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Calibrating Hull-White Model Using Market Data

The pricing of interest-rate derivative securities relies on models that describe the underlying process. These interest rate models depend on one or more parameters that you must determine by matching the model predictions to the existing data available in the market. In the Hull-White model, there are two parameters related to the short rate process: mean reversion and volatility. Calibration is used to determine these parameters, such that the model can reproduce, as close as possible, the prices of caps or floors observed in the market. The calibration routines find the parameters that minimize the difference between the model price predictions and the market prices for caps and floors.

For a Hull-White model, the minimization is two dimensional, with respect to mean reversion (α) and volatility (σ). That is, calibrating the Hull-White model minimizes the difference between the model’s predicted prices and the observed market prices of the corresponding caplets or floorlets.

Hull-White Model Calibration Example

Use market data to identify the implied volatility (σ) and mean reversion (α) coefficients needed to build a Hull-White tree to price an instrument. The ideal case is to use the volatilities of the caps or floors used to calculateAlpha(α) andSigma(σ). This will most likely not be the case, so market data must be interpolated to obtain the required values.

Consider a cap with these parameters:

Settle =' Jan-21-2008'; Maturity ='Mar-21-2011'; Strike = 0.0690; Reset = 4; Principal = 1000; Basis = 0;

The caplets for this example would fall in:

capletDates = cfdates(Settle, Maturity, Reset, Basis); datestr(capletDates')
ans = 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 21-Mar-2011

In the best case, look up the market volatilities for caplets with aStrike=0.0690, and maturities in each reset date listed, but the likelihood of finding these exact instruments is low. As a consequence, use data that is available in the market and interpolate to find appropriate values for the caplets.

Based on the market data, you have the cap information for different dates and strikes. Assume that instead of having the data forStrike=0.0690, you have the data forStrike1=0.0590andStrike2=0.0790.

Maturity Strike1 = 0.0590 Strike2 = 0.0790
21-Mar-2008 0.1533 0. 1526
21-Jun-2008 0.1731 0. 1730
21-Sep-2008 0. 1727 0. 1726
21-Dec-2008 0. 1752 0. 1747
21-Mar-2009 0. 1809 0. 1808
21-Jun-2009 0. 1809 0. 1792
21-Sep-2009 0. 1805 0. 1797
21-Dec-2009 0. 1802 0. 1794
21-Mar-2010 0. 1802 0. 1733
21-Jun-2010 0. 1757 0. 1751
21-Sep-2010 0. 1755 0. 1750
21-Dec-2010 0. 1755 0. 1745
21-Mar-2011 0. 1726 0. 1719

The nature of this data lends itself to matrix nomenclature, which is perfect for MATLAB®.hwcalbycaprequires that the dates, the strikes, and the actual volatility be separated into three variables:MarketStrike,MarketMat, andMarketVol.

MarketStrike = [0.0590; 0.0790]; MarketMat = {'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';'21-Mar-2011'};MarketVol = [0.1533 0.1731 0.1727 0.1752 0.18090.1800 0.1805 0.1802 0.1735 0.1757...0.1755 0.1755 0.1726;% First row in table corresponding to Strike10.1526 0.1730 0.1726 0.1747 0.1808 0.1792 0.1797 0.1794 0.1733 0.1751...0.1750 0.1745 0.1719];% Second row in table corresponding to Strike2

Complete the input arguments using this data forRateSpec:

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; 0.0792; 0.0793]; ValuationDate ='21-Jan-2008'; 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';....'21-Mar-2011';'21-Jun-2011'};Compounding = 4; Basis = 0; RateSpec = intenvset('ValuationDate', ValuationDate,...'StartDates', ValuationDate,'EndDates', EndDates,...'Rates', Rates,'Compounding', Compounding,'Basis'基础)
RateSpec = FinObj: 'RateSpec' Compounding: 4 Disc: [14x1 double] Rates: [14x1 double] EndTimes: [14x1 double] StartTimes: [14x1 double] EndDates: [14x1 double] StartDates: 733428 ValuationDate: 733428 Basis: 0 EndMonthRule: 1

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

Usehwcalbycapto calculate the values ofAlphaandSigmabased on market data. Internally,hwcalbycapcalls the functionlsqnonlin. You can customizelsqnonlinby passing an optimization options structure created byoptimoptionsand then this can be passed tohwcalbycapusing the name-value pair argument forOptimOptions. For example,optimoptionsdefines the target objective function tolerance as100*epsand then callshwcalbycap:

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™ functionlsqnonlindid 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) forhwcalbycap:

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

TheOptimOut.residualfield of theOptimOutstructure is the optimization residual. This value contains the difference between the Black caplets and those calculated during the optimization. You can use theOptimOut.residualvalue 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 ofAlphaandSigma.

