,.)
( t q )
[.] is taken under PT. and we have used theFloatBondi,i+l( t ) / P (t, I;.+]) and Li ( t ) are-
where the expectation E;"fact that both
1+1
martingales ,aswellastherelationshipofP ( t , T , + l ) ( l + J i L i ( t ) ) = P ( t , T , which,),in turn, assumes that the samefloating rate is used both for coupon and for discounting. In practice,one can structure a floating bond with different floating rates or indices,in which case one needs to model it accordingly.Through backward induction, we can generalize the above result to afloating-coupon bond with any numbers of coupons (from period i to N),i.e.
FloatBondi,,, ( t )= P(t,T,)
t 5 I;:.
(7.19)
In other words, the price of a (LIBOR flat) floating-coupon bond (orsimply floating bond) is the same as the price of a zero-coupon bondwith the principal payment on the first accrual start date of the floatingcoupon bond regardless of the number of coupons. For simplicity, herewe assume that the bond coupons all reset in the future and thus we donot need to handle the accrued interest. In other words, the bond pricewe have is really the forward bond price.From the above argument, we can also conclude that the duration ofa floating bond is very short, and thus has very little interest rate risk.This is one of the motivations for bond issuers to swap the fixed-couponbonds (or simply fixed bonds) that they issue to floating bonds as part ofthe liability management. More specifically, without such swap, if theinterest rate decreases or the market rallies, the PV of the bond issuer's
Sec. 7.5
Martingale Approach
261
liabilities increases. With such a swap, the bond issuer is hedged againstsuch liabilities increase. However, this may cause some volatilities inthe earnings of the bond issuer due to the volatility of the sort-term interest rate or the short-term interest expense, If the sort-term interest rateincreases significantly, then the short-term interest expense also increases significantly.With the martingale representation theorem presented in Chapter 2,we can obtain the following process
dL,(t)=cTldW,~+'.L' ( O I t I p ) ,
(7.20)
where W,T' + I ' L is a one-dimensional PT,,-Brownian motion for L,(t).Naturally, the volatility cr, should be zero on or after the reset date I ; R .The volatility 0,can be time dependent, state dependent, or stochastic.This is the essence of the martingale LIBOR market model. This is a1-factor model for any given forward LIBOR L,(t) of a particular maturity. However, since each forward LIBOR process can have a differentBrownian motion, it is a multi-factor model for the entire forwardLIBOR curve with as many factors or Brownian motions as there aredistinct maturities for the forward LIBORs. As such, we shall also termit as the full-dimensional LIBOR market model.These Brownian motions are normally not independent. The correlation of the incremental Brownian motions is the same as that of the incremental forward LIBOR rates.As usual, we rewrite the volatility term in the following form (whichby itself does not enforce lognormality)01
(4 '
= 2, (+I
(7.21)
where %(t) is the instantaneous percentage volatility of the LIBOR L,(t).Thus,dL, ( t )= A,( t )L, ( t ) dW,T
I+"
or
L
(0 I t I q R ) ,
(7.22)
262
The Interest Rate Market Model
L~( T )= L~(t)exp( -[+A:
Chapter 7
(s>ds + [ A (~ s ) d ~ q + l ~ i )
.
( t I T Iq R )
(7.23)
In general, A;(t) can be time dependent, state dependent, or stochastic. If A; is constant or only time dependent (not state dependent and notstochastic), then L;(t) is lognormal under the forward arbitrage-free. If A; is constant, thenmeasure
In general, we can define
-T. ,Liwhere W,,F1can be shown to be a lPT+]-Brownian motion starting
from time t, i.e., ,:+lJi
= 0 and
qf;T'l'Liis a Gaussian N ( O , f i )
variable, and & is the term percentage volatility or the Black volatility.Consequently,
L~( T )= Li( t exp
(-3;i;2(2, T )( T - t ) + & ( t ,T I( t I T I TR
)
?T;..lJi
1.
(7.26)
If the instantaneous percentage volatility Aiis constant, then the instantaneous and the term percentage volatilities are the same, i.e., 4 = IAil. If
(
needed, we can let Lj ( T ) = Li TiR)
T RIT .
