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; }
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); }
/** * 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); }