hwcalbycap

Calibrate Hull-White tree using caps

Syntax

[Alpha,Sigma,OptimOut] = hwcalbycap(RateSpec,MarketStrikeMarketMaturity,MarketVolatility)

[Alpha,Sigma,OptimOut = hwcalbycap(RateSpec,MarketStrikeMarketMaturity,MarketVolatility,Strike,Settle,Maturity)

[Alpha,Sigma,OptimOut] = hwcalbycap(___,Name,Value)

Description

[Alpha,Sigma,OptimOut] = hwcalbycap(RateSpec,MarketStrikeMarketMaturity,MarketVolatility) calibrates the Alpha (mean reversion) and Sigma (volatility) using cap market data and the Hull-White model using the entire cap surface.

The Hull-White calibration functions (hwcalbycap and hwcalbyfloor) support three models: Black (default), Bachelier or Normal, and Shifted Black. For more information, see the optional arguments for Shift and Model.

[Alpha,Sigma,OptimOut = hwcalbycap(RateSpec,MarketStrikeMarketMaturity,MarketVolatility,Strike,Settle,Maturity) estimates the Alpha (mean reversion) and Sigma (volatility) using cap market data and the Hull-White model to price a cap at a particular maturity/volatility using the additional optional input arguments for Strike, Settle, and Maturity.

Strike, Settle, and Maturity arguments are specified to calibrate to a specific point on the market volatility surface. If omitted, the calibration is performed across all the market instruments

For an example of calibrating using the Hull-White model with Strike, Settle, and Maturity input arguments, see Calibrating Hull-White Model Using Market Data.

[Alpha,Sigma,OptimOut] = hwcalbycap(___,Name,Value) adds optional name-value pair arguments.

Examples

Calibrate Hull-White Model from Market Data Using the Entire Cap Volatility Surface

This example shows how to use hwcalbycap input arguments for MarketStrike, MarketMaturity, and MarketVolatility to calibrate the HW model using the entire cap volatility surface.

Cap market volatility data covering two strikes over 12 maturity dates.

Reset = 4;
MarketStrike = [0.0590; 0.0790];

MarketMaturity = {'21-Mar-2008'; '21-Jun-2008'; '21-Sep-2008'; '21-Dec-2008';
    '21-Mar-2009'; '21-Jun-2009'; '21-Sep-2009'; '21-Dec-2009';
    '21-Mar-2010'; '21-Jun-2010'; '21-Sep-2010'; '21-Dec-2010'};
MarketMaturity = datenum(MarketMaturity);

MarketVolaltility = [0.1533 0.1731 0.1727 0.1752 0.1809 0.1800 0.1805 0.1802...
    0.1735 0.1757 0.1755 0.1755;
    0.1526 0.1730 0.1726 0.1747 0.1808 0.1792 0.1797 0.1794...
    0.1733 0.1751 0.1750 0.1745];

Plot market volatility surface.

[AllMaturities,AllStrikes] = meshgrid(MarketMaturity,MarketStrike);
figure;
surf(AllMaturities,AllStrikes,MarketVolaltility)
datetick
xlabel('Maturity')
ylabel('Strike')
zlabel('Volatility')
title('Market Volatility Data')

Set interest rate term structure and create a RateSpec.

Settle = '21-Jan-2008';
Compounding = 4;
Basis = 0;
Rates= [0.0627; 0.0657; 0.0691; 0.0717; 0.0739; 0.0755; 0.0765; 0.0772;
    0.0779; 0.0783; 0.0786; 0.0789];
EndDates = {'21-Mar-2008';'21-Jun-2008';'21-Sep-2008';'21-Dec-2008';...
    '21-Mar-2009';'21-Jun-2009';'21-Sep-2009';'21-Dec-2009';....
    '21-Mar-2010';'21-Jun-2010';'21-Sep-2010';'21-Dec-2010'};
RateSpec = intenvset('ValuationDate', Settle, 'StartDates', Settle,...
    'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding,...
    'Basis',Basis)

RateSpec = 

           FinObj: 'RateSpec'
      Compounding: 4
             Disc: [12x1 double]
            Rates: [12x1 double]
         EndTimes: [12x1 double]
       StartTimes: [12x1 double]
         EndDates: [12x1 double]
       StartDates: 733428
    ValuationDate: 733428
            Basis: 0
     EndMonthRule: 1

Calibrate Hull-White model from market data.

o = optimoptions('lsqnonlin','TolFun',1e-5,'Display','off');

[Alpha, Sigma] = hwcalbycap(RateSpec, MarketStrike, MarketMaturity,...
    MarketVolaltility, 'Reset', Reset,'Basis', Basis, 'OptimOptions', o)

Warning: LSQNONLIN did not converge to an optimal solution. It exited with exitflag = 3.
 