A cap or a floor consists of a series of cash flows, or caplets or floorlets. More specifically, the time t price CF(t,R) of a cap or a floor withthe strike rate R is the sum of the time t price all the caplets or floorlets,
Sec. 7.5
Martingale Approach
263
CFj(t,R)with the option expiration at the reset date q R i.e.,,
c
N-1
CF(t,R)=
CF,(t,R),
(7.27)
i=i( [)
where i(t) is defined by T,C)-, < t I
( T t = 0) . As usual, the above
equation prices cashflows with future resets and does not include the accrued interest.More specifically, each caplet or floorlet, CFi(t,R), essentially is acall or put option on a LIBOR Lj(t)(for the accrual period from Ti to Tj+1)with the strike rate R, option expiration at the reset date TiR, and payment at Ti+,. In other words, the terminal payoff (at the payment dateTj+l)of the caplet or floorlet with unit notional is given by
(
)
i
CF,(q+l,R = w ( ~ (qRi- R 6;)
3
(7.28)
where w is equal to 1 for a caplet and -1 for a floorlet.As usual, for pricing the caplet or floorlet, we utilize the fact thatCF; ( t , R ) / P (t,q+l) is a PT -martingale. First of all, we price as of the1+1
reset date q R i.e.,,
= P(
Thus, in general,
qR
w(Li( q R ) - R ) si)+
264
The Interest Rate Market Model
CF, ( t , R )= P(t,q+JE::+'
Chapter 7
CF;.(qR,R )
q+,) E:'41[( w(Li (Ti") - R )6.)+
=P (t,
]
If the LIBOR volatilities &(t)'s are only time dependent (not statedependent and not stochastic), then the above equation reduces to thefollowing Black's formula, as, in this case, not only is Li(t) a PI;.+1martingales, it is lognormal under Pq+l. In other words,
whered, =
In ( 4( t ) / R )- iq2( t,qR)(qR- t )
d, = d, +
&( t , l ; R ) J G
( t I q") ,
(7.32)
q.( t , q R ) , / 5
and w = 1 for a caplet and w = -1 for a floorlet.Market quotations essentially provide the implied term or Blackvolatilities to be used in the above Black's formula to back out the market prices. This is the essence of the market quoted Black volatilities.
Sec. 7.5
265
Martingale Approach
To obtain the instantaneous volatilities for general derivatives pricingand modeling, the model calibration processes (such as calibrations to allor selected caps/floors and European swaptions) are needed.In the plain vanilla capdfloors that we just discussed, the paymentdate is typically the same as the accrual end date, but, in general, it canbe any business date on or after the reset ate. If the payment date is thesame as the accrual start date, then we have the in-arrears caps/floors.More specifically, if the payment date is T, then, similar to Eq. (7.29)
and (2.51), the new caplet price C e ' ( t , R ) is given by
L
= P(
qR,T )(w (Li (q R -) R )Si )+
J
(7.34)
( t 5 T R< T )
andCF,' ( t ,R ) = P ( t , T )E:
(
CF,' qR,R)
p ( q",T )
!
(
)
P ( qR,T ) (w Lj ( q R - R ) S ,)+
= P(t,T)E:
p (T R , T )
I
.
(7.35)
( t 5 q RS T )
The only difference is that, in general, Li(t) is not a P,-martingales(unless T =T,+,). Typically, this can be handled by the convexity adjustment presented in Chapter 2.
