protected double blackPrice(
final double spot,
final double strike,
final double expiry,
final double rate,
final double carry,
final double vol,
final boolean isCall) {
final Function1D<Double, Double> intCon = ICP.getEuropeanPayoff(strike, isCall);
final ConvectionDiffusionPDE1DStandardCoefficients pde =
PDE.getBlackScholes(rate, rate - carry, vol);
double sMin = 0.0;
double sMax = spot * Math.exp(_z * Math.sqrt(expiry));
BoundaryCondition lower;
BoundaryCondition upper;
if (isCall) {
lower = new DirichletBoundaryCondition(0.0, sMin);
Function1D<Double, Double> upperValue =
new Function1D<Double, Double>() {
@Override
public Double evaluate(Double tau) {
return Math.exp(-rate * tau) * (spot * Math.exp(carry * tau) - strike);
}
};
upper = new DirichletBoundaryCondition(upperValue, sMax);
} else {
Function1D<Double, Double> lowerValue =
new Function1D<Double, Double>() {
@Override
public Double evaluate(Double tau) {
return Math.exp(-rate * tau) * strike;
}
};
lower = new DirichletBoundaryCondition(lowerValue, sMin);
upper = new DirichletBoundaryCondition(0.0, sMax);
}
final MeshingFunction tMesh = new ExponentialMeshing(0, expiry, _nTNodes, _lambda);
final MeshingFunction xMesh = new HyperbolicMeshing(sMin, sMax, spot, _nXNodes, _bunching);
final PDEGrid1D grid = new PDEGrid1D(tMesh, xMesh);
PDE1DDataBundle<ConvectionDiffusionPDE1DCoefficients> pdeData =
new PDE1DDataBundle<ConvectionDiffusionPDE1DCoefficients>(pde, intCon, lower, upper, grid);
PDEResults1D res = SOLVER.solve(pdeData);
// for now just do linear interpolation on price. TODO replace this with something more robust
final double[] xNodes = grid.getSpaceNodes();
final int index = SurfaceArrayUtils.getLowerBoundIndex(xNodes, spot);
final double w = (xNodes[index + 1] - spot) / (xNodes[index + 1] - xNodes[index]);
return w * res.getFunctionValue(index) + (1 - w) * res.getFunctionValue(index + 1);
}
private double logContactPriceFromPureSpot(final LocalVolatilitySurfaceMoneyness lv) {
final double fT = DIV_CURVES.getF(EXPIRY);
final double dT = DIV_CURVES.getD(EXPIRY);
final double dStar = dT / (fT - dT);
final ConvectionDiffusionPDE1DCoefficients pde =
PDE_PROVIDER.getLogBackwardsLocalVol(EXPIRY, lv);
final double theta = 0.5;
final double yL = -0.5;
final double yH = 0.5;
final ConvectionDiffusionPDESolver solver = new ThetaMethodFiniteDifference(theta, false);
final BoundaryCondition lower =
new NeumannBoundaryCondition(1 / (1 + dStar * Math.exp(-yL)), yL, true);
final BoundaryCondition upper = new NeumannBoundaryCondition(1.0, yH, false);
final MeshingFunction timeMesh = new ExponentialMeshing(0.0, EXPIRY, 100, 0.0);
final MeshingFunction spaceMesh = new ExponentialMeshing(yL, yH, 101, 0.0);
final PDEGrid1D grid = new PDEGrid1D(timeMesh, spaceMesh);
final PDE1DDataBundle<ConvectionDiffusionPDE1DCoefficients> db =
new PDE1DDataBundle<>(pde, PURE_LOG_PAY_OFF, lower, upper, grid);
final PDEResults1D res = solver.solve(db);
final int n = res.getNumberSpaceNodes();
final double val = res.getFunctionValue(n / 2);
return val;
}
public void testMixedLogNormalVolSurface() {
final double[] weights = new double[] {0.9, 0.1};
final double[] sigmas = new double[] {0.2, 0.8};
final MultiHorizonMixedLogNormalModelData data =
new MultiHorizonMixedLogNormalModelData(weights, sigmas);
final LocalVolatilitySurfaceStrike lv =
MixedLogNormalVolatilitySurface.getLocalVolatilitySurface(FORWARD_CURVE, data);
final LocalVolatilitySurfaceMoneyness lvm =
LocalVolatilitySurfaceConverter.toMoneynessSurface(lv, FORWARD_CURVE);
final double expected =
