-kdlD(t)lAt
(at IIliIl(X(l>))
@,(l)
,
(6.73)
where k, and kd are positive numbers that can be time- and statedependent.Figure 6.3 below shows an example of the tree grid construction withthe adaptive tree technique.
Figure 6.3. Example of tree grid construction with the adaptive tree techniquewith time- and state-dependent drift and volatility.
Once the above tree grid is constructed, then we are ready to construct the rest of the tree branches. A tree branch needs to start from agrid point and end at an grid point at the next tree time step. More specifically, in this case, Eq. (6.70) is reduced to
240
The Heath-Jarrow-Morton Framework
Pd&d
( t) +
Pd&d(f)’Pd +Pm
Chapter 6
<&,, ( t )= 0
+cGu(t)2=o(t)2At2,
+c=1
(6.74)
am
( t ) terms are missing due to the construction of the adapwhere thetive tree grid such that the middle tree branches exactly following the( t ) = 0 . This not only simplifies Eq. (6.70), but also,(local) drift oras we shall see, guarantees that the transition probabilities on the tree arenon-negative.Given G,,( t ) and a d ( t ), the amount of branching on the treegrid, the above equation can be easily solved with the following solutions,
urn
P, =
cr( t ) 2 At2
Mu(t)(uu
(t)-Gd
(t))
(6.75)
Note that, by design,(6.76)
a,,
Thus, as long as the absolute values of( t ) and a d ( t ) are largeenough (in terms of absolute values), the non-negative transition probabilities on the tree are easily guaranteed.In general, we can setk,, = k d =&.
(6.77)
If the drift and the volatility are constant, the following exercises arehelpful in understanding why Eq. (6.77) is used for an adaptive trinomialtree.
Sec. 6.4 Overview of Numerical Implementations of the RS and the IK Model
241
Exercise 6.4. Assuming k, = k, = k (this makes the adaptive trinomialtree symmetric and thus simplify the problem), where k is a positivenumber, and further assuming that the drift and the volatility are constant, verify that the local variance, and all the odd local moments of thevariable are matched.Exercise 6.5. Under the same condition as the previous exercise, expressthe kurtosis of X ( t ) as a function of k. Verify that adaptive trinomialtree matches the local kurtosis when Eq. (6.77) is satisfied.In other words, in addition to match the local variance and all theodd local moments, the adaptive trinomial tree can also match the localkurtosis, if the drift and the volatility are constant. This, in turn, helpsthe convergence. If the drift and the volatility are not constant, then wecan choose the solutions that approximately match the local kurtosis.
6.4.3
Overview of Applications of the Adaptive Trinomial TreeTechnique to the RS Model and the IK Model
The basic idea of the LRS implementation (Li, Ritchken, and Sankarasubramanian, 1995) is to build a tree for the process of the short rate r(t)and approximate the process of the cumulative variance &t). The rationale behind this idea is that the cumulative variance &(t) is a locally deterministic or a predictable process.It is important to point out that the process for &t) is path dependent.Thus, at each tree node, we need to carry a vector of &(t)’swith each f i t )determined by a path leading to that tree node. One subtlety is that thenumber of paths leading to a tree node increases exponentially respect tothe number of time steps to that tree node, and so does the number off i t ) ’ s in the f i t ) vector. A common approach for handling such path dependency is through state stratification or to keep a few (such as 3 or 5)selected f i t ) ’ s at each tree node (by sub-sampling of the distribution offit)). Then, during the backward induction, we need to perform interpolations on the payoffs respect to f i t ) .Since a recombining binomial tree when solving the RS model oftentimes does not converge, a trinomial tree is needed. A conventional recombining trinomial tree may not easily guarantee that the transitionprobabilities on the tree are non-negative. An adaptive trinomial tree can
242
The Heath-Jarrow-Morton Framework
Chapter 6
easily ensure that the transition probabilities on the tree are non-negativeon most of the tree nodes, except possibly for a very small number oftree nodes corresponding to extreme values of &t)’s (and thus are of verylow probabilities). Should negative transition probabilities occur, we cantruncate them to zero and normalize the total transition probability toone.As usual, the first step of constructing an adaptive trinomial tree is toconstruct the tree grid first. This is similar to the procedure described inthe previous section (see, e.g., Eq. (6.73)), excerpt that we need to properly handle the second state variable &). One simple approach is toconstruct the tree grid based on r(t) together with approximated averagef i t ) on each of the grid point. The approximated average f i t ) on any gridpoint can be obtained by approximating Eq. (6.28) by, e.g., using themost likely path of r(t) from time zero to that grid point (which essentially is a Brownian bridge problem).Then, we can construct the rest of the tree branches. Since there area vector of $(t)’s at each of tree node, the transition probabilities betweenthe states of r ( t ) ’ s actually depend on Kt). For the f i t ) that is used toconstruct the tree grid, the tree branches and the transition probabilitiescan be determined by the adaptive trinomial tree algorithm through Eq.(6.74). While all the &(t)’sshare the same tree branches at the same treenode, for other &t)’s, the transition probabilities are determined by Eq.(6.70).As we pointed out before, due to the stratification in &t) in the numerical procedure, during the backward induction, we need to performinterpolations on the payoffs respect to #t).For the multi-factor RS model or the IK model, if we use the furtherreduced volatility structure as indicated by Eq. (6.57), then a multi-factorIK model can be solved by solving independent 1-factor models and performing a convolution. This significantly simplifies the numerical implementations, particularly, recombining lattice implementations.
