LIBORVolatilityModelFourParameterExponentialFormIntegrated.java
/*
* (c) Copyright Christian P. Fries, Germany. Contact: email@christian-fries.de.
*
* Created on 21.07.2019
*/
package net.finmath.montecarlo.interestrate.models.covariance;
import java.util.Map;
import net.finmath.montecarlo.RandomVariableFactory;
import net.finmath.stochastic.RandomVariable;
import net.finmath.stochastic.Scalar;
import net.finmath.time.TimeDiscretization;
/**
* Implements the volatility model
* \[
* \sigma_{i}(t_{j}) = \sqrt{ \frac{1}{t_{j+1}-t_{j}} \int_{t_{j}}^{t_{j+1}} \left( ( a + b (T_{i}-t) ) exp(-c (T_{i}-t)) + d \right)^{2} \ \mathrm{d}t } \text{.}
* \]
*
* The parameters here have some interpretation:
* <ul>
* <li>The parameter a: an initial volatility level.</li>
* <li>The parameter b: the slope at the short end (shortly before maturity).</li>
* <li>The parameter c: exponential decay of the volatility in time-to-maturity.</li>
* <li>The parameter d: if c > 0 this is the very long term volatility level.</li>
* </ul>
*
* Note that this model results in a terminal (Black 76) volatility which is given
* by
* \[
* \left( \sigma^{\text{Black}}_{i}(t_{k}) \right)^2 = \frac{1}{t_{k} \int_{0}^{t_{k}} \left( ( a + b (T_{i}-t) ) exp(-c (T_{i}-t)) + d \right)^{2} \ \mathrm{d}t \text{.}
* \]
*
* @author Christian Fries
* @version 1.1
*/
public class LIBORVolatilityModelFourParameterExponentialFormIntegrated extends LIBORVolatilityModel {
private static final long serialVersionUID = -1613728266481870311L;
private final double[] coeffTaylorE1 = new double[] { 1, 1.0/2.0, 1.0/6.0, 1.0/24.0, 1.0/120.0 };
private final double[] coeffTaylorE2 = new double[] { 1, 2.0/3.0, 1.0/4.0, 1.0/15.0, 1.0/72.0 };
private final double[] coeffTaylorE3 = new double[] { 1, 3.0/4.0, 3.0/10.0, 1.0/12.0, 1.0/56.0 };
private final double[] coeffTaylorE17 = new double[] { 1, 1.0/2.0, 1.0/6.0, 1.0/24.0, 1.0/120.0, 1.0/720.0, 1.0/5040.0 };
private final double[] coeffTaylorE27 = new double[] { 1, 2.0/3.0, 1.0/4.0, 1.0/15.0, 1.0/72.0, 1.0/420.0, 1.0/2880.0 };
private final double[] coeffTaylorE37 = new double[] { 1, 3.0/4.0, 3.0/10.0, 1.0/12.0, 1.0/56.0, 1.0/320.0, 1.0/2160.0 };
private RandomVariableFactory randomVariableFactory;
private final RandomVariable a;
private final RandomVariable b;
private final RandomVariable c;
private final RandomVariable d;
private boolean isCalibrateable = false;
/**
* Creates the volatility model
* \[
* \sigma_{i}(t_{j}) = \sqrt{ \frac{1}{t_{j+1}-t_{j}} \int_{t_{j}}^{t_{j+1}} \left( ( a + b (T_{i}-t) ) \exp(-c (T_{i}-t)) + d \right)^{2} \ \mathrm{d}t } \text{.}
* \]
*
* @param randomVariableFactory The random variable factor used to construct random variables from the parameters.
* @param timeDiscretization The simulation time discretization t<sub>j</sub>.
* @param liborPeriodDiscretization The period time discretization T<sub>i</sub>.
* @param a The parameter a: an initial volatility level.
* @param b The parameter b: the slope at the short end (shortly before maturity).
* @param c The parameter c: exponential decay of the volatility in time-to-maturity.
* @param d The parameter d: if c > 0 this is the very long term volatility level.
* @param isCalibrateable Set this to true, if the parameters are available for calibration.
