
Sec. 7.9 The Nonexploding Bushy Tree Technique
309
Table 7.4 showsl-factor Bermudan payer swaption prices with 10%constant initial LIBOR and 20% constant volatility for quarterly LIBORand swaps obtained with a nonexploding bushy tree (compared withMCA Monte Carlo simulation results). The differences between thenonexploding bushy tree and those of the MCA technique are all withinthe 4*(standard deviation) of the MCA technique.Tables in the Appendix 7.13.1 show more numerical results obtainedwith the nonexploding bushy tree technique. Table 7.5 through Table 7.7show the ATM European and Bermudan quarterly swaption prices for 1through 3 factors (with volatility multiplier = 20% and the estimatedmarket correlation).
7.9.4.3 Effect of the Number of Factors
In general, models with different number of factors (even though calibrated to the same set of calibration instruments) can produce differentprices and hedge ratios (or sensitivities) for the same exotic derivativesinstrument, especially for those that are sensitive to the correlation or tovariables that forward volatilities, as these models have different modelcorrelations.A 1-factor model has an instantaneous correlation of 1, but it canhave term correlations (i.e., cumulative correlation over a period of time)of less then 1 due to the decorrelation effect of time dependent modelparameters. Some derivatives instruments are more sensitive to termcorrelations than the instantaneous correlation. Thus, a 1-factor modelmay be used for these derivatives instruments, though multi-factor models are better.Models with fewer factors tend not be able to accurately capture themarket correlation. Models with fewer, but greater than 1, factors caneither over estimate (e.g., at short term) or under estimate (e.g., at longterm) the market correlations. Thus, they can either over estimate or under estimate the price of correlation sensitive instruments.Here we present specific examples of the effect of the number of factors. Due to what we just discussed, the reader should not draw generalconclusions on the direction or magnitude of the effect of the number of
310
The Interest Rate Market Model
Chapter 7
Table 7.4. 1-factor Bermudan payer swaption prices with 10% constant initialLIBOR and 20% constant volatility for quarterly LIBOR and swaps obtainedwith a nonexploding bushy tree (compared with MCA Monte Carlo simulationresults). The differences between the nonexploding bushy tree results and thoseof the MCA technique are all within the 4* (standarddeviation) of the MCAtechnique.
a. A European swaption with the notion of ixj is an option with the expiration ofi years on a swap with the maturity of i+j years and start date equal to the exercise date of the option (plus a few business days depending on the convention).Here, a Bermudan swaption can be early exercised prior to the expiration of theEuropean swaption at the swap frequency. This type of the Bermudan swaptionsis not what is commonly traded in the market. Rather it is for the purpose ofcomparing results with what is available in the literature.b. Nonexploding bushy tree results.c. Markov Chain Approximation (MCA) Monte Carlo simulation results of Carrand Yang (1999).d. Percentage 4*(standard deviation) derived from the corresponding absolutevalues of Carr and Yang (1999).
Sec. 7.9
311
The Nonexploding Bushy Tree Technique
factors on the price of derivatives instruments based on specific examples. For more discussions, see, e.g., Rebonato (1999).More specifically, Figure 7.9 shows the effects of the number of factors (1 to 5 ) on various Bermudan swaptions. As can be seen from thefigure, in this particular case, models with fewer factors appear to underestimate the Bermudan swaption prices or the fbture expected payoffs,which can affect the early exercise strategies.lo7 In order not to introduce any artifacts from the calibration, these calculations were performedby using only one degree of freedom (the total variance) for the calibration to bring the European swaption values close to the corresponding 1factor values. Relative value of Bermudan swaption price is defined as(Bermudan swaption price)/(European swaption price), which is furthernormalized by the corresponding 1-factor value.
B g 1.1
!
=9:
$f
1.061.04
4 1.02
ZEP l
1
0.98 -I0
I2
4
6
Number of Factors
Figure 7.9. The effects of the number of factors on Bermudan swaptions.
The percentage uncertainties of the data in the above figure are normally less than 1% (as estimated from the put-call parity error and convergence).The computation time is roughly proportional to
(2n + l)m'[m( m/m'- 1)+ m/m'] for a nonexploding bushy tree with nfactors, m total tree steps, m' steps in each sub-bushy tree, 2n+l branchesper node, and one growth state per mode,
lo' In general, models with fewer factors can either under estimate or over estimate theBermudan swaption prices depending on the specifics of the Bermudan swaptions. Formore discussions, see, e.g., Rebonato (1999).
