edge low sell high”. An optimal hedging strategy can allow adealer “sell higher” by “hedging higher” or “buy lower” by “hedginglower”.More specifically, some of the adjustments in Eq. (4.135) may not beapplied in computing the hedges ratios. In other words, if one is able tobuy volatility cheaper (as in the case of Bermudan swaptions), one needsto use the secondary model calibration or adjustments to mark it to themarket. But, one may hedge without the secondary model calibration oradjustments or using higher volatility to achieve “buy low hedge high”.In other words, as we discussed before, sometimes there is a need of using two slightly different models, one for marking to the market (ormodel) and the other for hedging.Without a good understanding of this subject, one may lose somehidden profit in one’s portfolio.
4.6
Other Applications of Martingale Resampling
The martingale resampling technique starts with the motivation of improving numerical accuracy and precision (in both the valuation andhedging) by enforcing the required martingale relationship. However,the applications of the martingale resampling technique can be extended
204
Martingale Resampling and Interpolation
Chapter 4
to a much greater scope. For instance, as we have discussed, it can beused as (incremental) model calibrations and for creating non-parametricvolatility skew or smile models in both local volatility and stochasticvolatility model settings. It can also be applied with various controlvariate techniques, including addressing the inaccuracy and imprecisionfor credit charges resulting from the market process modeling or the scenario generation engine and the pricing engines not being fully consistent.In addition, we have also discussed the martingale interpolation thataddresses the situation that some models, such as the BGM/J model, require modeling of discrete quantities, such as LIBORs with discrete tenors, but the payoff of the trade being priced can be based on LIBORswith tenors not modeled by the BGM/J model. Of course, one can perform interpolations. But interpolations do not automatically guaranteethe required martingale relationships. The martingale interpolation is aninterpolation algorithm that guarantees the required martingale relationships.Here we present some other applications of the martingale resampling technique.
4.6.1
Modeling of Multiple Indices
In the interest rate modeling, we often need to model multiple interestrates (or indices) with different credit quality and other characteristic(such as liquidity and tax treatment). Aside from LIBOR, examples ofthese multiple interest rates (or indices) include the interest rate in Treasury debt, agency debt, municipal debt, commercial paper, and mortgageobligations. Different interest rates also arise from LIBOR of differentterms, as well as cross-currency basis spreads.One approach to modeling multiple indices is to specify and model aprimary index S, ( t ) such as the coupon index of an instrument or cashor the most liquid LIBOR in general, and then assume that all the indicesS,,, ( t ) are related to (or resampled from) the primary index with an additive and/or percentage spread. In other words,S,, ( t )=
( t ) S , ( t )+bi,,( t )
(i, j =1,2,3...) ,
(4.136)
where i indicates the terms of the index or the rate a n d j indicates themultiple indices or rates.If we do not need to price options on the spread between different
205
Sec. 4.6 Other Applications of Martingale Resampling
indices, then normally we do not need to explicitly handle the volatilityof the spread. In this case, ai,( t ) and bi, ( t ) are deterministic variablesand can be conveniently determined by the martingale resampling techniques discussed in the previous sections. The martingale targets are theprices of liquid underlying assets and options dependent on Si, ( t ).Equation (4.136) is good for normal diffusion processes and can beapplied to other diffusion processes as approximations. For lognormaldiffusion processes, a better resampling can be obtained by
In ( si,j ( t ) )= ai,j ( t )In (si ( t ) )f bi,j ( t )
(i, j = L2,3...)*
(4.137)
By simple shifting, the above equation can also be applied to shifted lognormal processes.The similar technique can be used to model the credit spread forcredit derivatives under certain circumstances.4.6.2
JLT Risk Neutralization of Credit Rating Transifion Process
When pricing credit derivatives based on credit rating transition probabilities, we can start with a historical credit rating transition matrix.One immediate issue is that the historical default probability is not thesame as (and typically less than) the implied or risk neural default probabilities. The JLT (Jarrow, Lando, and Turnbull, 1997) risk neutralization or calibration is aimed at addressing this issue and taking intoaccount the historical information.A simplified example of a I-year historical credit rating transitionmatrix is shown in Table 4.1, where the first column is the transitionfrom state (or rating) and the first row the transition to state (or rating), Dindicates the default state, and the numbers in the matrix are transitionprobabilities. The numbers in the last column are the historical defaultprobabilities. A more comprehensive credit rating transition matrix contains 15 states including half rating notches designated by +/-. Usually,the matrix also requires some data cleaning.
