Calibrating Floorlets Using the Normal (Bachelier) Model

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This example shows how to use hwcalbyfloor to calibrate market data with the Normal (Bachelier) model to price floorlets. Use the Normal (Bachelier) model to perform calibrations when working with negative interest rates, strikes, and normal implied volatilities.

Consider a floor with these parameters:

Settle = datetime(2016,12,30);
Maturity = datetime(2019,12,30);
Strike = -0.004075;
Reset = 2;
Principal = 100;
Basis = 0;

The floorlets and market data for this example are defined as:

floorletDates = cfdates(Settle, Maturity, Reset, Basis);
datestr(floorletDates')

ans = 6x11 char array
    '30-Jun-2017'
    '30-Dec-2017'
    '30-Jun-2018'
    '30-Dec-2018'
    '30-Jun-2019'
    '30-Dec-2019'

% Market data information
MarketStrike = [-0.00595; 0];
MarketMat = [datetime(2017,6,30) ; datetime(2017,12,30) ; datetime(2018,6,30) ; datetime(2018,12,30) ; datetime(2019,6,30) ; datetime(2019,12,30)];
MarketVol = [0.184 0.2329 0.2398 0.2467 0.2906 0.3348;   % First row in table corresponding to Strike 1 
             0.217 0.2707 0.2760 0.2814 0.3160 0.3508];  % Second row in table corresponding to Strike 2

Define the RateSpec using intenvset.

Rates= [-0.003210;-0.003020;-0.00182;-0.001343;-0.001075];
ValuationDate = datetime(2016,12,30);
EndDates = [datetime(2017,6,30) ; datetime(2017,12,30) ; datetime(2018,6,30) ; datetime(2018,12,30) ; datetime(2019,12,30)];
Compounding = 2;
Basis = 0;

RateSpec = intenvset('ValuationDate', ValuationDate, ...
'StartDates', ValuationDate, 'EndDates', EndDates, ...
'Rates', Rates, 'Compounding', Compounding, 'Basis', Basis);

Use hwcalbyfloor to find values for the volatility parameters Alpha and Sigma using the Normal (Bachelier) model.

format short
o=optimoptions('lsqnonlin','TolFun',100*eps);
warning ('off','fininst:hwcalbycapfloor:NoConverge')
[Alpha, Sigma, OptimOut] = hwcalbyfloor(RateSpec, MarketStrike, MarketMat,...
MarketVol, Strike, Settle, Maturity, 'Reset', Reset, 'Principal', Principal,...
'Basis', Basis, 'OptimOptions', o, 'model', 'normal')

Local minimum possible.
lsqnonlin stopped because the size of the current step is less than
the value of the step size tolerance.

Alpha = 
1.0000e-06

Sigma = 
0.3410

OptimOut = struct with fields:
     resnorm: 1.9233e-04
    residual: [5x1 double]
    exitflag: 2
      output: [1x1 struct]
      lambda: [1x1 struct]
    jacobian: [5x2 double]

The OptimOut.residual field of the OptimOut structure is the optimization residual. This value contains the difference between the Normal (Bachelier) floorlets and those calculated during the optimization. Use the OptimOut.residual value to calculate the percentual difference (error) compared to Normal (Bachelier) floorlet prices, and then decide whether the residual is acceptable. There is almost always some residual, so decide if it is acceptable to parameterize the market with a single value of Alpha and Sigma.

Price the floorlets using the market data and Normal (Bachelier) model to obtain the reference floorlet values. To determine the effectiveness of the optimization, calculate reference floorlet values using the Normal (Bachelier) formula and the market data. Note, you must first interpolate the market data to obtain the floorlets for calculation.

% MarketMatNum = datenum(MarketMat);
[Mats, Strikes] = meshgrid(MarketMat, MarketStrike);
MarketMat_T = yearfrac(Settle,Mats);
Mats_T = yearfrac(Settle,Maturity);
FlatVol = interp2(MarketMat_T, Strikes, MarketVol, Mats_T, Strike, 'spline');
% FlatVol = interp2(Mats, Strikes, MarketVol, datenum(Maturity), Strike, 'spline');

[FloorPrice, Floorlets] = floorbynormal(RateSpec, Strike, Settle, Maturity, FlatVol,...
'Reset', Reset, 'Basis', Basis, 'Principal', Principal); 
Floorlets = Floorlets(2:end)'

Floorlets = 5×1

    4.7637
    6.7180
    8.1833
    9.5825
   10.6090

Compare the optimized values and Normal (Bachelier) values, and display the results graphically. After calculating the reference values for the floorlets, compare the values analytically and graphically to determine whether the calculated single values of Alpha and Sigma provide an adequate approximation.

OptimFloorlets = Floorlets+OptimOut.residual;

disp('   ');

disp(' Bachelier   Calibrated Floorlets');

 Bachelier   Calibrated Floorlets

disp([Floorlets        OptimFloorlets])

    4.7637    4.7685
    6.7180    6.7263
    8.1833    8.1878
    9.5825    9.5795
   10.6090   10.6007

plot(MarketMat(2:end), Floorlets, 'or', MarketMat(2:end), OptimFloorlets, '*b');
xlabel('Floorlet Maturity');
ylabel('Floorlet Price');
ylim ([0 16]);
title('Bachelier and Calibrated Floorlets');
h = legend('Bachelier Floorlets', 'Calibrated Floorlets');
set(h, 'color', [0.9 0.9 0.9]);
set(h, 'Location', 'SouthEast');
set(gcf, 'NumberTitle', 'off')
grid on

Figure contains an axes object. The axes object with title Bachelier and Calibrated Floorlets, xlabel Floorlet Maturity, ylabel Floorlet Price contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Bachelier Floorlets, Calibrated Floorlets.

Calibrating Floorlets Using the Normal (Bachelier) Model

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