Package com.opengamma.strata.pricer.impl.volatility.smile

Class SabrHaganNormalVolatilityFormula

java.lang.Object

com.opengamma.strata.pricer.impl.volatility.smile.SabrHaganNormalVolatilityFormula

All Implemented Interfaces:

SabrVolatilityFormula

public final class SabrHaganNormalVolatilityFormula
extends Object
implements SabrVolatilityFormula

Formulas related to the SABR implied normal volatility function.

Only the "beta = 0" versions take strike/forward with negative values. The other formulas have a barrier at 0.

Reference: Hagan, P.; Kumar, D.; Lesniewski, A. & Woodward, D. "Managing smile risk", Wilmott Magazine, 2002, September, 84-108 Note: Formula references are related to the preprint version, e.g. available at https://www.researchgate.net/publication/235622441_Managing_Smile_Risk

Field Summary

Fields

Modifier and Type	Field	Description
static SabrHaganNormalVolatilityFormula	DEFAULT	Default implementation.

Method Summary

Modifier and Type	Method	Description
ValueType	getVolatilityType()	Gets the type of volatility returned by the SabrVolatilityFormula.volatility(double, double, double, double, double, double, double) method.
double	volatility(double forward, double strike, double timeToExpiry, double alpha, double beta, double rho, double nu)	Calculates the volatility.
ValueDerivatives	volatilityAdjoint(double forward, double strike, double timeToExpiry, double alpha, double beta, double rho, double nu)	Calculates volatility and the adjoint (volatility sensitivity to forward, strike and model parameters).
double	volatilityBeta0(double forward, double strike, double timeToExpiry, double alpha, double rho, double nu)	Calculates the normal implied volatility for the special case of beta=0.
ValueDerivatives	volatilityBeta0Adjoint(double forward, double strike, double timeToExpiry, double alpha, double rho, double nu)	Calculates the normal implied volatility and its derivatives (w.r.t.
double	volatilityBetaNonZero(double forward, double strike, double timeToExpiry, double alpha, double beta, double rho, double nu)	Returns the volatility using the generic formula with barrier at 0.
ValueDerivatives	volatilityBetaNonZeroAdjoint(double forward, double strike, double timeToExpiry, double alpha, double beta, double rho, double nu)	Returns the volatility using the generic formula with barrier at 0 at its derivatives.
protected double	zetaOverXhat(double zeta, double rho)	Computes the ratio zeta over xHat.
protected ValueDerivatives	zetaOverXhatAdjoint(double zeta, double rho)	Computes the ratio zeta over xHat and its derivatives.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

DEFAULT

public static final SabrHaganNormalVolatilityFormula DEFAULT

Default implementation.

Method Detail

getVolatilityType

public ValueType getVolatilityType()

Description copied from interface: SabrVolatilityFormula

Gets the type of volatility returned by the SabrVolatilityFormula.volatility(double, double, double, double, double, double, double) method.

Specified by:

getVolatilityType in interface SabrVolatilityFormula

Returns:

the type

volatility

public double volatility(double forward,
                         double strike,
                         double timeToExpiry,
                         double alpha,
                         double beta,
                         double rho,
                         double nu)

Description copied from interface: SabrVolatilityFormula

Calculates the volatility.

Specified by:

volatility in interface SabrVolatilityFormula

Parameters:

forward - the forward value of the underlying

strike - the strike value of the option

timeToExpiry - the time to expiry of the option

alpha - the SABR alpha value

beta - the SABR beta value

rho - the SABR rho value

nu - the SABR nu value

Returns:

the volatility

volatilityAdjoint

public ValueDerivatives volatilityAdjoint(double forward,
                                          double strike,
                                          double timeToExpiry,
                                          double alpha,
                                          double beta,
                                          double rho,
                                          double nu)

Description copied from interface: SabrVolatilityFormula

Calculates volatility and the adjoint (volatility sensitivity to forward, strike and model parameters).

By default the derivatives are computed by central finite difference approximation. This should be overridden in each subclass.

Specified by:

volatilityAdjoint in interface SabrVolatilityFormula

Parameters:

forward - the forward value of the underlying

strike - the strike value of the option

timeToExpiry - the time to expiry of the option

alpha - the SABR alpha value

beta - the SABR beta value

rho - the SABR rho value

nu - the SABR nu value

Returns:

the volatility and associated derivatives

volatilityBetaNonZero

public double volatilityBetaNonZero(double forward,
                                    double strike,
                                    double timeToExpiry,
                                    double alpha,
                                    double beta,
                                    double rho,
                                    double nu)

Returns the volatility using the generic formula with barrier at 0. Formula B.69a in the preprint version

Parameters:

forward - the forward value of the underlying

strike - the strike value of the option

timeToExpiry - the time to expiry of the option

alpha - the SABR alpha value

beta - the SABR beta value

rho - the SABR rho value

nu - the SABR nu value

Returns:

the volatility

volatilityBetaNonZeroAdjoint

public ValueDerivatives volatilityBetaNonZeroAdjoint(double forward,
                                                     double strike,
                                                     double timeToExpiry,
                                                     double alpha,
                                                     double beta,
                                                     double rho,
                                                     double nu)

Returns the volatility using the generic formula with barrier at 0 at its derivatives. Formula B.69a in the preprint version

Parameters:

forward - the forward value of the underlying

strike - the strike value of the option

timeToExpiry - the time to expiry of the option

alpha - the SABR alpha value

beta - the SABR beta value

rho - the SABR rho value

nu - the SABR nu value

Returns:

the volatility

volatilityBeta0

public double volatilityBeta0(double forward,
                              double strike,
                              double timeToExpiry,
                              double alpha,
                              double rho,
                              double nu)

Calculates the normal implied volatility for the special case of beta=0.

The case beta=0 removes the barrier for forward/strike at 0 and lead to significant simplification in the implementation.

Formula B.70a in the preprint version. Note that the preprint version has a typo (missing the zeta/xhat term).

Parameters:

forward - the forward value of the underlying

strike - the strike value of the option

timeToExpiry - the time to expiry of the option

alpha - the SABR alpha value

rho - the SABR rho value

nu - the SABR nu value

Returns:

the volatility

volatilityBeta0Adjoint

public ValueDerivatives volatilityBeta0Adjoint(double forward,
                                               double strike,
                                               double timeToExpiry,
                                               double alpha,
                                               double rho,
                                               double nu)

Calculates the normal implied volatility and its derivatives (w.r.t. to forward, strike and model parameters) for the special case of beta=0.

The case beta=0 removes the barrier for forward/strike at 0 and lead to significant simplification in the implementation.

Parameters:

forward - the forward value of the underlying

strike - the strike value of the option

timeToExpiry - the time to expiry of the option

alpha - the SABR alpha value

rho - the SABR rho value

nu - the SABR nu value

Returns:

the volatility and associated derivatives (forward, strike, alpha, beta, rho, nu)

zetaOverXhat

protected double zetaOverXhat(double zeta,
                              double rho)

Computes the ratio zeta over xHat. Special treatment is required for zeta close to 0 (ATM).

Parameters:

zeta - the zeta value

rho - the rho parameter value

Returns:

the ratio

zetaOverXhatAdjoint

protected ValueDerivatives zetaOverXhatAdjoint(double zeta,
                                               double rho)

Computes the ratio zeta over xHat and its derivatives. Special treatment is required for zeta close to 0 (ATM).

Parameters:

zeta - the zeta value

rho - the rho parameter value

Returns:

the ratio