Affine Pricing and Hedging of Collateralized Debt Obligations
AAffine Pricing and Hedging of Collateralized Debt Obligations
Zehra Eksi a , Damir Filipovi´c b, ∗ a Institute for Statistics and Mathematics, WU-University of Economics and Business, Welthandelsplatz 1, 1020Vienna, Austria b Swiss Finance Institute and Ecole Polytechnique Fédérale de Lausanne, Switzerland
Abstract
This study deals with the pricing and hedging of single-tranche collateralized debt obligations(STCDOs). We specify an affine two-factor model in which a catastrophic risk component isincorporated. Apart from being analytically tractable, this model has the feature that it capturesthe dynamics of super-senior tranches, thanks to the catastrophic component. We estimate thefactor model based on the iTraxx Europe data with six tranches and four different maturities,using a quasi-maximum likelihood (QML) approach in conjunction with the Kalman filter. Wederive the model-based variance-minimizing strategy for the hedging of STCDOs with a dynam-ically rebalanced portfolio on the underlying swap index. We analyze the actual performanceof the variance-minimizing hedge on the iTraxx Europe data. In order to assess the hedgingperformance further, we run a simulation analysis where normal and extreme loss scenarios aregenerated via the method of importance sampling. Both in-sample hedging and simulation anal-ysis suggest that the variance-minimizing strategy is most effective for mezzanine tranches interms of yielding less riskier hedging portfolios and it fails to provide adequate hedge perfor-mance regarding equity tranches.
Keywords: single-tranche CDO, affine term-structure of credit spreads, catastrophic risk,variance minimizing hedge
1. Introduction
A collateralized debt obligation (CDO) is a structured product that is backed by a portfolioof credit risky assets. A synthetic CDO is a special type of CDO in which the underlying creditrisky portfolio consists of single-name CDSs. Single tranche CDOs (STCDO), on the otherhand, make it possible to take an exposure on a specific segment of the underlying portfolio.A CDO long position holder is exposed to two types of risk. The default risk arises from thepossibility of a default of an obliger and the market or spread risk is associated with the changesin the credit qualities and the interest rates. Thus, a sound model for the pricing and hedging ofCDOs is expected to take both default and credit spread dynamics into account. ∗ Corresponding author
Email addresses: [email protected] (Zehra Eksi), [email protected] (DamirFilipovi´c )
November 23, 2020 a r X i v : . [ q -f i n . M F ] N ov verall, one can categorize the portfolio credit derivative models according to two ap-proaches. Under the bottom-up approach the fundamental objects to be modeled are the lossprocesses of the portfolio constituents whose sum gives the total portfolio loss. In contrast, the top-down approach aims to model the evolution of the aggregate portfolio loss process directly.For a systematic comparison of the top-down and bottom-up approaches we refer to (Bieleckiet al., 2010, Section 2) and Giesecke (2012). One may also classify the portfolio credit modelsas static or dynamic . In static models the particular interest is on the default time distributionsof constituents at a given point in time. This results in a model which does not allow for theconsistent pricing across different maturities. On the other hand, the dynamic models specifythe evolution of default time distribution or the total loss process depending on the top-down orbottom up framework that is followed. Prior to the financial crisis of 2008, copula-based mod-els (see, e.g., (Li, 2000)), which are static and focus on the default risk only, were the marketstandard due to the ease of implementation. The financial crisis 2008 showed that these modelswere inadequate. It highlighted the necessity for dynamic pricing and hedging models.In this paper, we propose an affine two-factor model within the framework of Filipovi´c et al.(2011) to dynamically price and hedge STCDOs. The proposed model is parsimonious andcapable of capturing the dynamics of senior tranches, thanks to the catastrophic risk compo-nent incorporated into the model. Moreover, the resulting loss given default distribution is aweighted mixture with stochastic weights: the first affine factor tunes the truncated exponentialdistribution, while the second factor drives the catastrophic component.We focus on the dynamic hedging of STCDOs with the underlying index swap by computingthe variance-minimizing hedging strategy corresponding to our model specification. Then, wetest the performance of the variance-minimizing hedge on the iTraxx Europe data. This analy-sis is complementary to Filipovi´c and Schmidt (2010) in the sense that it carries the theoreticalframework of that study into implementation. The novelty of this analysis is also due to the dataset we use. Although this data set does not include any default event, it covers a sufficientlylong time horizon which witnessed extreme market conditions such as the credit crisis in 2008.In order to investigate further on the performance of the hedging strategy, we run a simulationanalysis. Specifically, along with the normal scenarios we generate extreme scenarios by em-ploying the importance sampling technique. Both in-sample hedging and simulation analysissuggest that the variance-minimizing strategy is most effective for mezzanine tranches in termsof yielding less riskier hedging portfolios and it fails to provide adequate hedge performanceregarding equity tranches, which is in line with the results by Ascheberg et al. (2013).We estimate the factor model based on the iTraxx Europe data with six tranches and fourdifferent maturities, using a quasi-maximum likelihood (QML) approach in conjunction withthe Kalman filter. The QML approach necessitates the knowledge of the first two conditionalmoments of the factor process. Accordingly, we use the polynomial preserving property of affineprocesses and compute the conditional mean and variance of the factor process explicitly. Ourfindings suggest that apart from being analytically tractable, the two-factor affine model witha catastrophic component is capable of describing the dynamics of the tranche data simultane-ously. However, a two-factor affine model with the restriction of a zero catastrophic componentis not able to capture the dynamics of senior tranches, in particular during the crisis period (fora similar result in the context of a structural model, see (Collin-Dufresne et al., 2012)).2o our knowledge, Duffie and Gârleanu (2001), in which correlated intensities are con-structed for constituent names by using affine factor processes, is the first study addressing thedynamic framework for pricing CDOs. Schönbucher (2005), Sidenius et al. (2008), Arnsdorfand Halperin (2008), Frey and Backhaus (2010), Filipovi´c et al. (2011) and Cont and Minca(2013) are other examples of dynamic models for CDO pricing. Schönbucher (2005), Sideniuset al. (2008) and Filipovi´c et al. (2011) are very much in the same spirit that they aim to modelthe evolution of the forward distribution of the loss process. This allows for a consistent in-corporation of the dynamics of credit spreads in the modeling of multi-name credit derivatives.Inspired by the HJM framework ( see, (Heath et al., 1992)) for a default-free term structure,Filipovi´c et al. (2011) develop a dynamic no-arbitrage setting for the evolution of the forwardcredit spreads. Allowing for feedback and contagion effects, this framework provides a general-ization for the aforementioned top-down models. Furthermore, under this general framework ananalytically tractable class of doubly-stochastic affine term structure models is proposed.Frey and Backhaus (2010) study the dynamic hedging of STCDOs in a framework wherespread risk is incorporated along with the default contagion (for a static infectious default model ,see, (Davis and Lo, 2001). Reckoning with the incompleteness of the market arising from thepresence of spread and default risk, this study utilizes a variance-minimizing strategy for thehedging of STCDOs with underlying CDSs (see, e.g., (Schweizer, 1999) for the review ofvariance-minimizing and other quadratic hedging approaches). Notably, Frey and Backhaus(2010) show that the variance minimization provides a model-based endogenous interpolationbetween the hedging against spread risk and default risk. For the hedging of a STCDO positionwith a dynamically rebalanced portfolio on the index, Filipovi´c and Schmidt (2010) derive thevariance-minimizing hedging strategy based on a general top-down framework. Ascheberg et al.(2013) tests the performance of more than 10 credit models during the crisis period and find thatthe hedging performances of all models are unsatisfactory. In particular they showed that thetop-down models fail to hedge equity tranches.The remainder of this paper is structured as follows: Section 2 gives the basics of STCDOs,introduces the two-factor affine model as well as the corresponding variance minimizing strat-egy. Section 3 provides the estimation methodology in detail. Section 4, which is on the dataimplementation, presents the data set, discusses the practical issues and gives results on estima-tion, in-sample hedging and simulation. Section 5 summarizes the main findings and concludesthe paper. The technical appendix contains auxiliary results and proofs.