*/
public LIBORVolatilityModelFourParameterExponentialFormIntegrated(final RandomVariableFactory randomVariableFactory, final TimeDiscretization timeDiscretization, final TimeDiscretization liborPeriodDiscretization, final double a, final double b, final double c, final double d, final boolean isCalibrateable) {
super(timeDiscretization, liborPeriodDiscretization);
this.randomVariableFactory = randomVariableFactory;
this.a = randomVariableFactory.createRandomVariable(a);
this.b = randomVariableFactory.createRandomVariable(b);
this.c = randomVariableFactory.createRandomVariable(c);
this.d = randomVariableFactory.createRandomVariable(d);
this.isCalibrateable = isCalibrateable;
}
/**
* Creates the volatility model
* \[
* \sigma_{i}(t_{j}) = \sqrt{ \frac{1}{t_{j+1}-t_{j}} \int_{t_{j}}^{t_{j+1}} \left( ( a + b (T_{i}-t) ) \exp(-c (T_{i}-t)) + d \right)^{2} \ \mathrm{d}t } \text{.}
* \]
*
* @param timeDiscretization The simulation time discretization t<sub>j</sub>.
* @param liborPeriodDiscretization The period time discretization T<sub>i</sub>.
* @param a The parameter a: an initial volatility level.
* @param b The parameter b: the slope at the short end (shortly before maturity).
* @param c The parameter c: exponential decay of the volatility in time-to-maturity.
* @param d The parameter d: if c > 0 this is the very long term volatility level.
* @param isCalibrateable Set this to true, if the parameters are available for calibration.
*/
public LIBORVolatilityModelFourParameterExponentialFormIntegrated(final TimeDiscretization timeDiscretization, final TimeDiscretization liborPeriodDiscretization, final RandomVariable a, final RandomVariable b, final RandomVariable c, final RandomVariable d, final boolean isCalibrateable) {
super(timeDiscretization, liborPeriodDiscretization);
this.a = a;
this.b = b;
this.c = c;
this.d = d;
this.isCalibrateable = isCalibrateable;
}
/**
* Creates the volatility model
* \[
* \sigma_{i}(t_{j}) = \sqrt{ \frac{1}{t_{j+1}-t_{j}} \int_{t_{j}}^{t_{j+1}} \left( ( a + b (T_{i}-t) ) \exp(-c (T_{i}-t)) + d \right)^{2} \ \mathrm{d}t } \text{.}
* \]
*
* @param timeDiscretization The simulation time discretization t<sub>j</sub>.
* @param liborPeriodDiscretization The period time discretization T<sub>i</sub>.
* @param a The parameter a: an initial volatility level.
* @param b The parameter b: the slope at the short end (shortly before maturity).
* @param c The parameter c: exponential decay of the volatility in time-to-maturity.
* @param d The parameter d: if c > 0 this is the very long term volatility level.
* @param isCalibrateable Set this to true, if the parameters are available for calibration.
*/
public LIBORVolatilityModelFourParameterExponentialFormIntegrated(final TimeDiscretization timeDiscretization, final TimeDiscretization liborPeriodDiscretization, final double a, final double b, final double c, final double d, final boolean isCalibrateable) {
super(timeDiscretization, liborPeriodDiscretization);
this.a = new Scalar(a);
this.b = new Scalar(b);
this.c = new Scalar(c);
this.d = new Scalar(d);
this.isCalibrateable = isCalibrateable;
}
@Override
public RandomVariable[] getParameter() {
if(!isCalibrateable) {
return null;
}
final RandomVariable[] parameter = new RandomVariable[4];
parameter[0] = a;
parameter[1] = b;
parameter[2] = c;
parameter[3] = d;
return parameter;
}
@Override
public LIBORVolatilityModelFourParameterExponentialFormIntegrated getCloneWithModifiedParameter(final RandomVariable[] parameter) {
if(!isCalibrateable) {
return this;
}