6.5
Appendix
6.5.7
Closed-form Solutions for the RS Model
In this section we give the closed-form solutions for the zero-couponbond price and forward rate in the RS model.Recall that
Sec. 6.5
Appendix
243
SinceOf( s , T )=of( s , s ) k ( s , T )
=Of(s,s)k(s,t)k(t,T),
(6.79)
=of( s , t ) k ( t , T )
then,f ( t , T )=f ( 0 , T )
-k (t T )?
6 ( T )a; )(s, t
0, s,
?
(6.80)
+k(t,T)[ q ( s , t ) d KNow, we will try to eliminate the last integral in the above equationinvolving the incremental Brownian motion. Recall thatr ( t )= f ( t , t )=f ( O J ) -
6
0, (s,t)o>( s , t ) d s
lof (s,t)dW, ,
(6.81)
(s,t)c$- ( s , t ) d s ,
(6.82)
+
i.e.,
6
of(s,t)dW, = r ( t )- f ( 0 , t ) +
thus,
6..
The Heath-Jarrow-Morton Framework
244
Chapter 6
Now, let us evaluate the integral in the above equation. Recall that
(6.84)
P ( I , T ) = gTk(t,u)du,I
then,
-
1
=-
6
0, ( S J ) =
T
Of( s , u ) dtr
k ( s , u ) du ,
Of (3,s)
(6.85)
=-of ( S , S ) P ( S , T )
thus,
=-+,s)
p(s,u)du
1T
(6.86)
= - O f ( s , s ) k ( s , t ) k(t,u)du=-of ( s , t ) P ( t , T )
Therefore,
Consequently,
One way to obtain the zero-coupon bond price P(t,T) is through theforward ratef(t,T), i.e.,”
ny
Another equivalent way to obtain P(t,T) is through the short rate r(t) as we discussed
before, i.e., P ( t , T ) = E,“ [ e x P ( - f r ( . ) . ) ]
or
Sec. 6.5
Appendix
245
Since
we can evaluate the above integrals individually. In other words,
(6.94)
If ~ ( t= )K is a constant, then it is easy to verify that
246
The Heath-Jarrow-Morton Framework
Chapter 6
k ( t , ~ )exp (-K( T - t ) )1
1P ( ~ , T=-(I)- exp(-K(T - t ))) .
(6.95)
K
6.5.2
Closed-form Solutions for the IK Model
In this section we give the closed-form solutions for the zero-couponbond price and forward rate in the IK model.Recall that, with all the vectors expressed in their components,
Since
ai( s , T )=a,( s , s ) k ,( s , T )
=aj(s,s)kj(s,t)kj(t,T),= oj ( s , t ) k ,( t , T )
(6.97)
then,
Now, we will try to eliminate the last integrals in the above equationinvolving the incremental Brownian motions. Recall that
( s , t ) a ,(s,t)ds + goi(s,t)dW, ( t ) ,
(6.99)
[ai(s,t)dW,( t )= xi( t )+ [a; ( s , t ) q ( s , t ) d s .
