Interest Rate Products - Seoul National University · 2012. 12. 27. · 9.1. CHANGE OF NUMERAIRE AND THE INVARIANCE OF RISK NEUTRAL VALUATION 247 exists. Let Xbe any European contingent

Chapter 9

Interest Rate Products

Copyright c©2008–2011 Hyeong In Choi, All rights reserved.

9.1 Change of Numeraire and the Invarianceof Risk Neutral Valuation

The financial theory we have developed so far depends heavily on theprivileged role the riskless bond price Bt plays. Namely, suppose St isa tradable asset price, say, stock price. Then the martingale measureQ is defined to be the measure that makes the discounted stock price

processes S∗t =StBt

a martingale. In the parlance of finance, Bt plays

the role of the so-called numeraire, which, loosely speaking, providesa means of equating two different monetary values measured at twodifferent times by taking into account of the progression of time.

The special privilege of Bt may be justified, however feebly, if theshort rate is a constant or even a deterministic function of t. However,when term structure models—which are by nature stochastic—areused, Bt is not, in essence, any different from other bond pricesor even other tradable assets. For example, we may as well use

Ct =p(t, T )

p(0, T )as a numeraire for t ≤ T . In this section, we in fact do

just that, and see what is changed in finance. So suppose now thatSt is a price of some tradable security; let Bt be the price of risklessbond; and let Ct be another tradable asset price such that Ct > 0almost surely. For the purpose of normalization, we further assumethat C0 = 1. Then we have three assets; Bt, Ct, and St. We alsoassume that there is no arbitrage in this market so that a martingale

measure Q that makes 1,CtBt

, andStBt

simultaneously Q-martingale

9.1. CHANGE OF NUMERAIRE AND THE INVARIANCE OFRISK NEUTRAL VALUATION 247

exists. Let X be any European contingent claim with expiry T . ThusX ∈ FT , and by the Risk Neutral Valuation Principle its value Vt attime t is given by

Vt = BtEQ

[X

BT| Ft

].

On the other hand Ct may very well be used as a numeraire. As-suming so, let us now construct a new measure Q with respect to

whichBtCt

, 1, andStCt

are martingales. Since Bt is numeraire with

corresponding martingale measure Q,CtBt

is a Q-martingale. For any

t ≤ T , let us define the measure Qt on Ft by

dQt =CtBtdQ.

Then {Qt}t is a family of measures on {Ft} that are consistent in thefollowing sense: Let s < t, and let A ∈ Fs, then obviously A ∈ Ftalso. Now

Qs(A) =

∫AdQs =

∫A

CsBsdQ (∵ by definition of Qs)

=

∫AEQ

[CtBt| Fs

]dQ (∵

Ct

Btis a Q-martingale)

=

∫A

CtBtdQ (∵ by definition of conditional expectation)

= Qt(A).

Thus Qs and Qt coincide on Fs. This way this family of measurescan be pieced together to define a measure Q = QT on FT so thatQT |Ft= Qt. One can in fact extend this family to Q∞ on F∞ =

σ(⋃t≥0

Ft)

, but for our purpose QT is enough. Note also that we

have already seen this way of piecing together a consistent family of

measures. Let we now see thatStCt

is a Q-martingale. Let s < t, and

9.1. CHANGE OF NUMERAIRE AND THE INVARIANCE OFRISK NEUTRAL VALUATION 248

A ∈ Fs, Then∫A

SsCsdQ =

∫A

SsCs

CsBsdQ (∵ Q |Fs

= Qs)

=

∫A

SsBsdQ

=

∫AEQ

[StBt| Fs

]dQ (∵

St

Btis a Q-martingale)

=

∫A

StBtdQ (∵ by definition of conditional expectation)

=

∫A

StBt

BtCtdQ (∵ On Ft ⊃ Fs, dQ =

Ct

BtdQ)

=

∫A

StCtdQ.

AsSsCs∈ Fs, we have by the definition of conditional expectation,

EQ

[StCt| Fs

]=SsCs.

Similarly, one can easily see thatBtCt

is a Q-martingale. With this,

we are ready to prove the following important result.

Proposition 9.1. Let St, Bt, Ct, Q and Q be as defined above.Suppose X ∈ FT . Define

Vt = BtEQ

[X

BT| Ft

]Vt = CtEQ

[X

CT| Ft

].

Then Vt = Vt.

Proof. Let A ∈ Ft. Then∫A

CtBtEQ

[X

CT| Ft

]dQ

=

∫AEQ

[X

CT| Ft

]dQ (∵ Q |Ft

= Qt and dQt =Ct

BtdQ)

=

∫A

X

CTdQ (∵ by definition of conditional expectation)

=

∫A

X

CT

CTBT

dQ (∵ On FT ⊃ Ft, dQ =CT

BTdQ)

=

∫A

X

BTdQ.

9.2. FORWARD PRICE AND MEASURE 249

SinceCtBtEQ

[X

CT| Ft

]∈ Ft and the above equality holds for all

A ∈ Ft, we have, by definition of conditional expectation,

EQ

[X

BT| Ft

]=CtBtEQ

[X

CT| Ft

].

The above proposition was stated and proved for a particularmeasure Q. But there may be several martingale measures for thenumeraire Ct in case the market is not complete. However, even inthis case, we have shown in the previous chapters that the risk neu-tral valuation does not depend on any particular martingale measurechosen. Therefore we have the following important

Theorem 9.2. (Invariance of Risk-Neutral Valuation under Nu-meraire Change) Let Bt and Ct be tradable securities that are positivealmost surely. Furthermore assume that B0 = C0 = 1. Let Q (resp.Q) be a measure that makes all tradable securities discounted by Bt(resp. Ct) Q(resp. Q)-martingale. Let X ∈ FT be any Europeancontingent claim. Then

BtEQ

[X

BT| Ft

]= CtEQ

[X

CT| Ft

].