2. Modeling Framework
We fix a stochastic basis (Ω , F , ( F t ) , P ) satisfying the usual conditions and where P denotesthe historical probability measure. We consider a CDO pool of credits with the overall outstand-ing notional normalized to . The aggregate loss process, representing the ratio of CDO lossesrealized by time t , is indicated by L t . It is assumed that L t = (cid:88) s ≤ t ∆ L s is [0 , -valued, non-decreasing marked-point process with absolutely continuous P -compensator ν ( t, dx ) dt . 3e define, for any T > and x ∈ [0 , , the hypothetical ( T, x ) -bond which pays { L T ≤ x } at maturity T . That is, we have a defaultable zero-recovery zero-coupon bond, which pays ifthe realized loss fraction at T is less than or equal to x , and otherwise. The ( T, x ) -bond priceat time t ≤ T is denoted by P ( t, T, x ) . It follows that the risk-free zero-coupon bond price P ( t, T ) equals P ( t, T, . Throughout, we assume that the risk-free rate r is constant, so that P ( t, T ) = e − r ( T − t ) . ( T, x ) -bonds have the spanning property : any European contingent claim F ( L T ) with ab-solutely continuous payoff function F can be decomposed into the sum of ( T, x ) -bond payoffs, F ( L T ) = F (1) − (cid:82) F (cid:48) ( x )1 { L T ≤ x } dx . Hence the claim can be replicated by the static portfoliowith the value process F (1) P ( t, T ) − (cid:90) F (cid:48) ( x ) P ( t, T, x ) dx. (1)This, in particular, allows for the pricing of a STCDO via ( T, x ) -bonds. We recall that (see e.g.(Embrechts et al., 2005, Chapter 9)) a STCDO issued at time isspecified by a sequence of coupon payment dates < T < · · · < T n , a tranche with attachmentand detachment point x < x in [0 , , and a coupon rate κ ( x ,x ]0 . The attachment point x indicates the level at which losses in the underlying CDO pool begin to erode the notional of thetranche. At the detachment point x the full tranche is written down. Note that ( x , x ) = (0 , corresponds to the entire index . The holder of a long position in a STCDO• receives κ ( x ,x ]0 × H ( x ,x ( L T i ) at T i , i = 1 , , . . . , n ( coupon leg )• pays − ∆ H ( x ,x ( L t ) = H ( x ,x ( L t − ) − H ( x ,x ( L t ) at any time t ≤ T n where ∆ L t (cid:54) =0 ( protection leg )where we define H ( x ,x ( x ) := (cid:82) x x { x ≤ y } dy = ( x − x ) + − ( x − x ) + . It follows from (1)that the value at time t ≤ T n of the coupon leg is given by κ ( x ,x ]0 × S ( x ,x ( t ) , where we definethe annuity factor S ( x ,x ( t ) = (cid:88) t Let α , β y and β z be some non-increasing and càdlàg functions with α ( x ) = r and β y ( x ) = β z ( x ) = 0 for x ≥ . Then, under the above assumptions, there exists a lossprocess L which is unique in law such that P ( t, T, x ) = 1 { L t ≤ x } e − A ( T − t,x ) − B y ( T − t,x ) Y t − B z ( T − t,x ) Z t (7) defines an arbitrage-free ( T, x ) -bond market, where the functions A , B y and B z solve the Ric-cati equations ∂ τ A ( τ, x ) = α ( x ) + κ z θ z B z ( τ, x ) ,A (0 , x ) = 0 ,∂ τ B y ( τ, x ) = β y ( x ) − ( κ y + λ y ) B y ( τ, x ) − σ y B y ( τ, x ) ,B y (0 , x ) = 0 ,∂ τ B z ( τ, x ) = β z ( x ) + κ y B y ( τ, x ) − ( κ z + λ z ) B z ( τ, x ) − σ z B z ( τ, x ) ,B z (0 , x ) = 0 . Moreover, the compensator of L is given by ν ( t, (0 , x ]) = α ( L t − ) − α ( L t − + x )+( β y ( L t − ) − β y ( L t − + x )) Y t +( β z ( L t − ) − β z ( L t − + x )) Z t . Proof. Proof follows the same arguments as in Theorem 7.2 of Filipovi´c et al. (2011). That is, e − rt P ( t, T, x ) , ≤ t ≤ T , is a Q -martingale, for any T > and x ∈ [0 , . Corollary 2.1. The implied discounted gains process is a square-integrable Q -martingale withdynamics dG ( x ,x ] t = e − (cid:82) t r u du (cid:32) B ( x ,x ] t d (cid:102) W t + (cid:90) (0 , C ( x ,x ] t ( ξ )( µ ( dt, dξ ) − ν ( t, dξ ) dt ) (cid:33) where µ ( dt, dx ) denotes the integer-valued random measure associated to the jumps of the lossprocess L , and B ( x ,x ] t = (cid:90) ( x ,x ] (cid:40) κ ( x ,x ]0 (cid:88) t 3. Parameter Estimation and Filtering In the current framework the fundamental object modeled is the hypothetical term-structureof ( T, x ) -bonds, which is not directly observable in the market. However, given the marketobservable par coupon rates for all tranches and the index, the term-structure of ( T, x ) -bondscan be estimated via inverting the formula (2). More precisely, we assume there are J trancheswith attachment/detachment points x < x < · · · < x J = 1 . Denoting time to maturitywith τ = T − t , we first estimate the zero-coupon discount curve τ (cid:55)→ D ( t, τ, j ) = 1 x j − x j − (cid:90) x j x j − P ( t, t + τ, x ) dx (10)for all tranches ( x j − , x j ] . This, in turn, gives the implied zero-coupon spread curve R ( t, τ, j ) = − τ log D ( t, τ, j ) − r. (11)Finally, one can get the term structure of ( T, x ) -bonds via interpolating (11) in x .Suppose we are given a data set that has K time steps consisting of zero-coupon spreadsfor all available tranches and maturities. In each time step of the sample period, the data set isrepresented by an ( I × J ) matrix where I denotes the number of different time to maturities.Having the data set and the model, the estimation procedure comprises the specification of themodel parameters in such a way that the model describes the whole data series as much aspossible. Under the current modeling setup, the difficulty we face in estimation is due to theunobservability of the factor process. One way to overcome this problem is to use filtering. In aframework where the unobserved factor is a Gaussian process, a Kalman filter yields the exactlikelihood function via providing the prediction error and its variance (see Harvey (1990)). Whenusing non-Gaussian models, however, the exact likelihood function is not available in most cases.In such a situation, one can use a quasi-maximum likelihood (QML) approach in which the ideais to substitute the exact transition density of the non-Gaussian factor by a Gaussian densitywith mean and variance being equal to the first two true moments of the factor process. Thishas been a widely used method especially for the estimation of affine term structure models(see, for example, Geyer and Pichler (1999), Chen and Scott (2003) and Duffee and Stanton(2012)). Both in the one and two-factor models we presented above, the factor process is non-Gaussian. Hence, to estimate the model parameters and obtain the