
where p ( t ) is the correlation between dFT,O and dW:*F or d V * D ,i.e.,P ( t )=
ETSD[d&TT’DdyTT’F]- ET9D[ d y T T . D d q T * D ]dtdt.
(3.81)
( 0I t I T )
Plugging Eq. (3.80) into Eq. (3.79), we finally obtain
Thus, the SDE for the state variable F F ( t , T ) under the desired
142
Chapter 3
The Black-Scholes Framework and Extensions
probability measure of PT,D (or Eq. (3.72)) becomesd~ ( t ,T ) = -a(t,F ~oFFX) ( t ) p ( t ) d t+ a(t , F F)dW;T.D
.
(3.83)
( 0 1t I T )~fF ( t , ~ also) follows a lognormal process, or
(O<t<T)
then,
Thus, we can evaluate the forward value in Eq. (3.68) as
F F (s,T) = ET,D[F ( t ,T ) ]
I
F F (s,T)exp -JOF ( u ) g F F X( t ) p ( u ) d u .
[:
( 0 5s I t I T )
(3.86)
Therefore, for simple Quanto derivatives instruments such as European calls and puts, we can use the Black-Scholes formula with theabove modified forward value or convexity adjustment.
3.2
Black's Model
The Blacks model is an extension of the Black-Scholes model and canbe applied to, e.g., price options on forward contracts, futures options(options on futures), and some interest rate derivatives, such ascapdfloors and swaptions. In the Blacks model, the underlying assets
Sec. 3.3
Put-Call Parity Revised
143
(e.g., the futures) and variables (interest rates with different terms) areassumed to be lognormal with zero drift.Thus, the Black’s formula is the same as the Black-Scholes formulaexcept with the appropriate forward values. The Black formula is alsoused in the quadratic approximation whereby the price of a Europeanderivatives instrument is approximated by the first two moments, mean(or forward) and variance (or volatility), of its underlying asset or variable.The Black’s model is also a special case of the martingale models.
3.3
Put-Call Parity Revised
This is the case whereby different classes of assets or instruments need tobe funded with different numeraires, as discussed in Section 2.5.10.Recall that the price of an underlying asset U ( t ) and the price of aderivatives instrument V ( t ) follow the martingale relationships withdifferent numeraires ( N u and Nv ) and under different probabilitymeasures. In other words,(3.87)and(3.88)
Furthermore, let the derivatives instrument be plain vanilla put orcall options with expiry T and strike K and the time t values of the putand call options (P(t,T ,K ) and C ( t ,T ,K ) ,respectively) given by
144
The Black-Scholes Framework and Extensions
P ( t , T , K ) = N , (t)E;"Y
C ( t , T , K )= N , ( t ) E p
Chapter 3
max(K -U(T),O)
1
.
(3.89)
[ug{i;].
(3.90)
max(U(T)-K,O)
(OItIT)
Thus, the general put-call parity is given byC(t,T,K)-P(t,T,K)=N, (t)E;"Y(OItIT)
If the nwneraires are the same, i.e., Nu = Nv = N , then it is actuallythe case of the single numeraire. Thus, the above equation can be reduced to the simple and familiar put-call parity as
(0 I t I T )
where PN ( t ,T ) is the zero-coupon bond price corresponding to the numeraire N, i.e.,( 0 5t I T ) ,
(3.92)
is the forward price of the underlying asset. We have alsoPN ( t J )used the martingale relationshipand
331
u(t)= N ( t ) E Y [
(3.93)
(OItI T )
The above simple put-call parity needs to be revised in the general
Sec. 3.3
145
Put-Call Parity Revised
case whereby the numeraires are not the same. Similar to what we discussed in Section 2.5.10, in order to achieve consistent martingale modeling, we need to introduce a funding spread multiplier M F ( s , t ) suchthat the multiplier M ( s , t ) for the change of probability measure is givenby6'
(3.94)M , (SJ) = 1(0 I s I t )
In a way, this is similar to the modeling of FX derivatives.Thus, we have the following modified martingale relationship,
(3.95)
(Olslt)
Similarly,
Here we have implicitly assumed that the default events are not significant and theeffect of default, if any, is in the expected value sense. More details on the change ofprobability measure in the presence of default events can be found in Section 8.7.
