HestonModel.java
/*
* (c) Copyright Christian P. Fries, Germany. Contact: email@christian-fries.de.
*
* Created on 23.03.2014
*/
package net.finmath.fouriermethod.models;
import java.time.LocalDate;
import org.apache.commons.math3.complex.Complex;
import net.finmath.fouriermethod.CharacteristicFunction;
import net.finmath.marketdata.model.curves.DiscountCurve;
/**
* 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:
* <dl>
* <dt>\( S_{0} \)</dt> <dd>spot - initial value of S</dd>
* <dt>\( r^{\text{c}} \)</dt> <dd>the risk free rate (may be provided as a curve or a constant)</dd>
* <dt>\( \sigma \)</dt> <dd>the initial volatility level</dd>
* <dt>\( r^{\text{d}} \)</dt> <dd>the discount rate (may be provided as a curve or a constant)</dd>
* <dt>\( \xi \)</dt> <dd>the volatility of volatility</dd>
* <dt>\( \theta \)</dt> <dd>the mean reversion level of the stochastic volatility</dd>
* <dt>\( \kappa \)</dt> <dd>the mean reversion speed of the stochastic volatility</dd>
* <dt>\( \rho \)</dt> <dd>the correlation of the Brownian drivers</dd>
* </dl>
*
* @author Christian Fries
* @author Andy Graf
* @author Lorenzo Toricelli
* @version 1.0
*/
public class HestonModel implements CharacteristicFunctionModel {
private final LocalDate referenceDate;
private final double initialValue;
private final DiscountCurve discountCurveForForwardRate;
private final double riskFreeRate; // Constant rate, used if discountCurveForForwardRate is null
private final DiscountCurve discountCurveForDiscountRate;
private final double discountRate; // Constant rate, used if discountCurveForForwardRate is null
private final double volatility;
private final double theta;
private final double kappa;
private final double xi;
private final double rho;
/**
* Create a Heston model (characteristic function)
*
* @param referenceDate The date representing the time t = 0. All other double times are following {@link net.finmath.time.FloatingpointDate}.
* @param initialValue \( S_{0} \) - spot - initial value of S
* @param discountCurveForForwardRate The curve specifying \( t \mapsto exp(- r^{\text{c}}(t) \cdot t) \) - with \( r^{\text{c}}(t) \) the risk free rate
* @param discountCurveForDiscountRate The curve specifying \( t \mapsto exp(- r^{\text{d}}(t) \cdot t) \) - with \( r^{\text{d}}(t) \) the discount rate
* @param volatility \( \sigma \) the initial volatility level
* @param theta \( \theta \) - the mean reversion level of the stochastic volatility
* @param kappa \( \kappa \) - the mean reversion speed of the stochastic volatility
* @param xi \( \xi \) - the volatility of volatility
* @param rho \( \rho \) - the correlation of the Brownian drivers
*/
public HestonModel(final LocalDate referenceDate, final double initialValue, final DiscountCurve discountCurveForForwardRate, final DiscountCurve discountCurveForDiscountRate, final double volatility, final double theta, final double kappa, final double xi, final double rho) {
super();
this.referenceDate = referenceDate;
this.initialValue = initialValue;
this.discountCurveForForwardRate = discountCurveForForwardRate;
riskFreeRate = Double.NaN; // For safety
this.discountCurveForDiscountRate = discountCurveForDiscountRate;
discountRate = Double.NaN; // For safety
this.volatility = volatility;
this.theta = theta;
this.kappa = kappa;
this.xi = xi;
this.rho = rho;
}
/**
* Create a Heston model (characteristic function)
*
* @param initialValue \( S_{0} \) - spot - initial value of S
* @param discountCurveForForwardRate The curve specifying \( t \mapsto exp(- r^{\text{c}}(t) \cdot t) \) - with \( r^{\text{c}}(t) \) the risk free rate
* @param discountCurveForDiscountRate The curve specifying \( t \mapsto exp(- r^{\text{d}}(t) \cdot t) \) - with \( r^{\text{d}}(t) \) the discount rate
* @param volatility \( \sigma \) the initial volatility level
* @param theta \( \theta \) - the mean reversion level of the stochastic volatility
* @param kappa \( \kappa \) - the mean reversion speed of the stochastic volatility
* @param xi \( \xi \) - the volatility of volatility
* @param rho \( \rho \) - the correlation of the Brownian drivers
*/
public HestonModel(final double initialValue, final DiscountCurve discountCurveForForwardRate, final DiscountCurve discountCurveForDiscountRate, final double volatility, final double theta, final double kappa, final double xi, final double rho) {
this(null, initialValue, discountCurveForForwardRate, discountCurveForDiscountRate, volatility, theta, kappa, xi, rho);
}
/**
* Create a Heston model (characteristic function)
*
* @param initialValue \( S_{0} \) - spot - initial value of S
* @param riskFreeRate \( r^{\text{c}} \) - the risk free rate
* @param volatility \( \sigma \) the initial volatility level
* @param discountRate \( r^{\text{d}} \) - the discount rate