206
Martingale Resampling and Interpolation
Chapter 4
Table 4.1. A simlified example of a 1-year credit rating transition matrix wherethe first column is the transition from state (or rating) and the first row thetransition to state (or rating), D indicates the default state, and the numbers in the
matrix are transition probabilities. The numbers in the last column are thehistorical default probabilities
\AAAAAABBB BBBCCCDAAA 0.9081 0.0833 0.0068 0.0006 0.0012 0.0000 0.0000 0.0000AA 0.0070 0.9065 0.0779 0.0064 0.0006 0.0014 0.0002 0.0000A 0.0009 0.0227 0.9105 0.0552 0.0074 0.0026 0.0001 0.0006IBBB 0.0002 0.0033 0.0595 0.8693 0.0530 0.0117 0.0012 0.0018BB 0.0003 0.0014 0.0067 0.0773 0.8053 0.0884 0.0100 0.0106B 0.0000 0.0011 0.0024 0.0043 0.0648 0.8346 0.0407 0.0520CCC 0.0022 0.0000 0.0022 0.0130 0.0238 0.1124 0.6486 0.1979D 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.OOOO
I
In general, we denote PH as an n x n historical credit rating transition matrix (after some necessary data cleaning) and P the corresponding risk-neutralized matrix, where n denotes the default state.There are two requirements for the risk neutralization. The first oneis that we must reproduce today’s credit curve by setting the defaultprobabilities Pi,, ( i = 1,2,...,n ) to today’s market implied default probabilities (such as from those implied from CDS or risky bonds).76 Afterthis, we let P be a transform of the historical credit rating matrix andsatisfy the probability normalization, i.e.,n
c P i , j=1
(i=1,2)...,n ) .
(4.138)
j=l
This is the same as that the probability space span by the credit ratings iscomplete.It is easy to verify that the following transformation is a promisingone satisfying the above equation,
76 Here we have implicitly assumed that there is no correlation between the credit rating
transition process, the credit spread process, the recovery process, and other market processes. In the presence of such correlation, then the requirement of the risk neutralizationis to price CDS or risky bonds to the market.
Sec. 4.6
Other Applicafions of Martingale Resampling
207
(4.139)
(i, j = 1, 2,..., n - 1)
Pn,j = 0
We actually need to assume that the historical matrix is properly normalized after the data cleaning.In the JLT Markov Chain Model (Jarrow, Lando, and Turnbull,1997), a term structure of the historical credit rating transition matrix canbe obtained from one or more historical credit rating transition matricesthrough a bootstrapping and interpolation process. In other words, thehistorical forward credit rating transition matrix p H (tk ,t k + l ) for the timeinterval of (tk , t k + ] ) can be derived and used to reconstruct the historicalcredit rating transition matrix PH (0,T ) for the time interval of (0, T ) orthe term of T, i.e.,(4.140)
( t O = O , t, = T , T > O )Then, each of the historical forward credit rating transition matrixPH ( t , ,tk+l) can be risk-neutralized (using the JLT risk neutralization) toderive the risk-neutralized forward credit rating transition matrixP (t, ,tk+l) , from which the term risk-neutralized credit rating transitionmatrix can be derived, i.e.,N-1( O y T )
=
(tk ytk+l
)
k=O
(4.141)
( t ~ = O y tN=T, T>O)
In the JLT Markov Chain Model, it is essentially assumed that theforward credit rating transition matrix P (tk ,tk+]) is the same as the timeconditioned future credit rating transition matrix. This implies thatthe time-tk conditioned full future credit spread curve is available ateach of the future ratings.tk
208
Martingale Resampling and Interpolation
Chapter 4
This further implies that the JLT Markov Chain Model also predictsthe future volatilities of the credit spread curves. However, such volatilities are likely not to be consistent with the volatilities implied from thedefault swaption market or by other means.Such inconsistency in the credit spread volatilities is not an issue forusing the JLT Markov Chain Model to price instruments that depend oncredit ratings, but not on the credit spread volatilities, such as bonds withrating based variable coupons.However, such inconsistency in the credit spread volatilities is indeed an issue for using the JLT Markov Chain Model to price options onthe above instruments or any other instruments that depend on the creditspread volatilities. Important examples of these instruments are in thepricing of the counterparty credit risks which are default options.In the next section, we shall present the methodologies for addressing such inconsistency in the credit spread volatilities.