146
The Black-Scholes Framework and Extensions
Chapter 3
(3.96)( 0 Is I t )
Therefore, the general or revised put-call parity in Eq. (3.90) can berewritten as
c(t , T , K ) - P( t ,T ,K )
( 0 It I T )which can be further reduced to
c(t ,T , K ) - p ( t,T , K )
( 0It IT )
where PNu( t ,T ) and PNv( t ,7') are the zero-coupon bond prices corresponding to the numeraires NU and N , ,respectively, i.e.,
Sec. 3.4
147
Replication Model
(3.99)
As a special case, if
(3.100)and M F ( f , T ) has zero volatility, then Eq. (3.98) can be rewritten as thefollowing familiar form, i.e.,
(0 It 5T )where
Nu (l)is the forward price of the underlying asset.'Nu
3.4
(t~T)
Replication Model
With the simple buyhell and hold SFTS used in Section 2.5.2, we caneasily replicate the prices of some derivatives instruments and derivevarious arbitrage relationships without using any specific models. Forinstance, we can derive the forward price of an instrument, interest rateparity, put-call parity, and digital and barrier European options (as a portfolio of plain vanilla European options).Here we present a general replication methodology by replicating theterminal payoffs of a complicated derivatives instrument by a series ofplain vanilla European options. This replication model has the advantageof almost exactly matching the volatility skew and smile and has becomea standard model for pricing CMS caps/floors. It can also be used forpricing CMDS (Constant Maturity Default Swap) caps/floors.This replication model can also be extended to price American andBermudan style derivatives.
148
The Black-Scholes Framework and Extensions
Chapter 3
The replication model can be easily derived for the case of a singlestate variable, but can also be extended to the case of multiple state variables.Here we present a discussion of the simple case with a single statevariable. Let S, denote the time t value of the state variable that is amartingale under the probability measure P,,, with respect to numerairc
N ( t ) and V ( t ) denote the time t value of a European style derivativesinstrument with V ( t ) / N ( t ) being a P,,,-martingale. Furthermore, weassume that V ( t ) and N ( t ) are deterministic hnctions of S,, i.e.,V ( t )= V ( t , S , )
(3.102)
N ( t )= N ( t , S , ) ’
Since, at the expiration time T, the terminal value or payoffV(T,S,) of the derivatives instrument is known (as a deterministicfunction of S, ), then, as usual, its time t value V (t ) is given by
V ( t >= N ( t ) E r
[a]
(0 I t I T ) .
(3.103)
Our goal is to solve the above equation using replication when it cannot be readily reduced to the Black or Back-Scholes formula.More specifically, let’s assume that the price Ci( t ) of the plain vanilla European call option (with expiry T and strike Ki)is available fromthe market (or the SABR model), which means the volatility skew andsmile or the Black volatility at expiry T and at different strikes are available from the market (or the SABR model). More specifically,
ci( t )= E;“ [(s,- KJ+]
(0 I t I T )
(3.104)
Now, let’s try to replicate the terminal payoff V ( T ) by Mplain vanilla European call options as follows,
See. 3.5
149
Impact of Volatility Skews and Smiles
(3.105)
KO<... < Kj-1 < Ki < ... < K M - 1Recognizing that
0Kj-Ki
(jli)(i<j)’
(3.106)
we can easily approximate the weight wi as follows
(3.107)
WM-1
=o
Thus, the time t value V ( t ) of the derivatives instrument is given byM -1
V ( t ) = N ( t ) Cw i C i ( t )
(OltlT).
(3.108)
i=O
The accuracy of the replication model depends on the number ofplain vanilla European call options used and the ranges and distributionsof the strikes Ki.
3.5
Impact of Volatility Skews and Smiles on Hedge Ratios andHedging Strategies
As we pointed out before, it is important for us to evaluate the modelrisks or the implied views in the model before we can trust the PV andhedge ratios that the model produces. Computing the hedge ratios ismore subtle than computing the PV. As such, a model simply suitablefor PV computation may not be suitable for the hedge ratio computation.The local volatility model is such an example, as we discussed before.
