During lifetime of

the bond holder of the bond can sell it for lower bid price to someone else. Therefore real discount factor

over a period [ t 0 , T ] is related to ask quotes. Looking at the risk free coupon bearing bond one should to

note that the formula (0.3) should be adjusted to take into account trading spread. Hence

– B c ( t 0 , T ) +

n

1j

c B ask ( t 0 , t j ) + B ask ( t 0 , T ) = 0

Solution of the later equation should bring ask price. Nevertheless real world ask price can be lower than

it. Similarly the bid price defined by equation

– B c ( t 0 , T ) +

n

1j

c B bid ( t 0 , t j ) + B bid ( t 0 , T ) = 0

while the real world bid price can be higher than it. It looks quite likely that portfolio of bonds

[ n

1j

c B ( t 0 , t j ) ] [ B ( t 0 , T ) ]

is less liquid than correspondent coupon bond. Nevertheless middle price of the real bid-ask format can be

close to the single price model valuation.

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If coupon payments received from B do not immediately return to the bank and they are keeping until

expiration date then classical no arbitrage scheme should be adjusted to cover the real world problem. We

arrive at a new format of the pricing concept

a) borrow sum at risk free interest rate from bank to purchase risk free coupon bond

b) the total value of the cash flow at expiration date of the coupon bond should be also be equal to zero

Here total value of the cash flow is the date-T value of all payments received by A during lifetime of the

bond and the payment to the bank for the date- t 0 lending.

Note that the forward value of the CF is equal to

FV CF = – B c ( t 0 , T ) B – 1 ( t 0 , T ) +

n

1j

c B – 1 ( t j , T ) + 1 = 0 (0.4)

and from the equation (0.4) it follows that

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

n

1j

c B – 1 ( t j , T ) + 1 ] (0.4)

Note that values B ( t j , T ), j = 1, 2, … n are unknown at the date t 0 . Fundamental assumption of

pricing theory is that the forward rates are random variables, i.e.

B ( t j , T ) = B ( t j , T ; ω )

Formula (0.3) guarantees that PV of the cash flow (0.3) is equal to zero. Then formula (0.4) defines

coupon bond B(FV)

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

c ( t 0 , T ; ω ) as a random variable depending on market scenario ω.

This value guarantees that FV of the cash flow (0.4) is equal to zero too. Random value of the bond

B(FV)

c ( t 0 , T ; ω ) implies market risk in stochastic setting of the pricing problem. Generally speaking we

could not guarantee that face value of the cash flow does not equal to zero, i.e. equalities (0.2), ( 0.2) do

not take place for coupon bond in extended no arbitrage format. This observation leads us to extended

interpretation of the instrument pricing. In extended no arbitrage format pricing market risk is the attribute

of the spot price. We consider now risky pricing scheme in more details emphasizing our interest to

market risk. Define date-t 0 market implied forward bond price B ( t j , T ; t 0 ) by equality

B ( t 0 , T ) = B ( t 0 , t j ) B ( t j , T ; t 0 ) (0.5)

It shows that discount over the period [ t 0 , T ] can be presented as product of two discount factors over

forward period [ t j , T ] and its complementary spot period [ t 0 , t j ]. We use market implied forward

price of the bond as a market defined statistical estimate of the random future price B ( t j , T ; ω ). To

present calculation of the market risk we need to use a model of stochastic dynamics for the process

B ( t j , T ; ω ). Implied forward price presents no arbitrage estimate of the stochastic price. Indeed,

applying (0.5) we arrive at the equality

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B(FV)

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

n

1j

c B – 1 ( t j , T ; t 0 ) + 1 ] =

(0.6)

=

n

1j

c B ( t 0 , t j ) + B ( t 0 , T ) = B(PV)

c ( t 0 , T )

Note that no arbitrage pricing in stochastic setting does not eliminate market risk and does not protect an

investor from losses observed by investor at T.

Remark. In benchmark theory no arbitrage pricing is interpreted as equality PV of the cash flow to zero.

On the other hand modern pricing theory deals with the mathematical expectations of the stochastic future

transactions. This reduction ignores market risk and importance of the risk management in pricing.

