It is revised at 09/05/2013 version. I added introduction and made some insignificant changes.

1. 1
MARKET RISK AND MARK-TO-MARKET VALUATION OF THE CROSS CURRENCY
SWAP.
Ilya I. Gikhman
6077 Ivy Woods Court Mason,
OH 45040, USA
Ph. 513-573-9348
Email: ilyagikhman@yahoo.com
JEL : G12, G13
Key words. Discount factor, risk free bond, Libor, OIS, market risk, mark-to-market valuation, interest
rate swap, cross currency swap, stochastic Libor model.
Abstract. In this paper we discuss some popular notions of the fixed income pricing. We pay more
attention to formal side of the use such notions as discount factor and mark-to-market valuation of the risk
free cross currency swap.
Introduction. In the first section of this paper, we present arguments for using swap rate as a discount
factor. In the next sections, we continued development of the two pricing concept which was introduced
in [1]. One price of Interest Rate Swap contract is based on the implied swap rate known as swap fixed
rate or swap spread. In our interpretation it is an estimate of the other price that is based on the ‘market’
swap rate. This price is defined for each admissible market scenario. Such point of view interprets market
price as random process. Swap spread then is an estimate of the most probable or most expected outcome
of the random spread. The difference between deterministic swap spread and stochastic spread specifies
market risk of the swap rate. The contemporary finance deals with only implied swap rate and ignores the
market risk that stems from stochastic setting of the problem. In Cross Currency Swap section, we
construct stochastic and implied spreads. The problem is similar to one that was studied in [2], [3] where
the value of the spot spread was presented. On the other hand, we changed the setting format similar to
[3] and the stochastic and deterministic market implied basis of the swap is calculated. In [3] the two
discounting curves method was introduced. One discount curve is used for the presenting implied forward
rate, which is an estimate of the realized future rate. Other discount curve is served as discount all cash
flows. We pay attention to the latter discount factor that should guarantee swap value at par.

2. 2
0. Discount. First, we begin with remarks on discounting problem. Discount factor from date T to
date t, t ≤ T is a currency rate which presents at date t the value of the one unit of the currency at T.
There is no need to assume existing of other currencies and their discount rates. Last time the primary
discount rate in US has been changed from the government bond rates to Libor and then to OIS rates.
Initially discount factor comes to the existence with the risk free bond concept. One pays B ( t , T ) at t to
get one dollar at T. In this case the bond price B ( t , T ) < 1 at t is interpreted as discounted factor from
date T to date t. One can interpret bond purchase as a deal when government borrows B ( t , T ) from
investors at t and return them one dollar at T. In this case, the interest paid by the government is also
referred to as to funding cost. Coupon bond that pays periodic coupon c at dates t j , j = 1, 2, … n
can be represented as a cash flow
CF = 
n
1j
c χ ( t = t j ) 1 ( t j ) + F χ ( t = T ) 1 ( T )
Here, 1 ( t j ) denotes one dollar at the date t j , t n = T and F is called face value of the bond. The price
of the coupon bond at t 0 is interpreted as present value, PV 0 of the CF. In the single currency
environment the unique way to present one dollar in a future time is the risk free bond. In this case PV
can be interpreted as a functional defined on discrete time functions as
PV 0 χ ( t = t j ) 1 ( t j ) = B ( t 0 , t j ) 1 ( t 0 )
j = 1, 2, … n. Hence,
B c ( t 0 , T ) = PV 0 CF = 
n
1j
[ c B ( t 0 , t j ) + F B ( t 0 , T ) ] 1 ( t 0 )
Libor rate L ( t , T ) can also be interpreted as other type of the dollar denominated discount factor outside
of the USA. Regardless of the way of its construction, the rate can be used to get a fixed amount of
currency in a future moment. Indeed, in order to get one unit of a currency at T one should invest at t the
amount
[ 1 + L ( t , T ) ] – 1
in the Libor contract that expired at T. If Libor is set that there is no arbitrage opportunity between
funding in Libor and T-bond. Investors can use one of these two rates depending on location. One cannot
use bond rate outside of USA and Libor rate if there is no Libor contracts on a particular market. There is
also an economic difference between bond’s interest rate and Libor. The bond’s interest is USA
government borrowing rate while Libor is the market borrowing rate formed by investor demand for
trading. This economic difference between two primary rates represents itself in pricing of liquidity and
credit risks.
One of the latest developments in the fixed income market is a practice to use a swap rate as a discount
factor. In contrast to above two rates a swap rate does not provide fixed amount of dollars at a particular
date in the future and therefore it could not be used as a PV or discount factor. To illustrate this idea
consider overnight index swap (OIS). OIS rate is the rate of a fix-for-floating single currency swap

