Documentation

This is machine translation

Translated by Microsoft
Mouseover text to see original. Click the button below to return to the English verison of the page.

Note: This page has been translated by MathWorks. Please click here
To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

Calibrating Floorlets Using the Normal (Bachelier) Model

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 = 'Dec-30-2016';
Maturity = 'Dec-30-2019';
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')

% Market data information
MarketStrike = [-0.00595; 0];
MarketMat =  {'30-Jun-2017';'30-Dec-2017';'30-Jun-2018'; '30-Dec-2018';'30-Jun-2019'; '30-Dec-2019'};
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
ans =

  6×11 char array

    '30-Jun-2017'
    '30-Dec-2017'
    '30-Jun-2018'
    '30-Dec-2018'
    '30-Jun-2019'
    '30-Dec-2019'

Define the RateSpec.

Rates= [-0.003210;-0.003020;-0.00182;-0.001343;-0.001075];
ValuationDate = 'Dec-30-2016';
EndDates = {'30-Jun-2017';'Dec-30-2017';'30-Jun-2018';'Dec-30-2018';'Dec-30-2019'};
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 default value of the step size tolerance.




Alpha =

   1.0000e-06


Sigma =

    0.4833


OptimOut = 

  struct with fields:

     resnorm: 0.0015
    residual: [5×1 double]
    exitflag: 2
      output: [1×1 struct]
      lambda: [1×1 struct]
    jacobian: [5×2 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(MarketMatNum, MarketStrike);
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 =

    6.7497
    9.5487
   11.6158
   13.5871
   15.0412

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');
disp([Floorlets        OptimFloorlets])

plot(MarketMatNum(2:end), Floorlets, 'or', MarketMatNum(2:end), OptimFloorlets, '*b');
datetick('x', 2)
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
   
 Bachelier   Calibrated Floorlets
    6.7497    6.7675
    9.5487    9.5701
   11.6158   11.6269
   13.5871   13.5795
   15.0412   15.0174

Was this topic helpful?