Java tutorial
/** * 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.EuropeanVanillaOptionDefinition; import com.opengamma.analytics.financial.model.option.definition.OptionDefinition; import com.opengamma.analytics.financial.model.option.definition.SkewKurtosisOptionDataBundle; import com.opengamma.analytics.financial.model.option.definition.StandardOptionDataBundle; import com.opengamma.analytics.math.function.Function1D; /** * The Jarrow-Rudd option pricing formula extends the Black-Scholes-Merton * model for non-normal skewness and kurtosis in the underlying price * distribution. * <p> * The price of a call option is given by: * $$ * \begin{align*} * c = c_{BSM} + \lambda_1 Q_3 + \lambda_2 Q_4 * \end{align*} * $$ * $c_{BSM}$ is the Black-Scholes-Merton call price (see {@link BlackScholesMertonModel}) and * $$ * \begin{align*} * d_1 &= \frac{\ln(\frac{S}{K} + (r + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}}\\ * d_2 &= d_1 - \sigma\sqrt{T}\\ * Q_3 &= -\frac{(Se^{-rT})^3(e^{\sigma^2 T} - 1)^{\frac{3}{2}} e^{-rT}}{6}\frac{da(X)}{dS}\\ * Q_4 &= \frac{(Se^{-rT})^4(e^{\sigma^2 T} - 1)^2 e^{-rT}}{24}\frac{d^2a(X)}{dS^2}\\ * \lambda_1 &= \gamma_1(F) - \gamma_1(A)\\ * \lambda_2 &= \gamma_2(F) - \gamma_2(A)\\ * a(X) &= \frac{e^{-\frac{d_2^2}{2}}}{2\pi K\sigma\sqrt{T}} * \end{align*} * $$ * The skewness ($\gamma_1(A)$) and kurtosis ($\gamma_2(A)$) of a lognormal * distribution are $\gamma_1 = 3y + y^3$ and * $\gamma_2 = 16y^2 + 15y^4 + 6y^6 * + y^8$ where $y = \sqrt{e^{\sigma^2 T} - 1}$. * $\gamma_1(F)$ are $\gamma_2(F)$ are the observed skewness and kurtosis of * the underlying price distribution. * <p> * Put options are priced using put-call parity. */ public class JarrowRuddSkewnessKurtosisModel extends AnalyticOptionModel<OptionDefinition, SkewKurtosisOptionDataBundle> { private static final AnalyticOptionModel<OptionDefinition, StandardOptionDataBundle> BSM = new BlackScholesMertonModel(); /** * {@inheritDoc} */ @Override public Function1D<SkewKurtosisOptionDataBundle, Double> getPricingFunction(final OptionDefinition definition) { Validate.notNull(definition); final Function1D<SkewKurtosisOptionDataBundle, Double> pricingFunction = new Function1D<SkewKurtosisOptionDataBundle, Double>() { @SuppressWarnings("synthetic-access") @Override public Double evaluate(final SkewKurtosisOptionDataBundle 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 skew = data.getAnnualizedSkew(); final double kurtosis = data.getAnnualizedPearsonKurtosis(); final OptionDefinition callDefinition = definition.isCall() ? definition : new EuropeanVanillaOptionDefinition(k, definition.getExpiry(), true); final Function1D<StandardOptionDataBundle, Double> bsm = BSM.getPricingFunction(callDefinition); final double bsmCall = bsm.evaluate(data); final double d2 = getD2(getD1(s, k, t, sigma, b), sigma, t); final double sigmaT = sigma * Math.sqrt(t); final double a = getA(d2, k, sigmaT); final double call = bsmCall + getLambda1(sigma, t, skew) * getQ3(s, k, sigmaT, t, r, a, d2) + getLambda2(sigma, t, kurtosis) * getQ4(s, k, sigmaT, t, r, a, d2); if (!definition.isCall()) { return call - s * Math.exp((b - r) * t) + k * Math.exp(-r * t); } return call; } }; return pricingFunction; } private double getA(final double d2, final double k, final double sigmaT) { return Math.exp(-d2 * d2 / 2.) / k / sigmaT / Math.sqrt(2 * Math.PI); } private double getLambda1(final double sigma, final double t, final double skew) { final double q = Math.sqrt(Math.exp(sigma * sigma * t) - 1); final double skewDistribution = q * (3 + q * q); return skew - skewDistribution; } private double getLambda2(final double sigma, final double t, final double kurtosis) { final double q = Math.sqrt(Math.exp(sigma * sigma * t) - 1); final double q2 = q * q; final double q4 = q2 * q2; final double q6 = q4 * q2; final double q8 = q6 * q2; final double kurtosisDistribution = 16 * q2 + 15 * q4 + 6 * q6 + q8 + 3; return kurtosis - kurtosisDistribution; } private double getQ3(final double s, final double k, final double sigmaT, final double t, final double r, final double a, final double d2) { final double da = a * (d2 - sigmaT) / (k * sigmaT); final double df = Math.exp(-r * t); return -Math.pow(s * df, 3) * Math.pow(Math.exp(sigmaT * sigmaT - 1), 1.5) * df * da / 6.; } private double getQ4(final double s, final double k, final double sigmaT, final double t, final double r, final double a, final double d2) { final double sigmaTSq = sigmaT * sigmaT; final double x = d2 - sigmaT; final double da2 = a * (x * x - sigmaT * x - 1) / (k * k * sigmaTSq); final double df = Math.exp(-r * t); return Math.pow(s * df, 4) * Math.pow(Math.exp(sigmaTSq) - 1, 2) * df * da2 / 24.; } }