
Sec. 4.1
Martingale Interpolation
161
= M(s,T3)
Particularly, we can apply this technique to zero-coupon bond priceP( t , T ) . Recall that the ratio of the zero-coupon bond price P ( t ,T ) andthe money market account price p(t) is a martingale under the spot arbitrage-free measure Q , i.e.,
Therefore, the following interpolation,
(4.8)
always ensures the desired martingale relationship, i.e.,
This means that as long as we can model a set of zero-coupon bondprices (or forward rates) with discrete tenors in an arbitrage-free model,we can obtain zero-coupon bond prices (or forward rates) with continuous tenors in the same arbitrage-free model using this martingale interpolation technique.
162
Chapter 4
Martingale Resampling and Interpolation
It is interesting to point out that Eq. (4.8) does not explicitly dependon the numeraire in the martingale relationship. Thus, it is also valid forthe following martingale relationship under the terminal forward measurePT* with P ( t , T * ) as the numeraire, i.e.,
Moment matching and martingale resampling techniques that will bediscussed in the following sections should be used before the martingaleinterpolation is used. Martingale resampling techniques are still neededafter the martingale interpolation for the quantities being interpolated forachieving arbitrage-free at the second (or higher) order by matching thesecond (or higher) order (conditional) moments or option prices.Another type of interpolation (that is more subtle) is to interpolatealong time (as opposed to along term T presented above). More specifically, the question is that given martingale quantities M (tl ,T ) and
M ( t 2 , T ) satisfying Eq. (4.3), how do we find the martingale quantity
M ( t , T ) also satisfying Eq. (4.3)? An example of the need of this typeof interpolation is the situation when the look-up (or reset) interval is lessthan the time step in the numerical procedure. For instance, some of thefixed-income derivatives, such interest rate or credit range accruals orcallable range accruals, require daily look-up, and the numerical procedure normally cannot afford daily steps due to speed reasons.It turns out that, for this case, we cannot use the initial value of themartingale quantity, i.e., M ( s , T ), as the independent variable in theinterpolation. Instead, we can use time t as the independent variable inthe interpolation. Let’s consider the following interpolation,
M(t,T)=M(t,,T)+k(t,,t,,t)(M(t,,T)-M(t,,T))(slt,I t l t , )
.
(4.11)
It is easy to verify that for any deterministic variable k ( t , , t , , t )(conditioned on the information available up to time s), M ( t , T ) alwayssatisfies the martingale relationship or Eq. (4.3), i.e.
Sec. 4.1
163
Martingale Interpolation
=M(s,T)
(SIt, s t s t , )
Thus, we can use k ( t , , t 2 ,t ) to satisfy other requirements, such as therequirements on the second or higher moments. If we assume that theincremental variance is a linear fimction of time, i.e.,vur, ( M ( t , T )- M ( t ,, T ) ) =t-t- - - u k r , (M ( t 2,T) - M ( t , ,T)) .t2
(4.13)
-4
( s q I t < t , , t , a,)
ThenI
( s I t , Itlt2,t, <t2)
Also, if
corr, ( M ( t ,, T ) ,M ( t,, T ) - M(t1,T ) )= 0( s a , I t , , t , a,)
(4.15)
Then,corr, ( M ( t l ,r ),M ( t , T ) - M ( t l ,T ) ) = 0( s < t , < t < t , , t , a,)
Y
(4.16)
1
(4.17)
which is desirable. However,c o r r s ( M ( t , r ) , M ( 1 2 , T ) - M ( t , T#) )0( s q I t l t , , t , a,)
which is not desirable.
Martingale Resampling and Interpolation
164
Chapter 4
To address this issue, we can use a Brownian bridge in the interpolation.
