rates basics

8/10/2019 Rates Basics

1/22


2/22Electronic copy available at: http://ssrn.com/abstract=2194907

Curves and Term Structure Models. A Primer Christian Fries

Contents

1 Introduction 3

2 Foundations and Notation 42.1 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . 42.2 Currencies and Collateralization . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Valuation in Foreign Currency . . . . . . . . . . . . . . . . . 42.2.2 Valuation of Collateralization . . . . . . . . . . . . . . . . . 5

2.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Valuation of Linear Products . . . . . . . . . . . . . . . . . . . . . . 6

2.4.1 Discount Curves . . . . . . . . . . . . . . . . . . . . . . . . 62.4.2 Forward Curves . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Discount Curves and Forward Curves 83.1 Discount Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Forward Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2.1 Performance Index of a Discount Curve (or self-discounting) 9

4 Interpolation of Curves 114.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Interpolation Method and Interpolation Entities . . . . . . . . . . . . 11

4.2.1 Interpolation of Forward Curves . . . . . . . . . . . . . . . . 124.2.2 Interpolation of Forward Curve via Value Curve . . . . . . . . 12

5 Calibration of Curves 135.1 Global Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 135.2 Generalized Denition of a Swap . . . . . . . . . . . . . . . . . . . . 135.3 Calibration of Discount Curve to Swap Paying the Collateral Rate

(aka. Self-Discounted Swaps) . . . . . . . . . . . . . . . . . . . . . . 145.4 Calibration of Forward Curves . . . . . . . . . . . . . . . . . . . . . 15

5.4.1 Fix-versus-Float Swaps . . . . . . . . . . . . . . . . . . . . . 155.4.2 Float-versus-Float Swaps (Tenor Basis Swaps) . . . . . . . . 15

5.5 Calibration of Discount Curves when Payment and Collateral CurrencyDiffer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.5.1 Fixed Payment in Other Currency . . . . . . . . . . . . . . . 16

5.5.2 Float Payment in Other Currency . . . . . . . . . . . . . . . 165.6 Calibration of Discount Curves as Spread Curves . . . . . . . . . . . 175.7 Lack of Calibration Instruments (for Difference in Collateralization) . 18

6 Sample Code for Curve Calibration 19

7 Redening Forward Rate Market Models 20

References 21

c 2012 Christian Fries 2 Version 0.5 (20121227)http://www.christianfries.com/nmath/
http://www.christianfries.com/finmath/http://www.christianfries.com/finmath/


3/22


1 Introduction

In this note we discuss the denition, construction, interpolation and application of curves . We will discuss discount curves, a tool for the valuation of deterministic cash-ows and forward curves, a tool for the valuation of linear cash-ows of (possibly)stochastic indices.

With respect to the interpolation of (interest rate) forward curves, a commonmistake is to represent a forward curve in terms of (pseudo-)discount factors and applyan interpolation scheme on these discount factors. While this approach is in general notbacked by an economic concept it also introduces several (self-made) problems, e.g.,the interpolation of (so-called) overlapping instruments. Such complications will notoccur if curves are derived (dened) from the associated market instruments.

We derive discount curves and forward curves by considering valuation of the

associated market instruments. This will also motivate natural interpolation schemes,e.g., the interpolation of a forward value. 1

Once the curves and interpolations are dened, we are considering the problemof calibrating a set of curves to given market quotes. The value of an instrument is ingeneral determined by a whole collection of curves, e.g., one or two discount curvesand zero or more forward curves.

For the calibration we need the valuation formulas which express the value of anancial product with respect to curves and a calibration algorithm. Often curves aredetermined by the successive calibration to a single instrument, retaining the calibrationtwo previous calibration instruments, considering a specic ordering of the instruments(e.g., increasing in maturity). Such a method is often called a bootstrap. A bootstrap

algorithm may introduce difculties (like error propagation) and is to some extend notexible enough since it requires a certain ordering and independence of the calibrationproducts. Instead we consider a global calibration.

We consider a multi-variate optimization of the whole set of curves to a set of calibration instruments. The algorithm is numerical less performant than a classicalbootstrap, but more exible and robust.

In summary the calibration framework can calibrate many different types of curves(discount curves, forward curves, cross-currency discount curves) to a set of differentcalibration instruments (swaps, tenor-basis swaps, cross-currency swaps) using a simpleand generic implementation. A reference implementation is provided.

1 A forward value is the product of a forward and the associated discount factor .



4/22


2 Foundations and Notation

We will give a short review of the mathematical background, that is, risk neutralvaluation. This section is mainly to x notation. For an in depth introduction tomathematical nance see the references and references there in.

