Calibrate Shifted SABR Model Parameters for Swaption Instrument
Calibrate model parameters for a Swaption instrument when you use a SABR pricing method.
Load Market Data
% Zero curve ValuationDate = datetime("5-Mar-2016", 'Locale', 'en_US'); ZeroDates = datemnth(ValuationDate,[1 2 3 6 9 12*[1 2 3 4 5 6 7 8 9 10 12]])'; ZeroRates = [-0.33 -0.28 -0.24 -0.12 -0.08 -0.03 0.015 0.028 ... 0.033 0.042 0.056 0.095 0.194 0.299 0.415 0.525]'/100; Compounding = 1; ZeroCurve = ratecurve("zero",ValuationDate,ZeroDates,ZeroRates,'Compounding',Compounding)
ZeroCurve =
ratecurve with properties:
Type: "zero"
Compounding: 1
Basis: 0
Dates: [16×1 datetime]
Rates: [16×1 double]
Settle: 05-Mar-2016
InterpMethod: "linear"
ShortExtrapMethod: "next"
LongExtrapMethod: "previous"
% Define the swaptions. SwaptionSettle = datetime("5-Mar-2016", 'Locale', 'en_US'); SwaptionExerciseDate = datetime("5-Mar-2017", 'Locale', 'en_US'); SwaptionStrikes = (-0.6:0.01:1.6)'/100; % Include negative strikes SwapMaturity = datetime("5-Mar-2022", 'Locale', 'en_US'); % Maturity of underlying swap OptSpec = 'call';
Compute Forward Swap Rate by Creating Swap Instrument
Use fininstrument to create a Swap instrument object.
LegRate = [0 0]; Swap = fininstrument("Swap", 'Maturity', SwapMaturity, 'LegRate', LegRate, "LegType",["fixed" "float"], ... "ProjectionCurve", ZeroCurve, "StartDate", SwaptionExerciseDate)
Swap =
Swap with properties:
LegRate: [0 0]
LegType: ["fixed" "float"]
Reset: [2 2]
Basis: [0 0]
Notional: 100
LatestFloatingRate: [NaN NaN]
ResetOffset: [0 0]
DaycountAdjustedCashFlow: [0 0]
ProjectionCurve: [1×2 ratecurve]
BusinessDayConvention: ["actual" "actual"]
Holidays: NaT
EndMonthRule: [1 1]
StartDate: 05-Mar-2017
Maturity: 05-Mar-2022
Name: ""
ForwardValue = parswaprate(Swap,ZeroCurve)
ForwardValue = 7.3271e-04
Load the Market Implied Volatility Data
The market swaption volatilities are quoted in terms of shifted Black volatilities with a 0.8 percent shift.
StrikeGrid = [-0.5; -0.25; -0.125; 0; 0.125; 0.25; 0.5; 1.0; 1.5]/100;
MarketStrikes = ForwardValue + StrikeGrid;
Shift = 0.008; % 0.8 percent shift
MarketShiftedBlackVolatilities = [21.1; 15.3; 14.0; 14.6; 16.0; 17.7; 19.8; 23.9; 26.2]/100;
ATMShiftedBlackVolatility = MarketShiftedBlackVolatilities(StrikeGrid==0);Calibrate Shifted SABR Model Parameters
The Beta parameter is predetermined at 0.5. Use volatilities to compute the implied volatility.
Beta = 0.5; % Calibrate Alpha, Rho, and Nu objFun = @(X) MarketShiftedBlackVolatilities - volatilities(finpricer("Analytic", 'Model', ... finmodel("SABR", 'Alpha', X(1), 'Beta', Beta, 'Rho', X(2), 'Nu', X(3), 'Shift', Shift), ... 'DiscountCurve', ZeroCurve), SwaptionExerciseDate, ForwardValue, MarketStrikes); X = lsqnonlin(objFun, [0.5 0 0.5], [0 -1 0], [Inf 1 Inf]);
Local minimum possible. lsqnonlin stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance. <stopping criteria details>
Alpha = X(1); Rho = X(2); Nu = X(3);
Create SABR Model Using the Calibrated Parameters
Use finmodel to create a SABR model object.
SABRModel = finmodel("SABR",'Alpha',Alpha,'Beta',Beta,'Rho',Rho,'Nu',Nu,'Shift',Shift)
SABRModel =
SABR with properties:
Alpha: 0.0135
Beta: 0.5000
Rho: 0.4654
Nu: 0.4957
Shift: 0.0080
VolatilityType: "black"
Create SABR Pricer Using Calibrated SABR Model and Compute Volatilities
Use finpricer to create a SABR pricer object and use the ratecurve object for the 'DiscountCurve' name-value pair argument.
SABRPricer = finpricer("Analytic", 'Model', SABRModel, 'DiscountCurve', ZeroCurve)
SABRPricer =
SABR with properties:
DiscountCurve: [1×1 ratecurve]
Model: [1×1 finmodel.SABR]
SABRShiftedBlackVolatilities = volatilities(SABRPricer, SwaptionExerciseDate, ForwardValue, SwaptionStrikes)
SABRShiftedBlackVolatilities = 221×1
0.2978
0.2911
0.2848
0.2787
0.2729
0.2673
0.2620
0.2568
0.2518
0.2470
0.2423
0.2378
0.2335
0.2293
0.2252
⋮
figure; plot(MarketStrikes, MarketShiftedBlackVolatilities, 'o', ... SwaptionStrikes, SABRShiftedBlackVolatilities); h = gca; line([0,0],[min(h.YLim),max(h.YLim)],'LineStyle','--'); ylim([0.13 0.31]) xlabel('Strike'); legend('Market quotes','Shifted SABR', 'location', 'southeast'); title (['Shifted Black Volatility (',num2str(Shift*100),' percent shift)']);
Price Swaption Instruments Using Calibrated SABR Model and SABR Pricer
% Create swaption instruments. NumInst = length(SwaptionStrikes); Swaptions(NumInst, 1) = fininstrument("Swaption", ... 'Strike', SwaptionStrikes(1), 'ExerciseDate', SwaptionExerciseDate(1), 'Swap', Swap); for k = 1:NumInst Swaptions(k) = fininstrument("Swaption", 'Strike', SwaptionStrikes(k), ... 'ExerciseDate', SwaptionExerciseDate, 'Swap', Swap, 'OptionType', OptSpec); end Swaptions
Swaptions=221×1 Swaption array with properties:
OptionType
ExerciseStyle
ExerciseDate
Strike
Swap
Name
⋮
% Price swaptions using the SABR pricer. SwaptionPrices = price(SABRPricer,Swaptions); figure; plot(SwaptionStrikes, SwaptionPrices, 'r'); h = gca; line([0,0],[min(h.YLim),max(h.YLim)],'LineStyle','--'); xlabel('Strike'); title ('Swaption Price');