(6.100)
xi ( t )= -[a;
i.e.,
Sec. 6.5
Appendix
247
Thus,
26
(0;(s, T ) - 0;(s, I ) )oj(s, t )
-k ( I ,T )
i=I
Now, let us try to evaluate the integrals in the above equation. Recall that
then,
(6.103)
thus,
(6.104)
Therefore,
[( 0;( s , T )- a; (syt))oi( s , t ) d s = -pi (t,T)#i ( f ) .Consequently,
(6.105)
248
The Heath-Jarrow-Morton Framework
Chapter 6
(6.106)
Similar to the case of the RS model, the zero-coupon bond price P(t,T) isgiven byP ( ~ , T=) exp( - p f ( t , u ) d u )- P(0,t)
P(O,T)
.
(6.107)
Chapter 7
The Interest RateMarket Model
The interest rate market model of Brace, Gatarek, and Musiela (1997), ofMiltersen, Sandmann, and Sondermann (1 997), and of Jamshidian(1997), directly models market observable quantities, such as (discrete)forward LIBOR rates and (discrete) forward swap rates and has established itself as an important model." It is also known as the BraceGatarek-Musiela model, the BGM model, or BGWJ model.In this chapter, we shall discuss various forms of BGMIJ models indetails. Different from the BGM's original approach, we present another,yet equivalent, form of the BGM model based on the martingale modeling approach presented in Chapter 2. Various volatility structures, suchas CEV, shifted-lognormal, stochastic volatilities shall be discussed.We shall also present a multi-factor nonexploding bushy tree (NBT)technique:' which can solve the BGWJ and othzr multi-factor problemsmore efficiently than the conventional bushy tree technique. It is important to point out that in order to achieve desired accuracies in the numerical BGM/J model implementations, it is critical to apply the martingaleresampling or martingale control variate techniques presented in Chapter4.In addition, we shall discuss an interesting practical issue. Morespecifically, the BGM/J models require modeling of discrete quantities,such as LIBORs with discrete tenors. However, the payoff of the trade90 The original BGM/J model is for modeling default-free interest rate term structures, but
it can be extended to the modeling of defaultable interest rate term structures, which weshall discuss in the next chapter.''The initial version of the nonexploding bushy tree (NBT)technique was first reportedby Yi Tang and Bin Li at a RISK workshop on February 25, 1999.
250
The Interest Rate Market Model
Chapter 7
being priced can be based on LIBORs with tenors not modeled by theBGWJ model. This is particularly true for aged or existing trades. Ofcourse, one can perform interpolations. But interpolations do not automatically guarantee the required martingale relationships. This is also acriticism of the BGWJ model on a theoretical basis.92 This is addressedby the martingale interpolation technique presented in Chapter 4.
7.1
BGM Model versus HJM Model
The BGWJ LIBOR market model models the forward LIBOR rates (i.e.,uses the forward LIBOR rates as the state variables). The BGWJ swapmarket model models the forward swap rates (i.e., uses the forward swaprates as the state variables). The HJM model, on the other hand, modelscontinuous compounding forward rates.The BGWJ model specifically uses the concept of different arbitrage-free probability measures, such as the spot, forward, and annuityarbitrage-free measures. In the original BGM LIBOR market model,each forward LIBOR rate is a lognormal martingale diffusion process(i.e., with zero drift) under its own forward (up to the payment date) arbitrage-free measure (rather than the single spot arbitrage-free measure).In Jamshidian’s swap market model, each of the forward swap rate is alognormal martingale diffusion process (i.e., with zero drift) under itsown corresponding annuity arbitrage-free measure (rather than the singlespot arbitrage-free measure).Discrete HJM model is similar to BGWJ LIBOR model, except itmay not have closed-form solutions to the market caplet/floorlet prices.In a more general market model, each forward LIBOR rate and forward swap rate is a general martingale process under its own corresponding arbitrage-free measure. Such martingale process can be a diffusionprocess with various volatility structures including CEV, shifted lognormal, and stochastic volatilities (such as the SABR model) for modeling volatility skews and smiles. It can also be combined with a jumpprocess for modeling short expiry volatility smiles that tend to be morepronounced than what can be gracefully handled by a diffusion stochasticvolatility model.Consequently, the BGWJ LIBOR market model produces market92 HJM models do not suffer from such difficulty on a theoretical basis, as theoreticallythey can model forward rates with continuous tenors, although practically they need to beimplemented with forward rates with discrete tenors.
Sec. 7.1
BGM Model versus HJMModel
251
consistent closed-form solutions or approximations for liquid instruments, such as caps/floors and European swaptions. For caps/floors, theformulae from the lognormal LIBOR market model are the same as theBlack’s formulae, and for European swaptions, the Black’s formulae canbe used as good approximations to the formulae from the lognormalLIBOR market model. This, not only justifies the routine use of theBlack’s formula (Black, 1976) by market practitioners, but also significantly reduces the computation burden for model calibrations.The BGM’s original approach derives the LIBOR market modelfrom the zero-coupon bond price processes in the spot arbitrage-freemeasure, similar to the HJM model. In this aspect, the BGM model appears to be a sub-set of the HJM model (with different choice of the zerocoupon bond price volatilities). Jamshidian’s approach reveals that theBGM/J model is arbitrage-free even without the short rate (in which casethe market is incomplete). Jamshidian also developed the swap marketmodel, which, with lognormal volatility structures, produces the Black’sformulae for European swaptions. In these aspects, the BGM/J model isan extension of the HJM model.The numerical implementation of the BGMIJ models (and the general HJM models) for pricing Bermudan swaptions, and other non-vanilladerivatives still remains as a challenging task. This is because that, ingeneral, the BGM/J models (and the general HJM models) are nonMarkovian with respect to the underlying driving factors (although theyare Markovian with respect to all state variables) and, especially formulti-factor models, cannot be solved with the recombining lattice techniques.The conventional bushy tree, while capable of solving nonMarkovian models in principle, suffers from a severe practical limitation- its computation time grows exponentially as the number of tree timesteps increases (see Heath, Jarrow, and Morton, 1990 and Heath, Jarrow,Morton, and Spindel, 1993 for applications of the bushy tree technique tothe HJM model). A 2-factor conventional bushy tree is typically limitedto 18 tree steps. This renders the conventional bushy tree technique inaccurate for modeling derivatives with many event dates, such as cashflow dates, early exercise dates (for American and Bermudan style derivatives), and barrier monitoring dates.In this chapter, we shall also present a multi-factor nonexplodingbushy tree (NBT) technique, which breaks the computation time barrierof the conventional bushy tree, and allows over 100 tree steps. A nonex-
252
The Interest Rate Market Model
Chapter I
ploding bushy tree (NBT) typically consists of a number of small subbushy tree units with subsequent sub-bushy trees grown only from selected states of the preceding bushy trees. Such selected states can be themodes or sub-samplings of the state distribution of the preceding bushytrees.A 3-factor BGWJ LIBOR model has been implemented with thenonexploding bushy tree (NBT) technique, and accurate results havebeen obtained for caps/floors as compared with the closed-form solutions, and the European and Bermudan swaption results are comparablewith those of other techniques. The effects of the number of factors (upto 5 factors) on the Bermudan swaption prices were obtained throughpreliminary analyses. Sufficient conditions of convergence are also provided.
7.2
The Brace-Gatarek-MusielaOriginal Approach
Here we present simplified approaches along the line of reasoning of theBGM original approach. As with the HJM approach, the processes ofLIBOR can be derived from the processes of the zero-coupon bond priceP(t,r).Let Li(t) or Li (j=l,2, ..., N - l ) denote the time t forward LIBOR forthe accrual period from T to Tj+I and the payment at Ti+,,where T. is atenor structure satisfying: 0 < T, < T, < .,. < TN , and Sj = 6 (T, ,q+,) is thedepending onaccrual coverage or the year fraction between Tiand Ti+lthe day count basis.It is useful to mention, in passing, that, in general, in order to definea forward LIBOR or a floating interest rate or index in general, we need4 sets of dates, i.e., reset or fixing dates q R ,accrual start dates qs, accrual end dates q E ,payment dates qf. The reset or fixing date is theone when the floating rate or index is set or fixed (or becomes known).It is typically 2 business days prior to the corresponding accrual startdate, but this varies depending on the reset conventions in different currencies/countries. It is also when the variance or uncertainty of the rateor index stops accumulating. On or after the reset, the rate or index accrues an interest between the accrual start and end dates and the interestis paid on the payment date. The payment date is typically the same asthe accrual end date, but, in general, it can be any business date after thereset ate. If the payment date is the same as the accrual start date, then
Sec. 1.2
The Brace-Gatarek-Musiela Original Approach
253
the rate is termed as in-arrears (such as LIBOR in-arrears).For simplicity of disposition, we assume that
The reset date qR (which is typically 2 business days prior to the corresponding accrual start date) is when the variance or uncertainty of therate or index stops accumulating. Thus, on or after the reset date, theinstantaneous volatility of the rate (and the jump size, if we use a jumpmodel or jump-diffusion model) should be zero. For the same reason,the reset date is also used as the upper time limit in the variance integration for computing the equivalent or Black volatilities of the rate. Ifthere is an option on the rate, then the option expiry date is typically setat the reset date, though it can almost be any business date in general.When there is no conhsion, we may use qR and q interchangeably forthe simplicity of notation.Recall that the time t forward LIBOR and the time t zero-couponbond price P(t,T) with principal payment at time T (P(T,T)=l) are relatedthrough:l + s , L , ( t ) = P ( t , q ) / P ( t , ~ + , )( t I?).