The Interest Rate Market Model
266
7.5.2
Chapter 7
The Swap Market Model and the Black Formula for EuropeanSwaptions
A plain vanilla interest rate swap93essentially is a contract for exchanging a fixed coupon bond (with the fixed coupon rate R) with a floatingcoupon bond94(with the floating coupon rate equal to the correspondingLIBOR95of the same accrual dates). In order for the counterparty creditrisk to work out, there should be a netting agreement for the two bonds.A forward swap is the one that starts at a future date qs with fixed
coupon accrual start dates from qs to T i p l and accrual end and paymentdates from Ti!,
and T i , where T,'
is a tenor structure satisfling:
0 < qs < T; < ... < T i , and 6: = 6 qs,q+l is the accrual coverage or
(
the year fraction between Ts and qfl depending on the day count basis.Here the superscript S indicates the swap or the fixed leg of a swap.In general, one accrual period of the fixed leg of a swap (i.e., thefixed coupon bond without the principle payment) can correspond tomultiple accrual periods of the floating leg of a swap (i.e., the floatingcoupon bond without the principle payment) or a cap/floor. For instance,in U.S., the floating leg of a vanilla swap or a vanilla cap/floor has quarterly accrual periods and reset frequency, whereas the fixed leg of a vanilla swap has semi-annual accrual periods. Therefore, we use distinctnotations for the accrual periods for swaps and caps/floors.Let Swupi,rv(t,R)denote the time t price of such forward swap withthe fixed rate R and unit notional, which can be expressed as93 Here we focus on simple interest rate swaps. Aside from interest rate swaps, there aremany other types of swaps in other product areas, such as equity swaps, commodityswaps, cross-currency swaps, credit default swaps, equity default swaps, and hybridswaps cross product areas. Within interest rate swaps, there are also swaps involvingdifferent interest rate indices and other variables, such as inflation linked swaps, creditlinked swaps, treasury swaps, municipal (muni) swaps, basis swap, and prepaymentlinked and other mortgage swaps. Swaps can also be made very exotic by making thecoupon exotic (e.g., by making the coupon linked to the slope of a forward curve and tovolatilities or forward volatilities) and by adding features, such as callable features andtarget redemption features. In recent years, inflation linked swaps have gained strongpopularity for hedging the inflation risks due to the increased concerns about the inflationin the market.94 In order for the counterparty credit risk to work out, there must be netting agreementfor the two bonds.95 In the euro currency (EUR), for instance, the default interest rate index is EIBOR.
Sec. 7.5
267
Martingale Approach
swapi,,( t , R ) = u(FloatBondi,, ( t )- FixedBond,,, ( t , R ) ), (7.36)where F1oatBondi,, ( t ) is the time t price of the corresponding floatingcoupon bond in the swap, FixedBondi,, ( t , R ) is the time t price of thecorresponding fixed coupon bond in the swap, and u is equal to 1 for apayer swap (paying fixed coupon) and -1 for a receiver swap (receivingfixed coupon).As we pointed out before, through backward induction, the forwardprice of a floating-coupon bond is the same as the price of a zero-couponbond with the principal payment on the first accrual start date of thefloating-coupon bond regardless of the number of coupons, i.e.,
F l o a t B o n d i , N ( f ) = P ( t , ~ S ) t 5 qS .
(7.37)
On the other hand, the price of the fixed coupon bond is given byN
FixedBond,,,(t,R)=R
6y-lP(t,7',S)+f'(t,Ti)j=i+l
t
(7.38)
qS
Therefore, the time t price Swapi,rv(t,R)of the swap is given byswapi,N(t ,R ) =
where u = 1 for a payer swap (paying fixed) and w = -1 for a receiverswap (receiving fixed).This is the general arbitrage price of a swap and does not depend onany particular models. Thus, all arbitrage-free models should reproducethis result. This is often used as the target of a martingale resampling.As usual, unless otherwise indicated, we assume that the credit risk isabsent and all the counterparties are assumed to be able to issue LIBORflat floating coupon bonds (or borrow money at LIBOR flat). The implications of the credit risk on the prices of swaps (Litzenberger, 1992, Duf-
268
The Interest Rate Market Model
Chapter 7
fie and Huang, 1996) will be discussed later on.The time t (par) forward swap rate96Si,N(t)is defined as the fixed rateR such that the time t swap price is zero, i.e.,
where q,,,( t )=
.cN
d;-lP(t,Tf)
is an annuity, which is also a traded
j=i+l
asset. Thus, the time t swap price can be rewritten as
(7.41)
The reader may notice that the floating accrual and reset frequencydoes not come into the picture of the above swap related equations. Thisis due to the fact that what we discussed are forward swap prices andforward swap rates, which do include the accrued interests. When wecompute the accrued interests, the floating accrual and reset frequencydoes matter.A plain vanilla European swaption is a European call or put optionon an underlying swap. More specifically, a payer swaption is a calloption on the corresponding payer swap or an option to enter into thecorresponding payer swap. A receiver swaption is a put option on thecorresponding payer swap or an option to enter into the correspondingreceiver swap.A plain vanilla European swaption should be exercised only whenthe underlying swap ends up in-the-money at the option expiration date.In other words, the terminal payoff Swuptioni,N(Ti,R) of a Europeanswaption with the expiration date or the swap reset date of(which