Difference B ( t j , T ; ω ) – B ( t j , T ; t 0 ) is a factor of market risk of the bond implied by a particular

model representing stochastic dynamics B ( t , T , ω ). The set of scenarios

n

1j

{ ω : B ( t j , T , ω ) > B ( t j , T ; t 0 ) } , n

1j

{ ω : B ( t j , T , ω ) < B ( t j , T ; t 0 ) }

represent market risk of the bond seller buyer correspondingly. Assume for example market scenario ω

that implies inequality

B ( t j , T ; t 0 ) < B ( t j , T ; ω )

for all j. Then from (0.4) , (0.4) it follows that such scenarios FV CF A < 0 and

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

n

1j

c B – 1 ( t j , T ; ω ) + 1 ] = B c ( t 0 , T , ω )

Therefore the price of the bond that implied by the market scenario is higher than the spot price at the

date t 0 and ω { B c ( t 0 , T ; ω ) > B c ( t 0 , T ) }. The sets

Ω SR = { ω : B c ( t 0 , T ; ω ) > B c ( t 0 , T ) }

Ω BR = { ω : B c ( t 0 , T ; ω ) < B c ( t 0 , T ) }

define seller’s and buyer’s market risks correspondingly. Probabilities P ( Ω SR ) , P ( Ω SR ) denote values

of the market risk. Values

E B c ( t 0 , T ; ω ) χ ( Ω SR ) , E B c ( t 0 , T ; ω ) χ ( Ω BR )

represent average seller and buyer losses correspondingly.

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I. No arbitrage interest rate swap (IRS) valuation. Let us recall the well-known formulas of

the IRS valuation. Market value of an IRS is defined as the PV-reduction of the netted transactions if two

legs payments of the swap are scheduled on the same dates. Spot price IRS can be received by

replacement of the stochastic model rates by its implied forward estimates. This approach was used for a

FRA valuation in [MARKET RISK OF THE FIXED RATES CONTRACTS].

Generic IRS is a financial contract to exchange a fixed rate c for a floating rate L based on a notional

principal N. Let t = t 0 < t 1 < t 2 < … < t n = T be a known sequence of the reset dates and N notional

principal. The buyer of a IRS pays fixed rate payments cN to swap seller and receives the floating rate

payments N L ( t j – 1 , t j ) at the known sequence of date t j , j = 1, 2, … , n. If fixed and floating

transactions are scheduled on the same dates only the netted values of transactions are take place in real

world. Risk free floating rate L is usually one of the basic market rates such as Treasury rate, LIBOR, or

other similar rate. Future values L ( t j – 1 , t j ), j = 2, 3, … n are unknown at initiation date t 0 . Primary

assumption of the pricing theory is that the future rates are random variables

L ( t j – 1 , t j ) = L ( t j – 1 , t j , ω )

j = 2, 3, … n. Based on this assumption the real world cash flow from swap buyer perspective can be

represented in the form

CF ( t 0 , T ; L ) =

n

1j

[ L ( t j – 1 , t j , ω ) – c ] ∆ t j χ ( t = t j ) (1.1)

Positive terms on the right hand side in (1.1) correspond to payments to A while negative terms in (1.1)

are payments made by A to B. The IRS valuation problem is determination of the fixed rate

c = c ( t 0 , T ) that promises to counterparties A and B equality of their investment positions in the IRS

deal. The notion of equality should be refined based on no arbitrage principles. The PV reduction of the

right hand side of (1.1) leads us to equation

n

1j

[ L ( t j – 1 , t j , ω ) – c ] ∆ t j L ( t 0 , t j ) = 0

Solving this equation for the fixed rate we arrive at the value

C PV ( t 0 , T ; L , ω ) =

jj0

n

1j

j0jj1j

n

1j

t)t,t( L

)t,t(Lt )ω,t,t(L

(1.2)

Terms L ( t 0 , t j ) are interpreted as risk free discount factors from dates t j to t 0 . Recall the benchmark

formula for IRS valuation. Similar to (0.5) one defines implied market forward rates by the formula

l ( t j – 1 , t j ; t 0 ) = ]1)t,t(L

)t,t(L[

t

1

j0

1 -j0

j

(1.3)

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Stochastic forward rates in (1.2) imply market risk of the date-t 0 spot price used for buying IRS in the

spot market. Indeed replacing random rates L ( t j – 1 , t j ) in (1.2) by its market implied estimates

l ( t j – 1 , t j ; t 0 ) we arrive at the well known benchmark formula presenting the date- t 0 value of the IRS

c pv ( t 0 , T ) =

jj0

n

1j

j0j0j1j

n

1j

tΔ)t,t(L

)t,t(LtΔ)t;t,t(

l

=

jj0

n

1j

0

t)t,t(L

)T,t(L1

(1.2)

