hwcalbycap
Calibrate Hull-White tree using caps
Syntax
Description
[
calibrates the Alpha,Sigma,OptimOut] = hwcalbycap(RateSpec,MarketStrikeMarketMaturity,MarketVolatility)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.
[
estimates the Alpha,Sigma,OptimOut = hwcalbycap(RateSpec,MarketStrikeMarketMaturity,MarketVolatility,Strike,Settle,Maturity)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.
[
adds optional name-value pair arguments. Alpha,Sigma,OptimOut] = hwcalbycap(___,Name,Value)
Examples
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 = [datetime(2008,3,21) ; datetime(2008,6,21) ; datetime(2008,9,21) ; datetime(2008,12,21) ; ... datetime(2009,3,21) ; datetime(2009,6,21) ; datetime(2009,9,21) ; datetime(2009,12,21); datetime(2010,3,21) ; ... datetime(2010,6,21); datetime(2010,9,21) ; datetime(2010,12,21)]; 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) 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 = [datetime(2008,3,21) ; datetime(2008,6,21) ; datetime(2008,9,21) ; datetime(2008,12,21) ; ... datetime(2009,3,21) ; datetime(2009,6,21) ; datetime(2009,9,21) ; datetime(2009,12,21); datetime(2010,3,21) ; ... datetime(2010,6,21); datetime(2010,9,21) ; datetime(2010,12,21)]; 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: 1Calibrate 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 =
0.0146Compare 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
1.6919Setup 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
1.6921Plot Black prices against the calibrated Hull-White tree prices.
figure; plot(AllMaturities(:), BlkPrices, 'or', AllMaturities(:), HWPrices, '*b'); xtickformat; xlabel('Maturity'); ylabel('Price'); title('Black and Calibrated (HW) Prices'); legend('Black Price', 'Calibrated HW Tree Price','Location', 'NorthWest'); grid on
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 = datetime(2016,12,30); Maturity = datetime(2019,12,30); 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 = 6×11 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 = [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= [-0.002210;-0.002020;-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 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. <stopping criteria details>
Alpha = 1.0000e-06
Sigma = 0.3384
OptimOut = struct with fields:
resnorm: 1.5181e-04
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) 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.
[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'); [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(MarketMat(2:end), Caplets, 'or', MarketMat(2:end), OptimCaplets, '*b'); 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