> In hwcalbycapfloor>optimizeOverCapSurface at 232
  In hwcalbycapfloor at 79
  In hwcalbycap at 81 

Alpha =

    0.0943


Sigma =

    0.0146

Compare with Black prices.

BlkPrices = capbyblk(RateSpec,AllStrikes(:), Settle, AllMaturities(:),...
    MarketVolaltility(:),'Reset',Reset,'Basis',Basis);

BlkPrices =

    0.0604
         0
    0.2729
    0.0006
    0.6498
    0.0412
    1.1121
    0.1426
    1.6426
    0.3131
    2.1869
    0.4998
    2.7056
    0.6894
    3.2124
    0.8815
    3.7311
    1.0686
    4.2246
    1.2790
    4.7027
    1.4810
    5.1877
    1.6919

Setup Hull-White tree using calibrated parameters, alpha, and sigma.

VolDates    = EndDates;
VolCurve    = Sigma*ones(numel(EndDates),1);
AlphaDates  = EndDates;
AlphaCurve  = Alpha*ones(numel(EndDates),1);
HWVolSpec   = hwvolspec(Settle, VolDates, VolCurve, AlphaDates, AlphaCurve);

HWTimeSpec  = hwtimespec(Settle, EndDates, Compounding);
HWTree = hwtree(HWVolSpec, RateSpec, HWTimeSpec, 'Method', 'HW2000')

HWTree = 

      FinObj: 'HWFwdTree'
     VolSpec: [1x1 struct]
    TimeSpec: [1x1 struct]
    RateSpec: [1x1 struct]
        tObs: [0 0.6593 1.6612 2.6593 3.6612 4.6593 5.6612 6.6593 7.6612 8.6593 9.6612 10.6593]
        dObs: [733428 733488 733580 733672 733763 733853 733945 734037 734128 734218 734310 734402]
      CFlowT: {1x12 cell}
       Probs: {1x11 cell}
     Connect: {1x11 cell}
     FwdTree: {1x12 cell}

Compute Hull-White prices based on the calibrated tree.

HWPrices = capbyhw(HWTree, AllStrikes(:), Settle, AllMaturities(:), Reset, Basis)

HWPrices =

    0.0601
         0
    0.2788
         0
    0.6580
    0.0518
    1.1254
    0.1485
    1.6591
    0.3123
    2.2076
    0.5022
    2.7319
    0.6883
    3.2459
    0.8774
    3.7771
    1.0900
    4.2769
    1.2875
    4.7645
    1.4845
    5.2572
    1.6921

Plot Black prices against the calibrated Hull-White tree prices.

figure;
plot(AllMaturities(:), BlkPrices, 'or', AllMaturities(:), HWPrices, '*b');
datetick('x', 2)
xlabel('Maturity');
ylabel('Price');
title('Black and Calibrated (HW) Prices');
legend('Black Price', 'Calibrated HW Tree Price','Location', 'NorthWest');
grid on

Calibrating Caplets Using the Normal (Bachelier) Model

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

Consider a cap with these parameters:

Settle = 'Dec-30-2016';
Maturity = 'Dec-30-2019';
Strike = -0.001075;
Reset = 2;
Principal = 100;
Basis = 0;

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

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

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 = [-0.0013; 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

Define the RateSpec.

Rates= [-0.002210;-0.002020;-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 hwcalbycap 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] = hwcalbycap(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 = 0.3384

OptimOut = struct with fields:
     resnorm: 1.5181e-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) caplets and those calculated during the optimization. Use the OptimOut.residual value to calculate the percentual difference (error) compared to Normal (Bachelier) 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 the caplets using the market data and Normal (Bachelier) model to obtain the reference caplet values. To determine the effectiveness of the optimization, calculate reference caplet values using the Normal (Bachelier) formula and the market data. Note, you must first interpolate the market data to obtain the caplets for calculation.

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

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

Caplets = 5×1

    4.7392
    6.7799
    8.2609
    9.6136
   10.6455

Compare the optimized values and Normal (Bachelier) values, and display the results 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(' Bachelier   Calibrated Caplets');

 Bachelier   Calibrated Caplets

disp([Caplets        OptimCaplets])

    4.7392    4.7453
    6.7799    6.7851
    8.2609    8.2657
    9.6136    9.6112
   10.6455   10.6379

plot(MarketMatNum(2:end), Caplets, 'or', MarketMatNum(2:end), OptimCaplets, '*b');
datetick('x', 2)
xlabel('Caplet Maturity');
ylabel('Caplet Price');
ylim ([0 16]);
title('Bachelier and Calibrated Caplets');
h = legend('Bachelier Caplets', 'Calibrated Caplets');
set(h, 'color', [0.9 0.9 0.9]);
set(h, 'Location', 'SouthEast');
set(gcf, 'NumberTitle', 'off')
grid on

Input Arguments

RateSpec — Interest-rate specification for initial rate curve

structure

Interest-rate specification for initial rate curve, specified by the RateSpec obtained from intenvset. For information on the interest-rate specification, see intenvset.