Two sets of market scenarios

Q SR = { ω : C PV ( t 0 , T , ω ) > c pv ( t 0 , T ) }

Q BR = { ω : C PV ( t 0 , T , ω ) < c pv ( t 0 , T ) }

represent seller’s and buyer’s market risk scenarios correspondingly. Set Q SR defines scenarios for which

’market is underpriced’, i .e. buyer spot price of the IRS is less than it is implied by the market. On the

other hand scenarios Q BR are those for which ‘market is overpriced’ i.e. buyer of the IRS pays more than

it is suggested by the market. Now let buyer of the swap pays c pv ( t 0 , T ) for IRS. This price implies that

PV CF A ( t , T ; l ) = 0. This condition is equivalent to conclusion that market’s estimate PV of the

future transaction to a counterparty of the IRS to be equal to zero. This estimate of the spread does not

imply that the date-T future value of the same cash flow to the same counterparty would be also equal to

zero. Indeed

FV CF ( t 0 , T ; L ) =

n

1j

[ L ( t j – 1 , t j , ω ) – c pv ( t 0 , T ) ] ∆ t j L – 1 ( t j , T , ω )

On the other hand implied forward rates estimates one can see that

FV CF ( t 0 , T ; l ) =

n

1j

[ l ( t j – 1 , t j ; t 0 ) – c pv ( t 0 , T ) ] ∆ t j l – 1 ( t j , T ; t 0 )

Here c pv ( t 0 , T ) is defined by (1.2). In contrast to formula (1.2) one can see that the use implied

forward rates l ( t j – 1 , t j ; t 0 ) do not lead to the equality present and future values of the IRS cash flow

to zero. Therefore formula (1.2) does not present no arbitrage pricing. Solutions of the equations

PV CF ( t 0 , T ; l ) = 0

FV CF ( t 0 , T ; l ) = 0

usually are different and use the spread c pv ( t 0 , T ) as the spot value always implies the market risk. Let

us assume now that FV CF ( t 0 , T ; l ) > 0. Then market suggests higher chance of profit when buying

IRS. If FV CF ( t 0 , T ; l ) < 0 then it makes sense to sell swap. Note that real world bid and ask prices

can somewhat to balance chances of buying or selling IRS. Market risk of the IRS can be characterized by

the distribution

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P { );T,t(CFPV

);T,t(CFFV

)L;T,t(CFPV

)L;T,t(CFFV

0

0

0

0

l

l > x }

II. No arbitrage pricing of risky bond. Consider a corporate coupon bond at date t 0 that

promises to pay its buyer a fixed coupon payments c at known dates t j , j = 1, 2, … n and principal

payment equal to 1 at bond’s expiration date t n = T and assume that the chance of default of the bond

cannot be ignored. Suppose that default time > 0 can be interpreted as a random variable. Note that this

assumption can be either true or wrong. In theory in order to present valuation formulas we suppose that

distribution of the default time is known while given historical data statistics a particular model of

default implies the pricing risk. Note that the existence of the default represents credit risk while

complete consideration of the default depends on its probabilistic distribution. Deviation between model

implied time of default and its realized value represents pricing risk that is a component of the market risk

of the model. First consider a simple example that illustrates a risky pricing. Let us first study a simple

case when default can occur only at expiration date T of the zero coupon corporate bond. At default

corporation cannot guarantee bond’s face value $1 at expiration date. Event { ω : ( ω ) > T } denotes

no-default market scenarios. The cash flow CF from bond buyer perspective can be written as

CF ( t , ω ) = – R ( t 0 , T ) χ { t = t 0 } + [ δ χ { ( ω ) = T } + 1 χ { ( ω ) > T } ] χ { t = T }

where recovery rate δ [ 0 , 1 ) is assumed to be known. Equation PV CF ( t , ω ) = 0 can be presented

in the form

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

This is market price that represents price for each market scenario. If no information is available

regarding market risk one can suppose that expected value is a convenient estimate of the spot price, i.e.

R spot ( t 0 , T ) = E R ( t 0 , T ; ω ) = B ( t 0 , T ) [ δ P { ( ω ) = T } + P { ( ω ) > T } ]

Applying latter estimates for face value calculations we arrive at the formulas

FV CF ( t , ω ) = – B – 1 ( t 0 , T ) R ( t 0 , T ; ω ) + δ χ { ( ω ) = T } + χ { ( ω ) > T } = 0

FV E CF ( t , ω ) = – B – 1 ( t 0 , T ) R spot ( t 0 , T ) + δ P { ( ω ) = T } + P { ( ω ) > T } = 0

Thus we can conclude that calculated based on PV or FV reductions of the market and spot prices are

equal to zero. Therefore pricing of the risky bond does not admit arbitrage opportunity.