3. 3
denominated in dollars. The swap maturity is usually between one week and one year period. There are
no intermediate transactions between counterparties and the settlement of the OIS is netted value at
maturity of the swap, which represents the difference between daily compounded fixed and floating
overnight rates. The floating rate of OIS is Fed Funds Rate, FFR. This rate is charged for lending funds
by primary US financial institutions. The FFR of a current business day is issued next business day in the
morning. Here we present a formal definition of the OIS contract. Define fixed and floating interests due
to exchange at OIS maturity T by formulas
OIS fixed = N S ois ( T – t )
OIS floating = N { 
n
1j
[ 1 + ff ( j ) d j ] – 1 }
Here, OIS fixed is the interest over lifetime of the swap, N is notional principal of the OIS, S ois is the fixed
interest rate, ff ( j ) is the floating federal fund rate for the j period, ( T – t ) is lifetime of the swap in days
over 365 days in the year format, d j is number of days for the j term in 365 days format. The settlement
value of the OIS is the absolute value of the difference between the two values above.
Therefore, there is no fixed amount at maturity of the OIS. In other words S ois rate fails to answer: what is
the OIS’s present value at t of the $1 at a future moment T. Hence, the OIS or other swap rate could not
be interpreted as a discount rate. One can roll over $1 over a particular period by applying market fixed
rate but there is no fixed amount at t is waiting investors at maturity. The use of a rate from OIS-family as
discount factor for a an instrument might facilitate its valuation but this argument is not a sufficient to
interpret it as a discount rate. More correctly we can say that the OIS rate is a spot estimate of the risky
yield of the daily compounded FFR. As far as the future overnight FFR values are unknown at initiation
the swap contract holds a market risk with probably a small standard deviation. On the other hand, a FFR
but not a OIS rate is a convenient rate for mark-to market (MtM) daily valuations. We return to MtM
valuation later in this paper.
The heuristic idea to low discount rate can be viewed as a way to increase impression on market
participants of a better market stability that in turn may effect on trade volumes and from the other hand
as well as increase availability for funding. This would improve liquidity of the market. Hence, printing
money policy for main currencies can considered as a temporary support of the market nevertheless a
long term effect of QE policy leads to the destruction of the economy in the sense of changes prices either
for the goods or for the currencies. Other observations suggest that additional money do not are
distributed uniformly. More cash are coming to more economically developed countries while less
developed countries arrive at worst economic situation.
1. Interest Rate Swap, IRS. In this section, we outline floating-for-fixed IRS valuation. Suppose
for example that floating rate is a Libor rate. It is a common to use notation L ( t , T ) for the Libor at date
t which expired at T. This notation should be considered as a function of two variables t and T that is
determined for any fixed T and t  [ 0 , T ]. This definition corresponds to an assumption that a Libor
contract can be bought and sold during lifetime of the contract. Let counterparty A pays fixed rate
payments and receives floating rate payments from counterparty B at a sequence of reset dates of the
swap. Let t = t 0 denote initiation date of the swap. Assume that A makes fixed rate payments of N C at
dates s 1 < s 2 < … < s m = T to counterparty B and counterparty B makes floating rate payments

4. 4
N L ( t j – 1 , t j ) to A at t 1 < t 2 < < … < t n ≤ T. Then from the A perspective the cash associated
with IRS can be written in the form
CF A ( L ) = N [ 
n
1j
L ( t j – 1 , t j ) χ ( t = t j ) – C 
m
1i
χ ( t = s i ) ] (1)
N is a notional principal of the swap. IRS valuation problem is a determination value of the fixed rate C.
CF A is a stepwise function that changes its values at t = { t j , s i } , j = 0, 1, … n , i = 1, 2, … m.
Rates L ( t j – 1 , t j ) are unknown its values at initiation date t = t 0 and they can be modeled by a random
process [4]. The standard valuation approach for IRS valuation is that the present values, PVs of fixed and
floating legs of the swap should be equal. This approach implies that
PV CF A ( L ) = 0 (2)
Introduce an appropriate discount factor D ( t , T ). Then from (1) and (2) it follows that

n
1j
L ( t j – 1 , t j ,  ) D ( t , t j ) – C p 
m
1i
D ( t , s i ) = 0
and therefore
C p (  ) =
)s,t(D
)t,t(D)ω,t,t(L
i
m
1i
jj1j
n
1j





