## Calibrate the SABR Model

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`.

### Load Market Implied Black Volatility Data

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```

### Method 1: Calibrate Alpha, Rho, and Nu Directly

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 .

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.```

### Method 2: Calibrate Rho and Nu by Implying Alpha from At-The-Money Volatility

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 .

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

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 ```

### Use the Calibrated Models

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.

### References

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

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