Consider general case when default can occur at any time during lifetime of the corporate bond. Introduce

a discrete time approximation of the default time in the form

λ ( ω ) =

n

1k

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

Using discrete time default approximation cash flow from bond buyer perspective CF A can be written as

9

CF ( t , ω ) = – R ( t 0 , T ) χ { t = t 0 } +

n

1j

χ { λ ( ω ) = t j } [

1 - j

1k

c χ { t = t k } +

(2.1)

+ δ j χ { t = t j } ] + χ { λ ( ω ) > T } [

n

1k

c χ { t = t k } + 1 χ { t = T } ]

where δ j is recovery rate assigned to the event { λ ( ω ) = t j }. In theory recovery rates are assumed to

be known as well as their distributions P { λ ( ω ) = t j }. Right hand side of the formula (2.1) can be

interpreted as a formal definition of the risky coupon bond contract in a discrete time setting. PV 0

reduction of the cash flow (2.1) can be written in the following forms

PV 0 CF ( t , ω ) = – R ( t 0 , T ) +

n

1j

χ { λ ( ω ) = t j } [

1 - j

1k

c B ( t 0 , t k ) +

(2.2)

+ δ j B ( t 0 , t j ) ] + χ { λ ( ω ) > T } [

n

1k

c B ( t 0 , t k ) + B ( t 0 , T ) ]

The solution of the equation

PV 0 CF ( t , ω ) = 0

can be written as

R )PV(mkt

( t 0 , T , ω ) =

n

1j

χ { λ ( ω ) = t j } [

1 - j

1k

c B ( t 0 , t k ) +

(2.2)

+ δ j B ( t 0 , t j ) ] + χ { λ ( ω ) > T } [

n

1k

c B ( t 0 , t k ) + B ( t 0 , T ) ]

Randomness of the PV CF ( ω ) is stipulated by the default time that is a credit risk factor.

Future value of the cash flow is the future value specified by default time moment. It can be presented by

the formula

FV CF ( t , ω ) = –

n

1j

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

10

+ [

1 - j

1k

c B – 1 ( t k , t j ) + δ j ] } + χ { λ ( ω ) > T } R ( t 0 , T ) { B – 1 ( t 0 , T ) +

+ [

n

1k

c B – 1 ( t k , T ) + 1 ]

The solution of the equation FV CF ( t , ω ) = 0 can be written in the form

R )FV(

mkt ( t 0 , T , ω ) =

n

1j

B ( t 0 , t j ) [

1 - j

1k

c B – 1 ( t k , t j ) + δ j ] χ { λ ( ω ) = t j } +

(2.3)

+ B ( t 0 , T ) [

n

1k

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

Here B ( t k , T ) is a value of the bond issued at t k .

Applying implied forward bond price estimates we arrive at the formula

FV CF ( t , ω ) = –

n

1j

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

+ [

1 - j

1k

c B – 1 ( t k , t j ; t 0 ) + δ j ] + χ { λ ( ω ) > T } [

n

1k

c B – 1 ( t k , T ; t 0 ) + 1 ]

Here B ( t j – 1 , t j ; t 0 ) is implied forward discount factor at t 0 over period [ t j – 1 , t j ] determined in (0.5)

B ( t 0 , t k ) B ( t k , t j ; t 0 ) = B ( t 0 , t j )

k = 0 , 1, … n , j = k + 1, … n. Solving equation

FV CF ( t , ω ) = 0

and bearing in mind equality (0.5) we arrive at the formula

R )FV(mktimpl

( t 0 , T , ω ) =

n

1j

χ { λ ( ω ) = t j }

(2.3)

[

1 - j

1k

c B ( t 0 , t k ) + δ j ] + χ { λ ( ω ) > T } [

n

1k

c B ( t 0 , t k ) + 1 ]

Therefore

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R )PV(mkt

( t 0 , T , ω ) = R )FV(mktimpl ( t 0 , T , ω ) = R mkt ( t 0 , T , ω )

