// ------------------------------------------------------------------------- @Override public double volatility( double forward, double strike, double timeToExpiry, SabrFormulaData data) { ArgChecker.notNull(data, "data"); double alpha = data.getAlpha(); double beta = data.getBeta(); double rho = data.getRho(); double nu = data.getNu(); return volatility(forward, strike, timeToExpiry, alpha, beta, rho, nu); }
@SuppressWarnings("null") private double fdSensitivity( EuropeanVanillaOption optionData, double forward, SabrFormulaData sabrData, SabrParameter param, double delta) { Function<SabrFormulaData, Double> funcC = null; Function<SabrFormulaData, Double> funcB = null; Function<SabrFormulaData, Double> funcA = null; SabrFormulaData dataC = null; SabrFormulaData dataB = sabrData; SabrFormulaData dataA = null; Function<SabrFormulaData, Double> func = getVolatilityFunction(optionData, forward); FiniteDifferenceType fdType = null; switch (param) { case Strike: double strike = optionData.getStrike(); if (strike >= delta) { fdType = FiniteDifferenceType.CENTRAL; funcA = getVolatilityFunction(withStrike(optionData, strike - delta), forward); funcC = getVolatilityFunction(withStrike(optionData, strike + delta), forward); } else { fdType = FiniteDifferenceType.FORWARD; funcA = func; funcB = getVolatilityFunction(withStrike(optionData, strike + delta), forward); funcC = getVolatilityFunction(withStrike(optionData, strike + 2 * delta), forward); } dataC = sabrData; dataB = sabrData; dataA = sabrData; break; case Forward: if (forward > delta) { fdType = FiniteDifferenceType.CENTRAL; funcA = getVolatilityFunction(optionData, forward - delta); funcC = getVolatilityFunction(optionData, forward + delta); } else { fdType = FiniteDifferenceType.FORWARD; funcA = func; funcB = getVolatilityFunction(optionData, forward + delta); funcC = getVolatilityFunction(optionData, forward + 2 * delta); } dataC = sabrData; dataB = sabrData; dataA = sabrData; break; case Alpha: double a = sabrData.getAlpha(); if (a >= delta) { fdType = FiniteDifferenceType.CENTRAL; dataA = sabrData.withAlpha(a - delta); dataC = sabrData.withAlpha(a + delta); } else { fdType = FiniteDifferenceType.FORWARD; dataA = sabrData; dataB = sabrData.withAlpha(a + delta); dataC = sabrData.withAlpha(a + 2 * delta); } funcC = func; funcB = func; funcA = func; break; case Beta: double b = sabrData.getBeta(); if (b >= delta) { fdType = FiniteDifferenceType.CENTRAL; dataA = sabrData.withBeta(b - delta); dataC = sabrData.withBeta(b + delta); } else { fdType = FiniteDifferenceType.FORWARD; dataA = sabrData; dataB = sabrData.withBeta(b + delta); dataC = sabrData.withBeta(b + 2 * delta); } funcC = func; funcB = func; funcA = func; break; case Nu: double n = sabrData.getNu(); if (n >= delta) { fdType = FiniteDifferenceType.CENTRAL; dataA = sabrData.withNu(n - delta); dataC = sabrData.withNu(n + delta); } else { fdType = FiniteDifferenceType.FORWARD; dataA = sabrData; dataB = sabrData.withNu(n + delta); dataC = sabrData.withNu(n + 2 * delta); } funcC = func; funcB = func; funcA = func; break; case Rho: double r = sabrData.getRho(); if ((r + 1) < delta) { fdType = FiniteDifferenceType.FORWARD; dataA = sabrData; dataB = sabrData.withRho(r + delta); dataC = sabrData.withRho(r + 2 * delta); } else if ((1 - r) < delta) { fdType = FiniteDifferenceType.BACKWARD; dataA = sabrData.withRho(r - 2 * delta); dataB = sabrData.withRho(r - delta); dataC = sabrData; } else { fdType = FiniteDifferenceType.CENTRAL; dataC = sabrData.withRho(r + delta); dataA = sabrData.withRho(r - delta); } funcC = func; funcB = func; funcA = func; break; default: throw new MathException("enum not found"); } if (fdType != null) { switch (fdType) { case FORWARD: return (-1.5 * funcA.apply(dataA) + 2.0 * funcB.apply(dataB) - 0.5 * funcC.apply(dataC)) / delta; case BACKWARD: return (0.5 * funcA.apply(dataA) - 2.0 * funcB.apply(dataB) + 1.5 * funcC.apply(dataC)) / delta; case CENTRAL: return (funcC.apply(dataC) - funcA.apply(dataA)) / 2.0 / delta; default: throw new MathException("enum not found"); } } throw new MathException("enum not found"); }
/** * Computes the first and second order derivatives of the Black implied volatility in the SABR * model. * * <p>The first derivative values will be stored in the input array {@code volatilityD} The array * contains, [0] Derivative w.r.t the forward, [1] the derivative w.r.t the strike, [2] the * derivative w.r.t. to alpha, [3] the derivative w.r.t. to beta, [4] the derivative w.r.t. to * rho, and [5] the derivative w.r.t. to nu. Thus the length of the array should be 6. * * <p>The second derivative values will be stored in the input array {@code volatilityD2}. Only * the second order derivative with respect to the forward and strike are implemented. The array * contains [0][0] forward-forward; [0][1] forward-strike; [1][1] strike-strike. Thus the size * should be 2 x 2. * * <p>Around ATM, a first order expansion is used to due to some 0/0-type indetermination. The * second order derivative produced is poor around ATM. * * @param forward the forward value of the underlying * @param strike the strike value of the option * @param timeToExpiry the time to expiry of the option * @param data the SABR data. * @param volatilityD the array used to return the first order derivative * @param volatilityD2 the array of array used to return the second order derivative * @return the Black implied volatility */ @Override public double volatilityAdjoint2( double forward, double strike, double timeToExpiry, SabrFormulaData data, double[] volatilityD, double[][] volatilityD2) { double k = Math.max(strike, 0.000001); double alpha = data.getAlpha(); double beta = data.getBeta(); double rho = data.getRho(); double nu = data.getNu(); // Forward double h0 = (1 - beta) / 2; double h1 = forward * k; double h1h0 = Math.pow(h1, h0); double h12 = h1h0 * h1h0; double h2 = Math.log(forward / k); double h22 = h2 * h2; double h23 = h22 * h2; double h24 = h23 * h2; double f1 = h1h0 * (1 + h0 * h0 / 6.0 * (h22 + h0 * h0 / 20.0 * h24)); double f2 = nu / alpha * h1h0 * h2; double f3 = h0 * h0 / 6.0 * alpha * alpha / h12 + rho * beta * nu * alpha / 4.0 / h1h0 + (2 - 3 * rho * rho) / 24.0 * nu * nu; double sqrtf2 = Math.sqrt(1 - 2 * rho * f2 + f2 * f2); double f2x = 0.0; double x = 0.0, xp = 0, xpp = 0; if (DoubleMath.fuzzyEquals(f2, 0.0, SMALL_Z)) { f2x = 1.0 - 0.5 * f2 * rho; // small f2 expansion to f2^2 terms } else { if (DoubleMath.fuzzyEquals(rho, 1.0, RHO_EPS)) { x = f2 < 1.0 ? -Math.log(1.0 - f2) - 0.5 * Math.pow(f2 / (f2 - 1.0), 2) * (1.0 - rho) : Math.log(2.0 * f2 - 2.0) - Math.log(1.0 - rho); } else { x = Math.log((sqrtf2 + f2 - rho) / (1 - rho)); } xp = 1. / sqrtf2; xpp = (rho - f2) / Math.pow(sqrtf2, 3.0); f2x = f2 / x; } double sigma = Math.max(MIN_VOL, alpha / f1 * f2x * (1 + f3 * timeToExpiry)); // First level double h0Dbeta = -0.5; double sigmaDf1 = -sigma / f1; double sigmaDf2 = 0; if (DoubleMath.fuzzyEquals(f2, 0.0, SMALL_Z)) { sigmaDf2 = alpha / f1 * (1 + f3 * timeToExpiry) * -0.5 * rho; } else { sigmaDf2 = alpha / f1 * (1 + f3 * timeToExpiry) * (1.0 / x - f2 * xp / (x * x)); } double sigmaDf3 = alpha / f1 * f2x * timeToExpiry; double sigmaDf4 = f2x / f1 * (1 + f3 * timeToExpiry); double sigmaDx = -alpha / f1 * f2 / (x * x) * (1 + f3 * timeToExpiry); double[][] sigmaD2ff = new double[3][3]; sigmaD2ff[0][0] = -sigmaDf1 / f1 + sigma / (f1 * f1); // OK sigmaD2ff[0][1] = -sigmaDf2 / f1; sigmaD2ff[0][2] = -sigmaDf3 / f1; if (DoubleMath.fuzzyEquals(f2, 0.0, SMALL_Z)) { sigmaD2ff[1][2] = alpha / f1 * -0.5 * rho * timeToExpiry; } else { sigmaD2ff[1][1] = alpha / f1 * (1 + f3 * timeToExpiry) * (-2 * xp / (x * x) - f2 * xpp / (x * x) + 2 * f2 * xp * xp / (x * x * x)); sigmaD2ff[1][2] = alpha / f1 * timeToExpiry * (1.0 / x - f2 * xp / (x * x)); } sigmaD2ff[2][2] = 0.0; // double sigma = alpha / f1 * f2x * (1 + f3 * theta); // Second level double[] f1Dh = new double[3]; double[] f2Dh = new double[3]; double[] f3Dh = new double[3]; f1Dh[0] = h1h0 * (h0 * (h22 / 3.0 + h0 * h0 / 40.0 * h24)) + Math.log(h1) * f1; f1Dh[1] = h0 * f1 / h1; f1Dh[2] = h1h0 * (h0 * h0 / 6.0 * (2.0 * h2 + h0 * h0 / 5.0 * h23)); f2Dh[0] = Math.log(h1) * f2; f2Dh[1] = h0 * f2 / h1; f2Dh[2] = nu / alpha * h1h0; f3Dh[0] = h0 / 3.0 * alpha * alpha / h12 - 2 * h0 * h0 / 6.0 * alpha * alpha / h12 * Math.log(h1) - rho * beta * nu * alpha / 4.0 / h1h0 * Math.log(h1); f3Dh[1] = -2 * h0 * h0 / 6.0 * alpha * alpha / h12 * h0 / h1 - rho * beta * nu * alpha / 4.0 / h1h0 * h0 / h1; f3Dh[2] = 0.0; double[] f1Dp = new double[4]; // Derivative to sabr parameters double[] f2Dp = new double[4]; double[] f3Dp = new double[4]; double[] f4Dp = new double[4]; f1Dp[0] = 0.0; f1Dp[1] = f1Dh[0] * h0Dbeta; f1Dp[2] = 0.0; f1Dp[3] = 0.0; f2Dp[0] = -f2 / alpha; f2Dp[1] = f2Dh[0] * h0Dbeta; f2Dp[2] = 0.0; f2Dp[3] = h1h0 * h2 / alpha; f3Dp[0] = h0 * h0 / 3.0 * alpha / h12 + rho * beta * nu / 4.0 / h1h0; f3Dp[1] = rho * nu * alpha / 4.0 / h1h0 + f3Dh[0] * h0Dbeta; f3Dp[2] = beta * nu * alpha / 4.0 / h1h0 - rho / 4.0 * nu * nu; f3Dp[3] = rho * beta * alpha / 4.0 / h1h0 + (2 - 3 * rho * rho) / 12.0 * nu; f4Dp[0] = 1.0; f4Dp[1] = 0.0; f4Dp[2] = 0.0; f4Dp[3] = 0.0; double sigmaDh1 = sigmaDf1 * f1Dh[1] + sigmaDf2 * f2Dh[1] + sigmaDf3 * f3Dh[1]; double sigmaDh2 = sigmaDf1 * f1Dh[2] + sigmaDf2 * f2Dh[2] + sigmaDf3 * f3Dh[2]; double[][] f1D2hh = new double[2][2]; // No h0 double[][] f2D2hh = new double[2][2]; double[][] f3D2hh = new double[2][2]; f1D2hh[0][0] = h0 * (h0 - 1) * f1 / (h1 * h1); f1D2hh[0][1] = h0 * h1h0 / h1 * h0 * h0 / 6.0 * (2.0 * h2 + 4.0 * h0 * h0 / 20.0 * h23); f1D2hh[1][1] = h1h0 * (h0 * h0 / 6.0 * (2.0 + 12.0 * h0 * h0 / 20.0 * h2)); f2D2hh[0][0] = h0 * (h0 - 1) * f2 / (h1 * h1); f2D2hh[0][1] = nu / alpha * h0 * h1h0 / h1; f2D2hh[1][1] = 0.0; f3D2hh[0][0] = 2 * h0 * (2 * h0 + 1) * h0 * h0 / 6.0 * alpha * alpha / (h12 * h1 * h1) + h0 * (h0 + 1) * rho * beta * nu * alpha / 4.0 / (h1h0 * h1 * h1); f3D2hh[0][1] = 0.0; f3D2hh[1][1] = 0.0; double[][] sigmaD2hh = new double[2][2]; // No h0 for (int loopx = 0; loopx < 2; loopx++) { for (int loopy = loopx; loopy < 2; loopy++) { sigmaD2hh[loopx][loopy] = (sigmaD2ff[0][0] * f1Dh[loopy + 1] + sigmaD2ff[0][1] * f2Dh[loopy + 1] + sigmaD2ff[0][2] * f3Dh[loopy + 1]) * f1Dh[loopx + 1] + sigmaDf1 * f1D2hh[loopx][loopy] + (sigmaD2ff[0][1] * f1Dh[loopy + 1] + sigmaD2ff[1][1] * f2Dh[loopy + 1] + sigmaD2ff[1][2] * f3Dh[loopy + 1]) * f2Dh[loopx + 1] + sigmaDf2 * f2D2hh[loopx][loopy] + (sigmaD2ff[0][2] * f1Dh[loopy + 1] + sigmaD2ff[1][2] * f2Dh[loopy + 1] + sigmaD2ff[2][2] * f3Dh[loopy + 1]) * f3Dh[loopx + 1] + sigmaDf3 * f3D2hh[loopx][loopy]; } } // Third level double h1Df = k; double h1Dk = forward; double h1D2ff = 0.0; double h1D2kf = 1.0; double h1D2kk = 0.0; double h2Df = 1.0 / forward; double h2Dk = -1.0 / k; double h2D2ff = -1 / (forward * forward); double h2D2fk = 0.0; double h2D2kk = 1.0 / (k * k); volatilityD[0] = sigmaDh1 * h1Df + sigmaDh2 * h2Df; volatilityD[1] = sigmaDh1 * h1Dk + sigmaDh2 * h2Dk; volatilityD[2] = sigmaDf1 * f1Dp[0] + sigmaDf2 * f2Dp[0] + sigmaDf3 * f3Dp[0] + sigmaDf4 * f4Dp[0]; volatilityD[3] = sigmaDf1 * f1Dp[1] + sigmaDf2 * f2Dp[1] + sigmaDf3 * f3Dp[1] + sigmaDf4 * f4Dp[1]; if (DoubleMath.fuzzyEquals(f2, 0.0, SMALL_Z)) { volatilityD[4] = -0.5 * f2 + sigmaDf3 * f3Dp[2]; } else { double xDr; if (DoubleMath.fuzzyEquals(rho, 1.0, RHO_EPS)) { xDr = f2 > 1.0 ? 1.0 / (1.0 - rho) + (0.5 - f2) / (f2 - 1.0) / (f2 - 1.0) : 0.5 * Math.pow(f2 / (1.0 - f2), 2.0) + 0.25 * (f2 - 4.0) * Math.pow(f2 / (f2 - 1.0), 3) / (f2 - 1.0) * (1.0 - rho); if (Doubles.isFinite(xDr)) { volatilityD[4] = sigmaDf1 * f1Dp[2] + sigmaDx * xDr + sigmaDf3 * f3Dp[2] + sigmaDf4 * f4Dp[2]; } else { volatilityD[4] = Double.NEGATIVE_INFINITY; } } else { xDr = (-f2 / sqrtf2 - 1 + (sqrtf2 + f2 - rho) / (1 - rho)) / (sqrtf2 + f2 - rho); volatilityD[4] = sigmaDf1 * f1Dp[2] + sigmaDx * xDr + sigmaDf3 * f3Dp[2] + sigmaDf4 * f4Dp[2]; } } volatilityD[5] = sigmaDf1 * f1Dp[3] + sigmaDf2 * f2Dp[3] + sigmaDf3 * f3Dp[3] + sigmaDf4 * f4Dp[3]; volatilityD2[0][0] = (sigmaD2hh[0][0] * h1Df + sigmaD2hh[0][1] * h2Df) * h1Df + sigmaDh1 * h1D2ff + (sigmaD2hh[0][1] * h1Df + sigmaD2hh[1][1] * h2Df) * h2Df + sigmaDh2 * h2D2ff; volatilityD2[0][1] = (sigmaD2hh[0][0] * h1Dk + sigmaD2hh[0][1] * h2Dk) * h1Df + sigmaDh1 * h1D2kf + (sigmaD2hh[0][1] * h1Dk + sigmaD2hh[1][1] * h2Dk) * h2Df + sigmaDh2 * h2D2fk; volatilityD2[1][0] = volatilityD2[0][1]; volatilityD2[1][1] = (sigmaD2hh[0][0] * h1Dk + sigmaD2hh[0][1] * h2Dk) * h1Dk + sigmaDh1 * h1D2kk + (sigmaD2hh[0][1] * h1Dk + sigmaD2hh[1][1] * h2Dk) * h2Dk + sigmaDh2 * h2D2kk; return sigma; }