150
The Black-Scholes Framework and Extensions
Chapter 3
Here we provide some simple analysis on the impact of the volatilityskews and smiles on the hedge ratios, such as Delta due to Vega, andhedge strategies.As we discussed before, the volatility skew and smile refer to thedependency of the implied Black volatility on the strike and the value ofthe state variable. Eq. (3.47) gives an example of the volatility skewproduced by a CEV model. For instance, if the interest rate or the equityprice decreases, the associated volatility will increase. The situation ofthe FX rate is different in that if X is an exchange rate, 1/X is also an exchange rate with the same Black or percentage volatility, as we discussedin footnote 49 on page 57.The SABR stochastic volatility model (Hagan, Kumar, Lesniewski,and Woodward, 2002) provides examples of volatility smiles with nonmonotonic dependency of the implied Black volatility on the strike andthe value of the state variable in that if the interest rate or the equity priceincreases significantly, the associated volatility can also increase.An important factor for the impact of the volatility skew and smileon the hedging strategies is that a dealer typically cannot afford to hllyhedge Vega or maintain Vega neutrality. So, if one were to maintainDelta neutrality based on Delta computed from a model not consistentwith the volatility skew and smile, one actually could be quite away frombeing truly Delta neutral.As an example, if one is long in the interest rate market (or short inthe interest rate) and short in the (short-term) interest rate Vega, then oneis not as long in the market as predicated by a Black-like model (that assumes that the Back volatility is independent of the interest rate). This isdue to the fact that when the interest rate market rallies (or the interestrate decreases), the interest rate volatility tends to rally also, and onewould incur loss due to one’s short Vega position.This is the reason why some traders lost significant amounts ofmoney during the period of end of 2001 and 2002 when the USD shortterm interest rate started to decrease significantly due to the actions ofFOMC, which, in turn, led to a significant increase in the short term interest rate volatilities.More specifically, let V denote the PV of a derivatives instrument asof a function of the state variable S and the Black volatility o E .Then,
Sec. 3.5
Impact of Volatility Skews and Smiles
151
the infinitesimal change in V is given by6’(3.109)where the Black Delta A B is defined as(3.1 10)and the Black Vega V, is defined as(3.111)A better Delta A is the one that takes into account the dependency ofthe volatility on the state variable and is defined as
(3.112)
which we shall also term as skew Delta. The second term in the aboveequation is termed as Delta correction or Delta due to Vega and M is aset of model parameters that are kept constant in the partial differentiation.For a CEV model as described in Eq. (3.12), the skew ‘Deltais givenby(3.1 13)Based on Eq. (3.47), we haveOB(s,K,p)(sK)”-~’’* = Const.
(3.1 14)
69 For simplicity, we have ignored various other terms, such as the Gamma and Theta
terms.
152
The Black-Scholes Framework and Extensions
Chapter 3
So, assuming K is constant, we obtain( W O B
2s
(3.115)
Thus,(3.116)
With the SABR stochastic volatility model (Hagan, Kumar,Lesniewski, and Woodward, 2002), a different and oflen more marketconsistent Delta correction term can be obtained from Eq. (3.112).In general, Eq. (3.109) can be rewritten as
In other words, a part of the Vega P&L (the second term on the righthand side of the above equation) is hedged by Delta (which has lowerhedging cost than Vega hedging).
3.6
Other Extensions of Black-Scholes Framework
Other extensions of the Black-Scholes framework include the modelingof multiple state variables and/or multiple numeraires (for, e.g., crosscurrency derivatives and basket derivatives) and credit (and hybrid) derivatives such as default swaptions and defaultable derivatives.The SABR and other stochastic volatility models are also importantextensions of the Black-Scholes framework.We shall discuss more on some of these topics in later chapters.
Chapter 4
MartingaleResampling andInterpolation
Numerical procedures, such as grid (or finite difference), lattice (or recombining tree), bushy tree (or non-recombining tree), or Monte Carlosimulation for solving PDEs and SDEs, involve discretizations in bothtime and state dimensions (with possibly other approximations for speedimprovement), and, as such, both the accuracy and precision7' (or thebias and convergence) of the quantities that they model can be compromised. This situation is particularly severe for models involving multiplestate variables, such as full yield curve models, e.g., HJM and BGWJmodels, especially multi-factor models with stochastic volatilities, as theerrors of some state variables are from cumulative errors of other variables.The accuracy and precision of hedge ratios (or Greeks or sensitivities) can be even further compromised. This is particularly severe forderivatives with discontinuous payoffs, such as digital and barrier deriva~
70
The same as in Physics and other quantitative sciences, the accuracy of a result refersto the amount of systematic noise (or bias) in the result or how close the mean of theresult is to its true value, whereas the precision of a result refers to the amount of randomnoise (or convergence) in the result or how small the standard deviation of the result is.A result can be accurate, but not precise, which means that it contains little systematicnoise (or bias), but too much random noise. An example for this is an unbiased MonteCarlo simulation without reaching its convergence. On the other hand, a result can beprecise, but not accurate, which means that it contains little random noise, but too muchsystematic noise (or bias). An example for this is a biased Monte Carlo simulation reaching its convergence, but to a wrong value. This can be deceiving and thus dangerous.Ideally, we would like to have a result that is both accurate and precise.
