## Problem Setting

Generally the price of any security can be written as the net present value (NPV) of its discounted cash flows. Specifying the price of an MBS (here we consider only the pass-through MBS1) is as follows:

where P is the price of the MBS,

V is the value of the MBS, which is a random variable, dependent on the realization of the economic scenario,

PV(t) is the present value for cash flow at time t, d(t) is the discounting factor at time t, c(t) is the cash flow at time t,

M is the maturity of the MBS.

lA pass-through MBS is an MBS that passes through the principal and interest payments collected from a mortgage pool, minus the guaranty fee and servicing fee, to the MBS investor directly. This is in contrast to Collaterized Mortgage Obligations (CMOs), which have multiple tranches and pay the principal payments according to the seniorities of tranches. In this essay, we assume that mortgages in the MBS pool are homogenous.

Monte Carlo simulation is used to generate cash flows on many paths. By the strong law of large numbers, we have the following:

where V is the value calculated out in path i.

The calculation of d(t) is found from the short-term (risk-free) interest rate process, t-i t-i d (t) = d (0,1)d (1,2)d (t -1, t) = n exp(-r (i)At) = exp{-[Z r (i)]At>, (2.3)

where d(i, i+1) is the discounting factor for the end of period i+1 at the end of period i; r(i) is the short term rate used to generate d(i, i+1), observed at the end of period i;

At is the time step in simulation, generally monthly, i.e. At= 1 month. An interest rate model is used to generate the short term-rate r(i); then d(t) is instantly available when the short-term rate path is generated.

The difficult part is to generate c(t), the path dependent cash flow of MBS for month t, which is observed at the end of month t. From chapter 19 of Fabozzi , we have the following formula for c(t):

where MP(t): Scheduled Mortgage Payment for month t; TPP(t): Total Principal Payment for month t;

IP(t): Interest Payment for month t; SP(t): Scheduled Principal Payment for month t; PP(t): Principal Prepayment for month t. These quantities are calculated as follows:

WAC /12

WAC 12

PP(t) = SMM(t)(B(t -1) - SP(t)); B(t) = B(t -1) - TPP(t); SMM (t) = 1 -^ - CPR(t);

WAC2:

The principal balance of MBS at end of month t;

Weighted Average Coupon rate for MBS;

Weighted Average Maturity for MBS;

Single Monthly Mortality for month t, observed at the end of

Conditional Prepayment Rate for month t, observed at the end of

In Monte Carlo simulation, along the sample path, CPR(t) is the primary variable to be simulated. Everything else can be calculated out once CPR(t) is known. Different prepayment models offer different CPR(t), and it is not our goal to derive a new

2 WAC is the weighted average mortgage rate for a mortgage pool, weighted by the balance of each mortgage.

3 WAM is the weighted average maturity in month for a mortgage pool, weighted by the balance of each mortgage.

prepayment model or compare existing prepayment models. Instead, our concern is, given a prepayment model, how can we efficiently estimate the price sensitivities of MBS against parameters of interest? Generally different prepayment models will lead to different sensitivity estimates, so it is at the user's discretion to choose an appropriate prepayment function, as our method for calculating the "Greeks" is universally applicable.