312
The Interest Rare Market Model
Chapter 7
7.9.4.4 Convergence Analysis
Sufficient conditions of convergence are provided in the Appendix7.13.2, which are satisfied by the current implementation of the nonexploding bushy tree. Figure 7.10 shows an example of the convergence of2-factor swaption prices. The prices are normalized by those at the highest number of tree steps. More discussions of the sufficient conditions ofconvergence are provided in the Appendix 7.13.2.
ln
0.9960.9940.9923 0.99L
0
50
100
Number of Tree Steps
Figure 7.10. An example of the convergence of 2-factor swaption prices.
7.10
General Framework for Multi-factor Modeling for HybridMarket
So far, we have been mostly focused on modeling of a particular marketsector or a particular product area, such as the interest rate of a singlecurrency, the equity derivatives, and FX derivatives by themselves. Intoday's market, hybrid derivatives instruments have become more andmore popular. Examples of such derivatives instruments include interestrate and FX hybrid derivatives, such as PRDC (Power Reserve Dual Currency) notes or swaps and CPRDC (callable PRDC) notes or swaps, aswell as (callable) cross-currency swaps, interest rate and equity hybridderivatives, such as equity linked notes or swaps with callable or barrierfeatures, and interest rate and credit hybrid derivatives such as (callable)credit linked notes or swaps. Other features, such as snowball and/ortarget redemption, can also be incorporated. Convertible bonds are hybrid derivatives among equity, credit, and, to a lesser extent, interest rate.Interest rate derivatives with a weaker or lower credit rated counterpartyare often naturally an interest rate and credit hybrid derivatives instrument. Emerging market trades are often hybrid derivatives among FX,credit, interest rates, inflation.Thus, in order to value these derivatives correctly, we need to model
Sec. 7.10 General Framework for Multi-factor Modeling for Hybrid Market
313
several market sectors and product areas simultaneously and consistently. The basic idea is that the entire market is assumed to be driven byn common factors, or, n independent Brownian motionsw,’ ( j = 1,2,...n ) .The credit models naturally require jumps (such as jump to default).This can still be fitted into this framework, as this is typically handled byMerton’s model or variations of Merton’s model. The stochastic volatility models can be naturally fitted into this framework.Ideally, we also need some generic jump models to better model theshort term market movements and to handle, e.g., short term volatilitysmiles. Fitting jump models, particularly, hedgeable jump models, intothis framework is a part of the authors’ future research.More specifically, any state variable Sf ( i =1,2, ...m) (such as interest rates, volatilities, FX rates, equity prices) in the market is driven bythe same set of Brownian motions, i.e.,n
C
dS/ = (...)dt + 0; wydW,’ ( i = 1,2,...n) ,
(7.108)
i=l
where the weights or loadings wf can be obtained with PCA from thecorrelation matrix among a1 the state variables in the market.With this approach, if we have a 2-factor stochastic volatility interestrate model, then both the rates and volatility are driven by the same 2factors (but with different loadings). Similarly, if we have a 3-factorcross-currency interest rate model, then the two interest rates (in two currencies) and the FX rate can all be driven by the same 3 factors (but withdifferent loadings), if needed.This approach allows us to better capture the correlation and morestate variables with minimum number of factors.As we discussed before, in order to produce market consistent pricesfor exotic derivatives, secondary model calibration is needed to more orless estimate or account for the un-hedgeable variables.As we also discussed before, the exotic derivatives trading is drivenby asset management, liability management, and other financial activities. In the asset management area, the major driving factor is the desireof investors (such as various funds and financial institutions) to achievecoupon enhancement or return enhancement on their investment. Another driving factor is that some investors require principal protection on
The Interest Rate Market Model
314
Chapter 7
their investment. These motivate note issuers (such as government agencies, corporations, and financial institutions who need funding) to issuestructured notes (typically with exotic derivatives embedded in the coupon) to satisfy the needs of the investors and also to achieve lower funding rate for themselves. The note issuers then need to manage their liabilities, typically, by swapping the exotic coupons with dealers. Thedealers, in turns, try to hedge or replicate the exotic derivatives with simpler and more liquid instruments. However, they generally need to takepositions in the un-hedgeable variables, but can often partially rely on theportfolio offsetting and diversification benefits. The investors can alsoenter exotic swaps with dealers. A swap trade generally requires muchless fund than buying a note, and thus is also termed as unfunded trading(versus funded trading of buying a note). Unfunded trading can behighly leveraged and, thus, much riskier.