unobservable factor series weuse a QML approach based on a Kalman filter. Since the one-factor model is nested within thetwo-factor model, in what follows we will only give the estimation procedure for the later one.Here, we consider the partition t < t < · · · < t K = T on the interval [0 , T ] anddenote the value of the factor process at time t k by ( Y t k , Z t k ) . In Kalman filtering, there is the measurement (observation) equation expressing the observed data as the sum of a linear functionof the unobservable factor and a measurement error. The discrete time evolution of the unob-servable factor at t k is, in turn, expressed by the transition equation as linear in ( Y t k − , Z t k − ) .9nserting (7) into (10) reveals that R ( t k , τ, i ) is not linear in ( Y t k , Z t k ) . In order to obtain a linearmeasurement equation, we approximate the function β y as follows: β y ( x ) ≈ (cid:88) j =1 β j [ x j − ,x j ) ( x ) where the coefficients are given by β j = 1 x j − x j − (cid:90) x j x j − β y ( x ) dx. This immediately yields B z ( τ, x ) = B z ( τ, x j ) for x ∈ [ x j − , x j ) , implying that (cid:90) x j x j − e − A ( τ,x ) dx = e − κ z θ z (cid:82) τ B z ( s,x j ) ds (cid:90) x j x j − e − α ( x ) τ dx which finally provides the desired linear measurement equation . For the i th time to maturity, τ i , i = 1 , · · · , I , and tranche j , j = 1 , · · · , J , the measurement equation reads: R ( t k , τ i , j ) = C z ( τ i , j ) + 1 τ i ( B y ( τ i , x j ) Y t k + B z ( τ i , x j ) Z t k ) + (cid:15) ( t k , j ) where C z ( τ i , j ) = 1 τ i log( x j − x j − )+ 1 τ i κ z θ z (cid:90) τ i B z ( s, x j ) ds − τ i log (cid:16) (cid:82) x j x j − e − γ ( e − a x ∧ − e − a ) τ i dx (cid:17) and measurement errors, (cid:15) ( t k , j ) , are assumed to be i.i.d. N (0 , (cid:112) h j ) , for some h j > , showingthat the variance of the error depends on the tranche j only. Now we define a new index l ( i, j ) =( j − I + i and introduce R t k , the ( IxJ ) -dimensional vector which has R ( t k , τ i , j ) in its l ( i, j ) thentry. Furthermore, we denote by H the corresponding ( IxJ ) × ( IxJ ) diagonal covariancematrix of observation errors which, for i = 1 , · · · , I , has h j as the l ( i, j ) th diagonal entries.Let P ( Y t k , Z t k | Y t k − , Z t k − ) designate the transition density, which is the probability densityof the factor at time t k given its value at time t k − . In line with the QML approach, we substitutethe exact transition density of the factor by a Gaussian density, that is P ( Y t k , Z t k | Y t k − , Z t k − ) ∼ N ( µ t k , Q t k ) where the conditional mean µ t k and the covariance matrix Q t k are distributed in such a way thatthe first moments of the approximate Normal and the exact transition density are equal. As thenext step we compute µ t k and Q t k in the following proposition. This proposition mainly usesthe fact that the factor process is affine and provides the desired expressions by utilizing thepolynomial preserving property of affine processes. Proposition 3.1. Suppose the process ( Y, Z ) satisfies the dynamics given in (5) - (6) , then the P -conditional expectation of Y t and Z t is in the following form: E [ Y t | Y = y, Z = z ] = θ z κ z − κ y (cid:16) κ z (1 − e − κ y t ) − κ y (1 − e − κ z t ) (cid:17) + e − κ y t y + e − κ z t κ y κ z − κ y (cid:16) e t ( κ z − κ y ) − (cid:17) z,E [ Z t | Y = y, Z = z ] = θ z (1 − e − κ z t )+ e − κ z t z. oreover, the conditional variances V y , V z and the conditional covariance V yz are given by: V y ( t, y, z ) = (cid:32) e − (5 κ z +7 κ y ) t (cid:16) e (5 κ z +7 κ y ) t ( κ z − κ y )( κ z − κ y ) ( κ z ( κ z + κ y ) σ y + κ y σ z ) θ z − e (5 κ z +6 κ y ) t κ z ( κ z − κ y )( κ z − κ y ) σ y ( κ z ( θ z − y ) + κ y ( y − z )) + e (3 κ z +7 κ y ) t ( κ z − κ y ) κ y ( κ z + κ y ) σ z ( θ z − z + 2 e (4 κ z +7 κ y ) t κ y ( κ y − κ z )( κ z ( σ y − σ z ) + 2 κ y σ z ) × ( θ z − z ) − e (4 κ z +6 κ y ) t κ z ( κ z − κ y ) κ y σ z ( κ z θ z − ( κ z + κ y ) z ) + e κ z + κ y ) t κ z ( κ z + κ y ) × ( κ z σ y ( θ z − y ) − κ z κ y σ y ( θ z − y + z ) + 2 κ y (2 σ y y − σ y z + σ z z )+ κ z κ y ( − σ z θ z + σ y ( θ z − y + 4 z ))) (cid:17)(cid:33)(cid:44)(cid:32) κ z ( κ z − κ y )( κ z − κ y ) κ y ( κ z + κ y ) (cid:33) , (12) V z ( t, y, z ) = σ z e − κ z t ( e κ z t − e κ z t − θ z + 2 z )2 κ z , (13) V yz ( t, y, z ) = e − (2 κ z + κ y ) t σ z κ z − κ z κ y ) (cid:16) e (2 κ z + κ y ) t ( κ z − κ y ) κ y θ z − e κ y t κ y ( κ z + κ y )( θ − z ) − e ( κ z + κ y ) t ( κ z − κ y )( θ z − z ) + 2 e κ z t κ z ( κ z θ z − ( κ z + κ y ) z ) (cid:17) . (14) Proof. See Appendix.We are now ready to give the transition equation implied by the two-factor model. Denotethe time increment by δt = t k − t k − and define M ( t k ) = (cid:32) θ z κ z − κ y (cid:0) κ z (1 − e − κ y δt ) − κ y (1 − e − κ z δt ) (cid:1) θ z (1 − e − κ z δt ) (cid:33) ,M ( t k = (cid:32) e − κ y δt e − κ z δt κ y κ z − κ y (cid:0) e δt ( κ z − κ y ) − (cid:1) e − κ z δt (cid:33) . The unobserved state at time t k is evolved from the previous state according to: (cid:18) Y t k Z t k (cid:19) = M ( t k ) + M ( t k ) (cid:18) Y t k − Z t k − (cid:19) + v t k , where v t k are i.i.d. N (0 , Q ( t k )) with the covariance matrix Q ( t k ) = (cid:18) V y (cid:0) δt, Y t k − , Z t k − (cid:1) V yz (cid:0) δt, Y t k − , Z t k − (cid:1) V yz (cid:0) δt, Y t k − , Z t k − (cid:1) V z (cid:0) δt, Y t k − , Z t k − (cid:1) (cid:19) and V y , V z and V yz are as given in (12), (13) and (14) respectively.Letting t → ∞ in the conditional moments given in Proposition 3.1 yields the unconditionalmoments of the factor process which are given by the following corollary.11 orollary 3.1. Unconditional mean, variance and covariance of Y t and Z t is given by: µ y = θ z , µ z = θ z ,V y = σ y θ z κ y + κ y θ z σ z κ z + κ y ) κ z ,V z = σ z θ z κ z ,V yz = σ z θ z κ y κ z ( κ z + κ y ) = κ y ( κ z + κ y ) V z . Denoting the transpose of a matrix A by A (cid:62) , we use the notation (cid:0) Y t m | t n , Z t m | t n (cid:1) (cid:62) to repre-sent the estimate of the state of the factor process at t m given observations up to, and includingtime t n . In the same vein the estimate for the error covariance matrix, which serves as a measurefor the precision of the state estimate, will be denoted by P t m | t n . In what follows we shed lighton the filtering algorithm.Given the parameter set