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## Pricing Mortgage Backed Securities Using the Black-Derman-Toy Model

This example illustrates how the Financial Toolbox™ and Financial Instruments Toolbox™ are used to price a level mortgage backed security using the BDT model.

### Observe the Interest-Rate Tree

Visualize the interest rate evolution along the tree by looking at the output structure BDTTree. BDTTree returns an inverse discount tree, which you can convert into an interest-rate tree with the cvtree function.

BDTTreeR = cvtree(BDTTree);

Look at the upper branch and lower branch paths of the tree:

OldFormat = get(0, 'format');
format short

%Rate at root node:
RateRoot      = treepath(BDTTreeR.RateTree, 0)
RateRoot = 0.0399
%Rates along upper branch:
RatePathUp    = treepath(BDTTreeR.RateTree, [1 1 1 1 1])
RatePathUp = 6×1

0.0399
0.0397
0.0391
0.0383
0.0373
0.0360

%Rates along lower branch:
RatePathDown = treepath(BDTTreeR.RateTree, [2 2 2 2 2])
RatePathDown = 6×1

0.0399
0.0470
0.0550
0.0638
0.0734
0.0841

### Compute the Price Tree for the Non-Prepayable Mortgage

Let's say that we have a 3 year \$10000 level prepayable loan, with a mortgage interest rate of 4.64% semi-annually compounded.

MortgageAmount = 10000;
CouponRate = 0.0464;
Period = 2;
Settle='01-Jan-2007';
Maturity='01-Jan-2010';
Compounding = BDTTree.TimeSpec.Compounding;

format bank

Use the function amortize in the Financial Toolbox to calculate the mortgage payment of the loan (MP), the interest and principal components, and the outstanding principal balance.

NumPeriods = date2time(Settle,Maturity, Compounding)';

[Principal, InterestPayment, OutstandingBalance, MP] = amortize(CouponRate/Period, NumPeriods, MortgageAmount);

% Display Principal, Interest and Outstanding balances
PrincipalAmount = Principal'
PrincipalAmount = 6×1

1572.59
1609.07
1646.40
1684.60
1723.68
1763.67

InterestPaymentAmount = InterestPayment'
InterestPaymentAmount = 6×1

232.00
195.52
158.19
119.99
80.91
40.92

OutstandingBalanceAmount =OutstandingBalance'
OutstandingBalanceAmount = 6×1

8427.41
6818.34
5171.94
3487.35
1763.67
0.00

CFlowAmounts = MP*ones(1,NumPeriods);
% The CFlowDates are the same as the tree level dates
CFlowDates= {'01-Jul-2007' ,'01-Jan-2008' ,'01-Jul-2008' , '01-Jan-2009' , '01-Jul-2009' , '01-Jan-2010'} ;

% Calculate the price of the non-prepayable mortgage
[PriceNonPrepayableMortgage, PriceTreeNonPrepayableMortgage] = cfbybdt(BDTTree, CFlowAmounts, CFlowDates, Settle);
for iLevel = 2:length(PriceTreeNonPrepayableMortgage.PTree)
PriceTreeNonPrepayableMortgage.PTree{iLevel}(:,:)= PriceTreeNonPrepayableMortgage.PTree{iLevel}(:,:) - MP;
end

% Look at the price of the mortgage today tObs = 0
PriceNonPrepayableMortgage
PriceNonPrepayableMortgage =
10017.47

% The value of the non-prepayable mortgage is \$10017.47. This value exceeds
% the \$10000 amount borrowed since the homeowner received not only \$10000, but
% also a prepayment option.

% Look at the value of the mortgage on the last date, right after the last
% mortgage payment, is zero:
PriceTreeNonPrepayableMortgage.PTree{end}
ans = 1×6

0             0             0             0             0             0

% Visualize the price tree for the non-prepayable mortgage.
treeviewer(PriceTreeNonPrepayableMortgage)

### Compute the Price Tree of the Prepayment Option

% The Prepayment option is like a call option on a bond.
%
% The exercise price or strike will be equal to the outstanding principal amount
% which has been calculated using the function |amortize|.

OptSpec = 'call';
Strike = [MortgageAmount OutstandingBalance];
ExerciseDates =[Settle CFlowDates];
AmericanOpt = 0;
Maturity = CFlowDates(end);

% Compute the price of the prepayment option:
[PricePrepaymentOption, PriceTreePrepaymentOption] = prepaymentbybdt(BDTTree, OptSpec, Strike, ExerciseDates, AmericanOpt, ...
0, Settle, Maturity,[], [], [], ...
[], [], [],  [], 0, [], CFlowAmounts);

% Look at the price of the prepayment option today (tObs = 0)
PricePrepaymentOption
PricePrepaymentOption =
17.47

% The value of the prepayment option is \$17.47 as expected.

% Visualize the price tree for the prepayment option
treeviewer(PriceTreePrepaymentOption)

### Calculate the Price Tree of the Prepayable Mortgage.

% Compute the price of the prepayable mortgage.

PricePrepayableMortgage = PriceNonPrepayableMortgage - PricePrepaymentOption;

PriceTreePrepayableMortgage = PriceTreeNonPrepayableMortgage;

for iLevel = 1:length(PriceTreeNonPrepayableMortgage.PTree)
PriceTreePrepayableMortgage.PTree{iLevel}(:,:)= PriceTreeNonPrepayableMortgage.PTree{iLevel}(:,:) -  ...
PriceTreePrepaymentOption.PTree{iLevel}(:,:);
end

% Look at the price of the prepayable mortgage today (tObs = 0)
PricePrepayableMortgage
PricePrepayableMortgage =
10000.00

% The value of the prepayable mortgage is \$10000 as expected.

% Visualize the price and price tree for the prepayable mortgage
treeviewer(PriceTreePrepayableMortgage)

set(0, 'format', OldFormat);