2.1 Mathematical Background

Under some assumptions the valuation of a future cash ow can be written as anexpectation, that is

V (t ) = N (t ) EQN V (T )

N (T ) | F t , (1)

where V (T ) is the time T cash-ow, N is the value process of a traded asset which can

serve as a numraire and QN

is the equivalent martingale measure associated with N .If the above valuation formula holds, we have that the value of a linear function of future cash-ows is the linear function of the values of the single cash ows. In otherwords: we can represent the valuation of so called linear products by a basis consistingof the values of elementary products. This basis of elementary products is the set of curves.

Curves, like discount curves and forward curves are constructed for two reasons:

Valuation of linear instruments. This is performed by decomposing instrumentsinto the value of single cash-ows, which then allows to synthesize the valuationof linear function of the individual cash ows.

Valuation of a time T cash-ows as interpolation of valuations of cash ows atdiscrete times {T i }ni=0 .

Thus, curves are simply a methodology to interpolate on the cash-ows with respect totheir payment time .2

2.2 Currencies and Collateralization

2.2.1 Valuation in Foreign Currency

In the above valuation formula (1) it is assumed that V and N are expressed in thesame currency. If the two are in different currency, one of them has to be converted

by an exchange rate, which we will denote by F X . Let V be in currency U 2 and thenumraire N in currency U 1, then the valuation formula is given by

V (t) = F X U 2U 1 (t ) N (t ) EQ

N V (T )

F X U 2U 1 (T ) N (T )

| F t ,

2 This also applies to forward curve, see below, although in these cases there is also an associated xingtime of an index.



5/22


where F X U 2U 1 (t ) denotes the time t exchange rate for one unit of currency U 1 into one

unit of currency U 2. Furthermore

F X U 1U 2 = F X

U 2U 1

1.

2.2.2 Valuation of Collateralization

As discussed in [ 2], the valuation of a collateralized claim can be written as an expecta-tion with respect to a specic numraire, namely the collateral account N = N C . Wedenote the currency of the collateral numraire by [C ]. Let U denote the currency of thecash ow V (T ). Assume that the cash ow V (T ) is collateralized by units of N C . Inthis case the equation ( 1) holds with the numraire N = N C , i.e.,

V (t ) = F X U [C ] (t ) N C (t ) EQ

N C V (T )

F X U [C ] (T ) N C (T )

| F t (2)

(given that V (t ) is the collateral amount in the account N C ).

Remark 1 (Collateralization in Other Currency): From the above we see thatcollateralization in a different currency can be interpreted twofold:

1. We may consider a payment converted to collateral currency and valued withrespect to the collateral numraire N C , or, alternatively,

2. we may consider a payment in the currency U collateralized with respect to thecollateral account

N U, C := F X U [C ] N C .

We will adopt the latter interpretation, which will also make the valuation look moreconsistently 3

V (t ) = N U, C (t ) EQU,N

C V (T )N U, C (T )

| F t . (3)

Remark 2 (Funding and Collateralization): For an uncollateralized product therole of the collateral account is taken by the funding account and the correspondingnumraire is the funding account. Since the valuation formulas are identical to thecase of a special collateral account D (agreeing with the funding account), we willconsider an uncollateralized product as a product with a different collateralization.

2.3 Notation

In the following we use the notation U for the currency unit of a cash ow, i.e., youmay consider U = 1 e or U = 1$ . We will need this notation only when we considercross-currency basis swaps. The symbols V , N and P will denote value processesincluding the corresponding currency unit, e.g., V (t 0) = 0 .25e . The symbol V refers

3 As has been noted in [2], the measures agree, i.e., Q U,N C

= Q N C

.



6/22


to the value of the product under consideration, while N denotes the numraire, e.g.,the OIS accrued collateral account. The symbols X denotes a real number while I

denotes a real valued stochastic process, both can be considered as rates, i.e., unit-lessindices, e.g., X = 2 .5%. For example X is will denote the x rate in a swap, I willdenote the oating rate index in a swap, U denotes the currency unit of the two legs, N will be used to dene the discount factor and the value of the swap. The value of theswap is then denoted by V .

2.4 Valuation of Linear Products

2.4.1 Discount Curves

Consider a xed constant cash ow X , paid in currency U in time T , collateralized byan account C . The value of the cash ow is

V (t ) = N U, C (t ) EQN U,

C X U N U, C (T )

| F t .

Since X is a constant and the expectation operator is linear, we can express the valueV (t ) as

V (t ) = X P U,C (T ; t ), (4)

where

P U, C (T ; t ) = N U, C (t) EQN U,

C 1 U N U, C (T )

| F t

denes the value of a theoretical zero coupon bond. Note that equation (4) can be used

in two ways. First, for given market prices we may determine the P U,C

(T ; t ) - that iswe calibrate the curve

T P U,C (T ; t).