(3)
Random variable C p = C p ( t ,  ) depends on market scenario . It represents market fair value of the
IRS for each scenario  at t. Besides market price, there exists spot price. This price is applied for trades.
Spot price calculation uses implied market rates l ( t j – 1 , t j , t ) calculated at date t
l ( t j – 1 , t j ; t ) = ]1
)tt()t,t(L1
)tt()t,t(L1
[
1
1-j1-j
jj
j




where Δ j = t j – t j – 1 as estimates of the future random rates L ( t j – 1 , t j ,  ). Construction of the
cash flow CF A ( l ) follows (1) in which rate L is replaced by its implied estimate l . Thus, we arrive at
the formula
c p =
)s,t(D
)t,t(D)t;t,t(
i
m
1i
jj1j
n
1j





l
(4)
For example formulas

5. 5
P { C p (  ) > c p }, M seller = E C p (  ) χ { C p (  ) > c p },
V seller = E [ C p (  ) - c p ] 2
χ { C p (  ) > c p }
represent market risk, mean of the losses and correspondent volatility of the swap. Latter formulas are
primary characteristics of the risk management of the IRS valuation based on PV reduction. These
formulas use distribution of the random variable C p (  ). This variable is defined by the formula (3) that
is a weighted sum of the random variables L ( t j – 1 , t j ,  ). Hence in order to present distribution of the
random spread C p (  ) we need to know distribution of the future rates L ( t j – 1 , t j ,  ) , j = 2, 3, … .
We can use [4] either implied forward rate to present or other model for presentation the future rate
L ( v , v + h ,  )
l ( v , v + h ; s ) = l ( v , v + h ; t ) + 
s
t
 ( u ) l ( v , v + h ; u ) d u +
+ 
s
t
λ ( u ) l ( v , v + h ; u ) d w ( u )
Note that given current date t and putting s = v we note that l ( v , v + h ; v ) = L ( v , v + h ,  ).
Other approach [4] is a direct model of the future rate. The model can be presented in the form
L ( v , v + h ,  ) = L ( t , t + h ) + 
v
t
 ( u ) L ( u , u + h ,  ) d u +
+ 
v
t
 ( u ) L ( u , u + h ,  ) d w ( u )
where coefficients are deterministic or random functions, which satisfy some standard conditions.
Historical data usually used for estimation of the unknown coefficients of the equations. The solution of
the general linear stochastic differential equation can be presented in closed form [5]. Having the
distribution of the random variables L ( t j – 1 , t j ,  ), j = 2, 3, … one can calculate introduced above
market risk characteristics.
2. Cross Currency Swap, CCS.
The Problem Setting. Now we apply discount concept for valuation of floating-for-floating risk free
cross currency swap (CCS). Let t = t 0 denote initiation date and t 1 < t 2 < … < t n = T be a sequence
of reset dates of the swap. At initiation date the exchange of the two principals takes place. Counterparty
A sends to counterparty B amount of N 0 denominated in USDs and receives from B the equivalent
amount of N 1 of denominated in euro. During lifetime of the contract counterparty A pays B payments in
EUR based on 3M ( Libor + α ) and receives payments in USD based on 3M Libor from B. At the end of
the contract the initial amounts of N j , j = 0 , 1 are returned to their original owners. From the
counterparty B perspective cash flow associated with the CCS transactions can be presented in the form

6. 6
CF B = [ N 0 1 0 ( t 0 ) – N 1 1 1 ( t 0 ) ] χ ( t = t 0 ) + 
n
1j
{ N 1 [ L 1 ( t j – 1 , t j ) + α ] 1 1 ( t j ) –
– N 0 L 0 ( t j – 1 , t j ) 1 0 ( t j ) } χ ( t = t j ) + [ N 0 1 0 ( T ) – N 1 1 1 ( T ) ] χ ( t = T )
where low index 0 is assigned to USD and index 1 to the EUR. Denote L k ( t , T ) date t Libor rate for
j-currency with expiration at T , and 1 j ( t ) , j = 0 , 1 denotes one unit of j-currency. The exchange rate
q ( t ) at t is defined by equality
1 0 ( t ) = q ( t ) 1 1 ( t )
and therefore N 1 = N 0 q ( t 0 ). Note that cash flow CF B should be considered as a formal definition of
the CCS contract. The cash flow from the A perspective is CF A = – CF B . The CCS pricing problem is
derivation of the swap basis α that is also often referred to as to the cross currency swap spread. The cash
flow can be rewritten in equivalent form as
CF B = N 0 { [ 1 0 ( t 0 ) – q ( t 0 ) 1 1 ( t 0 ) ] χ ( t = t 0 ) +
+ 
n
1j
{ q ( t 0 ) [ L 1 ( t j – 1 , t j ) + α ] 1 1 ( t j ) – (5)
– L 0 ( t j – 1 , t j ) 1 0 ( t j ) } χ ( t = t j ) + [ 1 0 ( T ) – q ( t 0 ) 1 1 ( T ) ] χ ( t = T ) }
As far as the value N 0 does not effect on basis value without loss of generality one can put N 0 = 1.
Valuation. Ignoring credit and counterparty risks we note that first term on the right hand side (5) is
equal to zero. Similar to it the third term does not equal to zero as far as it represents transaction at T and
in general q ( T ) ≠ q ( t 0 ). Current approach to CCS valuation first defines primary and secondary
currencies. Then primary the USD Libor is used as primary discount rate for both sides of the contract
valuation. In case USD and EUR payments should be converted into correspondent to USD cash flow.
Then USD discount rate would be applied for calculations the PV equality of the swap legs. The EUR is
considered less liquid than USD and therefore it is chosen as secondary currency.
Consider the CCS defined by the cash flow (5). During the lifetime of the contract the Libor rates
L k ( t j – 1 , t j ) , k = 0 , 1 over the future periods [ t j – 1 , t j ] , j = 2, …, n are unknown at initiation date
t = t 0 and we assume that these rates can be interpreted as random variables. This assumption implies
that historical data are interpreted as a statistical population. This statistical point of view is called
randomization of the model. Using PV valuation we replace random future rates by its market implied
forward rates that known at t 0. Replacement of the random rates by its market estimates always presents
market risk. The inverse exchange of the original notional sums N j , j = 0, 1 occurs at maturity. In
formula (5) it is expressed by the third term. These final transactions is subject exchange rate risk. It
stems from the fact that exchange rate at t 0 in general does not equal to the real exchange rate at T, i.e.
equality
N 0 1 0 ( t 0 ) = N 1 1 1 ( t 0 )