4.6.3
Calibration of Credit Spread Processes
Let’s denote as $ ( t , T ) the time t future credit spread term structure atthe ifhcredit rating implied by the risk-neutralized credit rating transitionmatrix in the JLT Markov Chain Model and the recovery (which can berating based). Then, the first moment of the credit spread term structureis risk-neutralized already in the sense that the collection of credit spreadterm structures at all the credit ratings can reproduce today’s market implied default probabilities or today’s credit curve.However, as we pointed out before, the JLT Markov Chain Modelalso predicts the hture volatilities of the credit spread curves and suchvolatilities are likely not to be consistent with the volatilities impliedfrom the default swaption market or by other means.In this section, we shall present the methodologies for calibrating thecredit spread volatilities (to some pre-specified volatilities).A general approach is to introduce a diffusion component to the dynamics of the credit spread process using a credit spread model and calibrate the overall credit spread volatilities.Compared to the credit spread models based on the short hazard rateor default intensity, the risky or defaultable interest rate market model(Schonbucher, 2000 and 2003) provides a framework for easily calibrating the credit spread volatilities, though, being non-Markovian, it doespose various numerical challenges and we shall discuss how to address
Sec. 4.6
Other Applications of Martingale Resampling
209
some of these challenges in later chapters.Here, we adopt methodologies similar to those of the defaultableBGM model (see, e.g., Schonbucher (2000 and 2003)),77but we generalize it by incorporating it with the JLT Markov Chain Model. In this case,at any given credit rating, ( t ,T ) is a diffusion process.With this general framework, if needed, we can also allow stochasticrecovery, credit spread volatility skews and smiles, and correlation between the credit rating transition process, the credit spread process, therecovery process, and other market processes. The drift and volatilityterms of ji( t , T ) can be used in the calibration to price CDS or riskybonds to the market and match the implied or specified credit spreadvolatilities.One possibly way for the calibration is to find the credit spread termstructure S j ( t , T ) through the following resampling, i.e.,
si
Si ( t , T ) = aj ( t ,T )+ b( t , T ) i i ( t , T )
( i =1,2,...,n-1)
Y
(4.142)
where a; ( t ,T ) and b ( t , T ) are resampling parameters that can be used toprice CDS or risky bonds to the market and match the implied or specified credit spread volatilities. More specifically, ai ( t ,T ) is used to calibrate to the default probabilities conditioned on the credit rating i andtime t.This process can also be applied to risk neutralize credit spread processes of multiple obligors or counterparties, e.g., generated by a factormodel possibly with jumps due to, e.g., credit rating transition or creditcontagion.For instance, let’s denote the credit spread term structure for the jtficredit rating and kih industry as Sj,k( t , T ) , The credit spread term structure for ifh counterparty of thejfhcredit rating and k?‘ industry can be expressed as
These papers are downloadable from http://ssrn.com/abstract=26 1051, andhttp://www.schonbucher.de/papers/cdsoptions.pdf.
”
210
Martingale Resampling and Interpolation
Chapter 4
(4.143)
4.6.4
Risk Neutralization of Mortgage Prepayment Model
Mortgage prepayment models are often obtained through econometricsfor the purpose of forecasting the future prepayment rates in the real orstatistical probability measure. Thus, risk neutralization or calibration isneeded when pricing mortgage instruments. One possibility for risk neutralization is the use of OAS (option adjusted spreads). For instance, anOAS can be use to calibrate to a mortgage pool. Two OAS are needed tocalibrate to both a mortgage pool and the MBS (mortgage backed securities) based on the pool. When pricing mortgage derivatives, it may benecessary to use the first and the second moments in the risk neutralization or calibration process.
4.7
Accuracy and Precision Tests
Various tests can be easily performed to quantify the accuracy and precision improvements of various resampling techniques.For instance, we can apply these resampling techniques to MonteCarlo simulations to price a financial instrument and we can run the sameMonte Carlo simulations many times with different seeds for the randomnumber generator.The accuracy is related to the error of the overall mean of the priceof the financial instrument from all the simulations to the true value(which, in some cases, can be obtained from an analytical solution). Theprecision is related to the standard deviation of the mean of the price ofthe financial instrument from each simulation.The accuracy and precision improvement (or the reduction in theerror or the standard deviation) by more than a factor of 10 can be easilyachieved by using simple resampling techniques.
