public static double inv_g(double beta, double y)
{
if (beta == 0.0 || y >= 1.0)
{
return(Math.Log(y));
}
var expo = -(beta * beta - 2.0 * beta + y) / (beta * (beta - 1.0));
if (expo > 700.0)
{
//! This happens only for small beta
//! Need solution here - default to newton iteration?
//! For the time being we use an expansion
var l1 = expo + Math.Log((1 - beta) / beta); // no risk here, beta is small
var l12 = l1 * l1;
var l13 = l12 * l1;
var l14 = l12 * l12;
var l15 = l14 * l12;
var l2 = Math.Log(l1);
var l22 = l2 * l2;
var l23 = l22 * l2;
var l24 = l23 * l2;
var wwb = l1 - l2 + l2 / l1 + l2 * (-2.0 + l2) / (2.0 * l12)
+ l2 * (6.0 - 9.0 * l2 + 2.0 * l22) / (6.0 * l13)
+ l2 * (-12.0 + 36.0 * l2 - 22.0 * l22 + 3.0 * l23) / (12.0 * l14)
+ l2 * (60.0 - 300.0 * l2 + 350.0 * l22 - 125.0 * l23 + 12.0 * l24) / (60.0 * l15);
return(beta * wwb - wwb + beta - 2.0 + y / beta);
}
var aw = Math.Exp(expo) * (1.0 - beta) / beta;
var wb = SpecialFunction.wapr(aw);
return(beta * wb - wb + beta - 2.0 + y / beta);
}
public static double beta_black_pdf(double beta, double mu, double sigma, double x)
{
var gm1 = inv_g(beta, x);
var den = dg(beta, gm1);
return(SpecialFunction.pdf_normal((gm1 - mu) / sigma) / den / sigma);
}
public static double inv_m1_approx_smart(double beta, double f, double sigma)
{
// Compute some relevant bounds
var log_001 = -4.605170186;
var lo = log_001 * (1 - beta) - 0.5 * sigma * sigma;
var hi = Math.Min(37 - 0.5 * sigma * sigma, 0.5 * sigma * sigma); //! < 10
// Evaluate the function on those points
// Using the actual function to too costly
var zz = 0.0;
var vlo = m1_beta_black(beta, lo, sigma);
var dhi_dmu = 0.0;
var vhi = m1_beta_black_b(beta, hi, sigma, ref zz, ref dhi_dmu, ref zz, 1.0);
// Treat bounds
if (f < vlo)
{
return(inv_g(beta, f)); // no convexity
}
if (f > vhi)
{
return(inv_g(beta, f) - 0.5 * sigma * sigma); // full convexity
}
//! Middle region, fit approximately
// a + b x + c exp(d x)
var d = dhi_dmu / vhi;
var c = vhi / Math.Exp(dhi_dmu * hi / vhi);
var b = -(vlo - c * Math.Exp(d * lo)) / (hi - lo);
var a = -b * hi;
//! The invert in closed form
var aw = Math.Exp(-d * (a - f) / b) * c * d / b;
return(-(b * SpecialFunction.wapr(aw) + a * d - d * f) / (b * d));
}
public static double ibachelier(OptionType optType, double F, double K, double T, double premium, bool safe = false)
{
if (T <= 0.0)
{
throw FS.FailWith($"T must be postive, got {T}");
}
var epsilon = optType.ToEpsilon();
var intrinsic = Math.Max(epsilon * (F - K), 0.0);
if (premium <= intrinsic)
{
if (DoubleUtils.ApproximatelyEqual(premium, intrinsic, DoubleUtils.DoubleEpsilon))
{
return(0.0);
}
if (safe)
{
return(0.0);
}
throw FS.FailWith($"premium must be larger than intrinsic. Got premium = {premium}, vs intrinsic = {intrinsic}"); //!
