/**
* 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.twoasset;
import org.apache.commons.lang.Validate;
import com.opengamma.analytics.financial.model.option.definition.twoasset.ProductOptionDefinition;
import com.opengamma.analytics.financial.model.option.definition.twoasset.StandardTwoAssetOptionDataBundle;
import com.opengamma.analytics.math.function.Function1D;
import com.opengamma.analytics.math.statistics.distribution.NormalDistribution;
import com.opengamma.analytics.math.statistics.distribution.ProbabilityDistribution;
/**
*
* The value of a product option call is given by:
* $$
* \begin{eqnarray*}
* c = e^{-rT}\left(F N(d_2) - K N(d_1)\right)
* \end{eqnarray*}
* $$
* and a put by:
* $$
* \begin{eqnarray*}
* p = e^{-rT}\left(K N(d_1) - F N(d_2)\right)
* \end{eqnarray*}
* $$
* where
* $$
* \begin{eqnarray*}
* F &=& S_1 S_2 e^{\left(b_1 - b_2 + \sigma_2^2 + \rho \sigma_1 \sigma_2\right)T}\\
* \hat{\sigma} &=& \sqrt{\sigma_1 ^2 + 2 \rho\sigma_1\sigma_2}\\
* d_1 &=& \frac{\ln{\frac{F}{K}} + \frac{T\hat{\sigma}^2}{2}}{\hat{\sigma} \sqrt{T}}\\
* d_2 &=& d_1 - \hat{\sigma}\sqrt{T}
* \end{eqnarray*}
* $$
* and
* $$
* <ul>
* <li>$K$ is the strike</li>
* <li>$S_1$ is the spot value of the first asset</li>
* <li>$S_2$ is the spot value of the second asset</li>
* <li>$b_1$ is the cost-of-carry of the first asset</li>
* <li>$b_2$ is the cost-of-carry of the second asset</li>
* <li>$T$ is the time to expiry of the option</li>
* <li>$r$ is the spot interest rate for time $T$</li>
* <li>$\sigma_1$ is the annualized volatility of the first asset</li>
* <li>$\sigma_2$ is the annualized volatility of the second asset</li>
* <li>$\rho$ is the correlation between the returns of the two assets</li>
* <li>$N(x)$ is the CDF of the normal distribution $N(0, 1)$ </li>
* </ul>
*/
public class ProductOptionModel extends TwoAssetAnalyticOptionModel<ProductOptionDefinition, StandardTwoAssetOptionDataBundle> {
private static final ProbabilityDistribution<Double> NORMAL = new NormalDistribution(0, 1);
/**
* Gets the pricing function for a European-style product option
* @param definition The option definition
* @return The pricing function
* @throws IllegalArgumentException If the definition is null
*/
@Override
public Function1D<StandardTwoAssetOptionDataBundle, Double> getPricingFunction(final ProductOptionDefinition definition) {
Validate.notNull(definition, "definition");
return new Function1D<StandardTwoAssetOptionDataBundle, Double>() {
@SuppressWarnings("synthetic-access")
@Override
public Double evaluate(final StandardTwoAssetOptionDataBundle data) {
Validate.notNull(data, "data");
final double s1 = data.getFirstSpot();
final double s2 = data.getSecondSpot();
final double k = definition.getStrike();
final double b1 = data.getFirstCostOfCarry();
final double b2 = data.getSecondCostOfCarry();
final double t = definition.getTimeToExpiry(data.getDate());
final double r = data.getInterestRate(t);
final double sigma1 = data.getFirstVolatility(t, k);
final double sigma2 = data.getSecondVolatility(t, k);
final double rho = data.getCorrelation();
final double sigma = Math.sqrt(sigma1 * sigma1 + sigma2 * sigma2 + 2 * rho * sigma1 * sigma2);
final double sigmaT = sigma * Math.sqrt(t);
final double f = s1 * s2 * Math.exp(t * (b1 + b2 + rho * sigma1 * sigma2));
final double d1 = (Math.log(f / k) + t * sigma * sigma / 2) / sigmaT;
final double d2 = d1 - sigmaT;
final int sign = definition.isCall() ? 1 : -1;
return Math.exp(-r * t) * sign * (f * NORMAL.getCDF(sign * d1) - k * NORMAL.getCDF(sign * d2));
}
};
}
}