(7.2)
In comparison, the instantaneous forward rate At,T) in the HJMframework is defined asf ( t , T ) = -dlnP(t,T)/dT
( t I T ).
(7.3)
It is easy to verify that(7.4)Recall that under the spot arbitrage-free measure Q
d P ( t , T ) / P ( t , T ) = r ( r ) d t + o ( t , T ) d ~ @ I T ),
(7.5)
where W, is an n-dimensional (independent) Q -Brownian motion (n by 1vector) and o(t,T) (1 by n vector) is the percentage volatility of the zerocoupon bond price. This, in general, is an n-factor model.Applying It& lemma yields
254
Chapter 7
The Interest Rate Market Model
dln P( t , T ) = ( r( t )
-+la(t , T ) f ) d t + o(t,T)dW,
,
(7.6)
(tIT)where
] a ( f , T ) I=
Jm
( t IT )
(7.7)
and the superscript ' denotes the transpose of a vector or a matrix.Thus, we can obtain the following HJM modeldf(t,T)=-d
a In P ( t , T )aT
aao(t,T)=L -l~(t, T)l dt dKdT
aT
(7.8)(tIT)
Similarly, we can obtain the following process for the LIBOR
Applying ItB's lemma again yieldsd ( 1+ SiLi ( t ) )1+ SiLi ( t )
=
-+( ( t,q )r - ( t q+,)r)dt10
10
++)a(t,q)- ~ ( t , q +dt~)r+(a(t , q ) - 0( t ,q+,))dW,= -a(t , q+,)((9( t ,q.) - 0 l ( t , q+1))dt
+ (a(t , q )- a(t , ?+I)&
(tsq)In other words,
.
(7.10)
Sec. 1.2
The Brace-Gatarek-Musiela Original Approach
255
In order for the LIBOR to have the volatility structure in the form of
Lj(t)A(t,Tj)(with A(t,Tj) being a 1 by n vector), we need to have the following restrictions to the percentage volatility structures of the zerocoupon bond prices, i.e.,
or
(7.13)
(7.14)Consequently, the LIBOR process can be rewritten as
( t q )
It is important to point out that the above equation does not enforcethat the LIBORs have lognormal volatility structures, as the volatilityA(t,Tj) can be state dependent (i.e., dependent on the LIBORs) or even aseparate stochastic process.The above processes are under the spot arbitrage-free measure Q .As we will see in the section 7.4, it is often more convenient to formulatethese processes under the forward arbitrage-free measure Pi+lwith re-
256
The Interest Rate Market Model
Chapter 7
spect to the numeraire of P(t,7;.+,),the price of the zero-coupon bondwith principal payment time Ti+,.
7.3
Comparison Between HJM and BGM Models
Here we present a summary of the comparison between HJM and BGMmodels, as shown in the table below.Table 7.1 Comparison between HJM model and BGMIJ model.
HJM Model
BGWJ Model
~
State variable: f ( t , T )
( t IT )
State variable: Li( t )
(tI q )
f ( t , T ) is not market observable.