cS,R
96 It is also termed as par swap rate, swap par coupon, or breakeven coupon.
Sec. 7.5
2 69
Martingale Approach
is typically 2 business days before qs in U.S.) is given by97
the
We can try to use the same technique that we used for caps/floors inprevioussection,i.e.,thequantity
Swuptioni,, ( t , R ) / P (t,l;S,R) ( t I TiSyR) is a B S , R -martingale. Thus,T,
But this result does not lead to the Black‘s formula easily. One difference between caps/floors and swaptions is that a cap/floor is portfolio ofoptions (with each option given by a Black‘s formula) and a swaption isan option on a portfolio (of bonds).Recall that the time t forward swap rate S , d t ) is given by
)
Si,N ( t) = ( P ( t,TS - P ( t , T;
)) / Aj,, (f )
)
( t 5 qS ,
(7.44)
which is a relative price of traded assets with the annuity
or a forward starting swap can have the following variations,such as cash settlement, physical settlement, and cash discount settlement. In the physical settlement, the actual underlying swap contract is entered at the expiration date of aforward swap or a swaption, and for the swaption, such swap contract is only entered ifthe swap is in the money to the swaption holder. In the cash settlement, the market valueof the underlying swap at the expiration date is exchanged in cash, and such cash exchange only occurs if the swap is in the money to the swaption holder. In the cash discount settlement, it is the same as the cash settlement, except that the value of the underlying swap is determined by discounting all the future fixed-coupon cashflows by theswap rate at the expiration date. In the absence of the counterparty credit risk, the PVs ofthe swaption or the forward starting swap are the same or very close among these different settlements. However, in the presence of the counterparty credit risk, the PVs (ormore precisely, the default premium or CVA) can be drastically different. In general, thecounterparty credit risk is much more significant in the physical settlement than in thecash or cash discount settlement. Here we focus on the cash settlement or the physicalsettlement without the counterparty credit risk. The details of pricing the counterpartycredit risk shall be presented in the next chapter. The reader may recognize that a capletor a floorlet is actually a one period physically settled swaption.
97 The payoff of a swaption
The Interest Rate Market Model
270
Chapter 7
N
i$-,P(t,T,?)
Ai,N( t ) =
as the numeraire, and thus is a martingale
j=i+l
under the annuity arbitrage-free measure PAi,N.Swuptioni,, ( ~ , R ) / A ~( t, ), ( t I qs)
Since
is
also
a
P ~- ~ ,
martingale, the time t price Swuptioni,N(t,R)of the European swaptioncan also be evaluated under the annuity arbitrage-free measure PAi,Nthrough the following martingale relationship
where w is equal to 1 for a payer swaption (a call option on a payer swapor a put option on a receiver swap) and -1 for a receiver swaption (a calloption on a receiver swap or a put option on a payer swap).With the martingale representation theorem, we can obtain the following process (assuming that the forward swap rate process is continuous with finite variance)
p),
(7.46)
is a PAi,, -Brownian motion. The volatility D
~ can, be~
(l)= O i , N d K Ai,N YSi.NA.
where W,
‘9”
S.IsN
(0 I t I
time dependent, state dependent, or stochastic.This is the essence of the swap market model. This is a 1-factormodel for any given forward swap rate. However, since each forwardswap rate process can have a different Brownian motion, it is a multifactor model for the entire forward swap curve with as many Brownianmotions as there are distinct forward swap rates. Thus, it is also a fulldimensional model.These Brownian motions are normally not independent. The correlation of the incremental Brownian motions is the same as that of the in-
Sec. 7.5
Martingale Approach
271
cremental forward swap rates.As usual, we rewrite the volatility term in the following form (whichby itself does not enforce the lognormality)O i , N =‘i,N
(t)si,N
(7.47)
( t >’
where Aidt) is the instantaneous percentage volatility of the forwardswap rate &At).Thus,’i,N
(T , =S j , N (r)eXp(-r+A:N
(S)ds + r A j , N ( s ) d W : , N 7 s i , N
).