ϕ = ( κ z , κ y , θ z , λ z , λ y , σ z , σ y , a , γ, b , c , H ) , the Kalman filterconsists of prediction and updating steps which are applied for each time point in the data samplein the following way:1. Initialize the filter by using the unconditional moments: (cid:18) Y | Z | (cid:19) = (cid:18) θ z θ z (cid:19) , P | = σ y θ z κ y + κ y θ z σ z κ z + κ y ) κ z σ z θ z κ y κ z ( κ z + κ y ) σ z θ z κ y κ z ( κ z + κ y ) σ z θ z κ z Prediction: produce an estimate of the current state and covariance of the estimate via (cid:18) Y t k | t k − Z t k | t k − (cid:19) = M ( t k )+ M ( t k ) (cid:18) Y t k − | t k − Z t k − | t k − (cid:19) and P t k | t k − = M ( t k ) P t k − | t k − M ( t k ) (cid:62) + Q ( t k ) , respectively.3. Updating: for all i , i = 1 , · · · , I , and j , j = 1 , · · · , J , compute R ( t k | t k − , τ i , j ) = C z ( τ i , j )+ 1 τ i (cid:0) B y ( τ i , x j ) Y t k | t k − + B z ( τ i , x j ) Z t k | t k − (cid:1) , and form the corresponding IxJ -dimensional row vector R t k | t k − . Next, compute R t k and the innovation vector successively as e t k = R t k − R t k | t k − . ( IxJ ) × matrix, B , having B y ( τ i , x j ) τ i and B z ( τ i , x j ) τ i in the ( l ( i, j ) , stand ( l ( i, j ) , nd entries, respectively. Now, compute the innovation covariance matrix, F t k via F t k = BP t k | t k − B (cid:62) + H. After this, calculate the Kalman gain with the following: K t k = P t k | t k − B (cid:62) F − t k . Lastly, update the state vector and the estimate covariance by (cid:18) Y t k | t k Z t k | t k (cid:19) = (cid:18) Y t k | t k − Z t k | t k − (cid:19) + K t k e t k and P t k | t k = P t k | t k − − K t k BP t k | t k − , respectively.The Kalman filter provides the following likelihood function: log L ( R , R , ..., R N ; ϕ ) = − K π − K (cid:88) k =1 log | F t k | − K (cid:88) k =1 e (cid:62) t k F − t k e t k . Notice that L is a function of e t and F t which, eventually, depend on the parameter set ϕ . Thus,as the last step of the QML method, we choose ϕ in such a way that the likelihood functionis maximized. Finally, we want to point out that the observed data vectors may change sizeover the sample period. This is due to the unavailability of the data for some tranches and/ormaturities in some days of the sample period. To overcome this problem, we adjust the Kalmanfilter algorithm in such a way that it takes the changes in the size of the data into account. 4. Data Implementation In this section one and two-factor models described above are implemented on the real mar-ket data. Moreover, we perform an in-sample hedging analysis. We also run a simulation wherenormal and extreme loss scenarios are generated via method of importance sampling. Finallywe assess the hedging performance under these more general scenarios. The raw data comprises daily observations of iTraxx Europe from 30 August 2006 to 3August 2010. The stripped data, which has been sourced from Bank Austria , is the zero-coupon spreads, R ( t k , τ i , j ) , for four different time to maturities ( τ , · · · , τ ) = (3 , , , and six tranches j = 1 , ..., with standard attachment and detachment points , , , , , , . This corresponds to K = 972 observation days in each of which we have a × observation matrix. We thank Peter Schaller for providing the data. 13e illustrate the time series of 5-year zero-coupon spreads and the index spreads across fourdifferent maturities in Figure 1a and Figure 1b, respectively. Naturally, market conditions arereflected in the data set. The index and tranche data follow relatively stable pattern from thebeginning of the data period to July 2007, where the credit crunch erupted. In March 2008,we observe a spike in the spread data which stems from the panic due to the possibility of thecollapse of Bear Stearns. Furthermore, a drastic upward movement is observed starting fromSeptember 2008. This time period corresponds to the breakdown of the credit market due toevents such as the bankruptcy of Lehman Brothers. Moreover, Figure 1a and Figure 1b togethershow that the tranche data and the index data have the same up and downward trends during thetime period considered. One other feature of the data set we use is that there is no default eventduring the sample period. (a) iTraxx Europe 5-year zero-coupon spread data for alltranches. (b) iTraxx Europe index spread data for all maturities.Figure 1: iTraxx Europe data from 30 August 2006 to 3 August 2010 Running the estimation algorithm given in Section 3 we fit the one and two-factor models tothe iTraxx data. During the analysis the risk-free rate is considered to be constant at r = 0 . .In the following we discuss the results of the empirical analysis.As mentioned earlier, the QML approach makes it possible to estimate the model parametersand filter out the unobservable factor series simultaneously. We obtain the parameter estimatesgiven in Table 1. The likelihood values given in the table suggest that the corresponding like-lihood ratio test statistic (LRT) is highly significant and rejects the null hypothesis that theone-factor model is equivalent to the two-factor model.The filtered factor series are depicted in Figure 2. It is remarkable how the factor Z , whichdrives the catastrophic component, stays almost zero until the breakout of the credit crisis. LRT is given as the LL − LL ) where LL and LL denote the value of the log-likelihood for the nestedone-factor model and the two-factor model, respectively. θ z κ y - - 0.5223 0.0638 κ z σ y - - 0.3806 0.0163 σ z λ y - - -0.6671 0.0647 λ z a γ b c - - 0.0911 0.0096Log-likelihood 6.9365e+004 8.7842e+004 Table 1: Parameter estimates and corresponding standard errors (SE) for the one-factor and two-factor affine models.Figure 2: Filtered series of the factor Y and Z for the period 30 Aug 2006-3 Aug 2010. Taking − , − and − tranches as representatives for equity, mezzanineand senior tranches, respectively, we plot actual vs estimated zero-coupon spreads of 5-yearmaturity in Figure 3. We observe that the two-factor model yields a plausible fit across alltranches, outperforming the one-factor model. Also, it is remarkable how the one-factor modelestimates are far below the actual data whereas the two-factor model provides almost a perfectfit for the senior tranche. Here, we want to point out that a two-factor affine factor model withthe restriction of a zero catastrophic component, as the