L, ( t ) is market observable.
f ( t ,T ) = -a In P ( t , q p
1 + 44 ( t )= P ( t,q )/ P ( t,q.+1)
(tlT)
( t q )
Under the spot arbitrage-free measure Q ,the zero-coupon bond priceP(t,7') follows the following diffusion equations (for both models):
dP(t,T)=,(t)P(t,T)dt+P(t,T)a(t,T)dW, (tI T ) .Applying ItG's lemma yieldsdf (X,,t)=-dt+-dX,afJfat
ax,
+ t - J2f(dXJ2,
ax;
dtdt=O, dtdW, =o, dW:'dW,j =corr(dW:',dW,j)dt,d In P(t , T ) = ( r ( t )-+la(r,T)l2)dt + o(t , T )dW,
( t 5 T ).
Sec. 7.3
257
Comparison Between HJMand BGM Models
HJM Model
BGMIJ Model
d
d f ( t , T ) =--dlnP(t,T)dTd2do(t,T)= $-lo( t , T)I dt d W,dTdTT)(tI
d f ( t ,T ) = - ~ ( tT)?',( t ,T ) d t
dLj ( t )= -o(t,q+,)X' (t,I;.)dt
+Y( t,T)dW,
+ j ( t ,q ) d K
y(t,T)=o ( t , T )=
q t ,T )dT
-5 y(t,u)duT
t
+?+I)=-
c @(t,q)+ si ( )
j=i(r) 1
Li t
Nearly lognormal model
Lognormal model
r ( t ,T ) = r ( t ,T )min (f( t ,T ),M )
j ( t , q )= 2(t,?)Li ( t )
M. a large positive constant.
The Interest Rate Market Model
258
Chapter I
HJM Model
BGWJ Model
Markovian HJM models with respect to fewer state variables available.
Arbitrage-free without the short rateand the continuum of zero-couponbond prices.
Discrete HJM model is similar to BGWJ LIBOR model, except it may nothave closed-form solutions to the market capletlfloorlet prices.
7.4
Jamshidian’s Approach
Jamshidian’s approach reveals that the BGWJ model is arbitrage-free(among all the instruments that it can price) even without the short rate(in which case the market is incomplete).Jamshidian specifies the LIBOR market model in the forward arbitrage-free measure (up to the settlement date) of the longest LIBOR inthe model, instead of the spot arbitrage-free measure as in the case ofBGM’s approach. Thus, the n-factor LIBOR market model can be expressed as (Jamshidian, 1997):
(7.16)
under the forward arbitrage-free measure PTN of the longest maturity) with respect to P ( t , T N ) as the numeraire, where A(t,Tj)LIBOR L ~ - l ( toris the LIBOR percentage volatility (with A(t,Tj) being a 1 by n vector)and yTNis a standard n-dimensional PTN-Brownian motion (n by 1 vec-
tor).The Jamshidian’s approach for the LIBOR market model and theBGM approach are equivalent. They differ only in the probability measures under which the LIBOR processes are specified.Jamshidian also developed the swap market model. The swap mar-
Sec. 7.5
Martingale Approach
259
ket model with martingale approach shall be discussed in the next section.
7.5
Martingale Approach
The martingale approach, the BGM’s approach and Jamshidian’s approach for the LIBOR market model and the BGM approach are allequivalent. They differ in the probability measures under which theLIBOR processes are specified and in how the Brownian motions arespecified.7.5.1
The LIBOR Market Model and the Black Formula for CapslFloors
Recall that the LIBOR and the LIBOR zero-coupon bond price P(t,T)(P(T,T)=l)are related through:l + s , L , ( t ) = P ( t , ~ ) / P ( t , ~ , , )( t q),
(7.17)
which, according to the Harrison-Pliska no-arbitrage theorem presentedin Chapter 2, indicates that L,(t) is a martingale under the forward arbitrage-free measure PT+,with respect to P ( t ,q+l) as the numeraire.It is interesting to briefly discuss in passing the pricing of (LIBORflat) floating coupon bond (i.e., with the coupon rate of L j , the accrualcoverage or year fraction of Si , and the principal amount of 1). First ofall, it is easy to see that the time t price of the one-period or one-couponfloating-coupon bond (from period i to i+l and with coupon and principal payment at q+]) is independent of any particular model and is givenby
260
Chapter 7
The Interest Rate Market Model
FloatBondi,,l ( t )= P( t ,I;:+]) Ei+l
FloatBondi,i+l(T+I)
1
P(T+* T+lY
1
= P ( t , I;:+l)( 1+ JiLi ( t ) )
= P(t,T