7.1 1
Stochastic Volatility BGM Models
While the CEV and shifted lognormal BGM models can model the volatility skews, they cannot model the volatility smiles. By making thevolatility in the BGM model stochastic, one can model a large range ofvolatility smiles.One class of such models is as follows.
where W,J and Y k are independent Brownian motions under the spotarbitrage-free probability measure Q ,i.e.,(7.110)
and wf are normalized stationary or constant weights or loading (thatcan be obtained from PCA), i.e.,
tWfWf=1,
j=1
(7.111)
Sec. 7.1 1 Stochastic Volatility BGM Models
315
Furthermore, the volatilities follow the following lognormal stochastic process.n
d o i (t ) = a (f )oi ( t)
C w!dFj
(0 I t I
j=l
),
(7.1 12)
where w! are normalized stationary or constant weights or loadings, i.e.,(7.113)The correlation among the state variables (i.e., LIBOR and the volatilities) is given byCorr (dLi ( t) ,dL, ( t)) =
n~1i jw1kj
j=1
Y
(7.1 14)
(0 ~t I rnin(q,T,))n
....
Co..(dL,(t),do,(t))=Cw~w~
( O S t IT,),
(7.115)
j=1
j=1
(7.1 16)
(O<tIrnin(q,T,))The weights or loadings wy and w! can be obtained from the abovecorrelation using, e.g., PCA. The correlation can be obtained from historical data or implied from correlation sensitive market observablequantities, or the combination of both.In some ways, this model is similar to that of Andersen and Brotherton-Ratcliffe (2001), but this model is more general. This model is alsosimilar to the SABR model except that the SABR model models eachforward LIBOR or swap rate separately and, thus, is not a consistent termstructure model, whereas this model models all the forward LIBOR andswap rates in a consistent way as a term structure model. So, this modelcan be named as Term Structure SABR Market Model.
The Interest Rate Market Model
316
7.12
Chapter 7
Examples of Stochastic Volatility BGM Model Results
The following figures show example results of a stochastic volatilityBGM model. The model is based on the parameterization discussed inthe previous section with Monte Carlo simulations and martingale resampling. Not only can this model calibrate to the entire ATM cap andswaption volatility matrix, it can also calibrate well globally to the volatility skews and smiles at many expiries and terms. More details shall bereported elsewhere.
i
EUR Swap Vol(24mX12m)
0
MarketVol
-Stoch Vol BGM Model
-5% -4% -3% -2% -1% 0%
1% 2%
3% 4%
5%
3% 4%
5%
Skew
I
EUR Swap Vol(60mX60m)50%
-5% 4% -3% -2% -1% 0%
1% 2%
Skew
Figure 7.1 1. Examples of Black volatilities produced by a stochastic volatilityBGM model (Stoch Vol BGM Model) as compared with market volatilities(Market Vol) for 2 year (24m) into 1 year (12m) and 5 year (60m) into 5 year(60m) plain vanilla European swaptions in the EUR swap market.
Sec. 7.13 Appendix
7.13
Appendix
7.13.1 More Numerical Results Obtained With the NBT TechniqueTable 7.5. One-factor ATM European and Bermudan quarterly swaption prices(with volatility multiplier = 20%) obtained with a lognormal LIBOR marketmodel implemented with a nonexploding bushy tree (NBT).
* Computed from the BGM rank one approximation (Brace, Gatarek, andMusiela, 1997).
317
318
The Interest Rate Market Model
Chapter 7
Table 7.6. Two-factor ATM European and Bermudan SFV aption prices (withvolatility n11ultiplier = 20% and the estimated market correlation) obtained withobtained 1 ith a lognormal LIBOR market model in31plemented with anonexplodi nig bushy tree (NBT).