7. 7
is true at initiation date t 0 and it does not true at the swap maturity T. Hence, the valuation model should
keep in mind two types of the risks exchange and market risks. Each leg of the CCS deals with two
different currencies. Two currencies of the swap do not have equal liquidities and the credit ratings. From
B perspective the value of the market risk of the transaction at date t j , j = 1, 2, … n can be expressed in
the form
{ q ( t 0 ) [ L 1 ( t j – 1 , t j , ω ) + α ] 1 1 ( t j ) – L 0 ( t j – 1 , t j , ω ) 1 0 ( t j ) } χ ( Ω m B ( t j )) χ ( t = t j ) +
(1)
+ [ q ( t 0 ) 1 1 ( T ) – 1 0 ( T ) ] χ ( Ω q B ( T ) ) χ ( t = T )
Here, Ω m B ( t j ) = { ω : L 1 ( t j – 1 , t j , ω ) + α < L 0 ( t j – 1 , t j , ω ) $( t j ) } , j = 1 , 2 , … , n – 1.
Then Ω q B ( T ) = { ω : q ( t ) < q ( T , ω ) } represents the market scenarios for which cost in USD of
the one euro at T is lower than at initiation date t , i.e. 1 1 ( T ) < 1 1 ( t ).
Pricing problem from B perspective is two folds problem. One is to estimate the basis α and the second
is the value of the market risk that is the value of the chance that the basis value implied by the scenario
 will be lower than initially estimated , i.e. P {  : α (  ) < α }.
Risky scenario for A is a scenario when A pays higher price than spread value. The benchmark approach
presents the spot estimate α as the market implied value. Let us consider implied value of the swap in
stochastic market. Market price is the price assigned to the swap at the date t for each observable scenario.
The first step of the valuation problem is a model for the future rates L k ( t j – 1 , t j , ω ) and future
exchange rate q ( T , ω ). The second step is calculation of the spot price and presenting its risk analysis.
The benchmark approach uses the equation PV CF B = 0 to present a value of the basis swap value
α ( · , ω ). As far as the cash flow (5) consists from the two currencies the present value reduction should
be made cautiously. Pricing derivatives consist from fragmentation of the complex onto simple blocks.
There are different ways to present PV of the cash flow CF B . First, we need to choose single currency
for representation. Usually USD is used for swap valuation. Consider USD PV reduction of the one EUR
at t j , j = 1, 2, … n cash flow. Bearing in mind equalities 1 1 ( t j ) = q – 1
( t j ) 1 0 ( t j ) and the fact that
exchange rate at a future moments are unknown at initiation assume that exchange rate is a random
process q ( t ) = q ( t , ω ) , t ≥ t 0 . Valuation approach corresponds to the strategy of receiving a
payment in EUR and converting it immediately in USD. Such point leads us to the USD denominated
cash flow (1)

n
1j
{ q ( t 0 ) q – 1
( t j ) [ L 1 ( t j – 1 , t j , ω ) + α ] – L 0 ( t j – 1 , t j , ω ) } 1 0 ( t j ) χ ( t = t j ) +
(1)
+ [ q ( t 0 ) q – 1
( T ) – 1 ] 1 0 ( T ) χ ( t = T )
No arbitrage principle applied for a market scenario ω implies that PV 0 of the cash flow (1) leads to
the equation

