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finmath lib is a Mathematical Finance Library in Java.
It provides algorithms and methodologies related to mathematical finance.
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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 Merton jump diffusion model.
*
* The model is
* \[
* dS = \mu S dt + \sigma S dW + S dJ, \quad S(0) = S_{0},
* \]
* \[
* dN = r N dt, \quad N(0) = N_{0},
* \]
* where \( W \) is Brownian motion and \( J \) is a jump process (compound Poisson process).
*
* The process \( J \) is given by \( J(t) = \sum_{i=1}^{N(t)} (Y_{i}-1) \), where
* \( \log(Y_{i}) \) are i.i.d. normals with mean \( a - \frac{1}{2} b^{2} \) and standard deviation \( b \).
* Here \( a \) is the jump size mean and \( b \) is the jump size std. dev.
*
*
* @author Alessandro Gnoatto
* @version 1.0
*/
public class MertonModel 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 jumpIntensity;
private final double jumpSizeMean;
private final double jumpSizeStdDev;
/**
* Construct a Merton jump diffusion model with discount curves for the forward price (i.e. repo rate minus dividend yield) and for discounting.
*
* @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 volatility \( \sigma \) the initial volatility level
* @param jumpIntensity Coefficients of for the jump intensity.
* @param jumpSizeMean Jump size mean
* @param jumpSizeStdDev Jump size variance.
* @param discountCurveForDiscountRate The curve specifying \( t \mapsto exp(- r^{\text{d}}(t) \cdot t) \) - with \( r^{\text{d}}(t) \) the discount rate
*/
public MertonModel(final LocalDate referenceDate, final double initialValue,
final DiscountCurve discountCurveForForwardRate,
final DiscountCurve discountCurveForDiscountRate, final double volatility, final double jumpIntensity,
final double jumpSizeMean, final double jumpSizeStdDev) {
super();
this.referenceDate = referenceDate;
this.initialValue = initialValue;
this.discountCurveForForwardRate = discountCurveForForwardRate;
riskFreeRate = Double.NaN;
this.discountCurveForDiscountRate = discountCurveForDiscountRate;
discountRate = Double.NaN;
this.volatility = volatility;
this.jumpIntensity = jumpIntensity;
this.jumpSizeMean = jumpSizeMean;
this.jumpSizeStdDev = jumpSizeStdDev;
}
/**
* Construct a Merton jump diffusion model with constant rates for the forward price (i.e. repo rate minus dividend yield) and for the discount curve.
*
* @param initialValue \( S_{0} \) - spot - initial value of S
* @param riskFreeRate The constant risk free rate for the drift (repo rate of the underlying).
* @param volatility \( \sigma \) the initial volatility level
* @param jumpIntensity Coefficients of for the jump intensity.
* @param jumpSizeMean Jump size mean
* @param jumpSizeStdDev Jump size variance.
* @param discountRate The constant rate used for discounting.
*/
public MertonModel(final double initialValue, final double riskFreeRate,
final double discountRate,
final double volatility, final double jumpIntensity, final double jumpSizeMean,
final double jumpSizeStdDev) {
super();
referenceDate = null;
this.initialValue = initialValue;
discountCurveForForwardRate = null;
this.riskFreeRate = riskFreeRate;
this.volatility = volatility;
discountCurveForDiscountRate = null;
this.discountRate = discountRate;
this.jumpIntensity = jumpIntensity;
this.jumpSizeMean = jumpSizeMean;
this.jumpSizeStdDev = jumpSizeStdDev;
}
/**
* Construct a single curve Merton jump diffusion model.
*
* @param initialValue \( S_{0} \) - spot - initial value of S
* @param riskFreeRate The constant risk free rate for the drift (repo rate of the underlying). It is also used for discounting.
* @param volatility \( \sigma \) the initial volatility level
* @param jumpIntensity Coefficients of for the jump intensity.
* @param jumpSizeMean Jump size mean
* @param jumpSizeStdDev Jump size variance.
*/
public MertonModel(final double initialValue, final double riskFreeRate, final double volatility,
final double jumpIntensity, final double jumpSizeMean, final double jumpSizeStdDev) {
this(initialValue,riskFreeRate,riskFreeRate,volatility,jumpIntensity,jumpSizeMean,jumpSizeStdDev);
}
@Override
public CharacteristicFunction apply(final double time) {
final double logDiscountFactorForForward = this.getLogDiscountFactorForForward(time);
final double logDiscountFactorForDiscounting = this.getLogDiscountFactorForDiscounting(time);
final double transformedMean = jumpSizeMean - 0.5 * jumpSizeStdDev*jumpSizeStdDev;
return new CharacteristicFunction() {
@Override
public Complex apply(final Complex argument) {
final Complex iargument = argument.multiply(Complex.I);
final Complex exponent = (iargument.multiply(transformedMean))
.add(iargument.multiply(iargument.multiply(jumpSizeStdDev*jumpSizeStdDev/2.0)));
final Complex jumpTransform = ((exponent.exp()).subtract(1.0)).multiply(jumpIntensity*time);
final double jumpTransformCompensator = jumpIntensity*time*(Math.exp(transformedMean+jumpSizeStdDev*jumpSizeStdDev/2.0)-1.0);
return iargument
.multiply(
iargument
.multiply(0.5*volatility*volatility*time)
.add(Math.log(initialValue)-0.5*volatility*volatility*time-logDiscountFactorForForward))
.add(logDiscountFactorForDiscounting).add(jumpTransform.subtract(jumpTransformCompensator))
.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
*/
public LocalDate getReferenceDate() {
return referenceDate;
}
/**
* @return the initialValue
*/
public double getInitialValue() {
return initialValue;
}
/**
* @return the discountCurveForForwardRate
*/
public DiscountCurve getDiscountCurveForForwardRate() {
return discountCurveForForwardRate;
}
/**
* @return the riskFreeRate
*/
public double getRiskFreeRate() {
return riskFreeRate;
}
/**
* @return the discountCurveForDiscountRate
*/
public DiscountCurve getDiscountCurveForDiscountRate() {
return discountCurveForDiscountRate;
}
/**
* @return the discountRate
*/
public double getDiscountRate() {
return discountRate;
}
/**
* @return the volatility
*/
public double getVolatility() {
return volatility;
}
/**
* @return the jumpIntensity
*/
public double getJumpIntensity() {
return jumpIntensity;
}
/**
* @return the jumpSizeMean
*/
public double getJumpSizeMean() {
return jumpSizeMean;
}
/**
* @return the jumpSizeStdDev
*/
public double getJumpSizeStdDev() {
return jumpSizeStdDev;
}
@Override
public String toString() {
return "MertonModel [initialValue=" + initialValue + ", discountCurveForForwardRate="
+ discountCurveForForwardRate + ", riskFreeRate=" + riskFreeRate + ", discountCurveForDiscountRate="
+ discountCurveForDiscountRate + ", discountRate=" + discountRate + ", volatility=" + volatility
+ ", jumpIntensity=" + jumpIntensity + ", jumpSizeMean=" + jumpSizeMean + ", jumpSizeStdDev="
+ jumpSizeStdDev + "]";
}
}