public override Vector evolve(double t0, Vector x0, double dt, Vector dw)
{
Vector retVal = new Vector(2);
double vol, vol2, mu, nu, dy;
double sdt = Math.Sqrt(dt);
double sqrhov = Math.Sqrt(1.0 - rho_ * rho_);
switch (discretization_)
{
// For the definition of PartialTruncation, FullTruncation
// and Reflection see Lord, R., R. Koekkoek and D. van Dijk (2006),
// "A Comparison of biased simulation schemes for
// stochastic volatility models",
// Working Paper, Tinbergen Institute
case Discretization.PartialTruncation:
vol = (x0[1] > 0.0) ? Math.Sqrt(x0[1]) : 0.0;
vol2 = sigma_ * vol;
mu = riskFreeRate_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()
- dividendYield_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()
- 0.5 * vol * vol;
nu = kappa_ * (theta_ - x0[1]);
retVal[0] = x0[0] * Math.Exp(mu * dt + vol * dw[0] * sdt);
retVal[1] = x0[1] + nu * dt + vol2 * sdt * (rho_ * dw[0] + sqrhov * dw[1]);
break;
case Discretization.FullTruncation:
vol = (x0[1] > 0.0) ? Math.Sqrt(x0[1]) : 0.0;
vol2 = sigma_ * vol;
mu = riskFreeRate_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()
- dividendYield_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()
- 0.5 * vol * vol;
nu = kappa_ * (theta_ - vol * vol);
retVal[0] = x0[0] * Math.Exp(mu * dt + vol * dw[0] * sdt);
retVal[1] = x0[1] + nu * dt + vol2 * sdt * (rho_ * dw[0] + sqrhov * dw[1]);
break;
case Discretization.Reflection:
vol = Math.Sqrt(Math.Abs(x0[1]));
vol2 = sigma_ * vol;
mu = riskFreeRate_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()
- dividendYield_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()
- 0.5 * vol * vol;
nu = kappa_ * (theta_ - vol * vol);
retVal[0] = x0[0] * Math.Exp(mu * dt + vol * dw[0] * sdt);
retVal[1] = vol * vol
+ nu * dt + vol2 * sdt * (rho_ * dw[0] + sqrhov * dw[1]);
break;
case Discretization.NonCentralChiSquareVariance:
// use Alan Lewis trick to decorrelate the equity and the variance
// process by using y(t)=x(t)-\frac{rho}{sigma}\nu(t)
// and Ito's Lemma. Then use exact sampling for the variance
// process. For further details please read the Wilmott thread
// "QuantLib code is very high quality"
vol = (x0[1] > 0.0) ? Math.Sqrt(x0[1]) : 0.0;
mu = riskFreeRate_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()
- dividendYield_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()
- 0.5 * vol * vol;
retVal[1] = varianceDistribution(x0[1], dw[1], dt);
dy = (mu - rho_ / sigma_ * kappa_
* (theta_ - vol * vol)) * dt + vol * sqrhov * dw[0] * sdt;
retVal[0] = x0[0] * Math.Exp(dy + rho_ / sigma_ * (retVal[1] - x0[1]));
break;
case Discretization.QuadraticExponential:
case Discretization.QuadraticExponentialMartingale:
{
// for details of the quadratic exponential discretization scheme
// see Leif Andersen,
// Efficient Simulation of the Heston Stochastic Volatility Model
double ex = Math.Exp(-kappa_ * dt);
double m = theta_ + (x0[1] - theta_) * ex;
double s2 = x0[1] * sigma_ * sigma_ * ex / kappa_ * (1 - ex)
+ theta_ * sigma_ * sigma_ / (2 * kappa_) * (1 - ex) * (1 - ex);
double psi = s2 / (m * m);
double g1 = 0.5;
double g2 = 0.5;
double k0 = -rho_ * kappa_ * theta_ * dt / sigma_;
double k1 = g1 * dt * (kappa_ * rho_ / sigma_ - 0.5) - rho_ / sigma_;
double k2 = g2 * dt * (kappa_ * rho_ / sigma_ - 0.5) + rho_ / sigma_;
double k3 = g1 * dt * (1 - rho_ * rho_);
double k4 = g2 * dt * (1 - rho_ * rho_);
double A = k2 + 0.5 * k4;
if (psi < 1.5)
{
double b2 = 2 / psi - 1 + Math.Sqrt(2 / psi * (2 / psi - 1));
double b = Math.Sqrt(b2);
double a = m / (1 + b2);
if (discretization_ == Discretization.QuadraticExponentialMartingale)
{
// martingale correction
Utils.QL_REQUIRE(A < 1 / (2 * a), () => "illegal value");
k0 = -A * b2 * a / (1 - 2 * A * a) + 0.5 * Math.Log(1 - 2 * A * a)
- (k1 + 0.5 * k3) * x0[1];
}
retVal[1] = a * (b + dw[1]) * (b + dw[1]);
}
else
{
double p = (psi - 1) / (psi + 1);
double beta = (1 - p) / m;
double u = new CumulativeNormalDistribution().value(dw[1]);
if (discretization_ == Discretization.QuadraticExponentialMartingale)
{
// martingale correction
Utils.QL_REQUIRE(A < beta, () => "illegal value");
k0 = -Math.Log(p + beta * (1 - p) / (beta - A)) - (k1 + 0.5 * k3) * x0[1];
}
retVal[1] = ((u <= p) ? 0.0 : Math.Log((1 - p) / (1 - u)) / beta);
}
mu = riskFreeRate_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()
- dividendYield_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value();
retVal[0] = x0[0] * Math.Exp(mu * dt + k0 + k1 * x0[1] + k2 * retVal[1]
+ Math.Sqrt(k3 * x0[1] + k4 * retVal[1]) * dw[0]);
}
break;
case Discretization.BroadieKayaExactSchemeLobatto:
case Discretization.BroadieKayaExactSchemeLaguerre:
case Discretization.BroadieKayaExactSchemeTrapezoidal:
{
double nu_0 = x0[1];
double nu_t = varianceDistribution(nu_0, dw[1], dt);
double x = Math.Min(1.0 - Const.QL_EPSILON,
Math.Max(0.0, new CumulativeNormalDistribution().value(dw[2])));
cdf_nu_ds f = new cdf_nu_ds(this, nu_0, nu_t, dt, discretization_);
double vds = new Brent().solve(f, 1e-5, -x, theta_ * dt, 0.1 * theta_ * dt);
//double vds = new Brent().solve( boost::lambda::bind(&cdf_nu_ds, *this, boost::lambda::_1,
// nu_0, nu_t, dt, discretization_)-x,
// 1e-5, theta_*dt, 0.1*theta_*dt);
double vdw = (nu_t - nu_0 - kappa_ * theta_ * dt + kappa_ * vds) / sigma_;
mu = (riskFreeRate_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()
- dividendYield_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()) * dt
- 0.5 * vds + rho_ * vdw;
double sig = Math.Sqrt((1 - rho_ * rho_) * vds);
double s = x0[0] * Math.Exp(mu + sig * dw[0]);
retVal[0] = s;
retVal[1] = nu_t;
}
break;
default:
Utils.QL_FAIL("unknown discretization schema");
break;
}
return(retVal);
}
public cdf_nu_ds_minus_x(double _x0, HestonProcess _process, double _nu_0, double _nu_t, double _dt,
Discretization _discretization)
{
cdf_nu = new cdf_nu_ds(_process, _nu_0, _nu_t, _dt, _discretization);
x0 = _x0;
}