(7.48)
( t I T I Ts.,)
In general, A i d t ) can be time dependent, state dependent, or stochastic. If Ai,dt)is constant or only time dependent (not state dependent andnot stochastic), then Si,dt)is lognormal under the annuity arbitrage-free. If Ai& is constant, thenmeasure PA.I,NSi,N
( T )=
In general, we can define
&,T-Ai.N,Si,N
= iT,l,,N(S)dW,A’,N.si.N
(7.50)
( t I T 5 q‘?’)
where v A i , N , S i , N can be shown to be a P A i , N -Brownian motion startingt,Tfrom time t, i.e.,
T;J~’~J= 0 and
q$N>’i.N
is a Gaussian
N ( 0, f i )variable, and &,N ( t ,T ) is the term percentage volatility or
the Black volatility. Consequently,
272
The Interest Rate Market Model
Chapter 7
Si,N ( T )=
S ~ ,(f)exP(-iT,N2N(t,T)(T-t)+
4 ,( f~, T ) q $ N ’ s i ’ N) . (7.51)
( t I T I Tj’,’)
If the instantaneous percentage volatility /Z,N is constant, then theinstantaneous and the term percentage volatilities are the same, i.e.,&,N =l&,N1
*
If the swap rate volatilities &,’s are only time dependent (not statedependent and not stochastic), then the time t price Swuptioni,N(t,R)of theEuropean swaption reduces to the following Black’s formula, as, in thiscase, the swap rate SLN is lognormal under the annuity arbitrage-freemeasure P4, N . In other words,
where
and W = 1 for a payer swaption and W = -1 for a receiver swaption.Market quotations essentially provide the implied term or Blackvolatilities to be used in the above Black’s formula to back out the market price. This is the essence of the market quoted volatilities. To obtainthe instantaneous volatilities for general derivatives pricing and modeling, the model calibration processes (such as calibrations to all or selected caps/floors and European swaptions) are needed.
Sec. 7.6 Overview of Simultaneous and Globally Consistent Pricing and Hedging
7.6
273
Overview of Simultaneous and Globally Consistent Pricingand Hedging
It is important to point out that, theoretically, the lognormal LIBOR market model and the lognormal swap market model are mutually inconsistent, as they are closely related and specifying one to be lognormal precludes the other to be lognormal. In other words, in order to price andhedge caps/floors and swaptions consistently, if one specifies any volatility structure (such as lognormal, CEV, shifted lognormal, stochasticvolatilities, etc.) for the LIBOR market model, then one cannot independently specify the volatility structure for the swap market model, andvice versa Rather, one needs to rely on one model and use this model toderive the necessary parameters for the other model.A general approach is to solve the LIBOR market model with a common set of LIBORs that are needed for both caps/floors and theswaps/swaptions.For instance, in U.S., the cash LIBORs (i.e., the most liquid LIBORs), and the LIBORs in a floating leg of a plain vanilla swap and LIBORs in a plain vanilla cap/floor have a quarterly accrual and reset freq ~ e n c y .So~ ~we can model these LIBORs in a LIBOR market modeland, then, obtain zero-coupon bond prices (and swap rates) needed forpricing both caps/floors and swaptions.If the two sets of LIBORs have different accrual and reset frequencies (such as semiannual and quarterly), then we need to use the oneswith higher frequency (such as quarterly) to derive the ones with lowerfrequency (such as ~emiannual).’~It is more difficult to go the other wayaround, but it can be handled by the martingale interpolation (seeChapter 4).In general, one accrual period of the fixed leg of a swap can correspond to multiple accrual periods of the floating leg, which, in turn, cancorresponding to same or multiple accrual periods of the ash LIBORs orthe LIBORs for the caplfloor.We define a mapping function ic(i), which, given an index i of thefixed leg of a swap, returns the index i,(i) of the cash LIBORs or the98 It is possible for the LIBORs in the floating leg of a plain vanilla swap and LIBORs in
a plain vanilla cap/floor to have slightly different accrual and reset dates. In this case,interpolations are needed to obtain one set of LIBORs from anther one.99 There maybe a basis spread due to the liquidity of different accrual frequencies to beadded to the less liquid LIBORs.
274
The Interest Rate Market Model
Chapter 7
LIBORs for the capMoor with tenor matching that of the swap, i.e.,(7.55)For instance, in U.S., the cash LIBORs and the caps/floors havequarterly accrual period or reset frequency, whereas the swap fixed legshave semi-annual accrual period or frequency. In this case,
i, (i) = 2i .
(7.56)
This is also shown in the figure below.