9.2 Forward Price and Measure

Previously, we have discussed about the forward prices. In this sec-tion, we look at the forward contract involving general Europeancontingent claim. Let X be a European contingent claim with ex-piry T0. A forward contract at time t is an agreement entered at t tobuy or sell the contingent claim at time T < T0 at a price determinedat time t.

0 t T T0

↑ ↑ ↑ ↑present agreement entered;

price is fixed

actual delivery

and payment occur

expiry X

The question is what should the fair price be at time t, and theobvious answer to it should be the price K given by the formulabelow.

EQ

[X −KBT

| Ft]

= 0, (9.1)


where Bt is the usual riskless bond price, and Q is the martingalemeasure corresponding to the numeraire Bt. Solving (9.1) for K andmultiplying Bt on the numerator and the denominator, we have

K =BtEQ

[XBT| Ft

]BtEQ

[1BT| Ft

] =Vt

p(t, T ). (9.2)

The last equality is obtained by considering Vt = BtEQ

[X

BT| Ft

]is the value at t of the contingent claim X and BtEQ

[1

BT| Ft

]is

the value at t of the contingent claim that pays 1 at time T , whichis exactly p(t, T ). We call K the forward price at t of the contractentered at t for the delivery ofX at T and denote it by Ft = Ft(X;T )1

Suppose now we choose Ct =p(t, T )

p(0, T )the normalized bond price

as an another numeraire, and we let PT be the corresponding mar-tingale measure. This martingale measure PT is called a forwardmeasure. Then we have

Theorem 9.3.Ft = EPT [X | Ft] .

Proof. We showed that Ft =Vt

p(t, T ). By the invariance of risk neutral

valuation under numeraire change, we have

Vt = CtEPT

[X

CT| Ft

].

But CT =p(T, T )

p(0, T )=

1

p(0, T ), which is a known constant, and Ct =

p(t, T )

p(0, T ). Therefore we have

Vt = p(t, T )EPT [X | Ft] .

This theorem says that the forward price Ft is the expectationat time t of X under the forward measure FT . The argument in theproof is also restated as the following:

1If the binding agreement on this contract is made at an earlier time s < t, itsvalue Fs at s can be calculated by

Fs = EPT [X|Fs] = EPT [Ft|Fs].


Theorem 9.4. (Valuation of Contingent Claim with the Forwardmeasure) Let X ∈ FT , then its value Vt at t is given by

Vt = p(t, T )EPT [X | Ft] .

The merit of this theorem is that in order to value contingentclaims it is NOT necessary to know Bt. This gives tremendous ad-vantage because finding Bt using the general form of term structuremodel is by no means easy. This fact also comes in handy in theMonte Carlo simulation, the subject which we will not touch in de-tail here.

Let us now examine how the forward rate behaves under theforward measure PT . We have seen that under the usual martingalemeasure Q with numeraire Bt the bond price satisfies the followingstochastic differential equation

dp(t, T ) = p(t, T )[rtdt+ S(t, T )dWt],

for some S(t, T ), regardless of which term structure model was used.Let us also note that the forward measure PT is defined by piecingtogether measures Pt given by

dPt = ζtdQ,

where

ζt =p(t, T )

p(0, T )Bt.

Either using the fact that ζt is a Q-martingale or by direct compu-tation, it is easy to see that

dζt = ζtS(t, T )dWt = −ζt(−S(t, T ))dWt

ζ0 = 1.

Thus ζt is an exponential martingale that is used in the Girsanovmachinery, which says that the Brownian motion Wt correspondingto Pt is defined by

dWt = dWt − S(t, T )dt. (9.3)

Now it is easy to check that

d log p(t, T ) =

[rt −

1

2S2(t, T )

]dt+ S(t, T )dWt.

9.3. GENERALIZED BLACK-SCHOLES FORMULA 252

Taking∂

∂Tand interchanging the order of differential we have

df(t, T ) = −d ∂

∂Tlog p(t, T )

= − ∂

∂Td log p(t, T )

= S(t, T )∂S

∂T(t, T )dt− ∂S

∂T(t, T )dWt

= −∂S∂T

(t, T ) [dWt − S(t, T )dt]

= σ(t, T )dWt, (9.4)

where the last equality is due to (9.3) and the fact that

σ(t, T ) = −∂S∂T

(t, T ).

This stochastic differential equation (9.4) for f(t, T ) means thatf(t, T ) is a martingale under PT measure. Thus we have

Theorem 9.5.

(i) For each fixed T , f(t,T) is a PT -martingale, i.e.,

f(s, T ) = EPT [f(t, T ) | Fs] ,

(ii)f(t, T ) = EPT [rT | Ft].

The proof is obvious, but the interpretation of this theorem hassome interesting implication, namely, the forward rate f(t, T ) is theexpectation of the short rate seen at time t.

9.3 Generalized Black-Scholes Formula

A financial asset that can be bought and sold immediately in themarket in exchange for cash payment is called a spot aset. Typicalof such assets are stocks and bonds2. Let St be the price process ofa spot asset. If that asset is a stock, St is the usual price process;if it is a T1-bond3, St here stands for p(t, T1). As usual, we assumethat St is a semi-martingale written as

dSt = St(rtdt+ βtdWt). (9.5)

2It is customary to treat bonds as belonging the a separate asset class, i.e.,the fixed-income assets. But for our purpose we treat bonds as spot assets.

3T-bond is a bond with maturity T .


We furthermore assume σt is a deterministic function of t although itmay involve some deterministic parameter like T1 in the case of T1-bond. Here Wt is assumed to be the Brownian motion with respectto the martingale measure.

Let X be a European option on such asset with expiry T . Thenwe have, as usual,

X = (ST −K)+.

