A Top-Down Approach for Asset-Backed Securities: A Consistent Way of Managing Prepayment, Default and Interest Rate Risks

Abstract

We define a new approach to manage prepayment, default and interest rate risks simultaneously in some standard asset-backed securities structures. We propose a parsimonious top-down approach, by modeling directly the portfolio loss process and the amortization process. Both are correlated to interest rates. The methodology is specified for sequential- and pro-rata pay bonds (ABS, CMO, CDO of ABS), cash or synthetic. We prove analytical formulas to price all tranches, under and without the simplifying assumption that amortization occurs in the most senior tranche only. The model behavior is illustrated through the empirical analysis of an actual synthetic ABS trade.

Appendices

Appendix A: Proofs Under the Assumption (A)

A.1 Proof of Theorem 1

Under the s-forward neutral probability Q _s , the discount factor (B(t, s)) _{t,t ≤ s} process is the numeraire and we have, for all the tranches except the most senior one,

$$ E_1(s)= B(t,s)E_{t,Q_s}\left[ \left( K - EL(s,s) \right)^+ \right], $$

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and

$$ E_2(s,\bar s)=B(t,s) E_{t,Q_{s}}\left[ {\mathbf 1} \{ EL(s,s) \leq K \} EL(\bar s,\bar s) \right]. $$

A technical issue is coming from the fact we change one probability into another one every time. Formally, the processes (EL(t, T)) _t (for different T) have not the same laws under all these probabilities. Actually, only the drifts are changing. It is possible to state explicitly all these drifts. Remind that the drift of (EL(·, T)) under the risk neutral measure Q is zero. By classical arguments (see e.g., Brigo and Mercurio 2001), we obtain: Under the Q _s Forward measure, the drift of the process EL(t, T) is given by $- \rho EL(t,T)\sigma(t,T) \left(\sigma_{Q}(t) -\bar\sigma (t,s) \right),$ where σ _Q is the volatility of the usual numeraire. Since this usual numeraire is the money market account, it has no volatility. Thus, σ _Q (t) = 0 and, for every t, T, the latter drift is

$$ EL(t,T)\nu_{t,T,Q_{s}}= \rho EL(t,T)\sigma(t,T) \bar\sigma (t,s). $$

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Therefore, under any Forward neutral probability Q _s , the Expected Loss processes are still lognormal: $EL(dt,T)=EL(t,T). \left( \nu_{t,T,Q_s} dt + \sigma(t,T) dW_t \right)$.

To evaluate the expectation (Eq. 14), we can invoke the usual Black-Scholes formula:

$$\begin{array}{rll} E_1(s) &=& B(t,s)E_{t,Q_s}\left[ \left( K - EL(s,s) \right)^+ \right] \\ &=& B(t,s)E_{t,Q_s}\left[ \left( K - EL(t,s)\exp\left(\int_t^s \nu_{u,s,Q_s} \, du \right.\right.\right.\\ && \left.\left.\left.-\int_t^s \sigma^2(u,s)\, du/2 + \int_t^s \sigma(u,s) dW_u \right) \right)^+ \right] \\ &=& B(t,s)\exp\left(\int_t^s \nu_{u,s,Q_s} \, du\right) \\ && \cdot E_{t,Q_s}\left[\! \left(\! K_s^* - EL(t,s)\exp\left(\! -\! \int_t^s \sigma^2(u,s)\, ds/2 +\! \int_t^s \! \sigma(u,s) dW_u\! \right) \right)^+ \right]. \end{array}$$

So, we prove the formula for E ₁(s). To deal with $E_2(s,\bar s)$, choose now the numeraire $EL(\cdot,\bar s)$. Under the probability $Q_{\bar s}^E$ that is induced by this new numeraire,

$$ E_2(s,\bar s)=B(t,s) E_{t,Q_s}[EL(\bar s,\bar s)].E_{t,Q_{\bar s}^E}\left[ {\mathbf 1} \{ EL(s,s) \leq K \} \right], $$

or equivalently

$$ E_2(s,\bar s)=B(t,s) EL(t,\bar s) \exp\left(\int_t^{\bar s} \nu_{u,\bar s,Q_{s}} \, du\right).E_{t,Q_{\bar s}^E}\left[ {\mathbf 1} \{ EL(s,s) \leq K \} \right]. $$