Second, for given P U, C (T ; t ) we may value a cash ow.

2.4.2 Forward Curves

The same approach can now be applied to a payoff of a cash ow X I (T 1), paid incurrency U in time T 2, collateralized by account C , where X is a constant and I is astochastic index. Its value is

V (t ) = N U C (t ) EQ N U, C

X I (T 1) U N U,C (T 2) | F t .

Since X is a constant and the expectation operator is linear, we can express the valueV (t ) as

V (t ) = X F U,C

I (T 1, T 2; t ) P U, C (T 2; t ), (5)

where

F U,C

I (T 1, T 2; t ) = N U, C (t ) EQ

N U,C I (T 1) U

N U, C (T 2) | F t P U,

C (T 2; t ).



7/22


This denition allows us to derive F U,C

I (T 1, T 2; t ) from given market prices. Con-versely, given P U,C (T 2; t ) and F

U, CI (T 1, T 2; t ) we may value all linear payoff functions

of in I (T 1) paid in T 2.In (5) the forward depends on the xing time T 1 and the payment time T 2. However,

the offset of the payment time from the xing time d = T 2 T 1 can be viewed as aproperty of the index (a constant) and hence, the forward represents a curve

T F U,C

I (T, T + d; t).



8/22


3 Discount Curves and Forward Curves

We now take the derivation to give formal denitions for the discount curve and forwardcurves.

3.1 Discount Curves

Denition 3 (Discount Curve): Let P U, C (T ; t ) denote the time t value expressed in currency unit U of a unit cash-owof 1 unit of the currency U in T , collateralized by a collateral account C .

In this case we call T P U, C (T ; t ) the discount curve for cash ows in currencyU collateralized by the account C .

Remark 4 (Discount Curve): We assume that the value of a xed future cash-ow X is a linear function of its amount. In this case, we have that the time t value of a cashow X in T and currency U , collateralized with an account C is

X P U,C (T ; t ).

In other words, the discount curve allows us to valuate all xed cash ows in a givencurrency, collateralized by a given account.

3.2 Forward Curves

Denition 5 (Forward Curve):

Let t I (t ) denote an index, that is I is an adapted stochastic real valued process. LetV U,

C

I (T, T + d; t ) denote the time t -value of a payment of I (T ) is paid in T + d incurrency U , collateralized by an account C (where d 0).

Then we dene the forward of a payment of I (T ) paid in T + d in currency U ,collateralized by an account C as

F U,C

I (T ; t ) := V U,

C

I (T, T + d; t )P U,C (T + d; t )

.

Remark 6 (Forward Curve): The forward curve allows us to value a future payment

of the index I . In fact it is

V U,C

I (T, T + d; t ) = F U, CI (T ; t ) P

U,C (T + d; t).

We assume that the value of a oating future cash-ow X I (T 1), where X is areal valued constant, is a linear function of its amount X . In this case, we have that thetime t value of a cash ow X I (T ) paid in T + d and currency U , collateralized withan account C is

X F U,C

I (T ; t ) P U,C (T + d; t ).

In other words, the forward curve allows us to valuate all linear cash ows of an indexin a given currency, collateralized by a given account.



9/22


3.2.1 Performance Index of a Discount Curve (or self-discounting)

The OIS swap pays the performance of an account, accruing with the overnight rate,that is:

Denition 7 (Overnight Index Swap): Let N C (t ) denote the account accruing at the overnight rate, N C (0) = 1 , i.e.

N (t ) := (1 + r (t i ) t i ) exp t

0r (s )d s .

The overnight index swap pays x and receives the performance I i of the accrual accrualaccount, that is

I i := N C (T i+1 )

N C (T i ) 1.

in T i+1

with a quaterly tenor T 0, T

1, . . . .

The OIS swap is collateralized with respect to the account N C . Due to this, it issometimes called self-discounted. However, an alternative natural interpretation isthat the discount factor is given by the accrual account and that the swap pays a specialindex, namely one derived from the discount factors. In other words: we may dene aspecial type of forward curve from discount curve (and not the other ways around).

The forward of the index

I i := N C (T i+1 )

N C (T i ) 1

isP C (T i ; t ) P C (T i+1 ; t )

P C (T i+1 ; t ) .Hence, this is the same situation as for single curve interest rate theory swaps.