154
Martingale Resampling and Interpolation
Chapter 4
tives, and derivatives on a basket of underlying assets or state variables.These situations occur in all common product areas, such as interest ratederivatives, equity derivatives, credit derivatives, and FX derivatives.A fkndamental reason of such inaccuracy and imprecision is thatmartingale relationships required by arbitrage pricing theories are notexactly satisfied in these discrete numerical procedures. As for the hedgeratios, it is intriguing to see that, as we shall discuss later on, the accuracy of the hedge ratios computed by Maliavin calculus is directly relatedto the accuracy of some weighted moments of the underlying Brownianmotions, which are special cases of martingales.The martingale resampling techniques (also termed as martingalecontrol variate techniques) enforce such required martingale relationships in these discrete numerical procedures (or enforce the discrete martingale relationships) with simple numerical techniques and oftenthrough linear or approximately linear transformations, and thus can improve both accuracy and precision, i.e., reduce bias and improve convergence, or the computation speed for given accuracy and precision. Whilebeneficial to all the aforementioned numerical techniques, they are particularly effective in Monte Carlo simulations and bushy tree techniques.In some cases, the martingale resampling techniques can easily improve the accuracy and precision by a factor of 10 or more without increasing the number of paths (or states) and the number of time steps.Often times, without the martingale resampling techniques, the discretenumerical procedures, particularly, Monte Carlo simulations, do notyield satisfactory results, especially, satisfactory hedge ratios, or satisfactory speed.The first step of the martingale resampling techniques is to identifymartingale relationships and quantify how much the martingale relationships are off. This, by itself, can be a very valuable martingale testinginfrastructure.One can also use higher order discretization schemes (as opposed tothe Euler scheme) to improve the accuracy and precision. However, although higher order discretization schemes usually are effective in improving the accuracy of the first order martingale relationships (i.e.,those for the underlying assets or state variables), they may not beenough in improving the accuracy of the second or higher order martingale relationships (such as those for options on the underlying assets orstate variables). There are various other techniques for accuracy andprecision improvement, such as the predictor-corrector, state space
Sec. 4.1
Martingale Interpolation
155
smoothing, low-discrepancy sampling, importance sampling, and stratified sampling techniques. In many cases, the martingale resamplingtechniques are still needed. In some cases, these techniques and the martingale resampling techniques are complementary to one another and themartingale resampling techniques can be used in conjunction with thesetechniques. In some other cases, the martingale resampling techniquescan be used to replace some of these techniques, and the martingale resampling techniques often have advantages in accuracy and precision orin speed.In addition to numerical discretizations and approximations, somemodels, such as the BGWJ model, require modeling of discrete quantities, such as LIBORs with discrete tenors. However, the payoff of thetrade being priced can be based on LIBORs with tenors not modeled bythe BGM/J model. This is particularly true for aged or existing trades.Of course, one can perform interpolations. But interpolations do notautomatically guarantee the required martingale relationships. This is acriticism of the BGWJ model on a theoretical basis.71 Interpolationsusing proper martingale quantities and/or combined with martingale resampling techniques, that we shall term as martingale interpolation, canaddress this issue.Another dimension for the cause for such inaccuracy and imprecisionis for credit charges in that the market process modeling or the scenariogeneration engine and the pricing engines may not be fully consistent.For instance, the scenario generation can be implemented with theBGWJ model (with one or multiple factors), whereas the pricing enginescan be implemented with various short rate models (with one or multiplefactors). The martingale resampling techniques are efficient in significantly improving such consistencies by ensuring that the underlying instruments are priced correctly and thus significantly improving theaccuracy and precision of credit charges. This is very helpful in the implementation of a credit charge system that needs to bring together thescenario generation engine and the pricing engines from different tradingsystems. The same modeling and system architecture, in general, canalso be applied to pricing hybrid derivatives involving credit and anotherproduct area, especially, credit and interest rate hybrid derivatives.The martingale resampling techniques are essentially martingale con71 HJM models do not suffer from such difficulty on a theoretical basis, as theoretically
they can model forward rates with continuous tenors, although practically they need to beimplemented with forward rates with discrete tenors.