But, under the probability $Q_{\bar s}^E$, the Expected Loss processes EL(·, s) follows the diffusion equation

$$ EL(dt,s)=EL(t,s). \left( \sigma(t,s)\sigma(t,\bar s) dt + \sigma(t,s) dW_t \right). \nonumber $$

Then, we obtain an explicit expression for $E_2(s,\bar s)$.

A.2 Proof of Theorem 2

Under (A), we have

$$\begin{array}{rll} E_1^*(s) &=& B(t,s)E_{t,Q_s}\left[ 1- EL(s,s) - A(s,s) \right], \;\mbox{\rm and} \\ E_2^*(s,\bar s) &=& B(t,s)E_{t,Q_{s}}\left[ EL(\bar s,\bar s) \right]. \end{array}$$

By our change of measure, the processes (EL(t, T)) and (A(t, T)) are no more martingales under the new measures. Concerning the Expected Loss process, we had already found the Q _s -drift change (see Eq. 15). This implies

$$\begin{array}{rll} E_{t,Q_s}[EL(\bar s,\bar s)] &=& EL(t,\bar s) \exp\left( \int_t^{\bar s} \nu_{u,\bar s,Q_s} \, du \right)\\ &=& EL(t,\bar s) \exp\left( \rho\int_t^{\bar s} \sigma(u,\bar s) \bar\sigma(u,s) \, du \right). \end{array}$$

Similarly, we can deal with the amortization process A(., s) as with the Expected Loss process. Therefore, we have

$$ E_{t,Q_{s}}\left[ A(s,s) \right]=A(t,s) \exp\left( \tilde{\rho}\int_t^s \tau(u,s) \bar\sigma(u,s) \, du \right),$$

so the result.

A.3 Proof of Theorem 5

Under the Q _s Forward measure, the drift of the process $EL(t,\bar s)$ is given by $\nu_{t,\bar s,Q_s}$, with our previous notations. By some standard conditional expectation arguments, we have

$$\begin{array}{rll} F_2(s,\bar s) &=& E_t\left[ \exp\left( -\int_t^{\bar s} r_u \, du \right) B(\bar s,s) L(\bar s,s)\Delta(\bar s,s) O_K(\bar s) \right] \\ & =& E_{t}\left[ \exp\left( -\int_t^{\bar s} r_u \, du \right) B(\bar s,s) \left\{ \frac{1}{B(\bar s,s)} -1 \right\} O_K(\bar s) \right] \\ & =& E_{t}\left[ \exp\left( -\int_t^{\bar s} r_u \, du \right) \left\{ 1- B(\bar s,s)\right\} O_K(\bar s) \right] \\ &=& F_{1,1}(\bar s,\bar s) - E_{t}\left[ \exp\left( -\int_t^{\bar s} r_u \, du \right) E_{\bar s}\left[ \exp( -\int_{\bar s}^{ s} r_u \, du )\right] O_K(\bar s) \right] \\ &=& F_{1,1}(\bar s,\bar s) - F_{1,1}(s,\bar s). \; \end{array}$$

Appendix B: Semi-Analytical Formulas Without the Assumption (A)

In this appendix, we extend our formulas to remove the convenient previous Assumption (A). Now, the amortization process can reduce any tranche, possibly the most senior one. Closed-form formulas are no longer available, but we can rely on semi-analytical formulas instead. Broadly speaking, the method is simple: conditionally on the value of the amortization process at some time horizon, the “base case” formulas apply, by shifting the relevant strikes. Then, an integration w.r.t. the law of the expected amortized amounts provide the results.