In this section, we show how to derive pricing formula by utilizingthe machinery developed so far in this chapter.

Let Ft = F (t, T ) be the forward price of the forward contractwith the delivery at time T . Then by (9.2), we have

Ft =St

p(t, T ). (9.6)

Note that in the case of T1-bond (9.6) is

Ft =p(t, T1)

p(t, T )(9.7)

By Theorem 9.3, Ft is a PT -martingale, where PT is the forwardmeasure corresponding to the delivery time T . This fact can beargued directly as follows: first, St is the price process of a trad-able asset; second, (9.6) says that Ft is St discounted by p(t, T );third, PT is the martingale measure corresponding to the numerairep(t, T )/p(0, T ); therefore, Ft must be a martingale with respect toPT . Therefore by the Girsanov machinery (9.5) is transformed into

dFt = FtσtdWTt (9.8)

where W Tt is a PT -Brownian motion. As usual, let p(t, T ) satisfy the

following SDE

dp(t, T ) = p(t, T )[rtdt+ S(t, T )dWt]

where Wt is the Brownian motion with respect to the martingalemeasure. Then upon taking the logarithm of St/p(t, T ), then takingd and collecting the coefficient of dWt and making use of (9.3), wecan conclude that

σt = βt − S(t, T ).

(Be careful to distinguish this σt from the volatility σ(t, T ) of theHJM model.) Upon integrating (9.8), we have

Ft = F0 exp

[−1

2

∫ t

0σ2udu+

∫ t

0σudW

Tu

](9.9)


Let Q be the usual martingale measure for St. Then by our risk-neutral valuation principle, the value at t = 0 of X is given by

V0 = EQ[(ST −K)+/BT

]= EQ

[(ST −K)/BT · 1{ST≥K}

]= EQ

[ST /BT · 1{ST≥K}

]−KEQ

[1/BT · 1{ST≥K}

]= I1 − I2.

Note that

I1 =

∫{ST≥K}

ST /BTdQ.

If one uses St/S0 as a numeraire4, the corresponding measure Q =QT becomes

dQ = dQT =STS0BT

dQ.

Therefore I1 = S0dQ(ST ≥ K). On the other hand, the forwardmeasure Pt on Ft is defined using p(t, T )/p(0, T ) as the numeraire,so

dPt =p(t, T )

p(0, T )BtdQ.

In particular

dPT =1

p(0, T )BTdQ.

Therefore

I2 = K

∫{ST≥K}

1

BTdQ

= Kp(0, T )

∫{St≥K}

1

p(0, T )BTdQ

= Kp(0, T )PT (ST ≥ K).

Thus we have

V0 = S0Q(ST ≥ K)−Kp(0, T )PT (ST ≥ K) (9.10)

Now note that FT = ST . Thus ST ≥ K if and only if FT ≥ K, whichis equivalent to saying that

F0 exp

[−1

2

∫ T

0σ2t dt+

∫ T

0σtdW

Tt

]≥ K.

Taking logarithm and rearranging terms, it is equivalent to∫ T

0σtdW

Tt ≥ − log

F0

K+

1

2

∫ T

0σ2t dt (9.11)

4We need to assume St > 0 almost surely for all t.


Lemma 9.6. For deterministic σt,∫ Tt σtdW

Tt is a Gaussian random

variable with mean 0 and variance∫ T

0 σ2t dt .

Proof. Write ∫ T

0σtdW

Tt ≈

∑i

σti∆WTti ,

where 0 = t0 < t1 < · · · < tn = T is a partition of [0, T ]. Now each∆W T

ti has mean 0 and is independent of ∆W Ttj for i 6= j. Therefore

its variance is

EPT

[∑i

σti∆WTti

]2

=∑i

σ2ti∆ti.

The proof follows by passing to the limit as we take finer and finerpartitions.

Rewrite (9.11) by dividing by√∫ t

0 σ2t dt, we have

Z =

∫ T0 σtdW

Tt√∫ T

0 σ2t dt≥− log F0

K + 12

∫ To σ2

t dt√∫ t0 σ

2t dt

.

Since the left hand side of the above formula is a standard N(0, 1)random variable Z, we have

PT (ST ≥ K) = PT (FT ≥ K)

=

∫ ∞−d2

1√2πe−t

2/2dt,

where

d2 =log F0

K −12

∫ T0 σ2

t dt√∫ t0 σ

2t dt

.

ThereforePT (ST ≥ K) = N(d2),

which implies thatI2 = Kp(0, T )N(d2) (9.12)

Let us now look at I1. Note that

Q(ST ≥ K) = Q

(P (T, T )

ST≤ 1

K

)= Q

(GT ≤

1

K

),


where Gt = p(t,T )St

= 1Ft

. Recall that Q is the measure gotten by

using St/S0 as the numeraire. Thus Gt must be Q-martingale as itis p(t, T ) discounted5 by St. Now check that

dGt = dF−1t

= −F−1t σtdWt + (something)dt.

On the other hand, since Gt is a Q-martingale, the drift term mustdisappear when dGt is written in terms of dWt, where Wt is theQ-Brownian motion. Thus

dGt = −GtσtdWt (9.13)

Upon integrating (9.13), we have

GT = G0 exp

[−1

2

∫ T

0σ2t dt−

∫ T

0σtdWt

](9.14)

Thus it is easy to check that GT ≤ 1/K if and only if

−∫ T

0σtdWt ≤ log

(1

G0K

)+

1

2

∫ T

0σ2t dt.

Again dividing by√∫ T

0 σ2t dt and noting that Z = −

∫ T0 σdWt/

√∫ T0 σ2

t dt

is a N(0, 1) random variable, we can conclude that

Q(FT ≥ K) = Q(Z ≤ d1)

= N(d1)

where

d1 =log(F0K

)+ 1

2

∫ T0 σ2

t dt√∫ T0 σ2

t dt.