4.2
Brownian Bridge Interpolation
To avoid being overly complicated, here we focus on a 1-factor modelwith a constant effective volatility. In other words, we assume thatM ( t , T ) = f,( t , w ( t ) )
(s 5 t ) ,
(4.18)
or, more specifically,
M (4 T ) = f,(4 w ( 4 ))3
9
M ( t 2 , r )= f,( t , Fv ( t 2 ) ) .(s 5 t, < t,)9
(4.19)
The Brownian bridge problem is for given Brownian motions w(t,)and W (t, ) to find another Brownian motion p(t ) to bridge W (tl ) and
~ ( t , ) i.e.,,W ( t 1 ) = w(t1)
* ( t 2 ) = w(t,) *
(4.20)
(s I t , < t 2 )
We also require the usual properties of Brownian motion to hold, i.e.,Corr, (w(t, ), W ( t )-
w ( t ,)) = 0
Corr, (W ( t ) ,w ( t , ) -
w (t)) 0=
Corr, ( W ( 1 ) - w(t 1 ) , w ( t 2 )W (t ) )= 0
vur, (W ( t )- w (tl )) = t - t]vur, (w(t, ) - W ( t ) )= t, - t(s I t , 5 t 5 t,,t] < t 2 )Then, the interpolation is achieved through
(4.21)
Sec. 4.2
165
Brownian Bridge Interpolation
M ( t , T ) = f ,(t,@(t))
(4.22)
( S I t , I t I t , , t l <t,)This is also one way to perform Brownian bridge simulation.Using any Brownian motion W ' (t ) or incremental Brownian motion
AWf(t,,t2)=W'(t,)-W'(tl) (tl I t , ) , we can form the followingBrownian bridget - t,
BB ( t , t l,t2) = W ' (t ) - W'( tl ) - -(
W'( t 2 ) - W'(tl ))
t 2 -t1
,
t, -t ~ W ' ( t , , t ) - 3 A W ' ( t , t , )-t, - 4(tl ItIt,,t, < t,)
(4.23)
t2 - 4
which goes through zero at both time t, and t, .In order for @ (f ) to go through W (t,) and W (t, ) ,we let
@ ( t ) = W (t, ) + 2 A W (tl ,t, ) + BB ( t ,tl ,t 2 )t 2 - tl
9
(4.24)
(tl I t _ < t , , t , a,)where A W ( f , , t , ) = W ( t , ) - W ( t , ) (tl I t , ) ,
We hrther require that
A W ' ( t l , t ) and A W ' ( t , t , ) are independent o f W ( t l ) and W ( t , ) , particularly in numerical procedures.It is easy to show that the Brownian bridge is a normal random variable and it is easy to compute its mean and variance, i.e.,
166
Martingale Resampling and Interpolation
Chapter 4
= t - t , --( f - 4)’t2 -4(t, I t I t2,t1< t 2 )and it is easy to see that Eq. (4.20) and (4.21) are both satisfied.Thus, to perform a Brownian bridge simulation, we just need tosimulate a normal random variable with the correct mean and variance.Intuitively, the current Brownian bridge essentially inserts one time pointbetween t, and t, . We can also inert more time points if needed.Similarly, conditioned on @ ( t ) going through W ( t l) and W(t,) ,
(
Vuq1 @ ( t ) I P ( t , )= W(tl),@(tZ)= W ( t 2 ) )= V q l (BB(t,t,,t,))
(t, I t Itz,t, < t 2 )and the corresponding average variance is give by
>
(4.26)
167
Sec. 4.3 Moment Matching in One-factor Case
(4.27)
(tl I t l t 2 , t l a,)
As an example, the above martingale interpolation or simulation canbe applied to forward LIBORs under their prospective martingale measures (i.e., forward measures up to the LIBOR pay dates). In the case ofthe BGWJ LIBOR market model, for instance, the forward LIBORLi ( t ) is a Pi+]-martingale. Thus the following martingale relationshipcan be used for the martingale interpolation, i.e.,
L, (s) = E F [L, ( t ) ]
( s I t I T ).
(4.28)
The Brownian bridge can be applied to numerical algorithms withlong jumps (or large time step sizes). Sometimes, we do not need to perform additional numerical modeling for the Brownian bridge, rather, wecan find and rely on analytical solutions or approximations.The covariance resampling that we shall discuss later on can be applied to match the required incremental covariance or correlation.
4.3
Moment Matching in One-factor Case
Moment matching is best applied at the driving factor level, such as incremental Brownian motions, where the target values for all the momentsare known. Its variation can also be applied to state variables (such asstock prices, Ex rates, or interest rates) and other quantities with the target values provided by martingale relationships that will be detailed inSection 4.5.
168
4.3.7
Martingale Resampling and Interpolation
Chapter 4
Quadratic Resampling
Quadratic resampling) or moment matching for the first two moments,i.e., mean and variance or standard deviation, can be easily achievedthrough a simple linear or approximately linear transformation.Suppose that Xo is a set (or a vector) of samples of an incrementaldriving factor (e.g., an incremental Brownian motion) based on a numerical procedure, e.g., a tree or a Monte Carlo simulation, We can easily compute its mean and standard deviation as follows.