4.8
Examples of Numerical Results
The following figure shows some numerical results of resampling. Thereader can easily experiment with simple resampling methodologies in
211
Sec. 4.8 Examples of Numerical Results
Excel spread sheets and see the accuracy and precision improvements asa result of resampling.Number of Paths: 500020L
0
W0
5
15
E
g 10
-
20
9
5
Km
0
-3
-2
-1
0
1
2
3
No. of STD Away From Money
Figure 4.1. The improvement of (unconditional) quadratic resampling on theBrownian Motion and first moment martingale resampling on the state variableshown as the ratio of percentage errors in plain vanilla European option prices(with various strikes) with and without resampling. Kurtosis resampling can further reduce the errors in the option prices and conditional quadratic resamplingcan almost eliminate the errors in the option prices.
Chapter 5
Introduction toInterest RateTerm StructureModeling
In this chapter, we shall discuss the basics of the interest rate modelingand introduce some simple classical interest rate models. In the next twochapters, we shall introduce more advanced full yield curve interest ratemodels, including stochastic volatility models.
5.1
Interest Rate Models Classification
Interest rates (or yields) usually exhibit significant term structures. Inother words, interest rates at any given time can vary significantly depending on the time to maturity (term). For this reason, interest ratemodeling is often referred to as (interest rate) term structure modeling.For a normal yield curve, the yield is roughly an increasing function ofthe term. For an “inverted” yield curve, the yield is roughly a decreasingfunction of the term. There are economic reasons for both cases.Interest rate models are usually classified into the following interrelated categories based on, e.g., quantities or state variables being modeled (such as short rate or full yield curve), the processes used in themodeling (such as normal, lognormal, CEV processes, Markovian ornon-Markovian processes, and the number of factors), and how themodel parameters are specified (such stationary and no-stationary parameters), These classifications can have significant practical implications in terms of, e.g., model implied volatility skews and smiles, correla-
Set. 5.2
Short Rate Models
213
tion, forward volatilities, and forward volatility skews and smiles, andforward correlation.In the next few sections, we shall present some simple examples ofthese classifications.
Short Rate Models
5.2
The 1-factor short rate models typically relies on the following 0 - U(Ornstein-Uhlenbeck) processes,
=(Of-b,X,)dt+o,dW,
(5.1)
Y
where X, is a scalar, W, is a one dimensional Q -Brownian motion, 0, ,bt , crt are deterministic model parameters (i.e., not state dependent andnot stochastic).Furthermore, the short rate yt is a fbnction of X,, i.e.,
=f( X t ) ,
(5.2)and the (default-free) zero-coupon bond price P ( t , T ) (with P ( T , T ) = 1 )is given by the following martingale relationship,yt
[ (- f
P( f, T ) = EF exp
(5.3)rsds)]
The above 0-U process can be solved easily. First, let(5.4)
then,
dx,= -b,X,dt + exp
(5.5)
Thus,
dY, = 0,dt + o,dW,,
(5.6)
or,
=Yo + [exp( [b,du)B,ds+ Cexp( Ib,du)o,dW,Therefore,
.
(5.7)
214
Introduction to Interest Rate Term Structure Modeling
Chapter 5
Different 1-factor short rate models are obtained by specifying different functions in Eq. (5.2) and we shall discuss a few examples in thenext few sections.Multi-factor short rate models can be obtained by allowing X I to bea vector and/or by allowing some of the model parameters ( 0, , bf , of)to be stochastic themselves.5.2.1
Gaussian Short Rate Models
For a 1-factor Gaussian short rate model or the simplest interest ratemodel, we haveq =XI.(5.9)This includes Vasicek and Hull-White models and is also equivalentto Ho-Lee model. In this case, it can be shown that the (default-free)zero-coupon bond price P ( t , T ) given by Eq. (5.3) can be reduced to thefollowing closed-form solution, i.e.,(5.10)
(0 I t I T )where the instantaneous forward rate is defined asd In P(t,T )(5.1 1)f ( t , T ) f(01 t .5 T )dTand P ( t , T ) is a deterministic function of 0, , bl , ol , t , and T.Having a closed-form solution for the zero-coupon bond price (andpossibly other instruments) is a major advantage of the Gaussian shortrate models in terms of computation speed and numerical stability. Thedetailed derivation is left as an exercise to the reader.Having a closed-form solution for the zero-coupon bond price alsomakes a Gaussian short rate model equivalent to a Gaussian full yieldcurve model.