156
Martingale Resampling and Interpolation
Chapter 4
trol variate techniques with the martingale relationships as the controlvariate targets. The reason that they are also termed as martingale resampling techniques is that they are usually implemented with resampling techniques with resampling targets provided by the martingale relationships.The original resampling technique was developed for Monte Carlosimulations whereby the random samples are transformed or resampledto achieve the desired properties. Such resampling technique is commonly applied to perform moment matching for the underlying drivingfactors, such as the incremental Brownian motions, for which the targetmoments are known. For instance, when one performs a Monte Carlosimulation by drawing samples from N(0,l) (i.e., standard normal distribution with 0 mean and 1 standard deviation), one would find, e.g., that,due to the finite sampling, the sample mean most likely will not be 0, thesample standard deviation most likely will not be 1 and the sample kurtosis most likely will not be 3 as needed. The resampling technique isneeded to match these moments and possibly higher order moments.Similarly, for a simulation with multiple time steps and/or multipledriving factors, the resampling technique is needed to match the crossmoments, such as the correlation and the auto-correlation. For Brownianmotion driven simulations, the (non-overlapping) incremental Brownianmotions at different time steps and cross the simulation paths shouldhave zero ~orrelation.~’When we look across different time steps alonga given simulation path, these incremental Brownian motions shouldhave zero expected auto-correlation. Without enforcing the correct correlation and the expected auto-correlation through the resampling techniques, a simulation may not even be able to model Brownian motionscorrectly. The typical issue is that the cumulative variance of theBrownian motion is incorrect, even though the variance of all the corresponding incremental Brownian motions is correct.It is important to point out that the resampling is typically performedacross simulation paths (at given time steps) and not along a simulationpath. This requires all the simulation paths be available at the time whenthe resampling is performed. This is not a major issue. First of all, evenwithout resampling, in a Monte Carlo simulation, it is better to draw therandom samples across the simulation paths than along a simulation path.Secondly, in case keeping all the simulation paths in the computer memgeneral, for a Levy process, the (non-overlapping) incremental driving factorsshould be independent. It is more subtle to resample when jump processes are involved.
72 In
Sec. 4.1
Martingale Interpolation
157
ory is not feasible or efficient, it can be circumvented by running thesimulation and resampling in several separate batches.The resampling technique can be extended to a much greater scopeand applied not only to the deriving factors, but also to the state variables, the underlying asset prices, and the derivatives prices with the resampling targets provided by the martingale relationships.The essence of the resampling technique is to assume the true (or thetarget) value of the quantity to be resampled is a deterministic (often linear or approximately linear) function of its old value and then solve forthe parameters of this function.Another way to understand the significance of the moment matchingand martingale resampling is that our goal is to achieve weak convergence, i.e., the convergence in the sense of the expected values, and that,in turn, translates to the convergence of various (conditional and unconditional) moments.The martingale resampling technique is in a similar spirit as themodel calibration that forces the model to reprice some of the hedge instruments to their market prices. However, the martingale resamplingtechnique can enforce many more martingale relationships than themodel calibration. For instance, the model calibration does not take careof any martingale relationships not needed by the calibration instruments,whereas the martingale resample is able to take care of some thesemartingale relationships. One application of this is to use the martingaleresampling technique to calibrate to the volatility skews and smiles. Essentially, this procedure can convert various short rate models to approximated BGM models. Similarly, it can also convert short intensitybased for credit spread models to risky BGM models.Also, the martingale resampling technique can improve the accuracyand precision at the diving factor and the state variable level in terms ofthe accuracy and precision of various moments and cross moments (including correlation and auto-correlation) that the model calibration normally does not handle.Another difference between martingale resampling and calibration isthat theoretically, full yield curve models should automatically match theinitial yield curve (or the initial forward equity prices or initial forwardFX rates). For instance, the price of an FRA (Forward Rate Agreement)or a swap from the model should automatically match that obtained fromthe initial yield curve. However, due to the numerical errors discussedearlier, the initial yield curve (or the initial forward equity prices or ini-
158
Martingale Resampling and Interpolation
Chapter 4
tial forward FX rates) are not matched automatically in numerical implementations. But, for full yield curve models (and other models, suchas equity and FX models), the drift terms are completely determined bythe martingale arbitrage relationships, and, thus, there is no freedom inthe model to calibrate to the initial yield curve (or the initial forward equity prices or initial forward FX rates). Martingale resampling explicitlyrecognizes such numerical errors and allows additional freedom to matchthe forwards.More specifically, as an example, let’s consider the following diffusion process
dS, = p (t ,S, ) dt + o (t ,S, ) d K .