one-factor-model is not able to fit thesuper-senior tranche. There, the importance of the catastrophic risk component of the two-factormodel comes into play. That is, under the two-factor affine framework including the catastrophiccomponent becomes inevitable for a model fit in senior tranches.15 ug06 Jun07 Mar08 Dec08 Sep09 Jun100.050.10.150.2 S p r ead Equity Tranche, 5-Year Maturity Aug06 Jun07 Mar08 Dec08 Sep09 Jun100.010.020.030.040.05 S p r ead Mezzanine Tranche, 5-Year Maturity Aug06 Jun07 Mar08 Dec08 Sep09 Jun1012345678 S p r ead -3 Senior Tranche, 5-Year Maturity Figure 3: Actual vs Estimated Spread Series Given the parameter estimates and filtered factor series we simulate a loss trajectory. In-serting the parameter estimates, filtered factor series and the value of the loss trajectory intoformula (9), we obtain the implied expected loss given default series. Moreover, to investigatethe effect of the catastrophic component, we fix the catastrophic risk parameter at c = 0 andcompute the corresponding expected loss given default series. Figure 4 demonstrates how theimplied expected loss given default with and without catastrophic component change in the sam-ple period. Following from the fact that the factor Z stays very close to zero by August 2007,the series with and without catastrophic component coincide in the corresponding part of thesample period. This stipulates that the catastrophic risk component is needed during the crisistimes. Hence, we conclude that considering a non-zero catastrophic component provides moreflexibility in the modeling of expected loss given default in distressed markets.16 igure 4: Expected loss given default with and without catastrophic component We first specify a hedging time horizon [0 , T ] , and trading days t < t < · · · t K = T with δt := t k − t k − ≡ / .We denote by CP = { T , . . . , T n } the set of coupon payment dates and assume that anycoupon payment date satisfies T i ∈ { t , . . . , t K } for all i .Assume a time series of the loss process L t k , and the spot value processes Γ ( x ,x ] t k and Γ (0 , t k , k = 0 , . . . , K , are given. Consider a hedging strategy denoted by φ t k . The resulting nominalvalue process V t k of the self-financing hedging portfolio with zero initial capital, V t = 0 , isgiven by the recursive formula V t k = V t k − e rδt + P L (0 , t k where P L (0 , t k = φ t k − (cid:16) Γ (0 , t k − Γ (0 , t k − e rδt k +1 { t k ∈ CP } κ (0 , t H (0 , ( L t k ) − (cid:16) H (0 , ( L t k − ) − H (0 , ( L t k ) (cid:17)(cid:17) indicates the nominal daily profit and loss on ( t k − , t k ] . The discounted gains process G (0 , t k ofholding the index is then given by G (0 , t k = e − rt k V (0 , t k where V (0 , t k denotes the nominal valueprocess for the strategy φ t k ≡ . And following the same way one can obtain G ( x ,x ] t k .The hedging algorithm consists of computation and comparison of the nominal value pro-cess of the hedging portfolio and the gains process of the tranche position for each day of thehedging period. For the considered sample period, we implement the hedging algorithm for allattachment and detachment points. Taking − , − and − tranches as repre-sentatives for equity, mezzanine and senior tranches, respectively, Figure 5 depicts the hedgingstrategies, and the series for the gains process and nominal spot value of the STCDOs as wellas the hedging portfolio values. The change in φ at the beginning of the crisis around July2007 is observed to exhibit different patterns for each tranche: while for equity tranches φ isdecreasing in absolute value, indicating a reduction in insurance demand, for the senior tranchethere is the opposite behavior. We attribute this change to the changing correlation structurebetween tranches and the index: running correlation series indicate an upward jump towards Aug06 Jun07 Mar08 Dec08 Sep09 Jun10-30-25-20-15-10-505 Tranche Spot ValueTranche P&L ProcessHedging portfolio Aug06 Jun07 Mar08 Dec08 Sep09 Jun10-0.3-0.25-0.2-0.15-0.1-0.050 (a) Equity tranche Aug06 Jun07 Mar08 Dec08 Sep09 Jun10-8-7-6-5-4-3-2-101 Aug06 Jun07 Mar08 Dec08 Sep09 Jun10-0.075-0.07-0.065-0.06-0.055-0.05-0.045-0.04-0.035-0.03 (b) Mezzanine tranche Aug06 Jun07 Mar08 Dec08 Sep09 Jun10-2.5-2-1.5-1-0.500.511.5 Aug06 Jun07 Mar08 Dec08 Sep09 Jun10-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.10 (c) Senior trancheFigure 5: Hedging results for the sample period: gains process, nominal spot value and hedging portfolio value forequity, mezzanine and senior tranches (left); the hedging strategy φ (right). In order to better assess the performance of a given hedging strategy, Cont and Kan (2011)18uggest reduction in volatility criteria which measures the reduction in the dispersion of theprofit distribution with respect to the unhedged position. Formally, it is defined as the ratio ofthe volatility of daily profits from the hedged position to the volatility of the daily profits fromthe unhedged tranche position. According to this criteria, a hedging strategy performs betteras long as the related reduction in volatility value is smaller. We report the reduction in thevolatility of P&L in Figure 6. According to this criteria, the hedging strategy performs better forthe mezzanine tranches. Tranche % Figure 6: Reduction in volatility In the simulation analysis our objective is to elaborate on the performance of the varianceminimizing hedging strategy in a more general framework, where scenarios with nonzero lossesare permitted. Recall that in the current modeling setup we deal with three stochastic processes,namely the factors Y , Z and the loss process L . We use Euler discretization to approximate thediscrete time evolution of the factors Y and Z in Equation (5)-(6) on the equidistant time grid t < t < ... < t K = T with t k +1 − t k ≡ δt = 1 / .As the next step, to simulate the loss process L we use the simulated factors Y , Z , the setof parameter estimates and an additional parameter Ψ which we interpret as the importancesampling parameter. The reason why we need the parameter Ψ is as follows: there does notoccur any default during the sample period we consider. Hence, the parameter set coming fromthe in-sample analysis is not able to generate a remarkable number of jumps. Moreover, theMonte Carlo simulation is known to fail in generating rare events unless the number of simulatedscenarios is very large. Nevertheless, a frequently used technique in stress scenario generationis importance sampling (see, e.g. Boyle et al. (1997, Section 2.7) and references therein). In thiscontext, it is possible to manipulate the number of jumps via amplifying the jump intensity givenin (8) through the parameter Ψ . However, one should take care of the necessary measure changefor the adjustment of probabilities assigned to each scenario. This is necessary in particular forcomputing the empirical distribution function of the losses.We define the cumulative intensity process ¯Λ t = (cid:82) t Λ s ds and sketch the algorithm forsimulating a loss trajectory of length K as follows.19 lgorithm 4.1 (Loss process simulation) . 