Mean( X o )= E [X,],
where E [...] is the expectation based on the numerical procedure.These moments normally would differ from their target or true values, that are known and denoted as E [ X ] and St d(X) . Our task now isto find a set of samples X (as an approximately linear function of Xo)thatmatches the target moments. This can be easily achieved by the following approximately linear transformation.(4.30)
4.3.2
Moment Matching for All Odd Moments and Kurtosis
Matching all odd moment can be easily achieved by antithetic sampling,that, for each sample x, also includes -x with the same probability. This,of course, requires that the distribution being modeled is symmetric andcentered at zero (that is satisfied by the distribution of an incrementalBrownian motion).For matching the kurtosis (or kurtosis resampling), we actually neednon-linear transformations, but only in the tail region that we can control.For simplicity, we assume that Xo is a set of antithetic samples that hasbeen quadratically resampled. Then, the kurtosis resampled X for matching a given kurtosis E X4] can be obtained through the following equa-
[
tions, i.e.,
169
Sec. 4.3 Moment Matching in One-factorCase
(1x01
K)
where k is a positive constant and P(IXoI 2 K) = E
[
XoI241
. We also
require that X and XOare of the same sign. The non-linear transformationonly comes in the tail region roughly when (IXol 2 K ) , where K is apositive constant. The subtlety is that if K is too large r too small, theremay not be a solution or a stable solution. One criterion for a good solution is k being close to Std(Xo) .It is easy to verify that X and Xo have identical moments with ordersof less than 4. Particularly, since
thus,
(4.33)=E[X:]Equation (4.31) can be easily solved by first evaluating the fourthpower of X, i.e.,
170
Martingale Resampling and Interpolation
Chapter 4
2
And, then, its (conditional) kurtosis is given by
The target of this (conditional) kurtosis is given by
In arriving at the above equation, we have used the following relationship, i.e.,
Thus, the solution fork is given by
Sec. 4.3 Moment Matching in One-factor Case
171
If X is a standard normal distribution (or N(0,l) with mean of zeroand standard deviation of one), then (with detailed discussions presentedin the next section)
E[X4] = 3 .
(4.39)
Therefore, we can obtain the following resampling transformation byrewriting Eq. (4.31), i.e.,
x = xo
(1x01< K )
x =Sign(Xo)x
where
I
1 ( X o 4Sign(X0)= 0 (xo=o).-1 (XO<O)
(4.41)
An example of such moment-matched samples is the shocks of a trinomial tree, i.e., (-&,0, &) with probabilities of (1/6,2/3,1/6) , thatmatch all odd moments and up to 4* order even moments for a N(0,l)distribution.
Martingale Resampling and Interpolation
172
4.3.3
Chapter 4
Moment Matching for Higher Order Moments
So far we have been focusing on changing or resampling the values ofthe sample points. We can also change or resample the probabilities ofthe sample points and leave the values of the sample points unchanged.With this approach, with a sample of n+l distinct points (or simulation paths), we can easily match up to the nth moments. Unlike the previous sections, this approach can also match conditional moments, provided the target values of these moments are known.This is a problem of solving the following n+l simultaneous equations with n+l degrees for freedom in the probabilities,n+l
b
C p i x ; =mi= J’p(x)xjcixi=l
( j = O , l ...n ) ,
(4.42)
a
where xi’s are the coordinates of n+l distinct points with correspondingprobabilities of p i , and mj are the conditional moments computed fromthe true probability distributionp(x).The above equation can be easily solved. The key is to use the Lagrange representation of functions. More specifically, a function g(x)can be represented by the Lagrange functions 1: (x) as follows.
(4.43)j.#i
Obviously, :Z (x) is an nth order polynomial with the following properties.(4.44)
Thus,g (Xi ) = g, (Xi )
(i = 1,2...n + 1) .
(4.45)
If g(x) is a polynomial of the order of n or less, then g(x) and gdx) areidentical, as a polynomial of the order of n can have at most n distinct
173
Sec. 4.3 Moment Matching in One-jactor Case
roots. Otherwise, gkx) is one way for curve fit or interpolation of g(x).Now, let g (x)= x J , thenb
n+l
b
I p ( x ) x j d x = C X i j I p ( x)zi" ( x ) d xl=l
a
( j = O,l...n) .