ThereforeI1 = S0N(d1) (9.15)

Hence from (9.12) and (9.15), V0 becomes

V0 = S0N(d1)−Kp(0, T )N(d2).

Proposition 9.7. The value at t = 0 of the European call optionX = (ST −K)+ is given by

V0 = S0N(d1)−Kp(0, T )N(d2),

5Discounting has to be done by St/S0. But since S0 is a constant, the assertionis still valid.


where

d1 =log(

S0Kp(0,T )

)+ 1

2

∫ T0 σ2

t dt√∫ T0 σ2

t dt,

d2 =log(

S0Kp(0,T )

)− 1

2

∫ T0 σ2

t dt√∫ T0 σ2

t dt.

If we do everything from time t instead of 0, we have the following.

Theorem 9.8. The value Ct at t of the European call option X =(ST −K)+ is given by

Ct = StN(d1)−Kp(t, T )N(d2)

where

d1 =log(

StKp(t,T )

)+ 1

2

∫ Tt σ2

udu√∫ Tt σ2

udu,

d2 =log(

S0Kp(t,T )

)− 1

2

∫ Tt σ2

udu√∫ Tt σ2

udu.

To get the formula for the put option, one can utilize the put-callparity. Let Ct and Pt be the values at t of the call and put options,respectively. Then it is easy to check that

CT − PT = ST −K.

Thus by the usual risk neutral valuation,

BtEQ[CT /BT

∣∣Ft]−BtEQ [PT /BT ∣∣Ft]= BtEQ

[ST /BT

∣∣Ft]−KBtEQ [1/BT ∣∣Ft] ,where Q is the usual martingale measure. Therefore,

Ct − Pt = St −Kp(t, T ).

From this we have the following result.

Theorem 9.9. The value Pt at t of the European put option Y =(K − ST )+ is given by

Pt = −StN(−d1) +Kp(t, T )N(−d2).

9.4. POTPOURRI OF INTEREST-RATE PRODUCTS 258

9.4 Potpourri of Interest-rate Products

For the rest of this chapter, we briefly go over some of the moreimportant financial product related to the interest rate, or the termstructure of interest rate in general. In particular when we say p(t, T ),f(t, T ) and the likes, we always assume, though tacitly, the underly-ing term structure model. As a general rule, all of them are reducibleto the bond prices or the options thereof. For that reason, we expressthem in terms of the bond prices, which can be calculated, at leastin principle, via the term structure model.

9.4.1 Par bond with coupons

The bond with coupons is an instrument (contact) that pays fixedamount of cash as principal and the interest at maturity.

0 = T0 T1 T2 · · · Ti−1 Ti · · · TN

Fix a time t = T0 = 0. Suppose the time at which the bondpays interest is Ti (i=1, . . . , N), and let TN be the time of maturityand T0 = 0 is the present. At TN , the bond pays the interest andthe principal which is normalized to be 1. Suppose k is the nominal(coupon) interest rate of the bond, and let δ = 1

n , where n is thenumber of times the bond pays the interest per year. (We alwaysscale the time so that one year is set to be 1.) The interest paymentis kδ. For example, suppose the nominal interest rate is 10% andit pays the interest quarterly (four times a year), then the interestpayment is

kδ = 0.1× 0.25 = 0.025.

Since the bond also pays the principal at TN , the cash stream isas depicted below.

0 = T0 T1 T2 · · · Ti · · · TN

kδ kδ kδ kδ + 1

The easiest way of valuing this bond is to treat each payment attime Ti as separate zero coupon bond. Thus the value of the payment


at time Ti (i < N) has value kδp(t, Ti) at time t. Thus the total valueof this bond at time 0 is

p(0, TN ) + kδ

N∑i=1

p(0, Ti). (9.16)

The par bond is the bond whose value at time of issuance, i.e., t = 0is equal to the principal payment at maturity. Thus in order to doso, the issuer has to set the nominal interest rate k in such a way, thetotal value of the bond given by (9.16) is 1. Therefore, the nominalinterest rate k is calculated as

k =1− p(0, TN )

δ∑N

i=1 p(0, Ti).

9.4.2 Floating-rate bond

The floating-rate bond is the bond that is the same as the fixed-ratebond except that the interest payment varies according to pre-agreedarrangement. A typical manner in which the interest payment isdetermined is as follows: Let S < T . The interest payment at timeT is determined by the market rate at time S. Thus suppose oneinvests A at time S and gets paid B, then the annualized nominalinterest rate k during this period is

k =B −A

(T − S)A.

In particular, if A = p(S, T ), and B = p(T, T ) = 1, then

k =1− p(S, T )

(T − S)p(S, T )=

1

T − S

[1

p(S, T )− 1

]. (9.17)

This type of mechanism is generally adopted in the LIBOR6 deter-mination. Now suppose T1, · · · , TN are the times of interest paymentand TN is the time of maturity at which time the final interest andthe principal (=1) payments are made. Assume also that the pay-ment is done regularly so that Ti−Ti−1 = δ for some constant for alli. For simplicity let us also assume that t = T0

7 and T1 − T0 = δ.8

Let us denote the annualized nominal interest rate between Ti−1 andTi by L(Ti−1). Then (9.17) can be rewritten as

L(Ti−1) = k =1

δ

[1

p(Ti−1, Ti)− 1

](9.18)

6London Interbank Offered Rate7We do not assume T0 = 0.8It is not necessary to assume the regular payment, interval nor to set t = T0.

But for simplicity we make such assumptions. Everything done here works forirregular interval with minor modification.


for i = 1, · · · , N . Note that the interest payment at time Ti is

δL(Ti−1) =1

p(Ti−1, Ti)− 1.

Since it is a contingent claim at Ti, its value at T0 is given by

Vi = BT0EQ

[(1

p(Ti−1, Ti)− 1

)/BTi

∣∣∣∣FT0

].