8. 8

n
1j
{ q ( t 0 ) q – 1
( t j ) [ L 1 ( t j – 1 , t j , ω ) + α ] – L 0 ( t j – 1 , t j , ω ) } D 0 ( t 0 , t j ) +
+ [ q ( t 0 ) q – 1
( T ) – 1 ] D 0 ( t 0 , T ) = 0
It follows that
α ( · ; t 0 , ω ) = [ 
n
1j
q ( t 0 ) q – 1
( t j , ω ) B 0 ( t , t j ) ] – 1
{ 
n
1j
[ L 0 ( t j – 1 , t j , ω ) –
(6)
– q ( t 0 ) q – 1
( t j , ω ) L 1 ( t j – 1 , t j , ω )] D 0 ( t 0 , t j ) + [ 1 – q ( t 0 ) q – 1
( T , ω ) ] D 0 ( t 0 , T ) }
Equality (6) represents the market basis that is determined for each market scenario ω. At initiation of the
swap contract date t sellers and buyers draw a market estimate of the α ( · ; t , ω ). It is a market practice
to use market implied prices as the estimates of the contract components. The date t estimates of the
random future exchange rate and Libor rate q ( t j , ω ) , L ( t j – 1 , t j , ω ) are the non-random date t
forward exchange rate contract q ( t j , t ) and implied forward Libor rate l ( t j – 1 , t j , t ) correspondingly.
Then the market implied swap basis is an estimate of the random basis α ( · ; t , ω ). It can be presented in
the form
αˆ ( · , t 0 ) = [ 
n
1j
q ( t 0 ) q – 1
( t j , t ) B 0 ( t 0 , t j ) ] – 1
{ 
n
1j
{ [ l 0 ( t j – 1 , t j , t 0 ) –
(6)
– q ( t 0 ) q – 1
( t j , t 0 ) l 1 ( t j – 1 , t j , t 0 ) ] D 0 ( t 0 , t j ) + [ 1 – q ( t 0 ) q – 1
( T , t 0 ) ] D 0 ( t 0 , T ) }
Scenarios for which { ω : α ( · ; t , ω ) > α ( · , t ) } correspond to the risk of the counterparty A. For this
scenarios A pays a higher price than it is implied at t. Note that the basis valuation represented above is
applied by PV reduction
PV 0 ( t 0 ) CF B = 0
for valuations of the basis of the swap.
3. Mark-to-Market (MtM) valuation of the CCS with zero chance of default.
Given PV reduction of the cash flows let us outline the essence of the MtM valuation. Consider the first
trade day t 0 + 1 after swap initiation at t 0 and suppose that t 0 + 1 < t 1. At the date t 0 + 1 new
values of the implied Libor l k ( t j – 1 , t j , t 0 + 1 ), k = 0, 1 and forward exchange rates q ( t j , t 0 + 1 ),
j = 1, 2, … n are coming up that do not coincide with the corresponding data of these rates calculated at
previous date t 0 . New data effect on swap spread value αˆ ( t 0 + 1 ) which can go up or down with
respect to αˆ ( t 0 ). The market implied value of the CCS at the date t 0 + 1 can be presented in the form

9. 9
V B ( t 0 + 1 ) = 
n
1j
{ q ( t 0 ) q – 1
( t j , t 0 + 1 ) [ l 1 ( t j – 1 , t j , t 0 + 1 ) + αˆ ( t 0 + 1 ) ] –
– l 0 ( t j – 1 , t j , t 0 + 1 ) } D 0 ( t 0 + 1 , t j ) + [ q ( t 0 ) q – 1
( T , t 0 + 1 ) – 1 ] D 0 ( t 0 + 1 , T )
Recall that αˆ ( t 0 + 1 ) is calculated at t 0 + 1 and V B ( t 0 + 1 ) = V B ( t 0 + 1 , αˆ ( t 0 + 1 )) = 0.
The date-t 0 hypothetical forward transactions at t 0 + 1 is based on the basis spread αˆ ( t 0 ) can be
presented by the formula
N 1 [ L 1 ( t 0 , t 0 + 1 ) + αˆ ( t 0 ) ] 1 1 ( t 1 ) – N 0 L 0 ( t 0 , t 0 + 1 ) 1 0 ( t 1 ) =
= N 0 { q ( t 0 ) q – 1
( t 0 + 1 , t 0 ) [ L 1 ( t 0 , t 0 + 1 ) + αˆ ( t 0 ) ] – L 0 ( t 0 , t 0 + 1 ) } 1 0 ( t 0 + 1 )
while the real value of this transaction at the date t 0 + 1 is
N 0 { q ( t 0 ) q – 1
( t 0 + 1 ) [ L 1 ( t 0 , t 0 + 1 ) + αˆ ( t 0 ) ] – L 0 ( t 0 , t 0 + 1 ) } 1 0 ( t 1 )
The difference between realized value at date t 0 + 1 and its date t 0 estimate is equal to
δ ( t 1 ,  ) = N 0 q ( t 0 ) [ q – 1
( t 0 + 1 ) – q – 1
( t 0 + 1 , t 0 ) ] [ L 1 ( t 0 , t 0 + 1 ) + αˆ ( t 0 ) ]
The value δ ( t 1 ,  ) and date t 0 + 1 new values of the implied forward rates l k ( t j – 1 , t j , t 0 + 1 ),
k = 0, 1 will effect on change of the spread value at t 0 + 1. It is sufficient to make MtM calculations for
one of the cash flow either for ‘in’ or ‘out’. Assuming that t 0 + 1 < t 1 we note that
PV ( k ) ( CF B ,in ) = 
n
1j
q ( t 0 ) q – 1
( t j , k ) [ l 1 ( t j – 1 , t j , k ) + αˆ ( k ) ] D 0 ( k , t j ) +
+ q ( t 0 ) q – 1
( T , k ) D 0 ( k , T ) ,
where k = t 0 , t 0 + 1. Define inflow and outflow to counterparty B. Ignoring chance of default the
inflow and outflow to B can be represented correspondingly in the forms
CF B , in = 
n
1j
q ( t 0 ) q – 1
( t j ) [ L 1 ( t j – 1 , t j , ω ) + α ] 1 0 ( t j ) χ { t = t j } +
+ q ( t 0 ) q – 1
( T ) 1 0 ( T ) χ { t = T }
CF B , out = 
n
1j
L 0 ( t j – 1 , t j , ω ) 1 0 ( t j ) χ { t = t j } + 1 0 ( T ) χ { t = T }
Let
Δ PV ( t 0 ) ( CF B ,in ) = PV ( t 0 + 1 ) ( CF B ,in ) – PV ( t 0 ) ( CF B ,in ) > 0 (7)