}
if (F == K)
{
return(premium * Math.Sqrt(2.0 * Math.PI / T));
}
var sqrtT = Math.Sqrt(T);
var phiStar = -Math.Abs((premium - intrinsic) / (K - F));
var xBar = 0.0;
if (phiStar < -0.001882039271)
{
var g = 1.0 / (phiStar - 0.5);
var g2 = g * g;
var zeta = (0.032114372355 - g2 * (0.016969777977 - g2 * (2.6207332461e-3 - 9.6066952861e-5 * g2)))
/ (1.0 - g2 * (0.6635646938 - g2 * (0.14528712196 - 0.010472855461 * g2)));
xBar = g * (1.0 / Math.Sqrt(2.0 * Math.PI) + zeta * g2);
}
else
{
var h = Math.Sqrt(-Math.Log(-phiStar));
xBar = (9.4883409779 - h * (9.6320903635 - h * (0.58556997323 + 2.1464093351 * h)))
/ (1 - h * (0.65174820867 + h * (1.5120247828 + 6.6437847132e-5 * h)));
}
var xBar2 = xBar * xBar;
var nxBar = SpecialFunction.pdf_normal(xBar);
var phiTildaXBar = SpecialFunction.cdf_normal(xBar) + nxBar / xBar;
var q = (phiTildaXBar - phiStar) / nxBar;
var xStar = xBar + (3.0 * q * xBar2 * (2.0 - q * xBar * (2.0 + xBar2)))
/ (6.0 + q * xBar * (-12.0 + xBar * (6.0 * q + xBar * (-6.0 + q * xBar * (3.0 + xBar2)))));
return(Math.Abs((K - F) / (xStar * sqrtT)));
}
public static double beta_black_quick_delta_to_strike(double beta, double F, double T, double sigma, double qd)
{
if (qd < -1.0 || qd > 1.0)
{
throw FS.FailWith($"delta must be in (-1, 1), got {qd}");
}
var qdUsed = qd < 0.0 ? 1.0 + qd : qd;
return(g(beta, inv_g(beta, F) - SpecialFunction.cdfinv_normal(qdUsed) * sigma * Math.Sqrt(T)));
}
public static double beta_black_delta_to_strike(double beta, double F, double T, double sigma, double delta)
{
if (delta < -1.0 || delta > 1.0)
{
throw FS.FailWith($"delta must be in (-1, 1), got {delta}");
}
var mu = inv_m1_beta_black(beta, F, sigma * Math.Sqrt(T));
var usedDelta = delta < 0.0 ? -delta : 1.0 - delta;
return(g(beta, SpecialFunction.cdfinv_normal(usedDelta) * sigma * Math.Sqrt(T) + mu));
}
public static double beta_black_strike_to_quick_delta(double beta, double F, double T, double sigma, double strike)
{
var qd = SpecialFunction.cdf_normal((inv_g(beta, F) - inv_g(beta, strike)) / (sigma * Math.Sqrt(T)));
if (qd <= -0.5)
{
return(1.0 + qd);
}
if (qd > 0.5)
{
return(qd - 1.0);
}
return(qd);
}
public static double inv_m1_beta_black_approx_2(double beta, double f, double sigma)
{
//! Numerical
var expArg = -1.0 * (SpecialFunction.pow2(beta) - 2.0 * beta + f) / ((-1.0 + beta) * beta);
if (expArg > 37.0)
{
return(inv_g(beta, f));
}
var aw = -0.5000000000 * (2.0 * SpecialFunction.pow2(beta) + SpecialFunction.pow2(sigma) - 4.0 * beta + 2.0)
* Math.Exp(expArg)
/ ((-1.0 + beta) * beta);
var wb = SpecialFunction.wapr(aw);
return(beta * wb - wb + beta - 2.0 + f / beta);
}
public static double black76(OptionType optType, double F, double K, double sdev)
{