return new LIBORVolatilityModelFourParameterExponentialFormIntegrated(
super.getTimeDiscretization(),
super.getLiborPeriodDiscretization(),
parameter[0],
parameter[1],
parameter[2],
parameter[3],
isCalibrateable
);
}
@Override
public RandomVariable getVolatility(final int timeIndex, final int liborIndex) {
// Create a very simple volatility model here
final double timeStart = getTimeDiscretization().getTime(timeIndex);
final double timeEnd = getTimeDiscretization().getTime(timeIndex+1);
final double maturity = getLiborPeriodDiscretization().getTime(liborIndex);
if(maturity-timeStart <= 0) {
return new Scalar(0.0);
}
final RandomVariable varianceInstantaneous = getIntegratedVariance(maturity-timeStart).sub(getIntegratedVariance(maturity-timeEnd)).div(timeEnd-timeStart);
return varianceInstantaneous.sqrt();
}
private RandomVariable getIntegratedVariance(final double maturity) {
if(maturity == 0) {
return new Scalar(0.0);
}
/*
* Integral of the square of the instantaneous volatility function
* ((a + b * T) * Math.exp(- c * T) + d);
*/
/*
* http://www.wolframalpha.com/input/?i=integrate+%28%28a+%2B+b+*+t%29+*+exp%28-+c+*+t%29+%2B+d%29%5E2+from+0+to+T
* integral_0^T ((a+b t) exp(-(c t))+d)^2 dt = 1/4 ((e^(-2 c T) (-2 a^2 c^2-2 a b c (2 c T+1)+b^2 (-(2 c T (c T+1)+1))))/c^3+(2 a^2 c^2+2 a b c+b^2)/c^3-(8 d e^(-c T) (a c+b c T+b))/c^2+(8 d (a c+b))/c^2+4 d^2 T)
*/
final RandomVariable aaT = a.squared().mult(maturity);
final RandomVariable abTT = a.mult(b).mult(maturity*maturity);
final RandomVariable ad2T = a.mult(d).mult(2.0*maturity);
final RandomVariable bbTTT = b.squared().mult(maturity*maturity*maturity/3.0);
final RandomVariable bdTT = b.mult(d).mult(maturity*maturity);
final RandomVariable ddT = d.squared().mult(maturity);
final RandomVariable mcT = c.mult(-maturity);
final RandomVariable mcT2 = mcT.mult(2.0);
RandomVariable expA1 = mcT.expm1().div(mcT);
RandomVariable expA2 = mcT.sub(expA1.log()).expm1().div(mcT).mult(expA1).mult(2.0);
RandomVariable expB1 = mcT2.expm1().div(mcT2);
RandomVariable expB2 = mcT2.sub(expB1.log()).expm1().div(mcT2).mult(expB1).mult(2.0);
RandomVariable expB3 = mcT2.sub(expB2.log()).expm1().div(mcT2).mult(expB2).mult(3.0);
// Ensure that c is cut off from 0 (the term (exp(-x)-1)/x will have cancelations)
// 1 1/2 1/6 1/24 1/120 1/720 1/5040
// 1 2/3 1/4 1/15 1/72 1/420 1/2880
// 1 3/4 3/10 1/12 1/56 1/320 1/2160
final RandomVariable pA1 = polynom(mcT, coeffTaylorE1);
final RandomVariable pA2 = polynom(mcT, coeffTaylorE2);
final RandomVariable pB1 = polynom(mcT2, coeffTaylorE1);
final RandomVariable pB2 = polynom(mcT2, coeffTaylorE2);
final RandomVariable pB3 = polynom(mcT2, coeffTaylorE3);
final RandomVariable cCutOff1 = mcT.abs().sub(1E-12).choose(new Scalar(1.0), new Scalar(-1.0));
final RandomVariable cCutOff2 = mcT.abs().sub(1E-2).choose(new Scalar(1.0), new Scalar(-1.0));
final RandomVariable cCutOff3 = cCutOff2;
expA1 = cCutOff1.choose(expA1, pA1);
expA2 = cCutOff2.choose(expA2, pA2);
expB1 = cCutOff1.choose(expB1, pB1);
expB2 = cCutOff2.choose(expB2, pB2);
expB3 = cCutOff3.choose(expB3, pB3);
/*
integratedVariance = a*a*T*((1-Math.exp(-2*c*T))/(2*c*T))
+ a*b*T*T*(((1 - Math.exp(-2*c*T))/(2*c*T) - Math.exp(-2*c*T))/(c*T))
+ 2*a*d*T*((1-Math.exp(-c*T))/(c*T))
+ b*b*T*T*T*(((((1-Math.exp(-2*c*T))/(2*c*T)-Math.exp(-2*c*T))/(T*c)-Math.exp(-2*c*T)))/(2*c*T))
+ 2*b*d*T*T*(((1-Math.exp(-c*T))-T*c*Math.exp(-c*T))/(c*c*T*T))
+ d*d*T;
*/
RandomVariable integratedVariance = aaT.mult(expB1);
integratedVariance = integratedVariance.add( abTT.mult(expB2) );
integratedVariance = integratedVariance.add( ad2T.mult(expA1) );
