Record Class SviSlice

java.lang.Object

java.lang.Record

com.quantfinlib.pricing.SviSlice

Record Components:

a - variance level

b - wing slope scale, ≥ 0

rho - rotation, in (−1, 1)

m - horizontal shift (log-moneyness)

sigma - ATM curvature, > 0

public record SviSlice(double a, double b, double rho, double m, double sigma)
extends Record

One expiry of the raw SVI parametrisation (Gatheral 2004, "A parsimonious arbitrage-free implied volatility parameterization", Global Derivatives; Gatheral and Jacquier 2014, "Arbitrage-free SVI volatility surfaces", Quantitative Finance 14(1), equation 3.1):

  w(k) = a + b ( ρ (k − m) + √((k − m)² + σ²) )

where w is total implied variance σ_BS²·T and k = ln(K/F) is log-moneyness against the forward. The parameter set is admissible when b ≥ 0, |ρ| < 1, σ > 0 and a + bσ√(1−ρ²) ≥ 0, which keeps w non-negative everywhere (Gatheral-Jacquier, section 3.1).

Butterfly arbitrage. A slice is free of butterfly arbitrage if and only if the density factor g(k) = (1 − k w'/(2w))² − (w'²/4)(1/w + 1/4) + w''/2 is non-negative (Gatheral-Jacquier 2014, lemma 2.2) and d₊ → −∞ in the right wing. The Black density is g(k)/√(2πw)·exp(−d₋²/2); minDensityFactor(double, double, int) checks g on a grid.

Calibration (calibrate(double[], double[], double[])) is Levenberg-Marquardt on weighted total-variance residuals in a reparametrisation that keeps every admissibility bound and Lee's (2004) wing bound b(1+|ρ|) ≤ 4, plus an optional penalty residual on negative g over a grid that extends half a unit of log-moneyness past the quotes. Weights are the caller's: vega weights or the reciprocal of the bid-ask spread in variance terms are the usual choices. The fit is multi-start (27 starts) because the SVI objective has several local minima.

Field Summary

Fields

Modifier and Type	Field	Description
static final double	BUTTERFLY_PENALTY	Weight of the min(g, 0) penalty residuals in calibrate(double[], double[], double[]).

Constructor Summary

Constructors

Constructor	Description
SviSlice(double a, double b, double rho, double m, double sigma)	Creates an instance of a SviSlice record class.

Method Summary

Modifier and Type	Method	Description
double	a()	Returns the value of the a record component.
double	b()	Returns the value of the b record component.
static SviSlice	calibrate(double[] k, double[] w, double[] weights)	Calibrates a slice to total-variance quotes with the butterfly penalty on.
static SviSlice	calibrate(double[] k, double[] w, double[] weights, boolean butterflyPenalty)	Calibrates a slice; butterflyPenalty adds residuals BUTTERFLY_PENALTY·min(g, 0) on a 121-point grid over [min k − 0.5, max k + 0.5].
double	curvature(double k)	Second derivative d²w/dk².
double	densityFactor(double k)	Gatheral-Jacquier density factor g(k); negative means butterfly arbitrage.
final boolean	equals(Object o)	Indicates whether some other object is "equal to" this one.
final int	hashCode()	Returns a hash code value for this object.
double	impliedVol(double k, double t)	Implied volatility at log-moneyness k for expiry t years.
double	m()	Returns the value of the m record component.
double	minDensityFactor(double kMin, double kMax, int n)	Minimum of g over n equally spaced points of [kMin, kMax].
double	rho()	Returns the value of the rho record component.
double	rmse(double[] k, double[] w)	Root-mean-square total-variance error against quotes (unweighted).
double	sigma()	Returns the value of the sigma record component.
double	slope(double k)	First derivative dw/dk.
final String	toString()	Returns a string representation of this record class.
double	totalVariance(double k)	Total implied variance at log-moneyness k.

Methods inherited from class Object

clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Field Details

BUTTERFLY_PENALTY

public static final double BUTTERFLY_PENALTY

Weight of the min(g, 0) penalty residuals in calibrate(double[], double[], double[]).

See Also:

• Constant Field Values

Constructor Details

SviSlice

public SviSlice(double a,
 double b,
 double rho,
 double m,
 double sigma)

Creates an instance of a SviSlice record class.

Parameters:

a - the value for the a record component

b - the value for the b record component

rho - the value for the rho record component

m - the value for the m record component

sigma - the value for the sigma record component

Method Details

totalVariance

public double totalVariance(double k)

Total implied variance at log-moneyness k.

slope

public double slope(double k)

First derivative dw/dk.

curvature

public double curvature(double k)

Second derivative d²w/dk².

impliedVol

public double impliedVol(double k,
 double t)

Implied volatility at log-moneyness k for expiry t years.

densityFactor

public double densityFactor(double k)

Gatheral-Jacquier density factor g(k); negative means butterfly arbitrage.

minDensityFactor

public double minDensityFactor(double kMin,
 double kMax,
 int n)

Minimum of g over n equally spaced points of [kMin, kMax].

rmse

public double rmse(double[] k,
 double[] w)

Root-mean-square total-variance error against quotes (unweighted).

calibrate

public static SviSlice calibrate(double[] k,
 double[] w,
 double[] weights)

Calibrates a slice to total-variance quotes with the butterfly penalty on.

Parameters:

k - log-moneyness ln(K/F) of each quote

w - market total variance σ²T of each quote

weights - residual weights, or null for equal weights

calibrate

public static SviSlice calibrate(double[] k,
 double[] w,
 double[] weights,
 boolean butterflyPenalty)

Calibrates a slice; butterflyPenalty adds residuals BUTTERFLY_PENALTY·min(g, 0) on a 121-point grid over [min k − 0.5, max k + 0.5]. At least five quotes are required.

toString

public final String toString()

Returns a string representation of this record class. The representation contains the name of the class, followed by the name and value of each of the record components.

Specified by:

toString in class Record

Returns:

a string representation of this object

hashCode

public final int hashCode()

Returns a hash code value for this object. The value is derived from the hash code of each of the record components.

Specified by:

hashCode in class Record

Returns:

a hash code value for this object

equals

public final boolean equals(Object o)

Indicates whether some other object is "equal to" this one. The objects are equal if the other object is of the same class and if all the record components are equal. All components in this record class are compared with the compare method from their corresponding wrapper classes.

Specified by:

equals in class Record

Parameters:

o - the object with which to compare

Returns:

true if this object is the same as the o argument; false otherwise.

a

public double a()

Returns the value of the a record component.

Returns:

the value of the a record component

b

public double b()

Returns the value of the b record component.

Returns:

the value of the b record component

rho

public double rho()

Returns the value of the rho record component.

Returns:

the value of the rho record component

m

public double m()

Returns the value of the m record component.

Returns:

the value of the m record component

sigma

public double sigma()

Returns the value of the sigma record component.

Returns:

the value of the sigma record component