Now since

p(Ti−1, Ti) = BTi−1EQ

[B−1Ti| FTi−1

]and p(Ti−1, Ti) ∈ FTi−1 , we have

B−1Ti−1

= EQ

[p(Ti−1, Ti)

−1B−1Ti| FTi−1

].

Thus

EQ

[B−1Ti−1| FT0

]= EQ

[EQ

[p(Ti−1, Ti)

−1B−1Ti| FTi−1

]| FT0

]= EQ

[p(Ti−1, Ti)

−1B−1Ti| FT0

]Therefore, Vi can be written as

Vi = BT0EQ

[B−1Ti−1| FT0

]−BT0EQ

[B−1Ti| FT0

]= p(T0, Ti−1)− p(T0, Ti)

Hence the value V at T0 of this floating rate bond is

V = p(T0, TN ) +N∑i=1

(p(T0, Ti−1)− p(T0, Ti))

= p(T0, T0)

= 1.

Here the term p(T0, TN ) represents the value at T0 of the principalpayment 1 at TN .

The curious fact that the value at T0 of the floating rate bondequals the nominal (i.e., undiscounted) principal payment can bebetter understood if one considers the following trading strategy.Suppose one starts with one dollar at time T0 with which one buys


1

p(T0, T1)units of T1-bond whose price at time T0 is p(T0, T1). This

investment becomes worth1

p(T0, T1)dollars at time T1 as the value

of T1-bond at T1 is 1, i.e., p(T1, T1) = 1. Out of this one pays

δL(T0) =1

p(T0, T1)− 1 as the interest at T1. (Note that this interest

payment at time T1 is precisely the interest the floating rate bondpays at time T1.) It is clear that the remaining sum at T1 after theinterest payment is then 1. With it, one can repeat the investmentpattern. Namely at each time Ti the interest payment is

δL(Ti − 1) =1

p(Ti−1, Ti)− 1

and the sum available to invest is 1 with which one buys1

p(Ti, Ti+1)units of Ti+1-bond whose price at time Ti is p(Ti, Ti+1). Repeatingthe same way it is easy to see that at time TN one still has onedollar remaining after paying the interest, which will be given upas the principal payment. This way, this trading strategy preciselyreplicates the behavior of the floating rate bond.

9.4.3 Interest-rate swap

Swap is a generic financial term that stands for all sorts of contractsthat exchange one kind financial asset with another. The one wewill be discussing in this section is the so-called interest-rate swap.It works in principle as follows. Suppose A holds a bond that paysinterest in a fixed-rate, say k per annum. Suppose the time of interestpayment is Ti for i = 1, · · · , N and TN is the time of maturity atwhich it also pays 1 as the principal. For the sake of simplicity, let usassume T0 = t. Its cash stream consisting of interest only excludingthe principal is as in Figure 9.1.

t = T0 T1 T2 · · · Ti · · · TN

kδ kδ kδ kδ

Figure 9.1: Cash stream of interest rate only.

Hereδ = Ti+1 − Ti for i = 1, · · · , N − 1.


The value at T0 of income kδ paid at Ti is clearly kδp(T0, Ti). There-fore the total value at T0 of such cash stream is

kδN∑i=1

p(T0, Ti). (9.19)

Suppose now B holds a bond that pays the interest in floating rateas described in Subsection 9.4.2. Then its cash stream minus theprincipal payment of 1 at TN is as in Figure 9.2. Since the total

T0 T1 T2 · · · Ti · · · TN

δL(T0) δL(T1) δL(Ti−1) δL(TN−1)

Figure 9.2: Cash stream of floating rate.

value at T0 of the floating-rate bond at time T0 is 1 and the valueat T0 of the principal paid at TN is p(T0, TN ), the value at T0 of theinterest only cash stream must be

1− p(T0, TN ). (9.20)

The (interest-rate) swap is a contract that exchanges these two cashstreams. In the parlance of finance the payer of the swap is theone who pays out fixed-rate cash stream and receives the floating-rate cash stream. The position held by the payer is called the payerswap. Similarly, the receiver is the one who receives fixed-rate cashstream in lieu of the float-rate one, and his position is called thereceiver swap. It is easy to note that the payer/receiver designationis based on the fixed-rate cash stream.

When the swap contract is entered at T0, the values at T0 of bothstreams must coincide. Therefore

kδ

N∑i=1

p(T0, Ti) = 1− p(T0, TN ).

Solving for k, we have

k =1− p(T0, TN )

δ∑N

i=1 p(T0, Ti),

which is called the swap rate.


In practice, at each payment time both parties settle only thedifference between the fixed-rate income and the floating-rate incomewithout exchanging the whole amount. This practice is called thenetting.

This swap contract exchanges interest incomes only. But in orderto calculate the actual amount paid as interest one has to calculateit as if it is based on some principal. This figure representing suchfictitious principal is called the notional amount. When the organi-zations like BIS9 reports the amount of swap contracts outstandingworldwide, it reports this notional amount. As of December, 2010,BIS reports the total interest-rate swap outstanding worldwide is364 trillion dollars, which is really the notional amount. We willlater come back to the relevance of notional amount.

9.4.4 Forward (interest-rate) swap

Suppose there is a swap contract entered at time T0 as in the previoussubsection. The contract of the receiver, called the receiver swap, hasits value at time T0 equal to

X = kδN∑i=1

p(T0, Ti)− [1− p(T0, TN )]

= P (T0, TN ) + kδ

N∑i=1

p(T0, Ti)− 1.