(4.11
Rather than using simple Euler discretization scheme, the martingaleresampling technique (particularly the one that utilizes linear or approximately linear transformations) effectively uses the following discretization scheme
where A, (t , S , ) and A,, (t,S,) are the correction terms on the drift andthe volatility, respectively, for taking into account the numerical errors.These correction terms can be determined numerically with the martingale resampling techniques to be discussed in the following sections.In some cases, the martingale resampling technique can also be usedas an incremental model calibration for speed improvement. For instance, for some fdl-yield curve interest models, such as the generalHJM models, the calibration speed may not be fast enough for tradingand the end-of-day hedging requirement. One possible solution to this isto use the martingale resampling technique for an efficient incrementalmodel calibration based on a good full model calibration (which can beperformed, e.g., at the close of the market on each business day). This isalso an alternative to using closed-form solutions or approximations forcalibration speed improvement. In some cases, the incremental modelcalibration using the martingale resampling technique is faster than thefull model calibration using closed-form solutions or approximations.This enables us to use more general models without worrying muchabout the existence of closed-form solutions or approximations.Unlike the common calibration, the martingale resampling techniquenormally does not need to explicitly know or specify the model parame-
Sec. 4.1
Martingale Interpolation
159
ters, such as the drift and volatility. In this sense, the martingale resampling can be considered as a non-parametric calibration. The martingaleresampling can be used to create non-parametric models, e.g., local volatility or stochastic volatility models, for matching the volatility skewsand smiles by transformations on the state variables. In this case, themodel parameters are backed out from the distributions of the state variables.Another promising way of constructing a non-parametric model is tocombine historical simulation and martingale resampling. In otherwords, we first draw samples from historical data or derive incrementaldriving factors from historical data, and use the historical in the place ofthe Brownian motions, and then we perform various martingale resampling.It is important to ensure that the martingale resampling does notchange the state variables significantly, e.g., in terms of percentagechanges. Thus, it is important to start with a good prior solution or approximation, or to perform the martingale resampling with a few iterations.The typical usage of the martingale resampling techniques is to combine with various other numerical techniques as improvements. Thesetechniques include Monte Carlo simulation, simulation with low discrepancy sampling, simulation with stratified sampling, simulation with importance sampling, Latin hypercube, finite difference grid, recombiningor bushy tree, and nonexploding bushy tree (NBT).It is important to point out the different between the requirement onusing Monte Carlo and other sampling techniques for numerical integration and that for simulation. Numerical integration is a simpler situationin that it does not need to model the underlying dynamics of the timeevolution of the problem. Alternatively, it can be considered as a one(time) step simulation. The simulation, on the other hand, does need tomodel the underlying dynamics of the time evolution of the problem andthus imposes additional requirements, such as that the incremental driving factors at different time steps should have the required correlationand auto-correlation. The martingale resampling techniques can ensurethe correct correlation and the correct expected auto-correlation.
4.1
Martingale Interpolation
Interpolations are often needed in numerical implementations. However,
Martingale Resampling and Interpolation
160
Chapter 4
interpolations do not automatically guarantee the required martingalerelationships. Martingale interpolations are interpolations using propermartingale quantities possibly combined with martingale resamplingtechniques so as to guarantee the required martingale relationships.This is particularly useful for the BGWJ LIBOR market model, thatmodels LIBORs with discrete tenors, and the payoff of the instrumentbeing priced can be based on LIBORs with tenors not directly modeled.Let us take a look at the following martingale quantity and relationship,M ( s , T ) = E, [ M ( t , T ) ]
(s s t ),
(4.3)
where the martingale quantity M ( t ,T ) has a term structure representedby T.Suppose a numerical procedure gives accurate martingale quantitiesM ( t , T , ) and M ( t , T 2 ) ,and we need to use an interpolation (or extrapolation) to determine M ( t , T 3 ) so that we guarantee the required martingale relationship, i.e.,(3,
T3) = E, [
( 1 , T3 )]
(s 5 f ) ,
(4.4)
It is easy to see that if we use the term T as the independent variablein the interpolation (or extrapolation), then the above martingale relationship may not always hold.However, if we use the initial value of the martingale quantity itself,i.e., M ( s , T ) , as the independent variable in the interpolation (or extrapolation), then we can insure the above martingale relationship. Inother words, if we let
Then it is easy to verify that Eq. (4.4) is always satisfied, except for thecases when the denominators in Eq. (4.5)are zero. This can proved easily, as