(4.46)
a
Comparing the above equation with Eq. (4.42), we can obtainb
pi = I p ( x ) z ; ( x ) d x
(i=1,2...n + l ) .
(4.47)
a
One subtlety is that the pi's determined by the above equation maynot be in the interval of [0, 11. In this case, we will need to make the xi'sspread out more, possibly by using the quadratic and kurtosis resamplingfirst.If we also resample the xi's in addition to the pi's, then we can matchthe moments up to the order of 2n+l.Another way of handling the higher order moments is, after matchingthe first two moments, through matching of the moment generation function, such as the following, at various discrete points,g(il)=~[exp(ilx)]
(ilE
c').
(4.48)
It is easy to verify that the j" moment mi is given by
( j =O,1,2...)
It is easy to prove that if x has a normal distribution, then(4.50)If x is a standard normal distribution (or N (0,l) with mean of zeroand standard deviation of one), then
174
Martingale Resampling and Interpolation
Chapter 4
m2 j + ~= 0
(2j)!m2j =2’j!
(j=0,1,2 ...).
(4.5 1)
Particularly,m4 = 3
m6 =15 .ma = l o 5
(4.52)
With all these moments matching, the sample becomes a momentmatched series that is highly deterministic and correlated, and is morelike a multinomial tree or stratified sampling rather than the classicalMonte Carlo simulations. This is not a major issue for one-dimensionalsimulations. However, Latin hypercube methodology or the like needs tobe used for using these moment-matched samples or stratified samples inmulti-dimensional simulations.For valuations, such as VAR, that heavily depend on the trail distributions, one may need to resort to the extreme value theory (EVT), inaddition to the moment matching.4.3.4
Conditional Quadratic Resampling
Quadratic resampling) discussed previously can perform moment matching for the first two unconditional moments, i.e., unconditional mean andvariance or standard deviation. The conditional quadratic resampling, onthe other hand, can perform moment matching for the first two conditional moments. This is helpful in pricing options, as option prices aremore closely related to the conditional moments than to the unconditional moments.Define the iIh order conditional moment m( i, X,,K, ,K2) as
where p ( X) is the true probability density of the sample X. Equivalently,
Sec. 4.3 Moment Matching in One-factor Case
175
(4.54)
( i = 0 , 1 , 2...; Kl < K 2 )
If X is a normal distribution or an incremental Brownian motion,then all the above conditional moments can be obtained analytically ornumerically with high accuracy.The conditional quadratic resampling can be achieved through thefollowing transformation,
It is easy to verify that
[
E[ X’-
P(Kl <X <K2)
(4.56)
However, these are not exactly what we want for the conditional firstand second moment matching due to a subtlety that the condition is provided by Kl < X < K 2 and not by Kl I X ’ < K 2 . So, we need to requirethe following additional condition1{K11X’<K2} =1{K1<X<K2} ’
(4.57)
which may be automatically satisfied in some cases in discrete numerical
176
Martingale Resampling and Interpolation
Chapter 4
implementations,especially with large number of samples.If the above additional condition is not satisfied, then we may needto perform the conditional quadratic resampling in Eq. (4.55) in severaliterations andor make K, and K , more separated from each other. Theiterations can be performed by using Eq. (4.55) or, better yet, the following equations,
(4.59)
It is easy to verify that when the convergence is achieved, then
177
Sec. 4.3 Moment Matching in One-factor Case
It is also easy to verify that if all these conditional first and secondmoments are matched, then the unconditional first and second momentsare also matched. If matching only the first conditional moments arerequired, then we can set k = 1 in Eq. (4.55). This will preserve thehigher order conditional central moments.The conditional quadratic resampling technique is similar to thestratified sampling. Since the stratified sampling does not automaticallymatch the conditional moments, it is natural to combine the two techniques to achieve significant improvement in terms of accuracy and precision. In some extreme cases, the conditional quadratic resamplingtechnique converges to the stratified sampling technique. In general, theconditional quadratic resampling technique allows us to more or lesscontrol the randomness of the samples by controlling the distance between K, and K,.If X is N(0,l) (i.e., a standard normal distribution with 0 mean and 1standard deviation), thenm (0,x, q ,K , ) = N ( K , + x, ) - N ( K, +7
exp
m(l,O,K,,K,)=
(- 4K;) - exp (-4K , ~ )&
K, exp(
-4K,, ) - K , exp(-+ K: )
m(2,O,K,,K,) =
+ m (090,K , ,K , )In general,
x,)
6
(4.61)*
178
Martingale Resampling and Interpolation
Chapter 4
m (1, xo Kl K 2 ) = m (LO,Kl, 4 )- x o m (030, Kl ,K 2 )3
9
m (29x0, K1 , K 2 ) = m( 2,0,K1 , K 2 ) - 2Xom(1,O,K1, K 2 )
(4.62)
+x$m( o,o, K, ,K 2 )It is important to point out that even with one factor, one often needsto model many incremental Brownian motions at various time steps.Theoretically, these incremental Brownian motions should be independent from each other and thus have zero correlation. Looking along asimulation path, a Brownian motion should also have zero autocorrelation. These are important properties to be satisfied in order for usto perform simulations properly. But, practically, these properties wouldnot be achieved automatically in simulations, and this is where the covariance resampling comes in, which shall be discussed in the next section.
