net.finmath.fouriermethod.models

Class HestonModel

java.lang.Object

net.finmath.fouriermethod.models.HestonModel

All Implemented Interfaces:

CharacteristicFunctionModel, Model

public class HestonModel
extends Object
implements CharacteristicFunctionModel

Implements the characteristic function of a Heston model. The model is \[ dS(t) = r^{\text{c}}(t) S(t) dt + \sqrt{V(t)} S(t) dW_{1}(t), \quad S(0) = S_{0}, \] \[ dV(t) = \kappa ( \theta - V(t) ) dt + \xi \sqrt{V(t)} dW_{2}(t), \quad V(0) = \sigma^2, \] \[ dW_{1} dW_{2} = \rho dt \] \[ dN(t) = r^{\text{d}}(t) N(t) dt, \quad N(0) = N_{0}, \] where \( W \) is a Brownian motion. The model allows to specify two independent rate for forwarding (\( r^{\text{c}} \)) and discounting (\( r^{\text{d}} \)). It thus allow for a simple modelling of a funding / collateral curve (via (\( r^{\text{d}} \)) and/or the specification of a dividend yield. The free parameters of this model are:

\( S_{0} \)

spot - initial value of S

\( r^{\text{c}} \)

the risk free rate (may be provided as a curve or a constant)

\( \sigma \)

the initial volatility level

\( r^{\text{d}} \)

the discount rate (may be provided as a curve or a constant)

\( \xi \)

the volatility of volatility

\( \theta \)

the mean reversion level of the stochastic volatility

\( \kappa \)

the mean reversion speed of the stochastic volatility

\( \rho \)

the correlation of the Brownian drivers

Version:

1.0

Author:

Christian Fries, Andy Graf, Lorenzo Toricelli

Constructor Summary

Constructors

Constructor and Description
HestonModel(double initialValue, DiscountCurve discountCurveForForwardRate, double volatility, DiscountCurve discountCurveForDiscountRate, double theta, double kappa, double xi, double rho) Create a Heston model (characteristic function)
HestonModel(double initialValue, double riskFreeRate, double volatility, double theta, double kappa, double xi, double rho)
HestonModel(double initialValue, double riskFreeRate, double volatility, double discountRate, double theta, double kappa, double xi, double rho) Create a Heston model (characteristic function)
HestonModel(LocalDate referenceDate, double initialValue, DiscountCurve discountCurveForForwardRate, double volatility, DiscountCurve discountCurveForDiscountRate, double theta, double kappa, double xi, double rho) Create a Heston model (characteristic function)

Method Summary

Modifier and Type	Method and Description
CharacteristicFunction	apply(double time) Returns the characteristic function of X(t), where X is this stochastic process.

Methods inherited from class java.lang.Object

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

Constructor Detail

HestonModel

public HestonModel(LocalDate referenceDate,
                   double initialValue,
                   DiscountCurve discountCurveForForwardRate,
                   double volatility,
                   DiscountCurve discountCurveForDiscountRate,
                   double theta,
                   double kappa,
                   double xi,
                   double rho)

Create a Heston model (characteristic function)

Parameters:

referenceDate - The date representing the time t = 0. All other double times are following FloatingpointDate.

initialValue - \( S_{0} \) - spot - initial value of S

discountCurveForForwardRate - The curve specifying \( t \mapsto exp(- r^{\text{c}}(t) \cdot t) \) - with \( r^{\text{c}}(t) \) the risk free rate

volatility - \( \sigma \) the initial volatility level

discountCurveForDiscountRate - The curve specifying \( t \mapsto exp(- r^{\text{d}}(t) \cdot t) \) - with \( r^{\text{d}}(t) \) the discount rate

theta - \( \theta \) - the mean reversion level of the stochastic volatility

kappa - \( \kappa \) - the mean reversion speed of the stochastic volatility

xi - \( \xi \) - the volatility of volatility

rho - \( \rho \) - the correlation of the Brownian drivers

HestonModel

public HestonModel(double initialValue,
                   DiscountCurve discountCurveForForwardRate,
                   double volatility,
                   DiscountCurve discountCurveForDiscountRate,
                   double theta,
                   double kappa,
                   double xi,
                   double rho)

Create a Heston model (characteristic function)

Parameters:

initialValue - \( S_{0} \) - spot - initial value of S

discountCurveForForwardRate - The curve specifying \( t \mapsto exp(- r^{\text{c}}(t) \cdot t) \) - with \( r^{\text{c}}(t) \) the risk free rate

volatility - \( \sigma \) the initial volatility level

discountCurveForDiscountRate - The curve specifying \( t \mapsto exp(- r^{\text{d}}(t) \cdot t) \) - with \( r^{\text{d}}(t) \) the discount rate

theta - \( \theta \) - the mean reversion level of the stochastic volatility

kappa - \( \kappa \) - the mean reversion speed of the stochastic volatility

xi - \( \xi \) - the volatility of volatility

rho - \( \rho \) - the correlation of the Brownian drivers

HestonModel

public HestonModel(double initialValue,
                   double riskFreeRate,
                   double volatility,
                   double discountRate,
                   double theta,
                   double kappa,
                   double xi,
                   double rho)

Create a Heston model (characteristic function)

Parameters:

initialValue - \( S_{0} \) - spot - initial value of S

riskFreeRate - \( r^{\text{c}} \) - the risk free rate

volatility - \( \sigma \) the initial volatility level

discountRate - \( r^{\text{d}} \) - the discount rate

theta - \( \theta \) - the mean reversion level of the stochastic volatility

kappa - \( \kappa \) - the mean reversion speed of the stochastic volatility

xi - \( \xi \) - the volatility of volatility

rho - \( \rho \) - the correlation of the Brownian drivers

HestonModel

public HestonModel(double initialValue,
                   double riskFreeRate,
                   double volatility,
                   double theta,
                   double kappa,
                   double xi,
                   double rho)

Method Detail

apply

public CharacteristicFunction apply(double time)

Description copied from interface: CharacteristicFunctionModel

Returns the characteristic function of X(t), where X is this stochastic process.

Specified by:

apply in interface CharacteristicFunctionModel

Parameters:

time - The time at which the stochastic process is observed.

Returns:

The characteristic function of X(t).