First, let us consider the previous synthetic structure and the evaluation of risky principals and default legs of all tranches (including the most senior one). Recall that the risky principals of the equity tranche [0, K] are defined by $RP_{t,K}=E_t\left[ \int_t^{T^\ast} \exp\left( - \int_t^s r_u \, du \right) O_{K}(s) \, ds \right]$, and its default legs are

$$ DL_{t,K} = E_t\left[ \int_t^{T^\ast} \exp\left( - \int_t^s r_u \, du \right) {\mathbf 1}(L(s)+A_K(s) \leq K) L(ds) \right]. $$

As previously, we cover the case of sequential-pay bonds, for which $A_K(s) = [A(s) - (1-K)]^+$. But it should be noted that we deal with the case of pro-rata bonds too, for which A _K (s) is a fixed proportion of A(s), as in some stripped pass-through securities. To include these two reference situations explicitly,¹⁷ we assume the repaid principal of the tranche [0, K] at time s is a deterministic function of the portfolio repaid principal A(s) only: for every K and s, there exists a function ψ _K such that

$$A_K(s):= \psi_{K}(A(s))=\psi_{K}(A(s,s)).$$

To price the tranche [0, K], it is sufficient to evaluate all quantities like

$$ {\mathcal E}_1(s)= E_t\left[ \exp\left( - \int_t^s r_u \, du \right) \left( K - EL(s,s) - \psi_{K}(A(s,s)) \right)^+ \right], $$

and

$$ {\mathcal E}_2(s,\bar s)=E_t\left[ \exp\left( - \int_t^{s} r_u \, du \right) {\mathbf 1}\{EL(s,s)+ \psi_{K}(A(s,s)) \leq K\} EL(\bar s,\bar s)\right], $$

for every couple $(s,\bar s)$, $t\leq \bar s \leq s \leq T^\ast$. Actually, we will calculate first the quantity

$$ {\mathcal F}_1(s,\bar s)= E_t\left[ \exp\left( - \int_t^s r_u \, du \right) \left( K - EL(\bar s,\bar s) - \psi_{K}(A(\bar s,\bar s)) \right)^+ \right], $$

when $\bar s \leq s$. Indeed, note that ${\mathcal E}_1(s) = {\mathcal F}_1(s,s)$. Moreover, ${\mathcal F}_1$ is the same as the so-called term F ₁ that had been calculated in Section “Pricing of a Cash MBS/ABS Tranche” under the Assumption (A). We will need ${\mathcal F}_1$ for pricing coupon-bearing securities hereafter.

To fix the ideas, at time t, the event $A(\bar s,\bar s)=a$ will be identical to $\int_t^{\bar s} \tau(u,\bar s)\,d\tilde W_u = w(a)$, for some value w(a) that will depend on the spot curve A(t, ·). Clearly,

$$\begin{array}{rll}{\mathcal F}_1(s,\bar s) &=& B(t,s) E_{t,Q_s}\left[ \left( K - EL(\bar s,\bar s) - \psi_{K}(A(\bar s,\bar s)) \right)^+ \right] \\ &=& B(t,s) E_{t,Q_s}\left[ E_{t,Q_s}[ \left( K - EL(\bar s,\bar s) - \psi_{K}(a) \right)^+ | A(\bar s,\bar s)=a] \right], \end{array}$$

and the conditional expectation can be evaluated easily. Here, the conditioning event is

$$ a = A(t,\bar s)\exp \left( \tilde{\rho}\int_t^{\bar s} \bar \sigma(u,s)\tau(u,\bar s)\, du - \frac{1}{2}\int_t^{\bar s} \tau^2(u,\bar s)\, du + \int_t^{\bar s} \tau(u,\bar s)\, d\tilde W_u \right), $$

or equivalently

$$ \int_t^{\bar s} \tau(u,\bar s)\, d\tilde W_u = w(a).$$

But we can break down

$$ \int_t^{\bar s} \sigma(u,\bar s)\, d W_u = \xi_{\bar s} \int_t^{\bar s} \tau(u,\bar s)\, d\tilde W_u + \varepsilon,$$

where

$$ \xi_{\bar s} = \frac{\rho \tilde{\rho} \int_t^{\bar s} \sigma(u,\bar s)\tau(u,\bar s)\, du }{\int_t^{\bar s} \tau^2(u,\bar s)\, du}, $$

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and $\varepsilon \sim {\mathcal N}(0,\mu_{\bar s}^2)$, by setting

$$ \mu_{\bar s}^2 = \int_t^{\bar s} \sigma^2(u,\bar s)\, du - \xi_{\bar s}^2 \int_t^{\bar s} \tau^2(u,\bar s)\, du. $$

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Implicitly, note that the variance of ε depends on all the underlying volatility functions and arguments. Then, under Q _s and conditionally on $A(\bar s,\bar s)=a$, the random variable $EL(\bar s,\bar s)$ can be written

$$\begin{array}{rll} EL(\bar s,\bar s) &=& EL(t,\bar s)\exp \left( \int_t^{\bar s} \rho\sigma(u,\bar s)\bar\sigma(u,s)\, du \right.\\ &&\left.- \frac{1}{2}\int_t^{\bar s} \sigma^2(u,\bar s)\, du + \xi_{\bar s} w(a) + \varepsilon \right), \end{array}$$

and

$$\begin{array}{lll} &&E_{Q_s}[\left( K - EL(\bar s,\bar s) - \psi_K(a) \right)^+ ] \\ &&\quad=Put\left( EL(t,\bar s)\exp \left( \int_t^{\bar s} \rho\sigma(u,\bar s)\bar\sigma(u,s)\, du\right.\right.\\ &&\quad\quad\;\left.- \ \frac{1}{2}\int_t^{\bar s} \sigma^2(u,\bar s)\, du + \xi_{\bar s} w(a) + \frac{1}{2}\mu_{\bar s}^2 \right), \\ &&\quad\quad \left. K-\psi_K(a),\mu_{\bar s},\bar s-t\right) \cdot {\mathbf 1} ( K \geq \psi_K(a) ). \end{array}$$

It is sufficient to integrate the latter formula w.r.t. the r.v. $\int_t^{\bar s} \tau(u,\bar s)\, d\bar W_u$ to prove the result.

Theorem 7

Under (E), (IR) and (AM),

$$\begin{array}{rll} {\mathcal F}_1(s,\bar s) &=& B(t,s) \int Put\left(EL_w, K-\psi_K(a_1(w)),\mu_{\bar s},\bar s-t\right) \phi\left(\frac{w}{(\int_t^{\bar s} \tau^2(u,\bar s)\,du)^{1/2}}\right)\\ &&\cdot \frac{{\mathbf 1} ( K \geq \psi_K(a_1(w)) )}{(\int_t^{\bar s} \tau^2(u,\bar s)\,du)^{1/2}}dw, \end{array}$$

where

$$ EL_w := EL(t,\bar s)\exp \left( \int_t^{\bar s}\! \rho\sigma(u,\bar s)\bar\sigma(u,s)\, du - \frac{1}{2}\int_t^{\bar s}\! \sigma^2(u,\bar s)\, du + \xi_{\bar s} w + \frac{1}{2}\mu_{\bar s}^2 \right). $$

Moreover $\xi_{\bar s}$ (resp. $\mu_{\bar s}$ ) is given by Eq.  16 (resp. (Eq.  17 )) and

$$ a_1(w) := A(t,\bar s)\exp \left( \tilde{\rho}\int_t^{\bar s} \sigma(u,\bar s)\tau(u,s)\, du - \frac{1}{2}\int_t^{\bar s} \tau^2(u,\bar s)\, du+w\right).$$