Sec. 5.3
5.2.2
Affine Models and Quadratic Models
215
Lognormal Short Rate Models
The Black-Derman-Toy (BDT) model models the short rate r, in the following form(5.12)r, = UI e x p [ q K ] ,where ut and o, are time dependent, but not state dependent and not stochastic.By differentiating the above equation, it is easy to show that the BDTmodel can be expressed in the following SDE form,(5.13)The Black-Karakinski (BK) model has a more general form, i.e.,r, =lnx,,(5.14)in other words,(5.15)dlnr, = ( Q t - b l h r , ) d t + o l d y .Both BDT and BK models can be implemented by recombining trinomial trees to be presented in the next chapter.
5.2.3
Constant Elasticity of Variance Models
For a CEV (Constant Elasticity of Variance) model, we have1
r, = ( ( 1 - p ) x r ) i 7 ,
(5.16)
where p is a constant and the CEV power and satisfies 0 I p < 1 ,Since, in this case,dq = (...)dt + oIqpdW,,
(5.17)it can model the volatility skew (but not the volatility smile) in the interest rate.It is similar to and more general than the CIR (Cox-Ingersoll-Ross)model.
5.3
Affine Models and Quadratic Models
Duffe and Kan (1996) pioneered the affine model. Quadratic models areextensions of the affine model. More details can be found in Leippoldand Wu (2003).Basically, affine models and quadratic models are simple and re-
216
Introduction to Interest Rate Term Structure Modeling
Chapter 5
duced full yield models. More specifically, in an n-factor quadraticmodel, the SDE of the instantaneous forward rate is givenbyf ( t , T ) = f ( 0 , T )+ X I T A T X +, BTTX,(5.18)(OItIT)
2
where X,is an n-dimensional 0 - U process and an n x 1 vector, A, is ann x n matrix, and B, is an n x l vector. Both AT and BT are deterministic and only depend on the term T.If AT = 0 , then the above equation reduces to the affine model.
5.4
What Interest Rate Models Should One Use?
We have discussed several interest rate models in this chapter and shalldiscuss a few more in the next two chapters. A valid question is whatinterest rate models should one use or choose?In general, what model to use or choose depends on the trade beingpriced. Some trades, such as CMS spread trades, are highly sensitive tothe correlation, thus, one needs to use at least a 2-factor model. Sometrades, such as cancelable swaps, callable capped floater, callable inversefloaters, are more sensitive to the volatility skew and smile, thus, onemay need to use a CEV model. In order to price a cancelable swap moreaccurately, one often needs to calibrate the model to underlying European swaptions at the same strike as that of the cancelable swap.Digital options (such as those embedded in range accrual derivatives)are approximately call spreads and thus are sensitive to the slope of thevolatility skew and smile, which is more challenging to model accurately.Often times, one can use the SABR model to fit liquid volatilityskew and smile points and imply the entire volatility cube with the dimensions of time-to-expiry, term, and strike.For more complicated trades, such as callable, knock-out, or TARPRDC that requires a 3-factor model, one may consider using Gaussianmodels for some of the factors for better computation speed and betternumerical stability.Another interesting question is that since trades in the same portfoliocan be priced with very different models, how can one insure the consistency? The key to the consistency is that the model for each trade pricesaccurately to the market all (or most) of the hedge or replication instrument of that particular trade.
Sec. 5.4
What Interest Rate Models Should One Use?
217
As we pointed out previously, what model to use depends on the purpose that the model is being used for. Typically, models for EOD (EndOf Day) booking and riskhedging computation need to be parsimonious,and, thus, short rate models, or Gaussian full yield curve models, orMarkovian fill yield curve models (to be discussed in the next chapter)may be better choices. For structuring and for providing control variateto the parsimonious models, more complicated models with more factors,better volatility skew and smile handling, and, possibly, with stochasticvolatilities or jumps may be needed. Multi-factor BGM models withCEV or stochastic volatilities (to be discussed later on) are suchexamples. Various martingale resampling techniques are also needed toensure the accuracy and precision.As we discussed before, in addition to the primary model calibration,one also needs the secondary model calibration and the hedging calibration.
