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: [16x1 datetime] Rates: [16x1 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: [1x2 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.
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: [1x1 ratecurve] Model: [1x1 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
⋮
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');