Similarly and by leading the same changes of numeraire as in Theorem 1, we obtain

$$\begin{array}{rll}{\mathcal E}_2(s,\bar s)&=&B(t,s) EL(t,\bar s) \exp\left(\int_t^{\bar s} \nu_{u,\bar s,Q_{s}} \, du\right)\\ &&.E_{t,Q_{\bar s}^E}\left[ {\mathbf 1} \{ EL(s,s)+\psi_K(A(s,s)) \leq K \} \right] \\ &=& B(t,s) EL(t,\bar s) \exp\left(\int_t^{\bar s} \nu_{u,\bar s,Q_{s}} \, du\right) \\ && \cdot E_{t,Q_{\bar s}^E}\left[ E_{t,Q_{\bar s}^E}\left[{\mathbf 1} \{ EL(s,s)+\psi_K(a_2(w)) \leq K \} | \int_t^s \tau(u,s)\,d\tilde W_u = w\right]\right] \end{array}$$

where

$$ a_2(w) = A(t,s)\exp \left( \rho\tilde{\rho}\int_t^{\bar s} \sigma(u,\bar s)\tau(u,s)\, du - \frac{1}{2}\int_t^s \tau^2(u,s)\, du+w\right). $$

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Note that the latter function is slightly different from the previous one a ₁(w). Indeed, under $Q_{\bar s}^E$, the instantaneous drift of A(t, s) is now proportional to $\rho\tilde{\rho}\sigma(t,\bar s)\tau(t,s)$. We deduce

$$\begin{array}{lll} &&E_{t,Q_{\bar s}^E}\left[{\mathbf 1} \{ EL(s,s)+\psi_K(a_2(w))\leq K \}| A(s,s)= a_2(w) \right] \\ &&\quad={\mathbf 1}(K\geq \psi_K(a_2(w))). \Phi\left( \{ \ln( K - \psi_K(a_2(w))) \right. \\ &&\quad\quad\;\;\left. -\ln EL(t,s) - \int_t^{\bar s} \sigma(u,s)\sigma(u,\bar s)\, du + \frac{1}{2} \int_t^s \sigma^2(u,s)\, du - \xi_s w \}/\mu_s \right), \end{array}$$

where ξ _s and $\mu_s^2$ have been defined above.

Theorem 8

Under (E), (IR) and (AM), we have

$$\begin{array}{lll} &&{\mathcal E}_2(s,\bar s) = B(t,s) EL(t,\bar s) \exp\left(\int_t^{\bar s} \nu_{u,\bar s,Q_{s}} \, du\right) \\ &&\quad \cdot \int \Phi\left(\frac{1}{\mu_s} \{ \ln( K -\psi_K(a_2(w))) - \ln EL(t,s) - \int_t^s \sigma(u,s)\sigma(u,\bar s)\, du \right. \\ &&\quad \left. +\frac{1}{2} \int_t^s\! \sigma^2(u,s)\, du - \xi_s w \} \!\right) \cdot \phi\left(\!\frac{w}{(\int_t^s \tau^2(u,s)\,du)^{1/2}}\!\right) \frac{{\mathbf 1}(K\geq \psi_K(a_2(w)))dw}{(\int_t^s \tau^2(u,s)\,du)^{1/2}}, \end{array}$$

where a ₂ , ξ _. and μ _. are defined by the identities  18 ,  16 and  17 respectively.

Corollary 1

Let us consider a base tranche [0, K], K ∈ [0, 1], of a synthetic ABS structure. Under the assumptions of Theorems 7 and 8, its risky principal is $RP_{t,K}= \int_t^{T^\ast} {\mathcal F}_1(s,s) \, ds$ , and its default leg is $DL_{t,K}\simeq \sum_{i=1}^p \left[ {\mathcal E}_2\right.\!(T_i,T_i) - {\mathcal E}_2\left.\!(T_i,T_{i-1})\right]$.

