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;
}
static {
FORWARD_CURVE = new ForwardCurve(SPOT, DRIFT);
LOCAL_VOL =
new LocalVolatilitySurfaceMoneyness(ConstantDoublesSurface.from(FLAT_VOL), FORWARD_CURVE);
PDE = PDE_DATA_PROVIDER.getLogBackwardsLocalVol(EXPIRY, LOCAL_VOL);
INITIAL_COND = INT_COND_PROVIDER.getLogContractPayoffInLogCoordinate();
}
/**
* Prices a log-contract for a given local volatility surface by backwards solving the PDE
* expressed in terms of (the log of) the real stock price - this requires having jumps conditions
* in the PDE
*
* @param lv Local volatility
* @return Forward (non-discounted) price of log-contact
*/
private double logContractPriceFromSpotPDE(final LocalVolatilitySurfaceStrike lv) {
// Set up the PDE to give the forward (non-discounted) option price
final ConvectionDiffusionPDE1DCoefficients pde =
PDE_PROVIDER.getLogBackwardsLocalVol(0.0, -DRIFT, EXPIRY, lv);
final Function1D<Double, Double> initialCon =
INITIAL_COND_PROVIDER.getLogContractPayoffInLogCoordinate();
final double theta = 0.5;
final double yL = Math.log(SPOT / 6);
final double yH = Math.log(6 * SPOT);
final ConvectionDiffusionPDESolver solver = new ThetaMethodFiniteDifference(theta, false);
final BoundaryCondition lower1 = new NeumannBoundaryCondition(1.0, yL, true);
final BoundaryCondition upper1 = new NeumannBoundaryCondition(1.0, yH, false);
final MeshingFunction timeMesh1 =
new ExponentialMeshing(0, EXPIRY - DIVIDEND_DATE - 1e-6, 50, 0.0);
final MeshingFunction timeMesh2 =
new ExponentialMeshing(EXPIRY - DIVIDEND_DATE + 1e-6, EXPIRY, 50, 0.0);
final MeshingFunction spaceMesh = new ExponentialMeshing(yL, yH, 101, 0.0);
final PDEGrid1D grid1 = new PDEGrid1D(timeMesh1, spaceMesh);
final double[] sNodes1 = grid1.getSpaceNodes();
// run the PDE solver backward to the dividend date
final PDE1DDataBundle<ConvectionDiffusionPDE1DCoefficients> db1 =
new PDE1DDataBundle<>(pde, initialCon, lower1, upper1, grid1);
final PDETerminalResults1D res1 = (PDETerminalResults1D) solver.solve(db1);
// Map the spot nodes after (in calendar time) the dividend payment to nodes before
final int nSNodes = sNodes1.length;
final double[] sNodes2 = new double[nSNodes];
final double lnBeta = Math.log(1 - BETA);
for (int i = 0; i < nSNodes; i++) {
final double temp = sNodes1[i];
if (temp < 0) {
sNodes2[i] = Math.log(Math.exp(temp) + ALPHA) - lnBeta;
} else {
sNodes2[i] = temp + Math.log(1 + ALPHA * Math.exp(-temp)) - lnBeta;
}
}
final PDEGrid1D grid2 = new PDEGrid1D(timeMesh2.getPoints(), sNodes2);
final BoundaryCondition lower2 = new NeumannBoundaryCondition(1.0, sNodes2[0], true);
final BoundaryCondition upper2 = new NeumannBoundaryCondition(1.0, sNodes2[nSNodes - 1], false);
// run the PDE solver backward from the dividend date to zero
final PDE1DDataBundle<ConvectionDiffusionPDE1DCoefficients> db2 =
new PDE1DDataBundle<>(pde, res1.getTerminalResults(), lower2, upper2, grid2);
final PDETerminalResults1D res2 = (PDETerminalResults1D) solver.solve(db2);
final Interpolator1DDataBundle interpolDB2 =
INTEPOLATOR1D.getDataBundle(sNodes2, res2.getTerminalResults());
final double val2 = INTEPOLATOR1D.interpolate(interpolDB2, Math.log(SPOT));
return val2;
}
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
}
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);
}
/**
* Check the the log-contract is correctly prices using a backwards PDE expressed in terms of (the
* log of) the forward F(t,T) - this requires NO jumps conditions in the PDE
*/
@Test
public void backwardsDebugPDEtest() {
final double fT = DIV_CURVES.getF(EXPIRY);
final double lnFT = Math.log(fT);
final Function1D<Double, Double> payoff =
new Function1D<Double, Double>() {
@Override
public Double evaluate(final Double y) {
return y - lnFT;
}
};
// ZZConvectionDiffusionPDEDataBundle pdeBundle1 = getBackwardsPDEDataBundle(EXPIRY, LOCAL_VOL,
// payoff);
// ConvectionDiffusionPDE1DCoefficients pde =
// PDE_PROVIDER.getLogBackwardsLocalVol(FORWARD_CURVE, EXPIRY, LOCAL_VOL);
final ConvectionDiffusionPDE1DCoefficients pde =
PDE_PROVIDER.getLogBackwardsLocalVol(0.0, 0.0, EXPIRY, LOCAL_VOL_SPECIAL);
final double theta = 0.5;
final double range = Math.log(5);
final double yL = lnFT - range;
final double yH = lnFT + range;
final ConvectionDiffusionPDESolver solver = new ThetaMethodFiniteDifference(theta, false);
final BoundaryCondition lower = new NeumannBoundaryCondition(1.0, yL, true);
final BoundaryCondition upper = new NeumannBoundaryCondition(1.0, yH, false);
final MeshingFunction timeMesh = new ExponentialMeshing(0, EXPIRY, 100, 0.0);
final MeshingFunction spaceMesh = new ExponentialMeshing(yL, yH, 101, 0.0);
final PDEGrid1D grid = new PDEGrid1D(timeMesh, spaceMesh);
final double[] sNodes = grid.getSpaceNodes();
// run the PDE solver backward to the dividend date
// PDE1DDataBundle<ConvectionDiffusionPDE1DCoefficients> db1 = new
// PDE1DDataBundle<ConvectionDiffusionPDE1DCoefficients>(pde, initialCon, lower1, upper1,
// grid1);
final PDE1DDataBundle<ConvectionDiffusionPDE1DCoefficients> db1 =
new PDE1DDataBundle<>(pde, payoff, lower, upper, grid);
final PDETerminalResults1D res = (PDETerminalResults1D) solver.solve(db1);
final Interpolator1DDataBundle interpolDB =
INTEPOLATOR1D.getDataBundle(sNodes, res.getTerminalResults());
final double val = INTEPOLATOR1D.interpolate(interpolDB, lnFT);
assertEquals(0.41491529, Math.sqrt(-2 * (val) / EXPIRY), 5e-4); // Number from backwardsPDETest
// System.out.println(val + "\t" + Math.sqrt(-2 * val / EXPIRY));
}
/**
* 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);
}