4.4
Moment Matching in Multi-factor Case
Moment matching in the multi-factor case involves matching variouscross moments. For an n-dimensional incremental Brownian motion,A W ( t , A t ) = W ( t+ A t ) - W ( t )
( A t 2 0) ,
(4.63)
its cross moments Mi.::( t ,A t ) are given by the following moment generation function ~ ( h , t , ~ t ) ,G(h,t,At)~E,[exp(h.AW(t,At))]= exp(
[
$'~l~ A t )
Ad;.;: ( t ,A t ) = Et IV; ( t ,A t ) W p ( 1 , A t ) ]
( hE R " )(4.64)
The typical approach is the quadratic resampling or, in this case, covariance resampling, for matching the covariance of the factors, commonly, with the variance part determined by the implied volatilities and
Sec. 4.4
Moment Matching in Multi-factor Case
179
the correlation part determined by the historical data. This can be easilyachieved through the Choleski decomposition.More specifically, suppose that we need to simulate n random variables or, equivalently, one n-dimensional random variable X (nxl vector)with a given mean of M (nxl vector) and a given positive definite covariance of C (nxn matrix). Our goal is to find a method of resampling Xwith m samples (with m 2 n ) such that the sample mean and sample covariance are exactly the same as the given target mean M and target covariance C.Let XI (11x1vector) denote a sampling of X and .[XI] denote thecorresponding sample mean. Then, the corresponding sample covarianceis given by(4.65)
Since if m 2 n ,then there exist sampling methods such that the sample covariance is positive definite. Thus, we assume that C o v ( X l ) ispositive definite.Through the Choleski decomposition, we have
c = CICIT,
(4.66)
cov (XI) = c2c 2 T ,
(4.67)
where CI and C2are invertible nxn matrices.Now, we can define X2 through the following linear transformation(or a resampling) of XI,i.e.,
X 2 =C,C,-'(X, - E [ X J ) + M .
(4.68)
Then, it is easy to verify that X2 has the desired mean and covariance,i.e.,(4.69)
C0v(X2)=c .
(4.70)
The above approach is the essence of the quadratic resampling in thecase of multi-factors, which is also termed as covariance resampling.
180
Martingale Resampling and Interpolation
Chapter 4
It is desirable to ensure that the incremental Brownian motions atdifferent and non-overlapping time steps are independent. The zero correlation can be achieved by including all the incremental Brownian motions in the same covariance resampling procedure. This also ensuresthat, looking along a simulation path, a Brownian motion has zero autocorrelation in the expected value sense. These are very important for usin order to perform simulations properly.A common numerical difficulty in accuracy and precision for MonteCarlo simulations is in solving higher dimensional problems, such as inpricing derivatives on a basket of underlying assets or underlying variables, e.g., best or worst options in equity derivatives, nth-to-defaultorloss baskets in credit derivatives. More specifically, the difficulty is thatthe same basket can have different prices and hedge ratios depending onthe order in which the underlying assets or underlying variables are modeled. Or a basket of identical underlying assets or underlying variablescan have different hedge ratios with respect to each of the underlyingassets or underlying variables.This difficulty can be eliminated or significantly reduced if we firstresample each of the driving factors up to the kurtosis and then performsthe covariance resampling of all the incremental driving factors. Resampling to higher order moments can further help in this regard. We canalso resample the Brownian motion on each of the simulation path, ifneeded.The Latin hypercube methodology is a usefbl technique to be combined with the resampling techniques in multi-dimensional simulations.