To extend fully the results of the previous sections, it remains to tackle the case of cash structures. The next to last missing building block (to deal floating rate coupons) is

$$ {\mathcal F}_2(s,\bar s) = E_t\left[ \exp\left( -\int_t^{s} r_u \, du \right) L(\bar s,s)O_K(\bar s) \right], $$

But, invoking the same arguments as in the proof of Theorem 5, we obtain easily

$$ {\mathcal F}_2(s,\bar s) = B(t,\bar s) E_{t,Q_{\bar s}}\left[ O_K(\bar s) \right] - B(t,s) E_{t,Q_{ s}}\left[ O_K(\bar s) \right] = {\mathcal F}_1(\bar s,\bar s) - {\mathcal F}_1(s,\bar s) ,$$

and it is a known quantity. Thus, to evaluate principal paydowns in this case, the two last missing building blocks are the evaluation of

$$ {\mathcal A}_1(s,\bar s)=E_t\left[ \exp\left( - \int_t^{s} r_u \, du \right) {\mathbf 1}\{EL(\bar s,\bar s)+\psi_K(A(\bar s,\bar s)) \leq K\} A_K( s,s)\right], $$

and

$$ {\mathcal A}_2(s,\bar s)=E_t\left[ \exp\left( - \int_t^{s} r_u \, du \right) {\mathbf 1}\{EL(\bar s,\bar s)+\psi_K(A(\bar s,\bar s)) \leq K\} A_K( \bar s,\bar s)\right], $$

for every couples $(s,\bar s)$, $t\leq \bar s \leq s \leq T^\ast$. After some tedious calculations, it can be proved that:

Theorem 9

Under (E), (IR) and (AM), we have

$$\begin{array}{rll} {\mathcal A}_1(s,\bar s) &=& B(t,s) \int \Phi\left(\frac{1}{\mu_{{\mathcal A}_1}} \left\{ \ln( K - \psi_K(a_3(w))) - \ln EL(t,\bar s) \right.\right.\\ &&\qquad\qquad\quad\; \left.\left. - \int_t^{\bar s}\!\! \rho \sigma(u,\bar s)\bar\sigma(u, s)\, du \!+\! \frac{1}{2} \int_t^{\bar s}\!\! \sigma^2(u,\bar s)\, du \!-\! \xi_3 w \!-\! \xi_4 \tilde w \right\} \right) \\ && \cdot \phi_{\rho^\ast}\left(\frac{w}{v_{\bar s}},\frac{\tilde w}{v_{s}}\right) \cdot \psi_K(a_4(\tilde w)) \frac{{\mathbf 1}(K\geq \psi_K(a_3(w)) ) }{v_s v_{\bar s}}\,dw\, d\tilde w , \end{array}$$

where $\phi_{\rho^\ast}$ denotes the density of a bivariate random vector of standard Gaussian r.v. with correlation parameter ρ ^∗ , and where we have set

$$\begin{array}{rll} \xi_3 &:=& \rho\tilde{\rho}\frac{\int_t^{\bar s} \sigma(.,\bar s)\tau(.,\bar s).\int_t^s \tau^2(.,s)- \int_t^{\bar s}\sigma(.,\bar s)\tau(.,s).\int_t^{\bar s} \tau(.,s)\tau(.,\bar s)}{\int_t^{\bar s}\tau^2(.,\bar s) .\int_t^s\tau^2(.,s) - \left(\int_t^{\bar s}\tau(.,s)\tau(.,\bar s)\right)^2 },\\ \xi_4&:=& \rho\tilde{\rho}\frac{\int_t^{\bar s} \sigma(.,\bar s)\tau(.,s).\int_t^{\bar s} \tau^2(.,\bar s)- \int_t^{\bar s}\sigma(.,\bar s)\tau(.,\bar s).\int_t^{\bar s} \tau(.,s)\tau(.,\bar s)}{\int_t^{\bar s}\tau^2(.,\bar s) .\int_t^s\tau^2(.,s) - \left(\int_t^{\bar s}\tau(.,s)\tau(.,\bar s)\right)^2 }, \\ \mu_{{\mathcal A}_1}^2 &:=& \int_t^{\bar s} \sigma^2(.,\bar s) - \xi_3^2 \int_t^{\bar s} \tau^2(.,\bar s) - \xi_4^2\int_t^s \tau^2(.,s) -2\xi_3\xi_4 \int_t^{\bar s}\tau(.,s)\tau(.,\bar s),\\ a_3(w) &:=& A(t,\bar s)\exp \left( \tilde{\rho}\int_t^{\bar s} \tau(u,\bar s)\bar\sigma(u,s)\, du - \frac{1}{2}\int_t^{\bar s} \tau^2(u,\bar s)\, du+w\right),\\ a_4(\tilde w) &:=& A(t,s)\exp \left( \tilde{\rho}\int_t^{s} \tau(u,s)\bar\sigma(u,s)\, du - \frac{1}{2}\int_t^{s} \tau^2(u,s)\, du+\tilde w\right),\\ v_s^2 &:=& \int_t^s \tau^2(u,s)\,du, \;\;\; \rho^\ast := \frac{\int_t^{\bar s} \tau(.,s)\tau(.,\bar s)}{\left(\int_t^s \tau^2(.,s).\int_t^{\bar s} \tau^2(.,\bar s) \right)^{1/2} }\cdot\end{array}$$