From latter equality it follows in particular that

E PV 0 CF = E FV T CF = 0 (2.4)

is more correctly refer to as to unbiased estimate than no arbitrage price. Solution of the equation (2.4) is

usually associated with spot price. Hence

R spot ( t 0 , T ) =

n

1j

P { λ ( ω ) = t j } [

1 - j

1k

c B ( t 0 , t k ) +

(2.5)

+ δ j B ( t 0 , t j ) ] + P { λ ( ω ) > T } [

n

1k

c B ( t 0 , t k ) + B ( t 0 , T ) ]

Pricing risk can be estimated by probabilities

P { R spot ( t 0 , T ) < R mkt ( t 0 , T , ω ) } , P { R spot ( t 0 , T ) > R mkt ( t 0 , T , ω ) }

which represent correspondingly the values of the chance that spot price is undervalued and overvalued

by the market. As far as the randomness of the market value R ( t 0 , T , ω ) is formed by the time of

default these probabilities are characteristics of the credit risk. The fraction

} ) ω , T , t (R ) T , t ( R { P

} ) ω , T , t (R < ) T , t ( R { P

0mkt0spot

0mkt0spot

is a single parameter that characterizes credit risk.

Market risk of the spot price (2.2) stems from the fact that cash flow generated by the spot price

R spot differs from the R )FV(mktimpl

for a real world scenario. Indeed, suppose at least in theory that values

δ k are known and for some m, m ≤ n

m

1k

δ k B ( t 0 , t k ) ≤ R spot ( t 0 , T ) <

1 m

1k

δ k B ( t 0 , t k )

Then scenarios

Ω m = { ω : λ ( ω ) ≤ t m }

correspond to the market risk of the bond buyer and P { λ ( ω ) ≤ t m } is the value of the chance that

buyer of the bond will lose money. Buyer’s losses simultaneously define seller’s profit. On the other hand

the supplement set of scenarios

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m = { ω : λ ( ω ) > t m }

represents buyer profit.

III. The Marked to Market (MtM) is a new valuation approach. It represents periodically

adjusted account that reflect changes instrument prices over a specified time period. Marked to Market

valuation is composed by the two components. One represents spot price of the instrument. This is the

price at which buyer or seller can buy or sell it. Other component is the transactions which reflect netted

changes of the instrument values.

The goal of the Marked to Market valuation is to decrease losses due to credit risk. Let us consider first an

illustrative example. It is usually used as an introduction of reduced form bond pricing. Assume that risky

bond admits default only at expiration date and let δ [ 0 , 1 ) denote recovery rate of the bond. Define

stochastic market price of the bond as

R mkt ( t , T , ω ) = B ( t , T ) [ 1 χ ( > T ) + δ χ ( = T ) ] = B ( t , T ) [ 1 – ( 1 – δ ) χ ( = T ) ]

Here ( > T ) = ω 0 and ( = T ) = ω 1 – δ denote no default and default scenarios correspondingly.

Market price we call the ‘fair’ price as it uniquely is defined for each scenario ω. The spot price of the

bond is one that is used for buying or selling bond. Assume in theory for example that

R spot ( t , T ) = E R mkt ( t , T , ω ) (3.1)

Thus buyer of the risky bond pays R spot ( t , T ) and receives risky bond that might default only at the

expiration date T . The probability of default is a known number p. The value of the bond at default is

assumed to be a known constant δ [ 0 , 1 ). MtM valuation is specified by the transactions. The price

R spot ( t , T ) is the initial value of the MtM account which is under the bond seller supervision. The

supervision of the MtM account does not imply the complete ownership of the money in the MtM

account. Hence

MtM ( t ) = R spot ( t , T )

Then at expiration date R mkt ( T , T , ω 0 ) = 1 and the sum

R spot ( t , T ) [ 1 – r ( t , T ) 360

tT ] – 1

from MtM account goes to B while counterparty A receives the face value of the bond that is equal to 1.