Math.sqrt(weights[0] * sigmas[0] * sigmas[0] + weights[1] * sigmas[1] * sigmas[1]);
final double ft = FORWARD_CURVE.getForward(EXPIRY);
final double theta = 0.5;
// Review the accuracy is very dependent on these numbers
final double fL = Math.log(ft / 30);
final double fH = Math.log(30 * ft);
final ConvectionDiffusionPDESolver solver = new ThetaMethodFiniteDifference(theta, false);
final BoundaryCondition lower = new NeumannBoundaryCondition(1.0, fL, true);
final BoundaryCondition upper = new NeumannBoundaryCondition(1.0, fH, false);
final MeshingFunction timeMesh = new ExponentialMeshing(0, EXPIRY, 50, 0.0);
final MeshingFunction spaceMesh = new HyperbolicMeshing(fL, fH, (fL + fH) / 2, 101, 0.4);
final ConvectionDiffusionPDE1DStandardCoefficients pde =
PDE_DATA_PROVIDER.getLogBackwardsLocalVol(EXPIRY, lvm);
final PDEGrid1D grid = new PDEGrid1D(timeMesh, spaceMesh);
final PDE1DDataBundle<ConvectionDiffusionPDE1DCoefficients> db =
new PDE1DDataBundle<ConvectionDiffusionPDE1DCoefficients>(
pde, INITIAL_COND, lower, upper, grid);
final PDEResults1D res = solver.solve(db);
final int n = res.getNumberSpaceNodes();
final double[] values = new double[n];
for (int i = 0; i < n; i++) {
values[i] = res.getFunctionValue(i);
}
final Interpolator1DDataBundle idb = INTERPOLATOR.getDataBundle(grid.getSpaceNodes(), values);
final double elogS = INTERPOLATOR.interpolate(idb, Math.log(ft));
final double kVol = Math.sqrt(-2 * (elogS - Math.log(ft)) / EXPIRY);
assertEquals(
expected, kVol,
1e-3); // TODO Improve on 10bps error - local surface is (by construction) very smooth.
// NOTE: this has got worse since we improved the T -> 0
// behaviour of the mixed log-normal local volatility surface
}
public void testFlatSurface() {
final double theta = 0.5;
final double ft = FORWARD_CURVE.getForward(EXPIRY);
final double fL = Math.log(ft / 5.0);
final double fH = Math.log(5.0 * ft);
final ConvectionDiffusionPDESolver solver = new ThetaMethodFiniteDifference(theta, false);
final BoundaryCondition lower = new NeumannBoundaryCondition(1.0, fL, true);
final BoundaryCondition upper = new NeumannBoundaryCondition(1.0, fH, false);
final MeshingFunction timeMesh = new ExponentialMeshing(0, EXPIRY, 100, 0.0);
final MeshingFunction spaceMesh = new ExponentialMeshing(fL, fH, 101, 0.0);
final PDEGrid1D grid = new PDEGrid1D(timeMesh, spaceMesh);
final PDE1DDataBundle<ConvectionDiffusionPDE1DCoefficients> db =
new PDE1DDataBundle<>(PDE, INITIAL_COND, lower, upper, grid);
final PDEResults1D res = solver.solve(db);
final int n = res.getNumberSpaceNodes();
final double kVol = Math.sqrt(-2 * (res.getFunctionValue(n / 2) - Math.log(ft)) / EXPIRY);
assertEquals(FLAT_VOL, kVol, 1e-6);
final YieldAndDiscountCurve yieldCurve =
new YieldCurve("test", ConstantDoublesCurve.from(DRIFT));
final AffineDividends ad = AffineDividends.noDividends();
final EquityVarianceSwapBackwardsPurePDE backSolver = new EquityVarianceSwapBackwardsPurePDE();
final PureLocalVolatilitySurface plv =
new PureLocalVolatilitySurface(ConstantDoublesSurface.from(FLAT_VOL));
final double[] res2 = backSolver.expectedVariance(SPOT, yieldCurve, ad, EXPIRY, plv);
final double kVol2 = Math.sqrt(res2[0] / EXPIRY);
assertEquals(FLAT_VOL, kVol2, 1e-6);
}
/**
* Computes the price of a one-touch out barrier option in the Black-Scholes world by solving the
* BS PDE on a finite difference grid. If a barrier is hit at any time before expiry, the option
* is cancelled (knocked-out) and a rebate (which is often zero) is paid <b>immediately</b>. If
* the barrier is not hit, then a normal European option payment is made.