Suppose the both parties agree at time t < T0 to enter into the swapcontract at T0 with the rate k preset at time t. This agreement is

t T0 T1 · · · Ti · · · TN

agreemententer into

swap

Figure 9.3: Time of agreement.

called the forward swap. This X ∈ FT0 is a contingent claim withexpiry T0. Then applying the lemma below, its value Vt at t is given

9Bank for International Settlements.


by

Vt = BtEQ

[X/BT0

∣∣Ft]= BtEQ

[p(T0, TN )/BT0

∣∣Ft]+kδBt

N∑i=1

EQ

[p(T0, Ti)/BT0

∣∣Ft]−BtEQ

[1/BT0

∣∣F]= p(t, TN ) + kδ

N∑i=1

p(t, Ti)− p(t, T0) (9.21)

Lemma 9.10. For T1 < T2,

BtEQ[p(T1, T2)/BT1 |Ft

]= p(t, T2).

Proof. Intuitively, this is clear, since this is the value at t of thecontingent claim that pays T2-bond at T1, which simply must be theT2-bond even at t. Thus its value at t must be p(t, T2). To prove itanalytically, recall that

Bt/BT1 = exp

(−∫ T1

trsds

)and

P (T1, T2) = BT1EQ

[1/BT2

∣∣FT1

]= EQ

[exp

(−∫ T2

T1

rsds

) ∣∣∣FT1

].

Thus

BtEQ

[p(T1, T2)/BT1

∣∣Ft]= EQ

[Bt/BT1 · p(T1, T2)

∣∣Ft]= EQ

[exp

(−∫ T1

trsds

)EQ

[exp

(−∫ T2

T1

rsds

) ∣∣∣FT1

] ∣∣∣Ft]= EQ

[EQ

[exp

(−∫ T2

trsds

) ∣∣∣FT1

] ∣∣∣Ft](∵ −

∫ T1

trsds ∈ FT1

)= EQ

[exp

(−∫ T2

trsds

) ∣∣∣Ft]= p(t, T2).


As usual, the value Vt (as in (9.21)) of this forward swap contractmust be zero at time t. Therefore, upon setting Vt = 0 in (9.21) andsolving for k, we get

k =p(t, T0)− p(t, TN )

δ∑N

i=1 p(t, Ti),

which is the rate preset at t with which the swap contract is to beentered into at time T0. k is called the forward swap rate. Dividingthe denominator and the numerator by p(t, T0), we have

k =1− Ft(T0, TN )

δ∑N

i=1 Ft(T0, Ti),

where Ft(T0, Ti) = p(t, Ti)/p(t, T0) is the forward price at t for thepurchase of T0-bond in return for 1 at Ti.

9.4.5 Bond option

Let p(t, T1) be the price at t of the bond with maturity T1. In thissection, we study how to value the options on it. In particular, letX be the call option on it with strike price K at the expiry T ≤ T1.Then X ∈ FT and is given by

X =(p(T, T1)−K

)+.

We apply the generalized Black-Scholes formula in Section 9.3 toderive a valuation formula for X.

Let Ft = F (t, T ) be the price at t of the forward contract thatdelivers T1-bond at T ≤ T1. Then the forward price at t is given by

Ft = F (t, T ) =p(t, T1)

p(t, T ).

Assume, as we have done in Chapter 8, that p(t, U) satisfies thefollowing dynamics

dp(t, U) = p(t, U)[m(t, U)dt+ S(t, U)dWt

]for any maturity date U , where Wt is the Brownian motion withrespect to a physical measure. We furthermore assume S(t, U) is adeterministic function of two variables t and U . Then it is easy tocheck by brute force that

d

(p(t, T1)

p(t, T )

)=p(t, T1)

p(t, T )

[(S(t, T1)− S(t, T )

)dWt +

(|S(t, T )|2 − S(t, T1)S(t, T )

)dt],


where

S(t, T ) = −∫ T

tσ(t, u)du, (9.22)

as in Section 8.6. By the result of Section 9.2, the Brownian motionW Tt corresponding to the forward measure PT satisfies

dW Tt = dWt − S(t, T )dt.

Therefore we have

d

(p(t, T1)

p(t, T )

)=p(t, T1)

p(t, T )

[S(t, T1)− S(t, T )

]dW T

t .

In other words

dFt = Ft[S(t, T1)− S(t, T )

]dW T

t . (9.23)

This forward price dynamics can be also seen as follows. First, Ft isa martingale with respect to PT . Therefore Ft must be of the form

dFt = FtσtdWTt ,

for some σt. We then have to check what this σt has to be. For thatone takes the logarithm of Ft = p(t, T1)/p(t, T ), takes d, and collectsvolatility terms, i.e., the coefficients of dW T

t , to see that

σt = S(t, T1)− S(t, T )

= −∫ T1

Tσ(t, u)du,

where the last equality is due to (9.22). Invoking Theorem 9.8, wehave the following.

Theorem 9.11. Let X be the call option on the T1-bond. Assumethat the strike price of X is K and the expiry T ≤ T1. Assumefurther that for any U the bond price p(t, U) satisfies the SDE


],

where S(t, U) is a deterministic function of t and U , where Wt is theBrownian motion with respect to a physical measure. Then the valueCt at t of the call option is given by

Ct = p(t, T1)N(d1)−Kp(t, T )N(d2),


where

d1 =log(p(t,T1)Kp(t,T )

)+ 1

2

∫ Tt σ2

udu√∫ Tt σ2

udu,


)− 1

2

∫ Tt σ2

udu√∫ Tt σ2

udu,

and

σt = S(t, T1)− S(t, T ) = −∫ T1

Tσ(t, u)du.

By invoking Theorem 9.9, the put option formula can also be ob-tained.