integratedVariance = integratedVariance.add( bbTTT.mult(expB3) );
integratedVariance = integratedVariance.add( bdTT.mult(expA2) );
integratedVariance = integratedVariance.add( ddT );
return integratedVariance;
}
private RandomVariable polynom(final RandomVariable x, final double[] coeff) {
RandomVariable p = x.mult(coeff[coeff.length-1]).add(coeff[coeff.length-2]);
for(int i=coeff.length-3; i >= 0; i--) {
p = p.mult(x).add(coeff[i]);
}
return p;
}
@Override
public Object clone() {
return new LIBORVolatilityModelFourParameterExponentialFormIntegrated(
super.getTimeDiscretization(),
super.getLiborPeriodDiscretization(),
a,
b,
c,
d,
isCalibrateable
);
}
@Override
public LIBORVolatilityModel getCloneWithModifiedData(final Map<String, Object> dataModified) {
RandomVariableFactory randomVariableFactory = null;
TimeDiscretization timeDiscretization = this.getTimeDiscretization();
TimeDiscretization liborPeriodDiscretization = this.getLiborPeriodDiscretization();
RandomVariable a = this.a;
RandomVariable b = this.b;
RandomVariable c = this.c;
RandomVariable d = this.d;
boolean isCalibrateable = this.isCalibrateable;
if(dataModified != null) {
// Explicitly passed covarianceModel has priority
randomVariableFactory = (RandomVariableFactory)dataModified.getOrDefault("randomVariableFactory", randomVariableFactory);
timeDiscretization = (TimeDiscretization)dataModified.getOrDefault("timeDiscretization", timeDiscretization);
liborPeriodDiscretization = (TimeDiscretization)dataModified.getOrDefault("liborPeriodDiscretization", liborPeriodDiscretization);
isCalibrateable = (boolean)dataModified.getOrDefault("isCalibrateable", isCalibrateable);
if(dataModified.containsKey("randomVariableFactory")) {
a = randomVariableFactory.createRandomVariable(a.doubleValue());
b = randomVariableFactory.createRandomVariable(b.doubleValue());
c = randomVariableFactory.createRandomVariable(c.doubleValue());
d = randomVariableFactory.createRandomVariable(d.doubleValue());
}
if(dataModified.getOrDefault("a", a) instanceof RandomVariable) {
a = ((RandomVariable)dataModified.getOrDefault("a", a));
}else if(randomVariableFactory != null){
a = randomVariableFactory.createRandomVariable((double)dataModified.get("a"));
}else {
a = new Scalar((double)dataModified.get("a"));
}
if(dataModified.getOrDefault("b", b) instanceof RandomVariable) {
b = randomVariableFactory.createRandomVariable(((RandomVariable)dataModified.getOrDefault("b", b)).doubleValue());
}else if(randomVariableFactory != null){
b = randomVariableFactory.createRandomVariable((double)dataModified.get("b"));
}else {
b = new Scalar((double)dataModified.get("b"));
}
if(dataModified.getOrDefault("c", c) instanceof RandomVariable) {
c = randomVariableFactory.createRandomVariable(((RandomVariable)dataModified.getOrDefault("c", c)).doubleValue());
}else if(randomVariableFactory != null){
c = randomVariableFactory.createRandomVariable((double)dataModified.get("c"));
}else {
c = new Scalar((double)dataModified.get("c"));
}
if(dataModified.getOrDefault("d", d) instanceof RandomVariable) {
d = randomVariableFactory.createRandomVariable(((RandomVariable)dataModified.getOrDefault("d", d)).doubleValue());
}else if(randomVariableFactory != null){
d = randomVariableFactory.createRandomVariable((double)dataModified.get("d"));
}else {
d = new Scalar((double)dataModified.get("d"));
}
}
final LIBORVolatilityModel newModel = new LIBORVolatilityModelFourParameterExponentialFormIntegrated(timeDiscretization, liborPeriodDiscretization, a, b, c, d, isCalibrateable);
return newModel;
}
}