1. Initiate the jump time τ = 0 , the number ofjumps N = 0 , the loss process L t = 0 , and the cumulative arrival intensity ¯Λ t = 0 .2. Generate a number U from exponential distribution with parameter .3. While k < K and ¯Λ t k − ¯Λ τ < U calculate ¯Λ t k +1 via ¯Λ t k +1 = ¯Λ t k + Ψ( α ( L τ ) + β y ( L τ ) Y t k + β z ( L τ ) Z t k − r ) δt set L t k +1 = L t k , k (cid:55)→ k + 1 4. If ¯Λ t k − ¯Λ τ ≥ U , i.e., when a jump occurs generate a number s from the standard uniformdistribution. Compute the jump size via ∆ L t k = F − L ( L t k , Y t k , Z t k , s ) where F L is the cumulative loss given default distribution given by F L ( L t k , Y t k , Z t k , x ) = α ( L t k ) + β y ( L t k ) Y t k + β z ( L t k ) Z t k − α ( L t k + x ) α ( L t k ) + β y ( L t k ) Y t k + β z ( L t k ) Z t k − r − β y ( L t k + x ) Y t k + β z ( L t k + x ) Z t k α ( L t k ) + β y ( L t k ) Y t k + β z ( L t k ) Z t k − r Update the loss path, jump time and number of jumps L t k = L t k + ∆ L t k , τ = t k , N = N + 1 5. If k < K return to 2, else stop. Employing the methodology described above, we simulate scenarios, of whichare the normal scenarios and generated via taking importance sampling parameter Ψ = 1 . On theother hand, in order to simulate stress scenarios we set Ψ = 100 . We also set the probabilityestimate of each of the normal scenarios equal to q ( i ) = 1 / , i = 1 , , ..., , whilefor the stress scenarios we adjust the probability estimate of each scenario in the following way.Denote by τ n the n th jump time of the process L . Changing the jump intensity of theprocess L from Λ t to ΨΛ t and leaving the jump size distribution unchanged is tantamount to anequivalent change of measure where the measure P Ψ ∼ P is characterized by d P Ψ d P |F t = M t with the Radon-Nikodym derivative M t given by (see, e.g. Brémaud (1981, Chapter VIII, T10)) M t = (cid:89) n ≥ Ψ1 { τ n ≤ t } e (cid:82) t (1 − Ψ)Λ s ds = (cid:89) n ≥ Ψ1 { τ n ≤ t } e ¯Λ t (1 − Ψ) . (15)20ow suppose we generate the i th stress scenario from the distribution P Ψ . For this particularscenario we denote the total number of jumps realized in [0 , t K ] by N i . Then, according to (15)we define the corresponding weight of the scenario i under P by w ( i ) = e (Ψ − (cid:80) K − k =0 ( α ( L tk )+ β y ( L tk ) Y tk + β z ( L tk ) Z tk − r ) δt Ψ N i . Due to the law of large numbers we have (cid:80) i =1 w ( i )1000 ≈ . Nevertheless, we normalize the weight w ( i ) in an exact way and obtain the following estimate for the P -probability of scenario ip ( i ) = w ( i ) (cid:80) i =1 w ( i ) . Finally, to aggregate the scenarios we give equal weight to normal and stress scenarios, andset probabilities ¯ q ( i ) = q ( i ) / and ¯ p ( i ) = p ( i ) / , so that (cid:80) i =1 ¯ q ( i ) + ¯ p ( i ) = 1 , as it should be.Given the parameter estimates and the filtered factor series ( Y t k , Z t k ) we utilize formula (3) toconstruct series for the spot value of -year index and -year STCDO tranches. Then, we studythe hedging of STCDOs with the index. Moreover, we perform a conditional simulation analysisin which loss scenarios are generated conditional on the original filtered factor trajectories.The results are given below.We take the set of simulated scenarios and focus on the final date, T = 250 , of the simulationperiod. The empirical cumulative loss distribution function (under P ) at T is given in Figure 7.According to the figure, the simulation procedure is successful in the sense that it is able toproduce loss scenarios ranging between and . Figure 7: Empirical cumulative loss distribution at T = 250 for a total number of 2000 scenarios. In Figure 8 we depict the date T empirical cumulative distribution function of the totalhedging portfolio P&L. This figure implies that for mezzanine and senior tranches, variance-minimizing hedging strategies yield normalized total portfolio P&L values which are close tozero in most of the simulated trajectories. In other words, on average the hedging strategyperforms well for the mezzanine and senior tranches.21 80 -60 -40 -20 0 20 Profit c u m u l a t i v e p r obab ili t y EquityMezzanineSenior Figure 8: Empirical distribution of normalized total portfolio P&L at T . Next, for each scenario realization we compute the reduction in volatility for all tranchesand compute the descriptive statistics for reduction in volatility by taking the Radon-Nikodymdensities of the stress scenarios into account. Table 2 illustrates the results: variance minimizinghedge is observed to yield the greatest reduction in variance for the mezzanine tranches. Mean Median Std CV Max MinEquity 64.95 55.72 43.56 0.67 601.6 3.36Mezzanine 43.28 36.01 34.33 0.79 2223.6 10.26Senior 50.35 34.40 62.19 1.23 3261.6 15.98 Table 2: Descriptive Statistics of reduction in volatility for the variance-minimizing hedge We now present the results of the conditional simulation analysis. We fix the filtered factorseries given in Figure 2 and conditional on these trajectories we simulate 2000 loss scenariosagain with the importance sampling parameter values Ψ = 1 and Ψ = 100 . Conditional dis-tribution of the simulated loss process (under P ) at time T is depicted in Figure 9. One strikingresult is that, when compared with the loss distribution function given in Figure 7, conditionalloss distribution in Figure 9 gives higher probability to losses greater than . Moreover,simulation results suggest that for normal scenarios, Ψ = 1 , in 815 of 1000 simulated loss tra-jectories, there occurred a jump, that is, a default. These findings suggest that an actual defaultevent in the iTraxx was very much likely to occur.22 igure 9: Empirical conditional cumulative loss distribution function at T = 250 for a total number of 2000 scenarios. 