var num = optType.ToEpsilon();
if (sdev < 0.0)
{
throw FS.FailWith($"sdev must be positive. Got {sdev}");
}
if (sdev == 0.0 || K <= 0.0)
{
return(Math.Max(num * (F - K), 0.0));
}
else
{
var num2 = (Math.Log(F / K) + 0.5 * sdev * sdev) / sdev;
var num3 = num2 - sdev;
return(num * (F * SpecialFunction.cdf_normal(num * num2) -
K * SpecialFunction.cdf_normal(num * num3)));
}
}
public static double m1_beta_black(double beta, double mu, double sigma)
{
var betaMinusOne = beta - 1.0;
var betaMinusOneSq = betaMinusOne * betaMinusOne;
var sigma2 = sigma * sigma;
var sqrt2 = Math.Sqrt(2.0);
var sqrtPi = Math.Sqrt(Math.PI);
var t = -mu / sigma;
var t2 = t * t;
var one_over_two_betaMinusOneSq = 0.5 / betaMinusOneSq;
var e4 = Math.Exp(-0.5 * t2);
var a1 = t + sigma / betaMinusOne;
var group1 = -SpecialFunction.exp_x_N_y(-beta * mu / betaMinusOneSq + (sigma2 + 2.0 * mu) * one_over_two_betaMinusOneSq, a1);
var a2 = sigma - t;
var group2 = SpecialFunction.exp_x_N_y(mu + 0.5 * sigma2, a2);
var f0 = -1.0 / (2.0 * sqrtPi);
return(f0 * sqrt2 * e4 * beta * sigma
- betaMinusOneSq * group1
+ group2
- beta * (mu - beta + 2) * (SpecialFunction.cdf_normal(-t) - 1.0));
}
public static double beta_black_trunc(double beta, double mu, double sigma, double x)
{
var betaMinusOne = beta - 1.0;
var betaMinusOneSq = betaMinusOne * betaMinusOne;
var sigma2 = sigma * sigma;
var sqrt2 = Math.Sqrt(2.0);
var sqrtPi = Math.Sqrt(Math.PI);
var t = -mu / sigma;
var one_over_two_betaMinusOneSq = 0.5 / betaMinusOneSq;
var f0 = -1.0 / (2.0 * sqrtPi);
if (x < t)
{
var x2 = x * x;
var e2 = Math.Exp(-0.5 * x2);
var a4 = x + sigma / betaMinusOne;
var group4 = SpecialFunction.exp_x_N_y(-beta * mu / betaMinusOneSq + (sigma2 + 2.0 * mu) * one_over_two_betaMinusOneSq, a4);
return(f0 * sqrt2 * e2 * beta * sigma +
+betaMinusOneSq * group4
+ beta * SpecialFunction.cdf_normal(x) * (mu - beta + 2.0));
}
else
{
var t2 = t * t;
var e4 = Math.Exp(-0.5 * t2);
var a1 = t + sigma / betaMinusOne;
var group1 = -SpecialFunction.exp_x_N_y(-beta * mu / betaMinusOneSq + (sigma2 + 2.0 * mu) * one_over_two_betaMinusOneSq, a1);
var a2 = sigma - t;
var a3 = x - sigma;
var group2 = -SpecialFunction.exp_x_N_y(mu + 0.5 * sigma2, -a2);
var group3 = SpecialFunction.exp_x_N_y(mu + 0.5 * sigma2, a3);
return(f0 * sqrt2 * e4 * beta * sigma
- betaMinusOneSq * group1
+ (group2 + group3)
- beta * (mu - beta + 2) * (SpecialFunction.cdf_normal(-t) - 1.0));
}
}
public static double beta_black(double beta, OptionType optType, double F, double K, double T, double sigma)
{
if (T < 0.0)
{
throw FS.FailWith($"T < 0 not allowed, {T}");
}
if (sigma < 0.0)
{
throw FS.FailWith($"sigma < 0 not allowed {sigma}");
}
var stdev = sigma * Math.Sqrt(T);
if (stdev == 0.0)
{
