Numerical Example

2.4.1 Specification of Numerical Example

We need to specify two sets of data to price the mortgage: the mortgage data and the interest rate data, which includes the initial term structure and parameters for the interest rate model.

We price different mortgages to examine the different impacts that a term structure shift or change in volatility may have on different mortgage products.

The following data are fixed for all products:

Unpaid Balance/UPB =$4,000,000;

WAM =360 months.

Table 2.1 shows the difference between all the products. All the ARM products have the same subsequent adjustment period of 12 months, period cap/floor of 0.02, lifetime cap of initial WAC plus 0.06, and no lifetime floor.

Product

WAC

Index

Adjust First

FRM

0.07425

N/A

N/A

1 Year ARM

0.06425

Treasury 1 Year

12 month

3/1 FP1ARM

0.07425

Treasury 1 Year

36 month

5/1 FPARM

0.07425

Treasury 1 Year

60 month

7/1 FPARM

0.07425

Treasury 1 Year

84 month

10/1 FPARM

0.07425

Treasury 1 Year

120 month

1 Year ARM2

0.07425

Treasury 10 Year

12 month

Tab e 2.1 Product Specification for Mortgage Pricing

1 FP ARM refers to Fixed Period ARM, which keep the coupon rate constant for a certain period, and then adjust periodically, generally once a year. So All the FP ARM products are the same, except different Adjust First date, which is the first coupon reset date.

Tab e 2.1 Product Specification for Mortgage Pricing

1 FP ARM refers to Fixed Period ARM, which keep the coupon rate constant for a certain period, and then adjust periodically, generally once a year. So All the FP ARM products are the same, except different Adjust First date, which is the first coupon reset date.

2 This ARM is not a mortgage product in the market at present, and is constructed for illustration purpose only. The following sections will discuss why we introduce this product, and what nice properties it has.

We use the same parameters for all the different products in order to have comparable results. Thus we set all the products to have the same coupon rate, except the first 1 year ARM with index of Treasury 1 year rate, which has a 100 basis points (bps) teaser rate. All the ARM products have the same characteristics, except for the Adjust First date, which is the feature that distinguishes these products. Our initial term structure is the following: f(0,t) =ln(150+12t)/100, =0,1,.. .,360.

This will produce an upward-sloping curve increasing gradually from 5% to 8.7% along 30 year maturity, and R(0,t) is acquired by calculating the following:

which increases from 5%, to 7.78% gradually. Our assumptions for interest rate model parameters are the following: a=0.1; 0=0.1; J„=0.00025, «=0,1,2,3 (used in the FD gradient and gamma estimator calculation); ^0=0.00025, (used in the FD vega estimator calculation).

2.4.2 Comparison of PA and FD gradient estimators

In order to test whether our PA gradient estimators are accurate, and are within the error tolerance range, we calculate the finite difference (FD) gradient estimators at the same time during our pricing process. This section will demonstrate the accuracy of our PA estimators of delta, vega, and gamma for FRM, as well as the delta and gamma for t t

ARM.

Comparison of Modified Fourier Gradient Estimators for FRM

Figure 2.5 shows the FD estimator, PA estimator, their difference, and standard deviation of their difference for dd(t>. The four curves in each chart are specified as dA n following, which will be the convention for the rest of the paper: Blue: Modified Fourier Order 1; Green: Modified Fourier Order 2; Red: Modified Fourier Order 3; Cyan: Modified Fourier Order 4.

We can see that although these two estimators are pretty close, there exists a pattern in the difference of these two estimators. This will be explained later in the error analysis section.

Figure 2.6 shows the PA and FD gradient estimators for cash flow c(t): they are pretty close, and the difference behaves as random noise. Based on M0 and

80 80

, , dPV(t,0) .. c„ . , dPV(t) _ c „ , we can calculate-1—-, and figure 5.3 shows us the-—. Figure 5.4 shows the d0 dA n n

95% confidence interval for difference between PA and FD estimators of dPV (t >, and dA n we can see that 0 is generally contained in the 95% confidence interval.

FD estimator PA estimator

Figure 2.5 Gradient Estimator Comparison for ctt(t)/ cAn x 10

FD estimator x 10

FD estimator

0 100 200 300 400 x -jo6 difference of FD/PA

x 10

PA estimator x 10

PA estimator

0 100 200 300 400 x 107

0 100 200 300 400

0 100 200 300 400

0 100 200 300 400 x 106 difference of FD/PA

0 100 200 300 400

0 100 200 300 400 x-iq6 STD of difference

100 200 300 400

Figure 2.7 Gradient Estimator Comparison for dPV(t)/ dA„

x 10

0 100 200 300 400

x 10

Figure 2.8 95% Confidence Interval for dPV(t)/dAn

Comparison of Vega Estimators for FRM

In this section, we also compare the FD and PA estimators for the gradient w.r.t. interest rate volatility: Vega. Figure 2.9 shows the FD estimator, PA estimator, their difference, and standard deviation of their difference for dd(t). Also there exists a da pattern in the difference of these two estimators. This will also be explained later in the error analysis section. Figure 2.10 shows the gradient estimators for cash flow c(t): they are pretty close, and the difference behaves as random noise. Figure 2.11 shows us the dPV(t), and figure 2.12 shows the 95% confidence interval for dPV(t), and we can see da da that 0 is always contained in the 95% confidence interval.

FD estimator PA estimator

FD estimator

x 1Q7 PA estimator

100 200 300 ■|q5 difference of FD/PA

100 200 300 ■|q5 difference of FD/PA

0 100 200 300 400

x 1Q7 PA estimator

0 100 200 300 400 x iq6 STD of difference

0 100 200 300 400

0 100 200 300 400 x iq6 STD of difference

0 100 200 300 400

Figure 2.10 Gradient Estimator Comparison for cb(t)/ckr ■jq7 FD estimator x 107 PA estimator

Figure 2.11 Gradient Estimator Comparison for dPV(t)/da

Figure 2.12 95% Confidence Interval for dPV(t)/da Comparison of Gamma Estimators for FRM

• • n TA A A , 7. C d2 d (t) d2 c(t) d2 PV (t) . For gamma estimation, O=[A1 A2 A3 A4] . So ^ J, , _ , or-:— is a del dd2

4x4 matrix. If we want to estimate this matrix by the FD method, we would need 144 points to estimate 48 first order derivatives and to estimate 16 second order derivatives.

The following figures show the FD, PA estimators, the difference and STD of difference for diagonal gamma elements.

Gamma Hedging Example

Figure 2.13 gamma estimators for d2 d (t) .

Figure 2.13 gamma estimators for d2 d (t) .

Interest Rate Cap Gamma

Figure 2.14 gamma estimators for d2 CPR(t) .

Figure 2.14 gamma estimators for d2 CPR(t) .

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