5. Conclusion In this study, we propose an affine two-factor model for the pricing and hedging of STCDOs.The most distinguishing feature of this model lies in the fact that a catastrophic risk componentis considered as a tool for explaining the dynamics of the super-senior tranches. To test the realworld performance we estimate the affine factor model on the iTraxx Europe data covering aperiod which witnessed different market conditions such as the recent credit crisis. As the maintool for the estimation of the affine factor model we use quasi-maximum likelihood based on aKalman filter. This method requires the knowledge of conditional moments of the factor process.In this context, we utilize the polynomial preserving property for affine diffusion processes andcompute the first two conditional moments of the factor process explicitly. Estimation resultsshow that the two-factor model with the catastrophic component is successful in terms of fittingthe market data even for super-senior tranches. Next, we compute the variance-minimizing hedg-ing strategy based on the affine model. We investigate performance of the variance-minimizinghedging strategy on the data. We also ran a simulation analysis, in which the objective is to testthe performance of the hedging strategy under more general loss scenarios.Our findings suggest that within the data period, According to the reduction in volatilitycriteria, the variance minimizing strategy is effective in reducing the risk of mezzanine tranches.The simulation study yields results of the same direction.23 . Appendix Proof of Proposition 3.1. Suppose we are given the process X := ( Y, Z ) where Y and Z solve(5) and (6), respectively. This suggests that X is an affine diffusion process with the state space X = R (for a detailed information on affine diffusions see, e.g., Filipovi´c (2009, Chapter10)). Now let τ ≥ , k ∈ N , ( y, z ) ∈ X and denote the k th conditional cross moment of X T , T = t + τ , by f k ( τ, Y t , Z t ) = E [ Y pt + τ Z qt + τ | Y t = y, Z t = z ] p, q ∈ N , p + q = k. An affine diffusion has the property that the k th conditional moment always exists and is apolynomial of at most degree k of the current state ( y, z ) (see Theorem 2.5 in Filipovi´c andLarsson (2020) ). Being an affine diffusion, X also possesses the Markov property (see, forinstance Revuz and Yor (1999, Chapter III) for a detailed information on Markov processes). Inparticular, formally f k ( τ, y, z ) solves the Kolmogorov backward equation ∂∂τ f k ( τ, y, z ) = L f k ( τ, y, z ) ,f k (0 , y, z ) = y p z q (16)where L denotes the infinitesimal generator of the process X given by: L = κ y ( z − y ) ∂∂y + κ z ( θ z − z ) ∂∂z + 12 σ y y ∂ ∂y + 12 σ z z ∂ ∂z . To compute the conditional moments, taking the polynomial preserving property of process X into account, one can use a polynomial ansatz in equation (16). Then, matching the coefficientsyields a system of ordinary differential equations whose solution gives the coefficients of thepolynomial in the ansatz. In the following, we follow this procedure as the first step towards thecomputation of the moments up to and including order two: ( i) Let E [ Z t + τ | Y t = y, Z t = z ] =: g ( τ, Y t , Z t ) . Function g formally solves the Kolmogorovbackward equation, that is, ∂ τ g = κ y ( z − y ) ∂ y g + κ z ( θ z − z ) ∂ z g + 12 σ y y∂ yy g + 12 σ z z∂ zz g (17)Since X is an affine process, we have the polynomial property of moments, that is, g is in thefollowing form g ( τ, y, z ) = g ( τ ) + g y ( τ ) y + g z ( τ ) z (18)for some functions g , g y , g z . Plugging (18) into (17) gives ddτ g + ddτ g y y + ddτ g z z = κ y ( z − y ) g y + κ z ( θ z − z ) g z ddτ g = κ z θ z g z ,g (0) = 0 ,ddτ g y = − κ y g y ,g y (0) = 0 ,ddτ g z = κ y g y − κ z g z ,g z (0) = 1 . Solving above system we get g z ( τ ) = e − κ z τ , g y ( τ ) ≡ and g ( τ ) = θ z (1 − e − κ z τ ) implyingthat g ( τ, y, z ) = θ z (1 − e − κ z τ ) + e − κ z τ z ( ii) We set E [ Y t + τ | Y t = y, Z t = z ] =: h ( τ, Y t , Z t ) . Formally, h satisfies ∂ τ h = κ y ( z − y ) ∂ y h + κ z ( θ z − z ) ∂ z h + 12 σ y y∂ yy h + 12 σ z z∂ zz h (19)From the polynomial property of moments again we have h ( τ, y, z ) = h ( τ ) + h y ( τ ) y + h z ( τ ) z. (20)Plugging (20) into (19) gives ddτ h + ddτ h y y + ddτ h z z = κ y ( z − y ) h y + κ z ( θ z − z ) h z . Matching the coefficients we obtain ddτ h = κ z θ z h z ,h (0) = 0 ,ddτ h y = − κ y h y ,h y (0) = 1 ,ddτ h z = κ y h y − κ z h z ,h z (0) = 0 . Solving the system, we get h ( τ ) = θ z κ z − κ y ( κ z (1 − e − κ y t ) − κ y (1 − e − κ z τ )) h y ( τ ) = e − κ y τ , h z ( τ ) = e − κ z τ κ y κ z − κ y ( e τ ( κ z − κ y ) − h ( τ, y, z ) = θ z κ z − κ y ( κ z (1 − e − κ y τ ) − κ y (1 − e − κ z τ )) + e − κ y τ y + e − κ z τ κ y κ z − κ y ( e τ ( κ z − κ y ) − z ( iii) Let E [ Y t + τ Z t + τ | Y t = y, Z t = z ] =: f ( τ, Y t , Z t ) . f solves formally ∂ τ f = κ y ( z − y ) ∂ y f + κ z ( θ z − z ) ∂ z f + 12 σ y y∂ yy f + 12 σ z z∂ zz f. (21)Following exactly the same lines as above we have f ( τ, y, z ) = f ( τ ) + f y ( τ ) y + f z ( τ ) z + f z ( τ ) z + f zy ( τ ) zy + f y ( τ ) y . (22)Plugging (22) into (21) gives ddτ f + ddτ f y y + ddτ f z z + ddτ f z z + ddτ f zy zy + ddτ f y y = κ y ( z − y )( f y + 2 f y y + f zy z )+ κ z ( θ z − z )( f z + f zy y + 2 f z z )+ σ y yf y + σ z zf z . Thus we have ddτ f = κ z θ z f z ,ddτ f y = − κ y f y + κ z θ z f zy + σ y f y ,ddτ f z = κ y f y − κ z f z + (2 κ z θ z + σ z ) f z ,ddτ f z = κ y f zy − κ z f z ,ddτ f y = − κ y f y ,ddτ f zy = 2 κ y f y − ( κ y + κ z ) f zy , with f (0) = f z (0) = f y (0) = f z (0) = f y (0) = 0 , f zy (0) = 1 . f ( τ ) = e − (2 κ z + κ y ) τ θ z κ z κ y − κ z ) (cid:16) e κ z τ ( κ z + κ y ) κ z θ z + e κ y t ( κ z + κ y ) κ y ( σ z + 2 κ z θ z ) − e κ z τ κ z ( σ z + ( κ z + κ y ) θ z ) − e (2 κ z + κ y ) τ ( κ z − κ y )(2 κ z θ z + κ y ( σ z + 2 κ z θ z ))+ 2 e ( κ z + κ y ) τ ( κ z + κ y )( − κ y σ z + κ z θ z + κ z ( σ z − κ y θ z )) (cid:17) ,f y ( τ ) = θ z ( e − κ y τ − e − ( κ z + κ y ) τ ) ,f z ( τ ) = e − (2 κ z + κ y ) τ κ z ( κ z − κ y ) (cid:16) e κ z τ κ z κ y θ z + e κ y τ κ y ( σ z + 2 κ z θ z ) − e κ z τ κ z ( σ z + ( κ z + κ y ) θ z )+ e ( κ z + κ y ) τ ( − κ y σ z + κ z θ z + κ z ( σ z − κ y θ z )) (cid:17) ,f y ≡ , f zy = e − ( κ z + κ y ) τ , f z ( τ ) = κ y κ z − κ y ( e − ( κ z + κ y ) τ − e − κ z τ ) . Inserting