Call the calibration routine to find values for volatility parameters Alpha and Sigma

Use hwcalbycap to calculate the values of Alpha and Sigma based on market data. Internally, hwcalbycap calls the function lsqnonlin. You can customize lsqnonlin by passing an optimization options structure created by optimoptions and then this can be passed to hwcalbycap using the name-value pair argument for OptimOptions. For example, optimoptions defines the target objective function tolerance as 100*eps and then calls hwcalbycap:

o=optimoptions('lsqnonlin','TolFun',100*eps);

[Alpha, Sigma] = hwcalbycap(RateSpec, MarketStrike, MarketMat, MarketVol, ...
Strike, Settle, Maturity, 'Reset', Reset, 'Principal', Principal, 'Basis', ... 
Basis, 'OptimOptions', o)

Local minimum possible.

lsqnonlin stopped because the size of the current step is less than
the default value of the step size tolerance.

Warning: LSQNONLIN did not converge to an optimal solution. It exited with exitflag = 2.
 
> In hwcalbycapfloor at 93
  In hwcalbycap at 75 

Alpha =

   1.0000e-06


Sigma =

    0.0127

The previous warning indicates that the conversion was not optimal. The search algorithm used by the Optimization Toolbox™ function lsqnonlin did not find a solution that conforms to all the constraints. To discern whether the solution is acceptable, look at the results of the optimization by specifying a third output (OptimOut) for hwcalbycap:

[Alpha, Sigma, OptimOut] = hwcalbycap(RateSpec, MarketStrike, MarketMat, ...
MarketVol, Strike, Settle, Maturity, 'Reset', Reset, 'Principal', Principal, ...
'Basis', Basis, 'OptimOptions', o);

The OptimOut.residual field of the OptimOut structure is the optimization residual. This value contains the difference between the Black caplets and those calculated during the optimization. You can use the OptimOut.residual value to calculate the percentual difference (error) compared to Black caplet 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 caplets using market data and Black's formula to obtain reference caplet values

To determine the effectiveness of the optimization, calculate reference caplet values using Black’s formula and the market data. Note, you must first interpolate the market data to obtain the caplets for calculation:

[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');

Compute the price of the cap using the Black model:

[CapPrice, Caplets] = capbyblk(RateSpec, Strike, Settle, Maturity, FlatVol, ...
'Reset', Reset, 'Basis', Basis, 'Principal', Principal); 
Caplets = Caplets(2:end)';

Caplets =

    0.3210
    1.6355
    2.4863
    3.1903
    3.4110
    3.2685
    3.2385
    3.4803
    3.2419
    3.1949
    3.2991
    3.3750

Compare optimized values and Black values and display graphically

After calculating the reference values for the caplets, compare the values, analytically and graphically, to determine whether the calculated single values of Alpha and Sigma provide an adequate approximation:

OptimCaplets = Caplets+OptimOut.residual;

disp('   ');
disp('    Black76   Calibrated Caplets');
disp([Caplets                   OptimCaplets])

plot(MarketMat(2:end), Caplets, 'or', MarketMat(2:end), OptimCaplets, '*b');
xtickformat;
xlabel('Caplet Maturity');
ylabel('Caplet Price');
title('Black and Calibrated Caplets');
h = legend('Black Caplets', 'Calibrated Caplets');
set(h, 'color', [0.9 0.9 0.9]);
set(h, 'Location', 'SouthEast');
set(gcf, 'NumberTitle', 'off')
grid on

 Black76   Calibrated Caplets
    0.3210    0.3636
    1.6355    1.6603
    2.4863    2.4974
    3.1903    3.1874
    3.4110    3.4040
    3.2685    3.2639
    3.2385    3.2364
    3.4803    3.4683
    3.2419    3.2408
    3.1949    3.1957
    3.2991    3.2960
    3.3750    3.3663

Compare cap prices using the Black, HW analytical, and HW tree models

Using the calculated caplet values, compare the prices of the corresponding cap using the Black model, Hull-White analytical, and Hull-White tree models. To calculate a Hull-White tree based on Alpha and Sigma, use these calibration routines:

Price a portfolio of instruments using the calibrated HW tree

After building a Hull-White tree, based on parameters calibrated from market data, use HWTree to price a portfolio of these instruments:

To price this portfolio of instruments using the calibrated HWTree: