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This example shows how to use two different methods to calibrate
the SABR stochastic volatility model from market implied Black volatilities.
Both approaches use `blackvolbysabr`

.

This example shows how to set up hypothetical
market implied Black volatilities for European swaptions over a range
of strikes before calibration. The swaptions expire in three years
from the `Settle`

date and have 10-year swaps as
the underlying instrument. The rates are expressed in decimals. (Changing
the units affect the numerical value and interpretation of the `Alpha`

input
parameter to the function `blackvolbysabr`

.)

Load the market implied Black volatility data for swaptions expiring in three years.

Settle = '12-Jun-2013'; ExerciseDate = '12-Jun-2016'; MarketStrikes = [2.0 2.5 3.0 3.5 4.0 4.5 5.0]'/100; MarketVolatilities = [45.6 41.6 37.9 36.6 37.8 39.2 40.0]'/100;

At the time of `Settle`

, define the underlying
forward rate and the at-the-money volatility.

CurrentForwardValue = MarketStrikes(4) ATMVolatility = MarketVolatilities(4)

CurrentForwardValue = 0.0350 ATMVolatility = 0.3660

This example shows how to calibrate the` Alpha`

, `Rho`

,
and `Nu`

input parameters directly. The value of `Beta`

is
predetermined either by fitting historical market volatility data
or by choosing a value deemed appropriate for that market [1].

Define the predetermined `Beta`

.

Beta1 = 0.5;

After fixing the value of $$\beta $$ (`Beta`

), the parameters $$\alpha $$ (`Alpha`

), $$\rho $$ (`Rho`

), and $$\upsilon $$(`Nu`

) are all fitted directly. The
Optimization Toolbox™ function `lsqnonlin`

generates the parameter
values that minimize the squared error between the market volatilities and
the volatilities computed by `blackvolbysabr`

.

% Calibrate Alpha, Rho, and Nu objFun = @(X) MarketVolatilities - ... blackvolbysabr(X(1), Beta1, X(2), X(3), Settle, ... ExerciseDate, CurrentForwardValue, MarketStrikes); X = lsqnonlin(objFun, [0.5 0 0.5], [0 -1 0], [Inf 1 Inf]); Alpha1 = X(1); Rho1 = X(2); Nu1 = X(3);

Local minimum possible. lsqnonlin stopped because the final change in the sum of squares relative to its initial value is less than the default value of the function tolerance.

This example shows how to use an alternative
calibration method where the value of $$\beta $$ (`Beta`

) is
again predetermined as in Method 1.

Define the predetermined `Beta`

.

Beta2 = 0.5;

However, after fixing the value of $$\beta $$ (`Beta`

), the parameters $$\rho $$ (`Rho`

), and $$\upsilon $$ (`Nu`

) are fitted directly while $$\alpha $$ (`Alpha`

) is implied from the market
at-the-money volatility. Models calibrated using this method produce
at-the-money volatilities that are equal to market quotes. This approach is
widely used in swaptions, where at-the-money volatilities are quoted most
frequently and are important to match. To imply $$\alpha $$ (`Alpha`

) from market at-the-money
volatility ($${\sigma}_{ATM}$$), the following cubic polynomial is solved for $$\alpha $$ (`Alpha`

), and the smallest positive real
root is selected [2].

$$\frac{{(1-\beta )}^{2}T}{24{F}^{(2-2\beta )}}{\alpha}^{3}+\frac{\rho \beta \upsilon T}{4{F}^{(1-\beta )}}{\alpha}^{2}+\left(1+\frac{2-3{\rho}^{2}}{24}{\upsilon}^{2}T\right)\alpha -{\sigma}_{ATM}{F}^{(1-\beta )}=0$$

where:

$$F$$ is the current forward value.

$$T$$ is the year fraction to maturity.

To accomplish this, define an anonymous function as:

% Year fraction from Settle to option maturity T = yearfrac(Settle, ExerciseDate, 1); % This function solves the SABR at-the-money volatility equation as a % polynomial of Alpha alpharoots = @(Rho,Nu) roots([... (1 - Beta2)^2*T/24/CurrentForwardValue^(2 - 2*Beta2) ... Rho*Beta2*Nu*T/4/CurrentForwardValue^(1 - Beta2) ... (1 + (2 - 3*Rho^2)*Nu^2*T/24) ... -ATMVolatility*CurrentForwardValue^(1 - Beta2)]); % This function converts at-the-money volatility into Alpha by picking the % smallest positive real root atmVol2SabrAlpha = @(Rho,Nu) min(real(arrayfun(@(x) ... x*(x>0) + realmax*(x<0 || abs(imag(x))>1e-6), alpharoots(Rho,Nu))));

The function `atmVol2SabrAlpha`

converts at-the-money volatility into $$\alpha $$ (`Alpha`

) for a given set of $$\rho $$ (`Rho`

) and $$\upsilon $$ (`Nu`

). This function is then used in the
objective function to fit parameters $$\rho $$ (`Rho`

) and $$\upsilon $$ (`Nu`

).

% Calibrate Rho and Nu (while converting at-the-money volatility into Alpha % using atmVol2SabrAlpha) objFun = @(X) MarketVolatilities - ... blackvolbysabr(atmVol2SabrAlpha(X(1), X(2)), ... Beta2, X(1), X(2), Settle, ExerciseDate, CurrentForwardValue, ... MarketStrikes); X = lsqnonlin(objFun, [0 0.5], [-1 0], [1 Inf]); Rho2 = X(1); Nu2 = X(2);

Local minimum found. Optimization completed because the size of the gradient is less than the default value of the function tolerance.

The calibrated parameter $$\alpha $$ (`Alpha`

)
is computed using the calibrated parameters $$\rho $$ (`Rho`

) and $$\upsilon $$ (`Nu`

).

% Obtain final Alpha from at-the-money volatility using calibrated parameters Alpha2 = atmVol2SabrAlpha(Rho2, Nu2); % Display calibrated parameters C = {Alpha1 Beta1 Rho1 Nu1;Alpha2 Beta2 Rho2 Nu2}; CalibratedPrameters = cell2table(C,... 'VariableNames',{'Alpha' 'Beta' 'Rho' 'Nu'},... 'RowNames',{'Method 1';'Method 2'})

CalibratedPrameters = Alpha Beta Rho Nu ________ ____ _______ _______ Method 1 0.060277 0.5 0.2097 0.75091 Method 2 0.058484 0.5 0.20568 0.79647

This example shows how to use the calibrated models to compute new volatilities at any strike value.

Compute volatilities for models calibrated using Method 1 and Method 2 and plot the results.

PlottingStrikes = (1.75:0.1:5.50)'/100; % Compute volatilities for model calibrated by Method 1 ComputedVols1 = blackvolbysabr(Alpha1, Beta1, Rho1, Nu1, Settle, ... ExerciseDate, CurrentForwardValue, PlottingStrikes); % Compute volatilities for model calibrated by Method 2 ComputedVols2 = blackvolbysabr(Alpha2, Beta2, Rho2, Nu2, Settle, ... ExerciseDate, CurrentForwardValue, PlottingStrikes); figure; plot(MarketStrikes,MarketVolatilities,'xk',... PlottingStrikes,ComputedVols1,'b', ... PlottingStrikes,ComputedVols2,'r', ... CurrentForwardValue,ATMVolatility,'ok',... 'MarkerSize',10); xlim([0.01 0.06]); ylim([0.35 0.5]); xlabel('Strike', 'FontWeight', 'bold'); ylabel('Implied Black Volatility', 'FontWeight', 'bold'); legend('Market Volatilities', 'SABR Model (Method 1)',... 'SABR Model (Method 2)', 'At-the-money volatility');

The model calibrated using Method 2 reproduces the market at-the-money volatility (marked with a circle) exactly.

[1] Hagan, P. S., Kumar, D., Lesniewski, A. S. and Woodward,
D. E., *Managing smile risk*, Wilmott Magazine,
2002.

[2] West, G., “Calibration of the SABR Model in Illiquid
Markets,” *Applied Mathematical Finance*,
12(4), pp. 371–385, 2004.

`blackvolbysabr`

| `optsensbysabr`

| `swaptionbyblk`