/**
* Copyright (C) 2009 - present by OpenGamma Inc. and the OpenGamma group of companies
*
* Please see distribution for license.
*/
package com.opengamma.analytics.financial.model.option.pricing.analytic;
import org.apache.commons.lang.Validate;
import com.opengamma.analytics.financial.model.option.definition.CappedPowerOptionDefinition;
import com.opengamma.analytics.financial.model.option.definition.StandardOptionDataBundle;
import com.opengamma.analytics.math.function.Function1D;
import com.opengamma.analytics.math.statistics.distribution.NormalDistribution;
import com.opengamma.analytics.math.statistics.distribution.ProbabilityDistribution;
/**
* Pricing model for capped power options (see {@link com.opengamma.analytics.financial.model.option.definition.CappedPowerOptionDefinition}).
* <p>
* The price of this option is given by:
* $$
* \begin{align*}
* c &= S^ie^{[(i-1)(r + \frac{i\sigma^2}{2}) - i(r-b)]T}[N(d_1) - N(d_3)] - e^{-rT}[KN(d_2) - (C + K)N(d_4)]\\
* p &= e^{-rT}[KN(-d_2) - (C + K)N(-d_4)] - S^ie^{[(i-1)(r + \frac{i\sigma^2}{2}) - i(r-b)]T}[N(-d_1) - N(-d_3)]
* \end{align*}
* $$
* where
* $$
* \begin{align*}
* d_1 &= \frac{\ln\left(\frac{S}{K^{\frac{1}{i}}}\right) + (b + (i - \frac{1}{2})\sigma^2)T}{\sigma\sqrt{T}}\\
* d_2 &= d_1 - \sigma\sqrt{T}\\\\
* \end{align*}
* $$
* and
* $$
* \begin{align*}
* d_3 &= \frac{\ln\left(\frac{S}{(K + C)^{\frac{1}{i}}}\right) + (b + (i - \frac{1}{2})\sigma^2)T}{\sigma\sqrt{T}}\\
* d_4 &= d_3 - i\sigma\sqrt{T}
* \end{align*}
* $$
* for a call and
* $$
* \begin{align*}
* d_3 &= \frac{\ln\left(\frac{S}{(K - C)^{\frac{1}{i}}}\right) + (b + (i - \frac{1}{2})\sigma^2)T}{\sigma\sqrt{T}}\\
* d_4 &= d_3 - i\sigma\sqrt{T}
* \end{align*}
* $$
*/
public class CappedPowerOptionModel extends AnalyticOptionModel<CappedPowerOptionDefinition, StandardOptionDataBundle> {
private static final ProbabilityDistribution<Double> NORMAL = new NormalDistribution(0, 1);
/**
* {@inheritDoc}
*/
@Override
public Function1D<StandardOptionDataBundle, Double> getPricingFunction(final CappedPowerOptionDefinition definition) {
Validate.notNull(definition);
final Function1D<StandardOptionDataBundle, Double> pricingFunction = new Function1D<StandardOptionDataBundle, Double>() {
@SuppressWarnings("synthetic-access")
@Override
public Double evaluate(final StandardOptionDataBundle data) {
Validate.notNull(data);
final double s = data.getSpot();
final double k = definition.getStrike();
final double t = definition.getTimeToExpiry(data.getDate());
final double sigma = data.getVolatility(t, k);
final double r = data.getInterestRate(t);
final double b = data.getCostOfCarry();
final double power = definition.getPower();
final double inv = 1. / power;
final double cap = definition.getCap();
final boolean isCall = definition.isCall();
final double sigmaT = sigma * Math.sqrt(t);
final double x = t * (b + sigma * sigma * (power - 0.5));
final double d1 = getD(s / Math.pow(k, 1. / power), x, sigmaT);
final double d2 = d1 - power * sigmaT;
final double d3 = getD(isCall ? s / Math.pow(k + cap, inv) : s / Math.pow(k - cap, inv), x, sigmaT);
final double d4 = d3 - power * sigmaT;
final int sign = isCall ? 1 : -1;
final double df1 = Math.exp(-r * t);
final double df2 = Math.exp(t * ((power - 1) * (r + power * sigma * sigma * 0.5) - power * (r - b)));
final double term1 = Math.pow(s, power) * df2 * (NORMAL.getCDF(sign * d1) - NORMAL.getCDF(sign * d3));
final double term2 = df1 * (k * NORMAL.getCDF(sign * d2) - (k + sign * cap) * NORMAL.getCDF(sign * d4));
return sign * (term1 - term2);
}
};
return pricingFunction;
}
double getD(final double x, final double y, final double z) {
return (Math.log(x) + y) / z;
}
}