Let us consider a discount factor curve P U, C (T ; t ) as seen in time t . The curveallows the denition of a special index, namely the performance rate of the collateralaccount C in currency U over a period of period length d: Let

I (T i ) := 1 P U, C (T i + d; T i )

P U, C (T i + d; T i ) ,

where P U,C (T i + d; T i ) is the discount factor for the maturity T i + d as seen in time T i .The index I (T i ) is the payment we have to receive in T i + d collateralized with respect

to the collateral account C

, such that 1 + I (T i ) in T i+1 has the same value as 1 in T i .This index has a special property, namely that its forward can be expressed in terms of the discount factor curve P U,C too: The time t forward of I (T i ) is F (T i ; t ) where

F (T i ; t ) P U,C (T i + d; t ) = N U,

C (t ) EQN U,

C I (T i ) U N U, C (T i + d)

| F t

= N U, C (t ) EQN U,

C I (T i ) P U, C (T i + d; T i )N U,C (T i )

| F t

= N U, C (t ) EQN U,

C 1 U P U,C (T i + d; T i )N U, C (T i )

| F t

= P U, C (T i ; t ) P U,C (T i + d; t ).



10/22


Thus

F (T i ; t ) = P U, C (T i ; t ) P U,C (T i + d; t )

P U,C

(T i + d; t ) .

Consequently this index has the property that its forward can be expressed by the samediscount factors evaluated at a different time.

Denition 8 (Forward associated with a Discount Curve): Let P U,C (T i + d; t ) denote a discount curve. For a given period length d we dene theforward F d,U, C (T i ; t ) as

F d,U, C (T i ; t ) := P U,C (T i ; t) P U, C (T i + d; t )

P U, C (T i + d; t ) d . (6)

F d,U, C (T i ; t ) is the forward associated with the performance index of P U, C over aperiod of length d.

Remark 9 (Forwards from Discount Factors): The above denition relates a for-ward curve and discount factor curve. Note however, that we dene a forward from adiscount factor curve and that this denition is backed by a clear interpretation of theunderlying index. Conversely, we may dene a discount curve from a forward curveimplicitly such that the relation (6) holds. Note however, that a generalization of thisrelation should be considered with care, since the associated product may not exist.

Remark 10 (Self Discounting): A product like the OIS swaps are sometimes calledself-discounting since the discounting is performed on a curve corresponding to

the index they x. From the above, we nd an alternative (and maybe more natural)interpretation, namely that the swap pays the performance index of its collateral account,i.e., it pays the index associated with the discount curve.



11/22


4 Interpolation of Curves

In this section we consider a discount curve P U, C and an associated forward curve F U,C .To simplify notation we set D (T ) := P U, C (T ; t ) and F (T ) := F U, C (T ; t ).

4.1 Introduction

Forwards and discount factors are linked together by Denition 5, which says that thevalue of a forward contract V (T ) is the product of the forward F (T ) and the associateddiscount factor D (T + d), i.e., we have

V (T ) = F (T ) D (T + d). (7)

Note that V and D are value curves , i.e., for a xed T the quantities V (T ) and D (T )are values of nancial products. However, F (T ) is a derived quantity, the forward.

Since V and D represent values of nancial products, there is a natural interpretationfor a linear interpolation of different values V (T i ) and of different values of D (T i ),since this would correspond to a portfolio of such products. Note that dening aninterpolation method for V and D implies a (possible more complex) interpolationmethod of F .

On the other hand, it is common practice to dene an interpolation method for arate curve (both forward curve and discount factor curve) via zero rates, sometimeseven regardless of the nature of the curve, which then implies the interpolation of thevalue curves D and V . Some of these interpolations will result in natural interpolationson the value process V , others not. Examples are:

log-linear interpolation of the forward, log-linear interpolation of the discountfactor: the case is equivalent to log-linear interpolation of the value.

linear interpolation of the forward, log-linear interpolation of the discount factor:the case is equivalent with a linear interpolation of the value, with an interpolationweight being a function of the discount factor ratio.

For an in depth discussion of interpolation methods for rate curves see [ 5].

4.2 Interpolation Method and Interpolation Entities

The two basic ingredients for an interpolation are the interpolation method , e.g., linear

interpolation of interpolation points {(T i , x i )}, and the interpolation entity , that is, a(bijective) transformation from (T, x ) to the actual curve. For example, for discountcurves one might consider a linear interpolation of the zero rate. In this case theinterpolation method is linear interpolation and the interpolation entity is for T > 0

(T, x (T ) ) = ( T, log(D (T ))

T ),

where D denotes the discount curve. Given 0 < T i T T i+1 and discount factorsD (T j ), a linear interpolation of the zero rates would then imply the interpolation

D (T ) := exp T T iT i+1 T i

log(D (T i+1 ))T i+1

+ T i+1 T T i+1 T i

log(D (T i ))T i

T .