Floating IndexFixedIndexTime (Year):
,4,
0 1 2 3 4 5 6 7 81IIIII
0
I
0
1I
I
I
I
2I
I
I
3I
I
1
I
I
2
Figure 7.1. Example of the index of the floating leg of an interest rate swap andthe index of the fixed leg of the swap in U.S.
Some of the trades are structured referencing LIBORs with accrualperiods different than that of cash LIBORs. In this case, we introducethe general notation L(t,T,@ for the time t forward LIBOR with accrualstart at T and accrual period of 6. For instance, a swap can be structuredsuch that its floating leg has the same accrual periods with this fixed leg.In this case, the zero-coupon bond prices with accrual periods or frequency corresponding to the cash LIBOR or cap accrual period or frequency and the swap fixed leg accrual period or frequency are given byp T j + l ) = 1+ 8 j L( t ,Ti,dj ) ,P(t,Tj)('3
)
n
(7.57)
i, (i+1)-1
p ( t ,qS1 = 1 + G , s L ( t , q s , 6 f ) =( l + S , L ( t , T j , S , ) ) . (7.58)P(t$)j=ic(i)
Sec. 7.6
Overview of Simultaneous and Globally Consistent Pricing and Hedging 215
This shows that the cash LIBOR L(t,T,,6,)L(t,T,’,@)
and the LIBOR
with the swap fixed leg accrual period is related through
(aside from a basis spread)
(
n + ( ,Ti, ,
ic(i+I)-I
1+ 6; L t ,qs,6;) =
(1 6,L t
6;))
(7.59)
j=ic ( i )
where L ( t ,T, ,6,) is the same as L, ( t ) used in previous sections. For aforward starting swap, its PV is actually independent of its floating legfrequency (aside from a basis spread).7.6.7
Simultaneous Consistent Pricing Through Approximation
As pointed out by some authors, when the LIBOR processes are lognormal (or normal, CEV, or shifted lognormal), the swap rate processesare approximately lognormal (or normal, CEV, or shifted lognormal).Therefore, they can be used to simultaneously price LIBOR and swapderivatives in the same trading book as an approximation.A starting point of this approximation is to find the approximatedlognormal swap rate volatilities from the lognormal LIBOR volatilities.The motivation of specifying lognormal volatilities is to be consistentwith the market quoting conventions, which quote the prices ofcaps/floors and swaptions using the lognormal Black volatilities. This,by no means, enforces the processes followed by LIBOR or swap rates,which can be CEV, shifted lognormal, or stochastic volatility processes.In recent years, the interest rates are more like normal than lognormal inthe sense of the dependency of the volatilities on rates.Recall that the time t forward swap rate S,lY(t)is given by(tsqs),
(7.60)
N
a;-,P(t,T;)
where Ai,N( t ) =
is an annuity. This means that the
j=i+l
forward swap rate Si,N(t)is a function of Lj ( t ) with i, ( i ) I j I i, ( N )- 1 .With Itd’s lemma, we can obtain
276
The Interest Rate Market Model
dSi,N( t )= (...)dt +
Chapter 7
as,,,
iC(W
4
L
j=ic(i) dLj
j(t)
(7.61)
We used dSi,, ( t ) , rather than dln( t ), in the above because we assumed that the rates are more like normal thanOtherwise,using dln Si,N( t ) gives better approximation.Since the forward swap rate can be further re-written in terms of aportfolio of LIBORs as follows, i.e.,
and recognizing that
(7.63)
we compute
(‘) easily. If needed, we can also use the following
aLj
crude approximation,
We actually do not necessarily need to worry about the drift term ofthe forward swap rate Si,N(t),as we normally value derivatives under thecorresponding annuity probability measure, under which the forwardswap rate has zero drift.Recall from section 7.5.1 that the LIBOR processes are given by
loo By this statement, we do not mean that the (nominal) interest rates should be allowedto go negative. Rather, it mainly refers to the dynamics the interest rates in the positiveregion, particularly the functional dependency between the rate and its volatility.
Sec. 7.6
Overview of Simultaneous and Globally Consistent Pricing and Hedging 277
dLi ( t ) = Ai ( t ) L i(t)dW,q+19Li (0 5 t I q ),
(7.65)
where W,T.f+l'L . is a one-dimensional PT+,-Brownian motion