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We have seen the valuation of bonds with embedded option using binomial model. This method can be used when cash flows do not depend on how interest rates evolve. But, in case of some fixed income and derivative securities like mortgage-backed securities and asset-backed securities, cash flows received in one particular period are not only determined by the current and future interest rate levels but also on the path these interest rates take to reach the present level. This is termed as interest rate path dependency.  In pass-through securities, prepayments are path dependent because this month's prepayment rate would depend on whether there were any opportunities to refinance since the underlying mortgages were originated. In the case of adjustable rate pass-throughs. the prepayments are not the only ones that are path dependent but the periodic coupon rates depend on the history of the reference rate on which the coupon rate is fixed. In the case of Collateralized Mortgage Obligations (CMOs), the collateral prepayments are path dependent and the cash flow to be received in the current month by a CMO tranche depends on the outstanding balances of the other tranches in the deal. The history of prepayments is required to calculate those balances.

Monte-Carlo method is used to value securities in which cash flows are interest rate dependent. Though the valuation of pass-throughs using the Monte-Carlo method looks conceptually simple, it is practically complex. The reason is that it involves generating a set of cash flows based on simulated future mortgage refinancing rates, which imply simulated prepayment rates.

1. Use of Simulation to Generate Interest Rate Paths and Cash Flows:  Monte Carlo simulation is a model used for valuing mortgage-backed and asset-backed securities using the on-the-run Treasury issuer. In order to build an arbitrage free model, the interest rate paths are to be adjusted so that the model produces the correct value for the on-the-run Treasury issue. In the rest of the chapter wherever reference to interest rate paths is made it means adjusted interest rate paths.

The simulation works by generating many scenarios of future interest rate paths. For each month, a monthly interest rate and a mortgage refinancing rate are generated. While the monthly interest rates are used for discounting the projected cash flows, the mortgage refinancing rate is used for determining the cash flow because it represents the opportunity cost for the mortgagor.

If the refinancing rates are high relative to the mortgagor's contract rate, the latter will have less incentive to refinance or even a positive disincentive and if the refinancing rate is low relative to the mortgagor's contract rate, the latter will have an incentive to refinance. Prepayments are projected by feeding the refinancing rate and loan characteristics. Once the prepayments are projected, the cash flows along an interest rate path can be determined. Let us consider a newly issued mortgage pass-through security with a maturity of 120 months. Table 1 shows N simulated interest rate path scenarios.

Each scenario consists of a path of 120 simulated one-month future interest rates. Table 2 shows the paths of simulated mortgage refinancing rate corresponding to the scenarios shown in Table 1. Certain assumptions regarding Treasury rates and refinancing rates are made to build Table 2. The assumption is that there is a constant spread relationship between the refinancing rates and the one-month interest rates shown in Table 1.

Once we have mortgage refinancing rates, we can generate the cash flows on each interest rate path. For agency mortgage-backed securities, a prepayment model is required, and for asset-backed securities and non-agency mortgage-backed securities, we require prepayment model as well as a model for defaults and recoveries. Once we have the details of prepayments, defaults and recoveries, cash flow can be calculated. Table 3 depicts the resulting cash flows.

Table 1: Simulated Paths of One-Month Future Interest Rates

Internet Rate Path Numbera

### Month

1

2

3

...

n

...

N

1

f1(1)

f1(2)

f1(3)

...

f1(n)

...

f1(N)

2

f2(1)

f2(2)

f2(3)

...

f2(n)

...

f2(N)

3

f3(1)

f3(2)

f3(3)

...

f3(n)

...

f3(N)

...

...

...

...

...

...

...

...

t

ft(1)

ft(2)

ft(3)

...

ft(n)

...

ft(N)

...

...

...

...

...

...

...

...

118

f118(1)

f118(2)

f118(3)

...

f118(n)

...

f118(N)

119

f119(1)

f118(2)

f118(3)

...

f118(n)

...

f118(N)

120

f120(1)

f120(2)

f120(3)

...

f120(n)

...

f120(N)

1. Notation: ft(n), one-month future interest rate for month't' on path 'n'; 'N', total number of interest rate paths.

Table 2: Simulated Paths of Mortgage Refinancing Rates

Internet Rate Path Number a

### Month

1

2

3

...

n

...

N

1

r1(1)

r1(2)

r1(3)

...

r1(n)

...

r1(N)

2

r2(1)

r2(2)

r2(3)

...

r2(n)

...

r2(N)

3

r3(1)

r3(3)

r3(3)

...

r3(n)

...

r3(N)

...

...

...

...

...

...

...

...

t

rt(1)

rt(2)

rt(3)

...

rt(n)

...

rt(N)

...

...

...

...

...

...

...

...

118

r118(1)

r118(2)

r118(3)

...

r118(n)

...

r118(N)

119

r119(1)

r119(2)

r119(3)

...

r118(n)

...

r118(N)

120

r120(1)

r120(2)

r120(3)

...

r120(n)

...

r120(N)

1. Notation: rt(n), mortgage refinancing rate for month 't' on path 'n'; 'N', total number of interest rate paths.

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