these expressions into (22) we get f ( τ, y, z ) . ( iv) Set E [ Z t + τ | Y t = y, Z t = z ] =: g ( t, Y t , Z t ) . Then, g solves ∂ τ g = κ y ( z − y ) ∂ y g + κ z ( θ z − z ) ∂ z g + 12 σ y y∂ yy g + 12 σ z z∂ zz g. (23)Also, g is in the following form: g ( τ, y, z ) = q ( t ) + q y ( t ) y + q z ( t ) z + q z ( t ) z + q zy ( t ) zy + q y ( t ) y . (24)Inserting (24) into (23) gives ddτ q + ddτ q y y + ddτ q z z + ddτ q z z + ddτ q zy zy + ddτ q y y = κ y ( z − y )( q y + 2 q y y + q zy z )+ κ z ( θ z − z )( q z + q zy y + 2 q z z )+ σ y yq y + σ z zq z . which yields the system ddτ q = κ z θ z q z ,ddτ q y = − κ y q y + κ z θ z q zy + σ y q y ,ddτ q z = κ y q y − κ z q z + (2 κ z θ z + σ z ) q z ,ddτ q z = κ y q zy − κ z q z ,ddτ q y = − κ y q y ,ddτ q zy = 2 κ y q y − ( κ y + κ z ) q zy , q (0) = q z (0) = q y (0) = q y (0) = q zy (0) = 0 , q z (0) = . We solve this system of equations and get q ( τ ) = e − κ z τ ( e κ z τ − θ z ( σ z + 2 κ z θ z )2 κ z ,q z ( τ ) = e − κ z τ ( e κ z τ − σ z + 2 κ z θ z ) κ z ,q z ( τ ) = e − κ z τ ,q y ( τ ) ≡ q y ( τ ) ≡ q zy ( τ ) ≡ . Inserting above expressions into (24) yields the expression for g . ( v) Let E [ Y t + τ | Y t = y, Z t = z ] =: h ( τ, Y t , Z t ) . Formally, h satisfies the Kolmogorov’sbackward equation ∂ τ h = κ y ( z − y ) ∂ y h + κ z ( θ z − z ) ∂ z h + 12 σ y y∂ yy h + 12 σ z z∂ zz h. (25)From the polynomial property h is of the form h ( τ, y, z ) = p ( τ ) + p y ( τ ) y + p z ( τ ) z + p z ( τ ) z + p zy ( τ ) zy + p y ( τ ) y . (26)Plugging (26) into (25) gives ddτ p + ddτ p y y + ddτ p z z + ddτ p z z + ddτ p zy zy + ddτ p y y = κ y ( z − y )( p y + 2 p y y + p zy z )+ κ z ( θ z − z )( p z + p zy y + 2 p z z )+ σ y yp y + σ z zp z . which yields the following system of differential equations ddτ p = κ z θ z p z ,ddτ p y = − κ y p y + κ z θ z p zy + σ y p y ,ddτ p z = κ y p y − κ z p z + (2 κ z θ z + σ z ) p z ,ddτ p z = κ y p zy − κ z p z ,ddτ p y = − κ y p y ,ddτ p zy = 2 κ y p y − ( κ y + κ z ) p zy , p (0) = p z (0) = p y (0) = p z (0) = p zy (0) = 0 , p y (0) = 1 . Solving the system of linear ODEs yields p ( τ ) = e − (3 κ z + κ y ) τ θ z κ z ( κ z − κ y )( κ z − κ y ) κ y ( κ z + κ y ) (cid:16) e ( κ z + κ y ) τ ( κ z − κ y ) κ y ( κ z + κ y ) × ( σ z + 2 κ z θ z ) − e κ z τ κ z ( κ z − κ y )( κ z − κ y )( σ y + 2 κ y θ z ) − e κ z τ κ z × ( κ z − κ y ) κ y ( σ z + ( κ z + κ y ) θ z ) + e (3 κ z + κ y ) τ ( κ z − κ y )( κ z − κ y ) × ( κ z σ y + κ z κ y σ y + κ y σ z + 2 κ z κ y ( κ z + κ y ) θ z ) + e (3 κ z − κ y ) τ κ z ( κ z + κ y ) × ( κ y ( σ y − σ z ) + κ z ( σ y + 2 κ y θ z ) − κ z κ y ( σ y + 2 κ y θ z )) + 2 e (2 κ z + κ y ) τ κ y × ( κ y − κ z )(2 κ y σ z − κ z θ z + κ z ( σ y − σ z + 4 κ y θ z )) (cid:17) ,p y ( τ ) = e − κ y τ ((1 − e κ y τ ) κ z ( σ y + 2 κ y θ z ) + κ y (( e κ y τ − σ y + 2( e κ y τ − e ( κ y − κ z ) τ ) κ y θ z )) κ y ( κ y − κ z ) ,p z ( τ ) = e − (3 κ z + κ y ) τ κ z ( κ z − κ y )( κ z − κ y ) (cid:16) − e ( κ z + κ y ) τ ( κ z − κ y ) κ y ( σ z + 2 κ z θ z )+ e κ z τ κ z ( κ z − κ y )( κ z − κ y )( σ y + 2 κ y θ z ) + 2 e κ z τ κ ( κ z − κ y ) κ y × ( σ z + ( κ z + κ y ) θ z ) − e (3 κ z − κ y ) τ κ z ( κ y ( σ y − σ z ) + κ z ( σ y + 2 κ y θ z ) − κ z κ y ( σ y + 2 κ y θ z )) − e (2 κ z + κ y ) τ κ y ( κ y − κ z )(2 κ y σ z − κ z θ z + κ z ( σ y − σ z + 4 κ y θ z )) (cid:17) ,p zy ( τ ) = 2 κ y e − ( κ z + κ y ) τ ( e ( κ z − κ y ) τ − κ z − κ y ,p y ( τ ) = e − κ y τ ,p z ( τ ) = κ y e − κ z τ ( e ( κ z − κ y ) τ − ( κ z − κ y ) . Finally, inserting these coefficients into (26) gives h ( τ, y, z ) .Now we need to prove that the polynomial expressions in ( i ) − ( v ) actually solve (16), thatis, they provide the conditional moments of X t + τ . The next lemma gives a criteria for this tohold. Clearly, functions f , g and h appearing in ( i ) − ( v ) above are C , functions whose spatialderivatives satisfy the polynomial growth condition given in (27), meaning that the result ofLemma A.1 applies and this finishes the proof of the Proposition 3.1.29 emma A.1. Suppose u is a C -function on X , and u is a C , -function on R + × X whosespatial derivatives satisfy the polynomial growth condition (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ∂u∂y , ∂u∂z (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ K (1 + (cid:107) ( y, z ) (cid:107) ν ) , t ≤ T, ( y, z ) ∈ X (27) for some constant K = K ( T ) ≤ ∞ and some ν ≥ , for all T < ∞ .If u ( t, y, z ) satisfies the Kolmogorov backward equation ∂u∂t = κ y ( z − y ) ∂u∂y + κ z ( θ z − z ) ∂u∂z + 12 σ y y ∂ u∂y + 12 σ z z ∂ u∂z ,u (0 , y, z ) = u ( y, z ) (28) for all t ≥ and ( y, z ) ∈ X , then for all t ≤ T < ∞ u ( T − t, Y t , Z t ) = E [ u ( Y T , Z T ) | Y t , Z t ] Proof. Since u is assumed to be C , , in view of the Itô formula we get du ( T − t, Y t , Z t ) = (cid:32) − ∂u ( T − t, Y t , Z t ) ∂t + ∂u ( T − t, Y t , Z t ) ∂y κ y ( Z t − Y t )+ ∂u ( T − t, Y t , Z t ) ∂z κ z ( θ z − Z t ) + 12 σ y Y t ∂ u ( T − t, Y t , Z t ) ∂y + 12 σ z Z t ∂ u ( T − t, Y t , Z t ) ∂z (cid:33) dt + ∂u ( T − t, Y t , Z t ) ∂y σ y (cid:112) Y t dW yt + ∂u ( T − t, Y t , Z t ) ∂z σ z (cid:112) Z t dW zt (29)Now suppose u satisfies (28). Then, the drift term in (29) immediately vanishes, implying that u ( T − t, Y t , Z t ) is a local martingale with u (0 , Y T , Z T ) = u ( Y T , Z T ) . We now write du ( T − t, Y t , Z t ) = ∂u ( T − t, Y t , Z t ) ∂y σ y (cid:112) Y t dW yt + ∂u ( T − t, Y t , Z t ) ∂z σ z (cid:112) Z t dW zt In what follows our main objective is to show that under the assumptions of the lemma, u ( T − t, Y t , Z t ) is a true martingale.We have E (cid:34)(cid:90) T (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ∂u ( T − s, Y s , Z s ) ∂y , ∂u ( T − s, Y s , Z s ) ∂z (cid:19) (cid:20) σ y √ Y t σ z √ Z t (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) ds (cid:35) ≤ E (cid:34)(cid:90) T (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ∂u ( T − s, Y s , Z s ) ∂y , ∂u ( T − s, Y s , Z s ) ∂z (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) σ y Y t σ z Z t (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) ds (cid:35) ≤ K (cid:32) E (cid:34) sup s ≤ T (cid:107) ( Y s , Z s ) (cid:107) ν (cid:35)(cid:33) (30)30here the last inequality follows from assumption (27) and due to the fact that the diffusionparameter of the process X satisfies the linear growth condition. 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