10. 10
This represents a market scenario for which A should make larger hypothetical payment than it was
scheduled at t 0 . In this case counterparty A should make the first payment of Δ PV ( t 0 ) ( CF B ,in ) in
MtM account. The value of the payment represents risk exposure of the counterparty A. If the value (7) is
negative then the first MtM payment should be made by B. On the next day t 0 + 2 we repeat similar
calculation to present the value
Δ PV ( t 0 + 1 ) ( CF B ,in ) = PV ( t 0 + 2 ) ( CF B ,in ) – PV ( t 0 + 1 ) ( CF B ,in )
If Δ PV ( t 0 + 1 ) ( CF B ,in ) > 0 then A should add new payment Δ PV ( t 0 + 1 ) ( CF B ,in ) to MtM
account. To present value of the MtM account at t 0 + 2 we need to adjust the past date t 1 value of the
MtM account Δ PV ( t 0 ) ( CF B ,in ) for [ t 0 , t 0 + 1 ] period. If D 0 ( t 0 , t 0 + 1 ) represents primary
discount rate then date t 0 + 2 value of the past MtM account is
D 1
0

( t 0 , t 0 + 1 ) Δ PV ( t 0 ) ( CF B ,in )
Assume that Δ PV ( t 0 + 1 ) ( CF B ,in ) < 0. Then counterparty B due to MtM payment at the date
t 0 + 2. Consider the difference
Δ PV ( t 0 + 1 ) ( CF B ,in ) – D 1
0

( t 0 , t 0 + 1 ) Δ PV ( t 0 ) ( CF B ,in )
If the difference is positive then counterparty B should make the payment equal to the above value in
MtM account. If the difference is negative then
D 1
0

( t 0 , t 0 + 1 ) Δ PV ( t 0 ) ( CF B ,in ) – Δ PV ( t 0 + 1 ) ( CF B ,in )
should be returned to A. Similar calculations continue on next steps. Let t j , j = 1, 2, … n be a reset date
and Q ( t j – 1 ) denote the value of the MtM account at t j – 1 date. Consider the difference
Δ PV ( t j – 1 ) ( CF B ,in ) = PV ( t j ) ( CF B ,in ) – PV ( t j – 1 ) ( CF B ,in )
The value of the MtM account at the date t j is equal to
MtM act ( t j ) = Q ( t j – 1 ) D 1
0

( t j – 1 , t j ) + Δ PV ( t j – 1 ) ( CF B ,in ) (8)
If two terms on the right hand side (8) are positive then party A should pay to MtM account the sum of
Δ PV ( t j – 1 ) ( CF B ,in ). On the other hand if Δ PV ( t j – 1 ) ( CF B ,in ) < 0 and MtM act ( t j ) > 0
the sum Δ PV ( t j – 1 ) ( CF B ,in ) should be withdrawn from MtM account and return to party A. In case if two
terms on the right hand side (9) are negative then B should add the sum Δ PV ( t j – 1 ) ( CF B ,in ) to
MtM account.
Recall that in our construction we assumed that CCS with zero chance of default. In case of positive
chance of default MtM account should not be netted as MtM account should protect against
counterparties default.