BGM
*
II
NBT European
Payer Receiver % Parity Err0166.4 67.667.60.02% 77.9)98.1 99.599.40.02% 109.01156.5-0.08%1 x 5 154.4 156.2200.6 206.8206.30.13% 224.6
221.2
128.51-0.06% 168.21
128.61168.01
78.31109.21
=!=I
2x2
p
0.26%
I
III
NBT Bermudan
-0.09%
0.59%
115.41 115.71167.81 169.51261.11 268.11
115.81168.51266.51
-0.04%0.30%0.30%
5 x 2 121.9 122.95 x 3 177.3 179.35 x 5 276.6 286.5
122.1178.5283.1
0.31%0.22%0.60%
* Computed from the BGM rank one approximation (Brace, Gatarek, andMusiela, 1997).
Sec. 7.13 Appendix
319
Table 7.7. Three-factor ATM European and Bermudan swaption prices (withvolatility multiplier = 20% and the estimated market correlation) obtained withobtained with a lognormal LIBOR market model implemented with anonexploding bushy tree (NBT).
* Computed from the BGM rank one approximation (Brace, Gatarek, andMusiela, 1997).
7.13.2 Sufficient Conditions for ConvergenceHere we provide sufficient conditions for convergence in the sense ofweak convergence, i.e., the convergence of all conditional expected values to the corresponding continuous time limits. This is what is needed
320
Chapter 7
The Interest Rate Market Model
for derivatives pricing. Traded derivatives generally have piecewise infinitely differentiable conditional payoff functions with respect to theunderlying state variables. Thus, these conditional payoffs can be expressed as Taylor expansions in terms of the underlying state variables.Therefore, the conditional expected payoffs can be expressed as thesummations of all relevant conditional moments of the underlying statevariables. Consequently, the convergence of all the conditional expectedpayoffs is guaranteed by the convergence of all conditional moments ofthe underlying state variables.Let us first show that all the conditional moments of the incrementalBrownian motion modeled on a sub-bushy tree of a nonexploding bushytree converges to the continuous time limit as the total number of stepson the sub-bushy tree goes to infinity.Recall that the n-dimensional incremental Brownian motion (n by 1vector) is given byAW(t,At) = W ( t+ At) - W ( t )
(At 2 0).
(7.1 17)
The moment generating function G and moments M;$ are given byG ( A , t , A t ) = E , [ e x p ( A A W ( t , A t ) ) =exp(:IAI2]
At)
'
(7.118)
where A is a 1 by n vector.As can be seen, the moment generating function captures all themoments, i.e., if the generating function converges, then all the momentsconverge too.For the nonexploding bushy tree implementation, we build m' -stepsub-bushy trees for the time period from t to t+At. The correspondingmoment generating function on the tree with no interpolation (which isactually a conventional bushy tree) is given byG(A,t,At)=l?, [exp(AA@(t,At))],
(7.1 19)
where @(t,At) is the n-dimensional Brownian motion modeled on the
Sec. 7.13 Appendix
321
zi[.I
bushy tree andis the time t conditioned expected value obtained onthe tree.Using the iterated expectation theorem yields
=
Ei [exp( A A q ( t , y A t ) ) ]-
xEi+%Ai
[exp (AA # ( t + 9At,$$))]
7
(7.120)
The following condition was used in deriving the above equation.Condition 1 for the convergence:
is state independent. This may not be achievable for recombining trees.For convenience, we also assume the above quantity is time independent.Thus, the moment generating function can be expressed in terms of thesingle tree step moment generating function, i.e.,
G ( A , t , A t )={G(A,t,$$)]rn'.
(7.122)
The state shock matrix being state independent is sufficient for this condition to hold.Condition 2 for the convergence:A I (?(A,t,$) 5 B ,
(7.123)
where
(7.124)for some K E R, and K independent of m'.Since lim A"' = lim Brn'= exp($A)' At), thenrn'-tao
m'+m
322
The Interest Rate Market Model
lim e(A,t,At)= lim ( e ( A , t , $ ) }m'+x
m'+w
m'
=exp($I2At).
Chapter 7
(7.125)
The state shock matrix being independent of m' is sufficient for thiscondition to hold.With the above equations, we have provided sufficient conditionsunder which the moment generating function on a conventional bushytree converges to the continuous time limit.In the case of a nonexploding bushy tree, the key feature is that interpolations are performed at time t+At with respect to certain state variables (not with respect to time) based on the values at the growth nodes.moment generating function is also based onThus, the time t +*Atinterpolation (rather than being modeled directly).LetG(A,t +*At,$)denote such moment generating function. Therefore,the time t moment generating function on the nonexploding bushy tree isthus given byG (A,t , At) =
[exp(AA @ ( t , y A t ) ) (A,t +
At,$)]
. (7.126)
Condition 3 for the convergence:
lim E(A,t + y ~ t , $ )= 1.
m'+m
(7.127)
In this case, the moment generating function on the nonexploding bushytree converges to the continuous time limit, i.e.,mlim' 3 m G(A,t,At)= m'+mlim c(A,t,At)=exp(+lAl'At).