Data Types: struct

MarketStrike — Market cap strike

vector

Market cap strike, specified as a NINST-by-1 vector.

Data Types: double

MarketMaturity — Market cap maturity date

vector

Market cap maturity dates, specified as a NINST-by-1 vector.

Data Types: double

MarketVolatility — Market flat volatilities

matrix

Market flat volatilities, specified as a NSTRIKES-by-NMATS matrix of market flat volatilities, where NSTRIKES is the number of caplet strikes from MarketStrike and NMATS is the caplet maturity dates from MarketMaturity.

Data Types: double

Strike — Rate at which cap is exercised

decimal scalar

(Optional) Rate at which the cap is exercised, specified as a decimal scalar value.

Data Types: single

Settle — Settlement date of the cap

serial date number | date character vector

(Optional) Settlement date of the cap, specified as a scalar serial date number or date character.

Data Types: single | char

Maturity — Maturity date of the cap

serial date number | date character vector

(optional) Maturity date of the cap, specified as scalar serial date number or date character vector.

Data Types: single | char

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: [Alpha,Sigma,OptimOut] = hwcalbycap(RateSpec,MarketStrike,MarketMaturity,MarketVolaltility,'Reset',2,'Principal',100000,'Basis',3,'OptimOptions',o)

'Reset' — Frequency of payments per year

1 (default) | numeric

Frequency of payments per year, specified as the comma-separated pair consisting of 'Reset' and a scalar numeric value.

Data Types: double

'Principal' — Notional principal amount

100 (default) | nonnegative integer

Notional principal amount, specified as the comma-separated pair consisting of 'Principal' and a scalar nonnegative integer.

Data Types: single

'Basis' — Day-count basis used when annualizing the input forward rate

0 (actual/actual) (default) | integers of the set [0...13]

Day-count basis used when annualizing the input forward rate, specified as the comma-separated pair consisting of 'Basis' and a scalar value. Values are:

• 0 = actual/actual

• 1 = 30/360 (SIA)

• 2 = actual/360

• 3 = actual/365

• 4 = 30/360 (PSA)

• 5 = 30/360 (ISDA)

• 6 = 30/360 (European)

• 7 = actual/365 (Japanese)

• 8 = actual/actual (ICMA)

• 9 = actual/360 (ICMA)

• 10 = actual/365 (ICMA)

• 11 = 30/360E (ICMA)

• 12 = actual/365 (ISDA)

• 13 = BUS/252

For more information, see Basis.

Data Types: single

'LB' — Lower bounds

[0;0] (default) | numeric vector

Lower bounds, specified as the comma-separated pair consisting of 'LB' and a 2-by-1 vector of the lower bounds, defined as [LBSigma; LBAlpha], used in the search algorithm function. For more information, see lsqnonlin.

Data Types: double

'UB' — Upper bounds

[ ] (unbound) (default) | numeric vector

Upper bounds, specified as the comma-separated pair consisting of 'UB' and a 2-by-1 vector of the upper bounds, defined as [UBSigma; LBAlpha], used in the search algorithm function. For more information, see lsqnonlin.

Data Types: double

'XO' — Initial values

[0.5;0.5] (default) | numeric vector

Initial values, specified as the comma-separated pair consisting of 'XO' and a 2-by-1 vector of the initial values, defined as [Sigma0; Alpha0], used in the search algorithm function. For more information, see lsqnonlin.

Data Types: double

'OptimOptions' — Optimization parameters

structure

Optimization parameters, specified as the comma-separated pair consisting of 'OptimOptions' and a structure defined by using optimoptions.

Data Types: struct

'Shift' — Shift in decimals for shifted Black model

0 (no shift) (default) | positive decimal

Shift in decimals for the shifted Black model, specified as the comma-separated pair consisting of 'Shift' and a scalar positive decimal value. Set this parameter to a positive shift in decimals to add a positive shift to forward rate and Strike, which effectively sets a negative lower bound for forward rate and Strike. For example, a Shift value of 0.01 is equal to a 1% shift.

Data Types: single

'Model' — Indicator for model used for calibration routine

lognormal (Black model) (default) | values are normal and lognormal

Indicator for model used for calibration routine, specified as the comma-separated pair consisting of 'Model' and a scalar character vector with a value of normal or lognormal.

Data Types: char

Output Arguments

Alpha — Mean reversion value obtained from calibrating the cap using market information

scalar numeric

Mean reversion value obtained from calibrating the cap using market information, returned as a scalar value.

Sigma — Volatility value obtained from calibrating cap using market information

scalar numeric

Volatility value obtained from calibrating the cap using market information, returned as a scalar.

OptimOut — Optimization results

numeric structure

Optimization results, returned as a structure.