12/22


4.2.1 Interpolation of Forward Curves

For forward curves, a common approach is to consider an interpolation of the forwardas an independent entity (like for the discount curve). For interest rate forwards, apopular interpolation scheme (coming from the single curve interpretation of interestrates forwards) is to represent the forward in terms of synthetic discount factors. Thatis, if d denotes a period length associated with the forward and if F (T i ) is given forT i = i d, then one might consider interpolation of (pseudo-)discount factor

D F (T i ) :=i 1

k=0

(1 + F (T k ) p) 1,

possibly considering another transformation on D F (T ) to dene the actual interpolationentity.

It is obvious that this denition of the interpolation entity for forward curve iscomplex, results in problems for non-equidistant interpolation points and is - withoutfurther assumptions - not backed by a meaningful interpretation.

4.2.2 Interpolation of Forward Curve via Value Curve

The denition of the forward curve suggests an appealing alternative for the creationof an interpolated forward: Like a discount factor curve, the curve V (T ) = F (T ) D (T + d) represents the value of a nancial product. Hence, we may consider theinterpolation of V like we did for the curve D . For example, if we consider linearinterpolation of the value curve V , we interpolate the forward curve F by consideringthe interpolation entity F (T ) D (T + d) with a given discount curve D , i.e., we have

F (T ) :=1

D (T + d) T T iT i+1 T i

F (T i+1 )D (T i+1 + d) + T i+1 T T i+1 T i

F (T i )D (T i + d) .

for T i T T i+1 and given points F (T j ).



13/22


5 Calibration of Curves

5.1 Global OptimizationA curve (discount curve or forward curve) is used to encode values of market instru-ments. A forward curve together with its associated discount curve, allows to value alllinear products (linear payoffs) in the corresponding currency under the correspondingcollateralization.

The standard way to calibrate a curve is, hence, to obtain given market values of (linear) instruments (e.g., swaps). For each market value a single point in a singlecurve is calibrated. Hence the total number of curve calibrated interpolation points(aggregated across all curves) equals the number of market instruments.

By "sorting" and combining the calibration instruments, the corresponding equa-tions can be brought into the form of a system of a equations with a triangular structure,i.e., the value of the n -th calibration instruments only depends on the rst n curvepoints. This allows for an iterative construction of the curve.

However, in [ 3] we favor to calibrate the system of equations using a multi-variateoptimization algorithm, like the Levenberg-Marquard algorithm. This approach bringsserval advantages, e.g., the freedom to specify the calibration instruments and the abilityto extend the approach to over-determined systems of equations. This comes at the costof slower performance in terms of required calculation time.

What remains is to specify the valuation equations for the calibration instruments.

5.2 Generalized Denition of a Swap

Many of the following calibration instruments (from OIS swaps to cross-currencybasis-swaps) t under a generalized denition of a swap. The swap consists of two legs.Each leg consists of several periods [T i , T i+1 ]. We do not distinguish between periodstart time, period end time, xing time of the index and payment time. We assumethat for the period [T i , T i+1 ] index xing is in T i and payment is in T i+1 . This is donepurely to ease notation, the generalization to distinguished times is straight forward.

Denition 11 (Swap Leg): A swap leg pays the a multiple of the index I xed in T i plus some xed paymentX , both in currency unit U collateralized by the collateral account C and paid in timeT i+1 . Here and X are constants (possibly zero). The value of the swap leg can beexpressed in terms of forwards and discount factors as

V U,C

SwapLeg (I,X, {T i }ni=0 ; t ) =

n 1

i=0

F U, C (T i ) + X P U,C (T i+1 ),

where F U,C denotes the forward curve of the index I paid in currency U collateralizedwith respect to C and P U,C denotes the corresponding discount curve.

Denition 12 (Swap Leg with Notional Exchange): A swap leg with notional exchange has the payments as in Denition 11 together withan additional payment of 1 in T i and +1 in T i+1 . The value of the swap leg with



14/22


notional exchange can be expressed in terms of forwards and discount factors as

V U,C

SwapLeg (I,X, {T i }ni=0 ; t ) =

n 1

i=0F U,

C

(T i ) + X P U,C

(T i+1 )

+ P U, C (T i+1 ) P U,C (T i ) ,

where F U,C denotes the forward curve of the index I paid in currency U collateralizedwith respect to C and P U,C denotes the corresponding discount curve.

Denition 13 (Swap): A swap exchanges the payments of two swap legs, the receiver leg and the payer leg. Weallow that the legs have different indices, different xed payments, different paymenttimes, different currency units, but are collateralized with respect to the same account

C . The swaps receives a swap leg with value V U 1 ,C

SwapLeg ( 1I 1, X 1, {T 1i }

n 1i=0 ; t ) and pays

a leg with value V U 2 ,C

SwapLeg ( 2I 2, X 2, {T 2i }; t ). Since the currency unit of the two legs

may be different, the value of the swap in currency U 1 is

V Swap (t )

= V U 1 ,C

SwapLeg ( 1I 1, X 1, {T 1i }

n 1i=0 ; t ) V

U 2 ,CSwapLeg ( 2I 2, X 2, {T

2i }

n 2i=0 ; t ) F X

U 1U 2 (t )

=n 1 1

i=0

F U 1 ,C

1 (T 1i ) + X 1 P

U 1 ,C1 (T

1i+1 )

n 2 1

i=0

F U 2 ,C

2 (T 2i ) + X 2 P

U 2 ,C2 (T

2i+1 ) F X

U 1U 2 (t).

Denition 14 (Swap): A swap with notional exchange exchanges the payments of two swap legs with notionalexchange. The value of the swap in currency U 1 is

V Swap (t )

=n 1 1

i=0

F U 1 ,C

1 (T 1i ) + X 1 + 1 P

U 1 ,C1 (T

1i+1 ) P

U 1 ,C (T 1i )

n 2 1

i=0

F U 2 ,C2 (T 2i ) + X 2 + 1 P U 2 ,C2 (T 2i+1 ) P U 2 ,C (T 2i ) F X U

1U 2 (t ).

Many instruments can be represented (and hence valued) in this form. We will nowlist a few of them.

5.3 Calibration of Discount Curve to Swap Paying the Collateral Rate(aka. Self-Discounted Swaps)

Discount curves can be calibrated to swaps paying the performance index of theircollateral account. For example a swap as in Denition 13 where both leg pay in the



15/22


same currency U = U 1 = U 2. In a receiver swap the receiver leg pays a xed rate C ,and the payer leg pays an index I . Thus the value of the swap can be expressed in terms

of the discount factors P U,C

(T i+1 ; t ) only, which allows to calibrate this curve usingthese swaps. Overnight index swaps are an example.For the swap paying the performance of the collateral account we have

X 1 = C = const. = given, X 2 = 0 ,

F U 1 ,C

1 (T 1i ) = 0 , F

U 2 ,C2 (T

2i ) =

P U,C (T 2i ; t ) P U,C (T 2i+1 ; t )

P U,C (T 2i+1 ; t )(T 2i+1 T 2i ) ,

P U 1 ,C

1 = P U, C = calibrated, P U 2 ,

C

2 = P U,C = calibrated.

From one such swap we calibrate the time T discount factor P U,C (T ; t) with T =max( T 1n , T 2n ) (the last payment time).

5.4 Calibration of Forward Curves

5.4.1 Fix-versus-Float Swaps

Given a calibrated discount curve P U,C we consider a swap with payments in currencyU collateralized with respect to the account C , paying some index I and receiving somexed cash ow C . An example are swaps paying the 3M LIBOR rate. For such a swapwe have

X 1 = C = const. = given, X 2 = 0 ,

F U 1 ,C1 (T

1i ) = 0 , F

U 2 ,C2 (T

2i ) = F

U, C

(T 2i ) = calibrated,

P U 1 ,C

1 = P U,C = given, P U 2 ,

C

2 = P U, C = given.

From one such swap we calibrate the time T forward F U, C (T ) of I (T ) with T = T 2n 1)(the last xing time).

5.4.2 Float-versus-Float Swaps (Tenor Basis Swaps)

Given a calibrated discount curve P U,C and a calibrate forward curve F U,C

1 belongingto the index I 1, both in currency U and collateralized with respect to the account C , weconsider a swap collateralized with respect to the account C , paying some index I 2 = I

in currency U , receiving the index I 1 in currency U . An example are tenor basis swapspaying the 6M LIBOR rate, receiving the 3M LIBOR rate. For such a swap we have

X 1 = C 1 = const. = given, X 2 = C 2 = const. = given,

F U 1 ,C

1 (T 1i ) = F

U, C1 (T

1i ) = given, F

U 2 ,C2 (T

2i ) = F

U,C2 (T

2i ) = calibrated,

P U 1 ,C

1 = P U, C = given, P U 2 ,

C

2 = P U,C = given.

From one such swap we calibrate the time T forward F U,C

2 (T ) of I (T ) with T = T 2n 1

(the last xing time of index I 2).



16/22


5.5 Calibration of Discount Curves when Payment and Collateral Cur-rency Differ

5.5.1 Fixed Payment in Other Currency

Given a calibrated discount curve P U 1 ,C we consider a swap collateralized with respectto the account C , paying some index I 1 in currency U 1, and receiving some xed cashow C 2 in currency U 2. An example for such a swap is a cross-currency swap payingoating index I in collateral currency and receiving xed C 2 in a different currency .4

For such a swap we have


F U 1 ,C

1 (T 1i ) = F

U 1 ,C1 (T

1i ) = given, F

U 2 ,C2 (T

2i ) = 0 ,

P U 1 ,C

1 = P U 1 ,C = given, P U 2 ,

C

2 = P U 2 ,C = calibrated.