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:

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

帽后计算的参考价值lets, compare the values, analytically and graphically, to determine whether the calculated single values ofAlphaandSigmaprovide an adequate approximation:

OptimCaplets = Caplets+OptimOut.residual; disp(' '); disp(' Black76 Calibrated Caplets'); disp([Caplets OptimCaplets]) plot(MarketMatNum(2:end), Caplets,'or', MarketMatNum(2:end), OptimCaplets,'*b'); datetick('x', 2) 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') gridon
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 onAlphaandSigma, use these calibration routines:

  • Black model:

    CapPriceBLK = CapPrice;

  • HW analytical model:

    CapPriceHWAnalytical = sum(OptimCaplets);

  • HW tree model to price the cap derived from the calibration process:

    1. CreateVolSpecfrom the calibration parametersAlphaandSigma:

      VolDates = EndDates; VolCurve = Sigma*ones(14,1); AlphaDates = EndDates; AlphaCurve = Alpha*ones(14,1); HWVolSpec = hwvolspec(ValuationDate, VolDates, VolCurve,AlphaDates, AlphaCurve);

    2. Create theTimeSpec:

      HWTimeSpec = hwtimespec(ValuationDate, EndDates, Compounding);

    3. Build the HW tree using theHW2000method:

      HWTree = hwtree(HWVolSpec, RateSpec, HWTimeSpec,'Method','HW2000');

    4. Price the cap:

      Price = capbyhw(HWTree, Strike, Settle, Maturity, Reset, Basis, Principal); disp(' '); disp([' CapPrice Black76 ..................: ', num2str(CapPriceBLK,'%15.5f')]); disp([' CapPrice HW analytical..........: ', num2str(CapPriceHWAnalytical,'%15.5f')]); disp([' CapPrice HW from capbyhw ..: ', num2str(Price,'%15.5f')]); disp(' ');
      CapPrice Black76 ..........: 34.14220 CapPrice HW analytical.....: 34.18008 CapPrice HW from capbyhw ..: 34.14192

Price a portfolio of instruments using the calibrated HW tree

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

  • Two bonds

    CouponRate = [0.07; 0.09]; Settle =' Jan-21-2008'; Maturity = {'Mar-21-2010';'Mar-21-2011'};Period = 1; Face = 1000; Basis = 0;

  • Bond with an embedded American call option

    CouponRateOEB = 0.08; SettleOEB =' Jan-21-2008'; MaturityOEB ='Mar-21-2011'; OptSpec ='call'; StrikeOEB = 950; ExerciseDatesOEB ='Mar-21-2011'; AmericanOpt = 1; Period = 1; Face = 1000; Basis = 0;

To price this portfolio of instruments using the calibratedHWTree:

  1. Useinstaddto create the portfolioInstSet:

    InstSet = instadd('Bond', CouponRate, Settle, Maturity, Period, Basis, [], [], [], [], [], Face); InstSet = instadd(InstSet,'OptEmBond', CouponRateOEB, SettleOEB, MaturityOEB, OptSpec,...StrikeOEB, ExerciseDatesOEB,'AmericanOpt', AmericanOpt,'Period', Period,...'Face',Face,'Basis'基础);

  2. Add the cap instrument used in the calibration:

    SettleCap =' Jan-21-2008'; MaturityCap ='Mar-21-2011'; StrikeCap = 0.0690; Reset = 4; Principal = 1000; InstSet = instadd(InstSet,'Cap', StrikeCap, SettleCap, MaturityCap, Reset, Basis, Principal);

  3. Assign names to the portfolio instruments:

    Names = {'7% Bond';'8% Bond';'BondEmbCall';'6.9% Cap'};InstSet = instsetfield(InstSet,'Index',1:4,'FieldName', {'Name'},'Data', Names );

  4. Examine the set of instruments contained inInstSet:

    instdisp(InstSet)
    IdxType CoupRate解决成熟期EOMRul基础e IssueDate 1stCoupDate LastCoupDate StartDate Face Name 1 Bond 0.07 21-Jan-2008 21-Mar-2010 1 0 NaN NaN NaN NaN NaN 1000 7% Bond 2 Bond 0.09 21-Jan-2008 21-Mar-2011 1 0 NaN NaN NaN NaN NaN 1000 8% Bond IdxType CoupRate Settle Mature OptSpec Stke ExDate Per Basis EOMRule IssDate 1stCoupDate LstCoupDate StrtDate Face AmerOpt Name 3 OptEmBond 0.08 21-Jan-2008 21-Mar-2011 call 950 21-Jan-2008 21-Mar-2011 1 0 1 NaN NaN NaN NaN 1000 1 BondEmbCall Index Type Strike Settle Maturity CapReset Basis Principal Name 4 Cap 0.069 21-Jan-2008 21-Mar-2011 4 0 1000 6.9% Cap

  5. Usehwpriceto price the portfolio using the calibratedHWTree:

    formatbankPricePortfolio = hwprice(HWTree, InstSet)
    PricePortfolio = 980.45 1023.05 945.73 34.14

See Also

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