11. 11
Appendix.
Recall that PV concept came up with bonds valuations. By definition PV is an amount paid at initiation of
contracts in order to receive $1 at bond’s maturity. On the other hand an IRS is a contract for exchange
future rate payments N L ( t j – 1 , t j ,  ) unknown at initiation for the determined at t fixed rate payments
N c. Assume for example that t 1 < s 1. In this case counterparties obtain their payments and invest them
immediately at risk free interest rate D – 1
( t j , T ) , j = 1, 2, ... . Counterparty A account at maturity T
is equal to
FV A ( L ) = 
n
1j
L ( t j – 1 , t j ,  ) D – 1
( t j , T ) – C f 
m
1i
D – 1
( s i , T )
Note that the values D ( s i , T ) do not known at t and these values we assume can be interpreted as
random variables. No arbitrage principle extended on pricing for admissible market scenarios states that
counterparty starting with 0 account at initiation date t should arrive at 0 at maturity regardless of the
market scenario . Applying this idea, we arrive at the market spread value
C f (  ) =
)ω,T,s(D
)ω,T,t(D)ω,t,t(L
i
1
m
1i
j
1
j1j
n
1j







(3′)
Denote D ( t j , T ; t ) date t implied forward rate over the time period [ t j , T ]. Then we can define the
implied date-t spot value of the cash flow FV A ( l ) and calculate the spread implied by the market
c f =
)t;T,s(D
)t;T,t(D)t,t,t(
i
1
m
1i
j
1
j1j
n
1j






 l
(4′)
One can expect that in the most scenarios c p  c f . Assume for example that c p > c f . Then forward
value reduction suggests that the fixed leg rate c p is overvalued. Indeed, date t market analysis
demonstrates that swap spread that makes equal fixed and floating cash flows equal at T is less than
applied for the swap contract by using PV reduction. On the other hand inequality c p < c f implies the
opposite situation that suggests that floating leg payments dominates over the fixed one. A complete
valuation of the IRS cannot be reduced to a single FV or PV of the cash flows. Let us consider additional
characteristic of the IRS spread. Define market and market implied rate of return of the floating leg
RR ( t ,  ) =
)ω(PV
)ω(FV
- 1 =
)t,t(D)ω,t,t(L
)ω;T,t(D)ω,t,t(L
jj1j
n
1j
j
1
j1j
n
1j







- 1

12. 12
rr ( t ) =
PV
FV
- 1 =
)t,t(D)t,t,t(
)t;T,t(D)t,t,t(
jj1j
n
1j
j
1
j1j
n
1j







l
l
- 1
These are market estimates of the real rate of return of the IRS. The case c p > c f ( c p < c f )
corresponds to rr ( t ) < 0 ( rr ( t ) > 0 ). In stochastic environment investment can be characterized by
its PV and its expected market rate of return. Hence, in case rr ( t ) < 0 market suggests that that fixed
payment of the IRS is overstated while rr ( t ) > 0 means the opposite. The market estimate and
corresponding statement regarding value of the IRS’s spread is subject of risks. In the favorable for the
fixed rate payer’s case c p < c f the market risk can be defined as P { RR ( t ,  ) < rr ( t ) }. In the
favorable for fixed rate receiver case c p > c f the market risk can be measured by the probability
P { RR ( t ,  ) > rr ( t ) }.
Remark. The notation L ( t , T ) , 0 ≤ t ≤ T < + ∞ use above inexplicitly admitted that Libor contracts
can be traded over lifetime of the contract by using Libor rate. Indeed, for a particular contract with
expiration at T the function L ( t , T ) is defined for any t, t ≤ T. This function specifies the value of the
Libor contract at any time prior to T. On the other hand the real Libor rates are existed only for the fixed
maturities h equal to 1 day, 1 week, 2 weeks, and j = 1, …, 12 months. In such setting Libor rates are
defined with the help of the function L ( t , t + h ) = L h ( t ). Formulas (1) – (4) hold as they are take
place also for the fixed t and T. The stochastic equation for implied forward rate
l ( v , v + h ; s ) = l h ( v ; s ) , s > t which presents the future rate L h ( v ,  ) = L ( v , v + h ,  )
can be written as
l h ( v ; s ) = l h ( v ; t ) + 
s
t
 ( u ) l h ( v ; u ) d u +
+ 
s
t
λ ( u ) l h ( v ; u ) d w ( u )
Here, l ( v , v + h ; t ) = l h ( v ; t ) is deterministic number known at t. On the other hand we can use the
above direct model to define future Libor rate
L h ( v ,  ) = L h ( t ) + 
v
t
 ( u ) L h ( u ,  ) d u +
+ 
v
t
 ( u ) L h ( u ,  ) d w ( u )
Denote