We calibrate the discount factor P U 2 ,C (T ; t ) with T = T 2n (last payment time incurrency U 2).

5.5.2 Float Payment in Other Currency

If instead of a xed payment we have that an index I 2 is paid in an other currency U 2 wemay encounter the problem that the swap has two unknowns, namely the discount curveP U 2 ,C for payments in currency U 2 collateralized with respect to C and the forwardcurve F U 2 ,

C

2 of the index I 2 paid in currency U 2 collateralized with respect to C . Thetwo curves can be obtained jointly from two different swaps: rst a x-versus-oatswaps in currency U

2 collateralized by C (as in Section 5.4.1 ), and second a cross-

currency swap exchanging the index I 2 with an index I 1 in currency U 1 for which theforward F U 1 ,

C

1 is known. For the rst instrument we denote the xed payment by C 1,C 2. For the second instrument we denote the xed payment by s1, s2 (usually a spread).For the rst instrument we have


F U 1 ,C

1 (T 1i ) = 0 , F

U 2 ,C2 (T

2i ) = F

U 2 ,C2 (T

2i ) = calibrated,

P U 1 ,C

1 = P U 2 ,C2 = calibrated, P

U 2 ,C2 = calibrated.

for the second swap we have

X 1 = s1 = const. = given, X 2 = s2 = const. = given,F U 1 ,

C

1 (T 1i ) = F

U 1 ,C1 (T

1i ) = given, F

U 2 ,C2 (T

2i ) = F

U 2 ,C2 (T

2i ) = calibrated,

P U 1 ,C

1 = given, P U 2 ,C2 = calibrated.

We calibrate the discount factor P U 2 ,C (T ; t ) with T = T 2n and the forward F U 2 ,C2 (T)

with T = T 2n 1.Often market data is not available to calibrate the forward F U 2 ,

C

2 , but the forwardF U 2 ,

C 22 collateralized with respect to a different account C 2 is available. The two for-

wards differ by a possible convexity adjustment. One possible approximation (which4 Usually cross-currency swaps exchange two oating indices, we will consider this case below.



17/22


18/22


5.7 Lack of Calibration Instruments (for Difference in Collateralization)

The calibration of cross-currency curves (forward curve and discount curves for currencyU 2 with collateralization in currency U 1, see Section 5.5) and the calibration of un-collateralized curves (forward curves and discount curves for un-collaterlized products,see Section 5.6 ) may require market data which is not available, e.g., the forward of an index I paid in currency U 2 collateralized in a different currency or by a differentaccount.

In this case the curve can be obtained by adding additional model assumptions. Twosimple examples are:

the market rates are assumed to be independent of the type of collateralization, or

the forward rates are assumed to be independent of the type of collateralization.

The two assumptions lead to different results, since they imply different correlationswhich will lead to different (convexity) adjustments. Figure 1 shows the differencein the forward rates assuming identical market rates for 3M swaps collateralized inUSD-OIS or EUR-OIS, which is around 1 or 2 basis points (0.01%). For details on theexample see [ 3].

!"!!! #"!!! $"!!! %"!!! &"!!! '!"!!! '#"!!! '$"!!! '%"!!! '&"!!!!"!!(

'"!!(

#"!!(

)"!!(

$"!!(

/0122

/.132

/.122

/2132

2122

2132

!"#$%#& ()#*+,

*+,-.,/01230)4 *+,-.,/01230)405160782

/9:;,;


19/22


6 Sample Code for Curve Calibration

A reference implementation is available via [3] (source code and demo spreadsheet).We implement discount curves and forward curves in separate classes. This is

necessary since

the forward curve has an associated discount curves which describes the collater-alization of the index,

the interpolation entity of the two curves may be different.

Both classes inherit from a base class which may perform several interpolation methodson the interpolation entities.

The curve represent the independent parameters of the calibration solver. The

dependent values are represented by the generalized swap as in Denition 14. For eachsuch calibration instrument the curve to calibrate and the time to calibrate are specied.The solver performs a global optimization. Thus resolving interdependencies amongthe products (e.g., through a specic ordering) is not necessary.



20/22


7 Redening Forward Rate Market Models

Let N C denote an accrual account, i.e., N C is a process with N C

(t0) = 1

(e.g., acollateral account). Then N C denes a discount curve, namely the discount curveP C (T i+1 ; t ) of xed payments made in T , valued in t and collateralized by units of N C .