* @param theta \( \theta \) - the mean reversion level of the stochastic volatility
* @param kappa \( \kappa \) - the mean reversion speed of the stochastic volatility
* @param xi \( \xi \) - the volatility of volatility
* @param rho \( \rho \) - the correlation of the Brownian drivers
*/
public HestonModel(final double initialValue, final double riskFreeRate, final double volatility, final double discountRate, final double theta, final double kappa,
final double xi, final double rho) {
super();
referenceDate = null;
this.initialValue = initialValue;
discountCurveForForwardRate = null;
this.riskFreeRate = riskFreeRate;
discountCurveForDiscountRate = null;
this.discountRate = discountRate;
this.volatility = volatility;
this.theta = theta;
this.kappa = kappa;
this.xi = xi;
this.rho = rho;
}
public HestonModel(final double initialValue, final double riskFreeRate, final double volatility, final double theta, final double kappa,
final double xi, final double rho) {
this(initialValue, riskFreeRate, volatility, riskFreeRate, theta, kappa, xi, rho);
}
@Override
public CharacteristicFunction apply(final double time) {
final double logDiscountFactorForForward = this.getLogDiscountFactorForForward(time);
final double logDiscountFactorForDiscounting = this.getLogDiscountFactorForDiscounting(time);
return new CharacteristicFunction() {
@Override
public Complex apply(final Complex argument) {
final Complex iargument = argument.multiply(Complex.I);
final Complex gamma = iargument.multiply(rho * xi).subtract(kappa).pow(2)
.subtract(
iargument.multiply(iargument)
.add(iargument.multiply(-1)).multiply(0.5)
.multiply(2 * xi * xi))
.sqrt();
final Complex a = iargument
.multiply(rho * xi)
.subtract(kappa)
.subtract(gamma).multiply((-theta*kappa * time) / (xi * xi))
.subtract(iargument.multiply(rho * xi).subtract(kappa).subtract(gamma)
.multiply(new Complex(1).divide(gamma.multiply(time).exp()).subtract(1).divide(gamma))
.multiply(0.5).add(new Complex(1).divide(gamma.multiply(time).exp())).log()
.add(gamma.multiply(time)).multiply((2 * theta*kappa) / (xi * xi)));
final Complex b = iargument.multiply(iargument).add(iargument.multiply(-1)).multiply(-1)
.divide(iargument.multiply(rho * xi).subtract(kappa)
.add(gamma.multiply(new Complex(1).divide(gamma.multiply(time).exp()).add(1)
.divide(new Complex(1).divide(gamma.multiply(time).exp()).subtract(1)))));
return a.add(b.multiply(volatility*volatility)).add(iargument.multiply(Math.log(initialValue) - logDiscountFactorForForward)).add(logDiscountFactorForDiscounting).exp();
}
};
}
/**
* Small helper to calculate rate off the curve or use constant.
*
* @param time Maturity.
* @return The log of the discount factor, i.e., - rate * time.
*/
private double getLogDiscountFactorForForward(final double time) {
return discountCurveForForwardRate == null ? -riskFreeRate * time : Math.log(discountCurveForForwardRate.getDiscountFactor(null, time));
}
/**
* Small helper to calculate rate off the curve or use constant.
*
* @param time Maturity.
* @return The log of the discount factor, i.e., - rate * time.
*/
private double getLogDiscountFactorForDiscounting(final double time) {
return discountCurveForDiscountRate == null ? -discountRate * time : Math.log(discountCurveForDiscountRate.getDiscountFactor(null, time));
}
/**
* @return the referenceDate
*/
@Override
public LocalDate getReferenceDate() {
return referenceDate;
}
/**
* @return the initialValue
*/
@Override
public double getInitialValue() {
return initialValue;
}
/**
* @return the discountCurveForForwardRate
*/
@Override
public DiscountCurve getDiscountCurveForForwardRate() {
return discountCurveForForwardRate;
}
/**
* @return the riskFreeRate
*/
@Override
public double getRiskFreeRate() {
return riskFreeRate;
}
/**
* @return the discountCurveForDiscountRate
*/
@Override
public DiscountCurve getDiscountCurveForDiscountRate() {
return discountCurveForDiscountRate;
}
/**
* @return the discountRate
*/
@Override
public double getDiscountRate() {
return discountRate;
}
/**
* @return the volatility
*/
public double getVolatility() {
return volatility;
}
/**
* @return the theta
*/
public double getTheta() {
return theta;
}
/**
* @return the kappa
*/
public double getKappa() {
return kappa;
}
/**
* @return the xi
*/
public double getXi() {
return xi;
}
/**
* @return the rho
*/
public double getRho() {
return rho;
}
/* (non-Javadoc)
* @see java.lang.Object#toString()
*/
@Override
public String toString() {
return "HestonModel [referenceDate=" + referenceDate + ", initialValue=" + initialValue
+ ", discountCurveForForwardRate=" + discountCurveForForwardRate + ", riskFreeRate=" + riskFreeRate
+ ", discountCurveForDiscountRate=" + discountCurveForDiscountRate + ", discountRate=" + discountRate
+ ", volatility=" + volatility + ", theta=" + theta + ", kappa=" + kappa + ", xi=" + xi + ", rho=" + rho
+ "]";
}
}