(7.128)
The following are sufficient conditions to Condition 3 that are easyto realize in numerical implementations.Condition 3.1: There are no extrapolations.Condition 3.2: The distance d between any adjacent growth nodes usedfor interpolations satisfieslim d = O .m'-+m
Since, in this case, the interpolation error e is given by
(7.129)
323
Sec. 7.13 Appendix
e=O(d)
(m’+w),
(7.130)
thus the Condition 3 is satisfied, asI
1jm e ( ~ , t + + ~ t =,$)
m +-
1jm
m -m
e ( ~ , t + y ~ tm1jm,+-3O)(+e)=l
(7.131)
All the above sufficient conditions of convergence are easy to satisfyand are satisfied by the current implementation of the nonexplodingbushy tree. Particularly, the no extrapolation condition is satisfied byalways including the outermost tree branches as growth nodes. Furthermore, in the current implementation, the number of growth nodes in asub-bushy tree is of the order of O(m’) , and the distance d between anyadjacent growth nodes used for interpolations is of the order of
With these conditions, it is easy to show that all the conditional moments of any incremental square integrable It6 process (from t to t+&)converge to the continuous time limits, as such incremental It8 process issimply a linear function of an incremental Brownian motion (with coefficients independent of the incremental Brownian motion). Furthermore,if we take the limit m’ + 00 first and then take the limit At + 0 , we caneasily show that all the conditional moments of any square integrable It6process converge to the continuous time limits.
7.13.3 Application of Girsanov’s Change of Measure Theorem toDerivation of the Martingale or Full-dimensional LIBOR MarketModelHere we present the application of Girsanov’s change of measure theorem to the derivation of the martingale or full-dimensional LIBOR market model. This requires that we rewrite the above change of measuremultiplier (closely related to the Radon-Nikodfm derivative) in a suitable form, i.e., an exponential martingale form.Let
q ( t) = 1+ S,L, ( t)then
( 0It 5 ) ,
(7.132)
324
The Interest Rate Market Model
Chapter 7
dq ( t ) = SidLj( t )
Thus,
r, ( t )= r, (0)
Therefore,
(tI T I T . )Changing the measure with the above change of measure multiplieressentially changes the drift of a Brownian motion. Applying the Girsanov change of measure theorem yields
(0 I t I qJwhere W,T.' + I ' L.'--ISimilarly,
is a one-dimensional Pc+,-Brownian motion for L;-,(t).
325
Sec. 7.13 Appendix
( 0 5 t q )Applying the Girsanov change of measure theorem recursively yields
(ostsq)Since the instantaneous correlation is measure invariant (with respectto Girsanov's change of measure), i.e.,
( O I t IT)then
(OItII;)Consequently, we can obtain the following LIBOR process, i.e.,
dL, ( f ) = oidW,q+l>Li
+aidyTN ' i
( 0 5 t q )which, valid for all LIBORs, is under the forward arbitrage-free measure
The Interest Rate Market Model
326
Chapter 7
PTN of the longest maturity LIBOR L ~ - l ( t )Note.that~ ~ (1)) =aioj~orr,covar, ( d (t),d~,~
(
(0 ~t Imin(q,Tj))
Idt.
(7.142)
For simplicity, we can assume that all the q's are positive. In thiscase,
con; (dLi(t),dLj(t))=corr, (dW,TNJi,dW,TN,L')(7.143)
(0It I min (q , T ~ ) )Therefore, the above LIBOR process can be rewritten as
dLi ( t )= c7idW,T+13LI--
N-l
c
oiojsj
j=r+l l+SjLj(t)
corr, (dLi (t),dLJ(t))dt
.
(7.144)
+aidFTN,Li(OItIT)
This is the martingale or full-dimensional LIBOR market model in ageneral form and under the same probability measure ( PTN). The correlation among all LIBORs, Corr(dLi (t),dLj ( t ) ), is an input to themodel. As we pointed out before, the volatility oican be time dependent, state dependent, or stochastic.