13. 13
c p ( h ) =
)s,t(D
)t,t(D)t;t,t(
i
m
1i
jj1jh
n
1j





l
Putting  c p ( h ) = c p ( h 1 ) – c p ( h 2 ) , h 1 > h 2 we note that
 c p ( h ) =
)s,t(D
)t,t(D)t;t,t(
)s,t(D
)t,t(D)t;t,t(
i
m
1i
jj1j2
n
1j
i
m
1i
jj1j1
n
1j
2
2
1
1











ll
Here l j = l jh . We used for simplicity the same sequence of reset dates for the shorter rate l 2 as for the
longer rate l 1 . This is implied market estimate of the basis spread. Investment in basis spread is risky.
The value of the risk from buyer perspective is measured by the probability
P {  C f ( h ,  ) <  c p ( h ) }
where  C f ( h ,  ) = C f ( h 1 ,  ) - C f ( h 2 ,  ) and
C p ( h q ,  ) =
)s,t(D
)t,t(D)ω,t,t(L
i
m
1i
jj1jq
n
1j
q
q





q = 1, 2. It presents a value of the chance that buyer pays more than it implies by the market scenario.
Here we present alternative approach to CCS valuation. We call two investments are equal at a moment of
time if their instantaneous rates of return are equal and if two investments are equal at any moment on a
time interval then we called them equal at this interval [1].
Payment received at t j and denominated in EURs should be at t j converted in USD and invested in US
Treasury bonds. Alternatively they can be invested in EUR denominate bonds and then converted into
USDs at the swap maturity T. In either case the pricing should specify the lowest swap spread. Note that
it is possible that the pricing solution of the CCS can be obtained by converting a EUR payment
immediately in USD while other EUR payment is better to invest in EUR bond and convert it at a later
time or at the maturity. This remark highlight the fact that spread calculation is the sufficiently risky
procedure.
The PV denominated in USD at t and FV at T can written in the forms

14. 14
PV ( t ) ( CF B ,in ) = 
n
1j
q ( t 0 ) q – 1
( t j , ω ) [ L 1 ( t j – 1 , t j , ω ) + α ] D 0 ( t , t j ) +
+ q ( t 0 ) q – 1
( T , ω ) D 0 ( t , T )
FV ( T ) ( CF B , in ) = 
n
1j
q ( t 0 ) q – 1
( t j , ω ) [ L 1 ( t j – 1 , t j , ω ) + α ] F 0 ( t j , T, ω ) +
+ q ( t 0 ) q – 1
( T , ω )
PV ( T ) ( CF B , out ) = 
n
1j
L 0 ( t j – 1 , t j , ω ) D 0 ( t , t j ) + D 0 ( t , T )
FV ( T ) ( CF B , out ) = 
n
1j
L 0 ( t j – 1 , t j , ω ) F 0 ( t j , T , ω ) + 1
Here D 0 ( t , T ), F 0 ( t , T ) denote an appropriate USD discount and forward rates. One should presents
the models for the future rates D , l , q , F and their market implied forward rates should be modeled to
present a numerical solution of the swap pricing problem. The equation representing equality of the rates
of return of the market implied cash flows can be written as
)CF(VP
)CF(VF
in,B
in,B
=
)CF(VP
)CF(VF
out,B
out,B
(9)
Solving (9) for α ( · , ω ) we arrive at the market stochastic market swap basis values. Thus
α ( · , ω ) =
=
)ω,T,t(F)ω,t(q)t(q)ω,T,t(Λ)t,t(D)ω,t(q)t(q
])t,t(D)ω,T,t(F)ω,T,t(Λ[)ω,t,t(L)ω,t(q)t(q
j0j
1
0
n
1j
0j00j
1
0
n
1j
j00j00j1-j1j
1
0
n
1j










–
)ω,T,t(F)ω,t(q)t(q)ω,T,t(Λ)t,t(D)ω,t(q)t(q
])T,t(D)ω,T,t(Λ[)ω,t(q)t(q
j0j
1
0
n
1j
0j00j
1
0
n
1j
00j
1
0





 


where

15. 15
Λ 0 ( t 0 , T ,  ) =
1)t,t(D)ω,t,t(L
1)ω,T,t(F)ω,t,t(L
j00j1-j0
n
1j
j0j1-j0
n
1j






Next applying market implied deterministic estimates we can get implied by the market value α ( · , t ).
The valuation formula is similar to above formula in which random rates q ( t j ,  ), L k ( t j – 1 , t j ,  ),
F k ( t j , T ,  ) are replaced by its market estimates q ( t j , t 0 ) , l k ( t j – 1 , t j ,  ), f k ( t j , T , t 0 ),
k = 0, 1. We used notation D 0 ( t , t j ), j = 1, 2, … n for USD discount factor. The real world examples
of the discount factor are US T-bond or Libor rates. In section 1 we argued that OIS rate by its
construction does not guarantee at t = t 0 a fixed amount at any future moments t j , j = 1, 2, … n.

16. 16
Bibliography.
1. Gikhman I. Alternative Derivatives Pricing: Formal Approach. LAP LAMBERT Academic
Publishing, 2010, p. 164.
2. R. Jarrow, S Turnbull. Derivatives Securities, South-Western College Publishing, 2ed. 2000, 684.
3. W. Boenkost, W. Schmidt. Cross Currency Swap Valuation, 2005, 11.
4. I. Gikhman. Fixed Rates Modeling.
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2287165.
5. I. Gikhman , A. V. Skorohod. Stochastic Differential Equation Springer-Verlag 1972.