*
* <p>If the barrier is above the spot it is assumed to be an up-and-out barrier (otherwise it
* would expire immediately) otherwise it is a down-and-out barrier As there are exact formulae
* for this case (see {@link BlackBarrierPriceFunction}), this is purely for testing purposes.
*
* @param spot The current (i.e. option price time) of the underlying
* @param barrierLevel The barrier level
* @param strike The strike of the European option
* @param expiry The expiry of the option
* @param rate The interest rate.
* @param carry The cost-of-carry (i.e. the forward = spot*exp(costOfCarry*T) )
* @param vol The Black volatility.
* @param isCall true for call
* @param rebate The rebate amount.
* @return The price.
*/
public double outBarrier(
final double spot,
final double barrierLevel,
final double strike,
final double expiry,
final double rate,
final double carry,
final double vol,
final boolean isCall,
final double rebate) {
final Function1D<Double, Double> intCon = ICP.getEuropeanPayoff(strike, isCall);
final ConvectionDiffusionPDE1DStandardCoefficients pde =
PDE.getBlackScholes(rate, rate - carry, vol);
final boolean isUp = barrierLevel > spot;
final double adj = 0.0; // _lambda == 0 ? ZETA * vol * Math.sqrt(expiry / (_nTNodes - 1)) : 0.0;
// System.out.println("adjustment: " + adj);
double sMin;
double sMax;
BoundaryCondition lower;
BoundaryCondition upper;
if (isUp) {
sMin = 0.0;
sMax =
barrierLevel
* Math.exp(-adj); // bring the barrier DOWN slightly to adjust for discrete monitoring
if (isCall) {
lower = new DirichletBoundaryCondition(0.0, sMin);
} else {
Function1D<Double, Double> lowerValue =
new Function1D<Double, Double>() {
@Override
public Double evaluate(Double tau) {
return Math.exp(-rate * tau) * strike;
}
};
lower = new DirichletBoundaryCondition(lowerValue, sMin);
}
upper = new DirichletBoundaryCondition(rebate, sMax);
} else {
sMin =
barrierLevel
* Math.exp(adj); // bring the barrier UP slightly to adjust for discrete monitoring
sMax = spot * Math.exp(_z * Math.sqrt(expiry));
lower = new DirichletBoundaryCondition(rebate, sMin);
if (isCall) {
Function1D<Double, Double> upperValue =
new Function1D<Double, Double>() {
@Override
public Double evaluate(Double tau) {
return Math.exp(-rate * tau) * (spot * Math.exp(carry * tau) - strike);
}
};
upper = new DirichletBoundaryCondition(upperValue, sMax);
} else {
upper = new DirichletBoundaryCondition(0.0, sMax);
}
}
final MeshingFunction tMesh = new ExponentialMeshing(0, expiry, _nTNodes, _lambda);
final MeshingFunction xMesh = new HyperbolicMeshing(sMin, sMax, spot, _nXNodes, _bunching);
final PDEGrid1D grid = new PDEGrid1D(tMesh, xMesh);
PDE1DDataBundle<ConvectionDiffusionPDE1DCoefficients> pdeData =
new PDE1DDataBundle<ConvectionDiffusionPDE1DCoefficients>(pde, intCon, lower, upper, grid);
PDEResults1D res = SOLVER.solve(pdeData);
// for now just do linear interpolation on price. TODO replace this with something more robust
final double[] xNodes = grid.getSpaceNodes();
final int index = SurfaceArrayUtils.getLowerBoundIndex(xNodes, spot);
final double w = (xNodes[index + 1] - spot) / (xNodes[index + 1] - xNodes[index]);
return w * res.getFunctionValue(index) + (1 - w) * res.getFunctionValue(index + 1);
}