Chapter 8
Credit RiskModeling andPricing
As we pointed out before, in essence, pricing the counterparty credit riskis the same as pricing defaultable OTC derivatives or pricing the premium of the embedded default options in OTC derivatives, which is arelatively new area in derivatives modeling.These default options, though embedded, are economically realtrades, and need to be priced and hedged properly. Just as a CDS is fortrading credit risks in a defaultable bond, a default option in a defaultableOTC derivatives instrument is for trading credit risks in the corresponding defaultable OTC derivatives instrument.To price the counterparty credit risk or to compute the creditchargehenefit or CVA, it typically starts with modeling of the loss givendefault (LGD) and the default probabilities. More specifically, it involves modeling the market exposures (including the collateral strategies) and credit spread or default probabilities, and, ideally, also creditrating transitions, default recoveries, as well as correlation between themarket processes and the credit processes.We shall start with pricing various defaultable instruments, and thenaddress some subtleties in pricing the counterparty credit risk.A common subtlety is that pricing the counterparty credit risk isreally pricing options on a basket of defaultable derivatives in a counterparty portfolio due to the netting agreement and the asymmetric treatment of default (except for extinguishing trades). As we pointed outpreviously, the prevailing trade-level based derivatives models (or micromodels) normally do not and often cannot price the specific counterparty
328
Credit Risk Modeling and Pricing
Chapter 8
credit risk, even though they can price the expected market risks (whichinclude the credit risks of reference obligors in credit derivatives) andsome generic counterparty credit risk by discounting at LIBOR. Therefore, additional portfolio-based derivatives models (or macro models) areneeded for pricing the specific counterparty credit risk.Furthermore, the trades in a counterparty portfolio can be modeledby different pricers, booked in different systems, from different desks, oreven from different product areas. This presents significant challenges tothe counterparty credit risk modeling and the system.
8.1
Pricing Simple Defaultable Instruments
Here we present an example on how to price a simple defaultable instrument, such as a defaultable zero-coupon bond. The pricing of defaultable coupon bonds and CDS follows a similar procedure. We shall present more general discussions in the next section.A common approach to modeling the default process is through asimplified Merton or structural model whereby there is a real or hypothetical underlying asset value process X,and the default event happensif the asset value is at or below a default barrier (or a hypothetic liabilitylevel) H I .Io8 In other words, the default time process z is given by thefollowing stopping or first exit time, i.e.,r=inf(tlX, S H ~ }
(~<t).
(8.1)
For the simplicity of exposition, we have assumed that no default hasoccurred at or before time 0, which can be easily relaxed.Let 9 ( t ) denote the time t (survival) value of a simple risky or defaultable instrument (if no default occurs at or before time t ) with a single positive (survival) payoff of V( T ) at time T if no default occurs at orbefore time T and a recovery of E?(z) if default occurs at time z. Thus,'Ox Another category of credit models is the reduced-form model that directly models thedefault probability or the default intensity, and does not start with the modeling of thedefault events (as does a structural model). The structural model, on the other hand, hasan advantage of conveniently specifying default correlation (by specifying the correlationamong the asset value processes). Merton's original paper can be found at Merton(1974). As we shall see, we can possibly use a structural model and a reduced-formmodel together to jointly model the default event and the default intensity processes.
Sec, 8.1
Pricing Simple Defaultable Instruments
329
by applying the Harrison-Pliska martingale no-arbitrage theorem,lo9weobtain
(OIt I T )
where the expectation is taken under the forward probability measure PTwith respect to a default-free zero-coupon bond P ( t , T ) as the numerairewith P ( T , T ) = 1 and conditioned on all the information up to time t. Forthe simplicity of exposition, we have assumed that the recovery r? ( r ) isimmediately determined and settled at default time r , which can be relaxed.The above equation is very intuitive whereby the first term on theright-hand side represents the non-default or survival payoff and the second term represents the default payoff.It is interesting to ask in passing whether or not the following familiar-looking martingale relationshipV(t) ?
EF [V ( T ) ]P(t,T)(0I t I T )
-=
(8.3)
holds.It turns out that, in general, it does not hold, as 17 ( t ) or ( T ) do notfully capture the cashflows of the instruments. More specifically, theyonly capture the non-default or survival payoffs, but do not capture thedefault payoffs. As a matter of fact, P ( t , T ) E T [ f ( T ) ]