4.5
Martingale Resampling
The martingale resampling technique extends the moment matchingbased resampling technique and can be applied not only to the derivingfactors, but also to the state variables (such as stock prices, FX rates, orinterest rates), the underlying asset prices, and the derivatives instrumentprices with the resampling targets provided by the martingale relationships.The martingale resampling technique enforces the required martingale relationships in various numerical procedures. In other words, theresampling targets are provided by the martingale relationships.
Sec. 4.5
4.5.1
181
Martingale Resampling
UnconditionalMartingale Resampling at the State Variable Level
One way to test whether or not a state variable variables (such as a stockprice, an FX rate, or an interest rate) S ( t ) is modeled accurately is tofind its martingale measure and then test whether or not it satisfies themartingale relationship, i.e.,
(4.71)
(SIt).
Even if after the moment matching at the driving factor level as described in the previous sections, the martingale relationship at the statevariable level may not be satisfied with desired accuracy, particularly, inthe case of state dependent drift or lognormal or shifted lognormal models due to various numerical errors, such as the discretization error inItB’s Lemma.More generally, martingale resampling includes matching the forward values or prices or the initial forward yield curve (that we shallterm as the first moment or first order martingale resampling) and matching option prices (that we shall term as the higher moment or higher order martingale resampling). By unconditional martingale resampling atthe state variable level, we mean the resampling or transformation of theunconditional moments of the underlying state variables.If the state variable S ( t ) is a normal process, then we can apply additive resampling, i.e.,S‘(t)=S(t)-E,[S(t)]+S(s)
(SIt).
(4.72)
If the state variable S ( t ) is a lognormal process, then we can applymultiplicative resampling, i.e.,(4.73)
It is obvious that the required martingale relationship is satisfied, i.e.,S’(s)=E,[S’(t)]
(SIt).
(4.74)
The reason that we use additive resampling for normal state variablesand multiplicative resampling for lognormal state variables is to correctthe first moment while preserving the central distribution of the state
182
Martingale Resampling and Interpolation
Chapter 4
variables by preserving the higher order central moments of the normalstate variables and the logarithms of lognormal state variables. Anotherbenefit of these resampling choices is that they translate into simple linear transformations on the underlying Brownian motions (or other typesof underlying driving factors).More specifically, for normal state variables, the above martingaleresampling ensures:
Similarly, for lognormal state variables, the above martingale resampling ensures:
(4.76)
If the state variable S ( t ) is a shifted lognormal process, Le.,S( t ) + As ( t ) is a lognormal process (under its martingale measure) with
As ( t ) possibly being a stochastic variable, then we can apply the following martingale resampling
(SIt)
It is easy to verify that the required martingale relationship Eq. (4.74)is satisfied. Similarly, the higher order central moments are also preserved, i.e.,
and the resampling translates into simple linear transformations on theunderlying Brownian motions (or other types of underlying driving fac-
Sec. 4.5
183
Martingale Resampling
tors).If the state variable S ( t ) follows other processes, then we can applya linear resampling, i.e.,S'( t ) = U S ( t ) + b .
(4.79)
Recall that in the case of the BGM/J LIBOR market model, for inis thestance, the forward LIBOR Li ( t ) is a Pi+lmartingale, wheremeasure with respect to the numeraire P ( t , T c l ) . Thus, the followingmartingale relationship can be obtained,
Li(s) = Er' [Li( t ) ]
(s It I
11:).
(4.80)
In general, the numerical procedures may not be implemented underthe martingale measure of the state variable, e.g., the forward LIBORLi( t ) , but rather under the measure lPN with respect to a numeraireN ( t ) , which is the price of a traded asset with no intermediate cashflows. In this case, the martingale relationship involving the forwardLIBOR Li( t ) can be derived from the martingale relationship of theprice of the floating leg and the fixed leg of an FRA (Forward RateAgreement) contract under the measure PN, i.e.,
(4.81)
(s I t 511:)
Particularly, the following martingale relationship can be obtained,i.e.,
184
Martingale Resampling and Interpolation
Chapter 4
(s I t 5 q )
and
(4.83)(sIt5q)
Similar relationship can be derived for other state variables, such asswap rates. We shall discuss more details on