Let {T i } denote a given tenor discretization. As shown in Section 3.2.1 the period-[T i , T i+1 ] performance index I (T i , T i+1 ) of the an accrual account, i.e.,

I (T i , T i+1 ; T i ) := N C (T i+1 )

N C (T i ) 1

has the property that its time t forward (of a payment of I (T i , T i+1 ) made in T i+1 ,collateralized in units of N C ) (following the denition of a forward from Section 3.2 )is given as

I (T i , T i+1 ; t ) := P C (T i ; t ) P C (T i+1 ; t )

P C (T i+1 ; t )

This relation allows us to create a term-structure model for the curve P C which hasthe same structural properties as a standard single curve (LIBOR) market model. Thismodel is given by

A joint modeling of the processes L i (t ) := I (T i ,T i +1 ;t )T i +1 T i , e.g., as log-normalprocesses under the measure Q N

C

.

The additional assumption that the process P C (T i ; t ) is deterministic on its shortperiod t (T i 1, T i ].

From these two assumptions it follows that the processes L i have the structure of astandard LIBOR market model and Q N

C

corresponds to the spot measure. Indeed wehave

i 1

j =0

1 + L j (T j ) (T j +1 T j ) =i 1

j =0

1 + I (T j , T j +1 ; T j )

=i 1

j =0

N C (T j +1 )N C (T j )

= N C (T i ).

What we have described is how to use the standard LIBOR market model as a termstructure model for the collateral account N C (e.g., the OIS curve). Now, modelingall other rates (including LIBOR) can be performed by modeling (possibly stochastic)spreads over this curve. This is analog to a defaultable market model.



21/22


References

The literature on multi-curve modeling, discounting, funding and collateral-ization is huge. We just give a few references related to the construction of curves.

[1] CLARK , JUSTIN : Constructing a Non-USD Swap Discounting Curve withUSD Collateral.http://www.edurisk.ie

[2] FRIES , C HRISTIAN P.: Funded Replication: Fund Exchange Process andthe Valuation with Different Funding-Accounts (Cross-Currency Analogy toFunding Revisited). July 2012.http://papers.ssrn.com/abstract=2115839 .

[3] FRIES , C HRISTIAN P.: Curve Calibration. Object oriented reference imple-mentation.http://www.finmath.net/topics/curvecalibration .

[4] FRIES , C HRISTIAN P.: LIBOR Market Model. Object oriented referenceimplementation.http://www.finmath.net/topics/libormarketmodel .

[5] HAGAN , PATRICK S.; W ES T , G RAEME : Interpolation Methods for CurveConstruction. Applied Mathematical Finance , Vol. 13, No. 2, 89-129, June2006

[6] M ERCURIO , FABIO : LIBOR Market Models with Stochastic Basis, March2010.

http://www.edurisk.ie/http://papers.ssrn.com/abstract=2115839http://papers.ssrn.com/abstract=2115839http://www.finmath.net/topics/curvecalibrationhttp://www.finmath.net/topics/curvecalibrationhttp://www.finmath.net/topics/libormarketmodelhttp://www.finmath.net/topics/libormarketmodelhttp://www.christianfries.com/finmath/http://www.christianfries.com/finmath/http://www.finmath.net/topics/libormarketmodelhttp://www.finmath.net/topics/curvecalibrationhttp://papers.ssrn.com/abstract=2115839http://www.edurisk.ie/


22/22


Notes

Suggested CitationFRIES , CHRISTIAN P.: Curves and Term Structure Models.http://ssrn.com/abstract=2194907http://www.christian-fries.de/finmath/curves

Classication

Classication: MSC-class : 65C05 (Primary), 68U20, 60H35 (Secondary).ACM-class : G.3; I.6.8.JEL-class : G13.

Keywords: Curves, Discount Curve, Forward Curve, Calibration, Bootstrapping,

Multi-Curve, Tenor-Basis, Cross-Currency-Basis, Collateralization, Funding,OIS Discounting, Funding Curve, Spread Curve

22 pages. 1 gures. 0 tables.

http://ssrn.com/abstract=2194907http://www.christian-fries.de/finmath/curveshttp://www.ams.org/msc/http://www.acm.org/class/1998/ccs98.htmlhttp://www.acm.org/class/1998/ccs98.htmlhttp://www.aeaweb.org/journal/jel_class_system.htmlhttp://www.aeaweb.org/journal/jel_class_system.htmlhttp://www.christianfries.com/finmath/http://www.christianfries.com/finmath/http://www.aeaweb.org/journal/jel_class_system.htmlhttp://www.acm.org/class/1998/ccs98.htmlhttp://www.ams.org/msc/http://www.christian-fries.de/finmath/curveshttp://ssrn.com/abstract=2194907

rates basics

Documents