Chapter 6
6.1
The HeathJarrow-MortonFramework
The Heath-Jarrow-Morton Model
Heath, Jarrow, and Morton (Heath, Jarrow, and Morton, 1992) developeda general framework for multi-factor no-arbitrage fill yield curvemodels7*,which includes many other no-arbitrage interest rate models asspecial cases.In the Heath-Jarrow-Morton (HJM) framework, the full yield curveis represented by the continuous compounding instantaneous forwardrate, At,r), the rate as of time t for the accrual period starting from T( T 2 t ) and maturing instantaneously. In other words,At,T) is defined as(6.1)or
f ( t , T )= -dln P(t,T)/dT ,
(6.2)
where P(t,T) is the time t price of a T maturity of a (default-free) zerocoupon bond with P(T,T) = 1.Furthermore, At,T) is modeled by the following n-factor process for
78
The original Heath-Jarrow-Morton (HJM) model is for modeling default-free interestrate term structures, but it can be extended to the modeling of defaultable interest rateterm structures.
Sec. 6.1
219
The Heath-Jarrow-MortonModel
any given constant T (T 2 t )
’’n
df(t,T)=p(t,T)dt+Co;.(t,T)dW,(t)y
i=l
where p(t,T) is the drift and c$(t,T) are the volatilities of the forward rate,are the independent one-dimensional Brownian motions underand Wi(t)the risk neutral measure or the spot arbitrage-free measure Q .Heath, Jarrow, and Morton identified the necessary and sufficientconditions for the absence of arbitrage for the HJM model. An importantnecessary condition for no-arbitrage is that, under the spot arbitrage-freemeasure Q , ~ ’
Using vector notation, we can re-write the above two equations in asimpler form
where of ( t , T ) is an n-dimensional vector (a 1 by n vector) of o)( t , u ) ,W, an n-dimensional vector (an n by lvector) of Wi(t), and o > ( f , u ) isthe transpose of of( t , ~. )Based on the martingale modeling presented in Chapter 2, essentially, the HJM model can be derived from the stochastic process of thezero-coupon bond price P(t,T). Recall that P ( t , T ) / P ( t ) (for a constantT ( T 2 t ) ) is a Q -martingale, and if is continuous and square-integrable,then79 In general, the drift term and the volatility term can depend on the present and past
rates and bond prices. For the simplicity of the notion, no extra parameters are introduced for these dependencies. Rather, these dependencies are dealt with implicitly. Wealso make the usual mathematical assumptions.8o This condition plus other technical conditions form the necessary and sufficient conditions for no-arbitrage. For details, see Heath, Jarrow, and Morton (1992).
220
The Heath-Jarrow-Morton Framework
d P ( t , T ) / P ( t , T ) = r ( t ) d t + of ( t , T ) d W , ,
Chapter 6
(6.6)
where o,(t,T)is an n-dimensional vector (a 1 by n vector) of the bondprice volatilities, r(t) =f(t,t) is the short rate, and W, an n-dimensional Q-Brownian motion (an n by Ivector).Using It6's lemma, we can re-write the above equation asd l n P( t , T ) = ( r ( t )-;lof (t,T)I2)dt+ op( t , T ) d y ,
(6.7)
where 1.1 is the standard norm of a vector. Thus
- - - L ( - - - I1o p ( t , T ) r d tdT
+op(t,T)dW,
2
Consequently,
(6.9)
d o , ( t ,T )of(t , T ) = aT
uIT ) , as P(u,u) = 1, i.e., the bond priceNote that op( u , u ) = 0 ( t Iat the maturity u is deterministically one. Soo p
(t ,T ) = -
T
of(t ,u )
,
(6.10)
and one can easily verify the no-arbitrage conditionP ( t , T ) =of( I J ) fo;
(t,u)du *
(6.11)
Consequently,
d f ( t , T ) = -0-p (t , T ) o > ( t , T ) d t +
(t,T)dW,.
(6.12)
Using the It6's integral, we can re-write the forward rate process in a
Sec. 6.1
The Heafh-Jarrow-MorfonModel
221
slightly different form
Generally, one can obtain different models under the HJM framework by specifying different yield or price volatility structures, i.e