From A perspective transactions at expiration date T can be represented as following

– R spot ( t , T ) [ 1 – r ( t , T ) 360

tT ] – 1 + 1 (3.2)

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Here r ( t , T ) denotes a risk free interest rate which can be applied in the real world for investing the

sum R spot ( t , T ) at t. No default scenario implies that the total sum MtM ( T , ω 0 ) goes directly to the

bond seller. For default scenario value of the bond is equal to δ

R mkt ( T , T , ω 1 – δ ) = δ < 1 (3.3)

MtM account is designed to eliminate credit losses due to default. It can de performed by the buyer and

seller agreement on a ‘fair’ recovery rate frr. Define fair recovery rate by the formula

frr ( T , t ) = R spot ( t , T ) [ 1 – r ( t , T ) 360

tT ] – 1

Then for the default scenario

MtM ( T , ω 1 – δ ) = { frr ( T , t ) + [ δ – frr ( T , t ) ] } χ { δ < frr ( T , t ) } +

(3.3)

+ frr ( T , t ) χ { δ ≥ frr ( t , t ) }

In common format A purchases the bond and makes the cash payment to B at t. Formula (3.3) shows

that if δ [ 0 , frr ( T , t ) ) then value of the MtM account at T is the sum of two terms. The MtM

format the cash for bond is coming to B at default moment or at expiration date T, i.e. bond is going to A

at t while cash goes to B through the MtM account at T. Right hand side of the equality (3.3) represents

two terms assigned to two transactions at T. The first term frr ( T , t ) is the payment to B while the

second term [ frr ( T , t ) – δ ] is the payment that goes to A. Latter payment represents the falling price

of the bond. On the other hand given that δ [ frr ( T , t ) , 1 ] counterparty B receives frr ( T , t ) while

A receives δ for bond. Thus from bond buyer A perspective the value of the MtM account at T can be

written in the form

When δ = 0 the MtM sum should go to A and the net value of payments to A is zero. In general situation

when 0 < δ < frr ( T , t ) the credit losses should be returned to A while remainder goes to B. On the

other hand if frr ( T , t ) < δ ≤ 1 counterparty A does not receive any payments from MtM account and

its total value goes to B at T. Hence the value of the bond at expiration date T is defined by the formula

δ [ 1 – r ( t , T ) 360

tT ] – R spot ( t , T ) , if 0 ≤ δ < frr ( T , t )

R mkt ( t , T , ω 1 – δ ) = {

δ [ 1 – r ( t , T ) 360

tT ] , if frr ( T , t ) < δ ≤ 1

The latter equality can be rewritten as

R mkt ( t , T , ω 1 – δ ) = δ [ 1 – r ( t , T ) 360

tT ] – R spot ( t , T ) χ { 0 ≤ δ < frr ( T , t ) } (3.3)

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From this formula it follows that in case when δ exceeds fair value then amount

R spot ( t , T ) [ 1 – r ( t , T ) 360

tT ] – 1

goes to B and amount δ goes to A at T. The problem is given recovery rate and risk free interest rate to

define values R mkt ( t , T , ω ) and R spot ( t , T ) = E R mkt ( t , T , ω ).

Bearing in mind that there is no upfront payments and that spot price is the expected present value, EPV

then EPV of the all transactions to a counterparty is equal to zero. On the other hand no arbitrage pricing

suggests that zero value of all transaction to the counterparty at initiation implies zero value of the all

transactions at expiration date T. Define no arbitrage pricing in the stochastic market and spot price by

equations

MtM ( T , ω ) = 0 , E MtM ( T , ω ) = 0

correspondingly. From (3.2) it follows that

R mkt ( t , T , ω ) = 1 χ ( ω = ω 0 ) + R mkt ( t , T , ω ) χ ( ω = ω 1 – δ ) (3.4)

The second term on the right hand side is defined by equality (3.3). Hence, if δ ( frr ( T , t ) , 1 ] then

R spot ( t , T ) = E R mkt ( t , T , ω ) = ( 1 – p ) + δ [ 1 – r ( t , T ) 360

tT ] (3.5)

On the other hand

R spot ( t , T ) = ( 1 – p ) + { δ [ 1 – r ( t , T ) 360

tT ] – R spot ( t , T ) } p

if 0 ≤ δ < frr ( T , t ). Solving later equation we arrive at the formula

R spot ( t , T ) = p1

1

{ ( 1 – p ) + δ [ 1 – r ( t , T )

360

tT ] } (3.5)

Formulas (3.5) - (3.5) represent spot price of the risky bond in MtM pricing format. Hence, if corporate

bond’s recovery rate is sufficiently high δ ( frr ( T , t ) , 1 ] then MtM pricing format of the risky bond

is equivalent to the standard reduced form model valuation. If recovery rate of the corporate bond satisfies

condition 0 ≤ δ < frr ( T , t ) then formula (3.5) shows that the price in MtM format is lower than one

that is represented by the reduced form. The adjustment factor is ( 1 + p ) – 1 < 1. Recall that higher price

of the bond suggests lower interest rate.