Note that the previous term ${\mathcal A}_1$ involves a two-dimensional integration. At the opposite, the term ${\mathcal A}_2(s,\bar s)$ is simpler and similar to ${\mathcal E}_2$. We prove easily

Theorem 10

Under (E), (IR) and (AM), we have

$$\begin{array}{rll} {\mathcal A}_2(s,\bar s) &=& B(t,s) \int \Phi\left(\frac{1}{\mu_{{\mathcal A}_2}} \left\{ \ln( K - \psi_K(a_5(w))) - \ln EL(t,\bar s) \right.\right. \\ &&\qquad\qquad\quad\; \left. \left. - \int_t^{\bar s} \rho \sigma(u,\bar s)\bar\sigma(u, s)\, du + \frac{1}{2} \int_t^{\bar s} \sigma^2(u,\bar s)\, du - \xi_{\bar s} w \right\} \right) \\ && \cdot \ \phi\left(\frac{w}{v_{\bar s}}\right) \cdot \psi_K(a_5( w)) \frac{{\mathbf 1}(K\geq \psi_K(a_5(w)))}{v_{\bar s}}\,dw , \end{array}$$

where

$$\mu_{{\mathcal A}_2}^2 = \int_t^{\bar s} \sigma^2(.,\bar s) - \xi_{\bar s}^2 \int_t^{\bar s} \tau^2(.,\bar s),$$

and

$$a_5(w) = A(t,\bar s)\exp \left( \tilde{\rho}\int_t^{\bar s} \bar \sigma(.,s)\tau(.,\bar s) - \frac{1}{2}\int_t^{\bar s} \tau^2(.,\bar s)+w\right).$$

Thus, the previous theorems allow us to evaluate cash structures, as described in Section “Pricing of a Cash MBS/ABS Tranche”.

Corollary 2

Under the Assumptions (E), (IR), (AM) and (FC) and with our previous notations, the cash bond price of Section “ Pricing of a Cash MBS/ABS Tranche ” is

$$ P_t = \sum\limits_{i,T_i \geq t} \left\{ C_{0} \Delta_i {\mathcal F}_1(T_i,T_{i-1}) + {\mathcal A}_1(T_i,T_{i-1}) - {\mathcal A}_2(T_i,T_{i-1}) \right\} + {\mathcal E}_1(T_p). $$

Under the Assumptions (E), (IR), (AM) and (FlC), the related cash bond price is

$$ P_t = \sum\limits_{i,T_i \geq t} \left\{ \Delta_i {\mathcal F}_2(T_i,T_{i-1}) + {\mathcal A}_1(T_i,T_{i-1}) - {\mathcal A}_2(T_i,T_{i-1}) \right\}+ {\mathcal E}_1(T_p) . $$