Theorem 9.12. Let X be the put option on the T1-bond. Assumethat the strike price of X is K and the expiry T ≤ T1. Assumefurther that for any U the bond price p(t, U) satisfies the SDE


]where S(t, U) is a deterministic function of t and U ,where Wt is theBrownian motion with respect to a physical measure. Then the valuePt at t of the put option is given by

Pt = −p(t, T1)N(−d1) +Kp(t, T )N(−d2),

where


)+ 1

2

∫ Tt σ2

udu√∫ Tt σ2

udu,


)− 1

2

∫ Tt σ2

udu√∫ Tt σ2

udu,

and

σt = S(t, T1)− S(t, T ) = −∫ T1

Tσ(t, u)du.

9.4.6 Option on coupon bond

In this section, we study options on coupon bonds. Unlike the zero-coupon bond we have studied in the previous section, the presenceof the stream of cash payments complicates the problem. Let Ti, fori = 1, · · ·n, be the time of interest payment and Tn the maturity of


T1 Ta−1 T Ta Tn· · · · · ·

kδ kδ kδ kδ + 1

Figure 9.4: Timeline and cash flows.

this bond. Let T be the expiry of the option, which falls in the timeinterval [Ta−1, Ta), i.e., Ta−1 ≤ T < Ta. Let X be the call optionon this bond with the strike price K at the expiry T . Then as theowner of X is entitled, at T , to the whole of subsequent cash flows,X as an FT -random variable must be of the form

X =

(p(T, Tn) + kδ

n∑i=a

p(T, Ti)−K

)+

.

As K is related to this set of cash flows, the result of the pre-vious subsection cannot be directly applied, which makes it a verydifficult problem to value X in a general setting. However in a cer-tain case it can be done without too much complication, say, if thebond price p(t, T ) can be given as a function p(t, T, r) of three de-terministic variables t, T, r, where r stands for the short rate. Notethat p(t, T,∞) = 0, which conforms to the intuition that the bondshould have no value if the interest rate (hence the discount rate)is infinite, i.e., the future cash flows should amount to zero presentvalue. Define

V (T, r) = p(T, Tn, r) + kδ

n∑i=a

p(T, Ti, r).

ClearlyV (T, 0) = 1 + kδ(n− a+ 1).

Since no one in the right frame of mind will pay V (T, 0) or more at Tfor the remaining subsequent cash flow of this bond, the strike priceK should have been set as smaller than V (T, 0). Hence it is veryreasonable to assume that

V (T,∞) = 0 < K < V (T, 0).

Therefore, by continuity, there must be r∗ such that V (T, r∗) = K.Let Ki be such that

Ki = p(T, Ti, r∗).


Then

V (T, r∗) = p(T, Tn, r∗) + kδ

n∑i=a

p(T, Ti, r∗)

= Kn + kδn∑i=a

Ki.

Furthermore, using the fact that p(t, T, r) and hence V (T, r) is amonotone decreasing function of r, it is not hard to check that for anyr, V (T, r) ≥ K if and only if p(T, Ti, r) ≥ Ki for every i = a, · · · , n.Therefore we can write

X =(p(T, Tn)−Kn

)++ kδ

n∑i=a

(p(T, Ti)−Ki

)+.

Once X is broken down this way, it can be valued by valuing each(p(T, Ti)−Ki

)+separately, say, using the method introduced in the

previous subsection.The same argument applies to the put option on coupon bond.

9.4.7 Cap and floor

9.4.7.1 Cap

Floating-rate interest payment is a common form of fixed-incomeinvestment. The problem with this kind of arrangement is that thepayer of floating-rate interest may be exposed to unexpectedly largeinterest payment. To hedge against such contingency, one may enterinto a contract that limits (caps) the maximum of interest rate thepayer is liable for. Suppose the cash stream is as in Figure 9.5.Suppose the interest-rate cap is k. According to this arrangement, if

t = T0 T1 T2 · · · Ti · · · TN

δL(T0) δL(T1) δL(Ti−1) δL(TN−1)

Figure 9.5: Timeline and cash stream.

the interest rate L(Ti−1) at Ti is less than k, the payer pays δL(Ti−1)as usual. But according to this contract, if L(Ti−1) > k, then thepayer is obligated to pay only kδ, which amounts to the payer’s


paying δL(Ti−1) but receiving the difference δL(Ti−1)− kδ from thereceiver. In other words, the cap contract entitles payer to receive

X = δ(L(Ti−1)− k

)+(9.24)

at each time Ti. This individual component to be executed at Ti iscalled the caplet, and the entirety is called the cap. In this subsectionwe study how to value it. Let pi = p(Ti−1, Ti). Then using (9.18) wehave

X = δ

[1

δ

(1

pi− 1

)− k]+

=

(1

pi− (1 + kδ)

)+

= (1 + kδ)p−1i (K − pi)+,

where K = (1 + kδ)−1. Therefore the value Vt at t of this caplet Xis

Vt = BtEQ

[B−1TiX∣∣Ft]

= (1 + kδ)BtEQ

[B−1Tip−1i (K − pi)+

∣∣Ft] . (9.25)

Now

EQ

[B−1Tip−1i (K − pi)+

∣∣Ft]= EQ

[EQ[B−1Tip−1i (K − pi)+

∣∣FTi−1

]∣∣Ft]= EQ

[p−1i (K − pi)+ EQ

[B−1Ti

∣∣FTi−1

]∣∣Ft](∵ pi ∈ FTi−1)

= EQ

[p−1i (K − pi)+B−1

Ti−1BTi−1 EQ

[B−1Ti

∣∣FTi−1

]∣∣Ft]= EQ

[B−1Ti−1

(K − pi)+∣∣Ft] .

(∵ BTi−1EQ

[B−1Ti−1

∣∣FTi−1

]= pi)

Therefore

Vt = (1 + kδ)BtEQ

[B−1Ti−1

(K − p(Ti−1, Ti)

)+∣∣Ft] .Note that

BtEQ

[B−1Ti−1

(K − p(Ti−1, Ti)

)+∣∣Ft]is the value of the put option an Ti-bond with the strike price Kat the expiry Ti−1, which can be evaluated by the method, say, inSubsection 9.4.5.


9.4.7.2 Floor

If a cap is like a call option, a floor is a contract that works like aput option. so the floor contract holder receives at Ti

Y = δ(k − L(Ti−1)

)+.

As before, each individual contract executed at each time is called thefloorlet and the entirety of such floorlets is called the floor. Althoughit is entirely possible to evaluate Y mimicking what was done forcaplet, it is more illustrative to look at the so-called floor-cap parity.Let Caplet(t) be the value of the caplet at t and Floorlet(t) that ofthe floorlet at t. Then it is trivial to see that

Floorlet(Ti)− Caplet(Ti) = δ(k − L(Ti−1)

)= (1 + kδ)− 1

pi.

Thus, upon apply the risk neutral valuation principle, we have

Floorlet(t)− Caplet(t)

= (1 + kδ)BtEQ

[B−1Ti

∣∣Ft]−BtEQ [p−1i B−1

TI

∣∣Ft] .Now

EQ

[p−1i B−1

Ti

∣∣Ft] = EQ

[EQ[p−1i B−1

Ti

∣∣FTi−1

]∣∣Ft]= EQ

[p−1i EQ

[B−1Ti

∣∣FTi−1

]∣∣Ft](∵ pi ∈ FTi−1)

= EQ

[p−1i B−1

Ti−1BTi−1EQ

[B−1Ti

∣∣FTi−1

]∣∣Ft]= EQ

[B−1Ti−1

∣∣Ft](∵ BTi−1EQ

[B−1Ti

∣∣FTi−1

]= pi)

Therefore

Floorlet(t)− Caplet(t) = (1 + kδ)p(t, Ti)− p(t, Ti−1), (9.26)

from which and from the formula for Caplet(t), Floorlet(t) can befound. Formula (9.26) is called the floor-cap parity.

9.4.8 Swaption and etc.

Swaption is an option to enter into a swap contract at a future dataa fixed rate10 k. Figure 9.6 shows the timeline. The swap contract

10The interest rate k is not the swap rate as described in 9.4.3. It is simply afixed number agreed upon when the swaption contract is entered into.


t T0 T1 · · · Ti · · · TN

Figure 9.6: Timeline.

is to be entered at T0 with the fixed rate k, and the cash flow occursat times Ti, i = 1, 2, · · · , n. The value at T0 of the receiver swap is

p(T0, Tn) + kδ

n∑i=1

p(T0, Ti)− 1,

and the payoff at T0 is(p(T0, Tn) + kδ

n∑i=1

p(T0, Ti)− 1

)+

.

This is exactly a call option on the coupon bond with coupon rate kwhose strike price is 1 and expiry T0. The valuation of this contractcan be done using the method of 9.4.6.

Remark 9.13. Options on caps and floors are also very popularcontracts. An option on a cap is called a caption, and similarly anoption on a floor is called a floortion. However, the valuation of suchintstruments is more complicated and is beyond the scope of thislecture. The readers are referred to many advanced textbooks.

EXERCISES 273

Exercises

9.1. Let Wt be a Brownian motion and for each t let Xt be a randomvariable given by

Xt =

∫ t

0e2u+3dWu

(a) Find the mean and variance of Xt.

(b) For fixed t, is Xt a Gaussian random variable? Justify youranswer.

(c) Regarding Xt as a stochastic process, is Xt a martingale? Ifso, why; if not, why not?

9.2. Suppose a floating rate bond X pays out interest δL(Ti−1) attime Ti for i = 1, ..., n such that

δL(Ti−1) =1

p(Ti−1, Ti)− 1

where p(t, T ) is the price at t of the zero-coupon bond paying 1 at T.Assume also this bond X pays out the principal 1 at time Tn. Whatshould its value at T0 be? Justify your answer.

9.3. Answer the following questions.

(a) Let p(t, T ) be the price at t of the zero-coupon bond paying 1at T . Let T < T1. Let F (t, T ) be the forward price at t of theforward contract delivering the T1-bond (i.e., the zero-couponbond that pays 1 at T1) at T . Write down F (t, T ) in terms ofzero-coupon bond prices p(t, T ) (for various T ).

(b) Explain why Ft = F (t, T ) as a stochastic process has to bea martingale with respect to PT where PT is the T -forwardmeasure.

9.4. Let Xt be a stochastic process given by dXt = a dt + b dWt,where a and b are constants and Wt is a Brownian motion. Let Ybe a random variable defined by Y =

∫ T0 e

T−tdXt, where T is a fixedpositive number.

(a) Prove that Y is a Gaussian random variable.

(b) Calculate the mean and the variance of Y.

EXERCISES 274

9.5. Let t < T1 < T2, and let St be a stock price process.

(a) What is the forward price F (T1, T2) at T1 of the contract todeliver this stock at T2?

(b) Let X ∈ FT2 be a contingent claim that pay F (T1, T2) at timeT2. What is the value of X at t?

9.6. Let T0 < T1 < · · · < TN be given such that Ti − Ti−1 = δ fori = 0, · · · , N . The LIBOR rate L(Ti−1) between Ti−1 and Ti is givenby

L(Ti−1) =1

δ[

1

p(Ti−1, Ti)− 1],

where p(t,T) is the value at t of the zero-coupon bond paying 1 atT.

(a) The caplet(Ti−1) at Ti−1 with the cap rate k is defined by

caplet(Ti−1) = δ(L(Ti−1)− k)+.

Describe the floorlet contract foorlet(Ti−1) with the floor ratek.

(b) State and prove the floor-cap parity formula.

Documents

Interest Rate Products - Seoul National University · 2012. 12. 27. · 9.1. CHANGE OF NUMERAIRE AND THE INVARIANCE OF RISK NEUTRAL VALUATION 247 exists. Let Xbe any European contingent