return(Math.Max(optType.ToEpsilon() * (F - K), 0.0));
}
var mu = inv_m1_beta_black(beta, F, stdev);
var d = (inv_g(beta, K) - mu) / stdev;
var t1 = K * SpecialFunction.cdf_normal(d);
var t2 = beta_black_trunc(beta, mu, stdev, d);
var put = t1 - t2;
var intrinsic = Math.Max(optType.ToEpsilon() * (F - K), 0.0);
if (optType == OptionType.Put)
{
return(Math.Max(put, intrinsic));
}
else if (optType == OptionType.Call)
{
return(Math.Max(put + F - K, intrinsic));
}
else
{
throw FS.E_CASE(optType);
}
}
public static double m1_beta_black_b(double beta, double mu, double sigma, ref double beta_b, ref double mu_b, ref double sigma_b, double res_b)
{
var betaMinusOne = beta - 1.0;
var betaMinusOneSq = betaMinusOne * betaMinusOne;
var sigma2 = sigma * sigma;
var sqrt2 = Math.Sqrt(2.0);
var sqrtPi = Math.Sqrt(Math.PI);
var t = -mu / sigma;
var t2 = t * t;
var one_over_two_betaMinusOneSq = 0.5 / betaMinusOneSq;
var e4 = Math.Exp(-0.5 * t2);
var a1 = t + sigma / betaMinusOne;
var b1 = -beta * mu / betaMinusOneSq + (sigma2 + 2.0 * mu) * one_over_two_betaMinusOneSq;
var dmgroup1_db1 = 0.0;
var dmgroup1_da1 = 0.0;
var group1 = -SpecialFunction.exp_x_N_y_b(b1, a1, ref dmgroup1_db1, ref dmgroup1_da1, 1.0);
var a2 = sigma - t;
var muOneHalfS2 = mu + 0.5 * sigma2;
var dgroup2_dmuOneHalfS2 = 0.0;
var dgroup2_da2 = 0.0;
var group2 = SpecialFunction.exp_x_N_y_b(muOneHalfS2, a2, ref dgroup2_dmuOneHalfS2, ref dgroup2_da2, 1.0);
var f0 = -1.0 / (2.0 * sqrtPi);
var dnmt_dmt = 0.0;
var nmt = SpecialFunction.cdf_normal_b(-t, ref dnmt_dmt, 1.0);
var res = f0 * sqrt2 * e4 * beta * sigma
- betaMinusOneSq * group1
+ group2
- beta * (mu - beta + 2) * (nmt - 1.0);
var e4_b = res_b * f0 * sqrt2 * beta * sigma;
beta_b += res_b * f0 * sqrt2 * e4 * sigma;
beta_b += -res_b * (mu - 2.0 * beta + 2.0) * (nmt - 1.0);
sigma_b += res_b * f0 * sqrt2 * e4 * beta;
var betaMinusOneSq_b = -res_b * group1;
var group1_b = -res_b * betaMinusOneSq;
var group2_b = res_b;
mu_b += -res_b * beta * (nmt - 1.0);
var nmt_b = -res_b * beta * (mu - beta + 2);
var t_b = -nmt_b * dnmt_dmt;
var a2_b = group2_b * dgroup2_da2;
var muOneHalfS2_b = group2_b * dgroup2_dmuOneHalfS2;
mu_b += muOneHalfS2_b;
sigma_b += muOneHalfS2_b * sigma;
sigma_b += a2_b;
t_b += -a2_b;
var b1_b = -group1_b * dmgroup1_db1;
var a1_b = -group1_b * dmgroup1_da1;
beta_b += -b1_b * mu / betaMinusOneSq;
mu_b += b1_b * (-beta / betaMinusOneSq + 2.0 * one_over_two_betaMinusOneSq);
sigma_b += b1_b * 2.0 * sigma * one_over_two_betaMinusOneSq;
betaMinusOneSq_b += b1_b * beta * mu / betaMinusOneSq / betaMinusOneSq;
var one_over_two_betaMinusOneSq_b = b1_b * (sigma2 + 2.0 * mu);
t_b += a1_b;
var betaMinusOne_b = -a1_b * sigma / betaMinusOne / betaMinusOne;
sigma_b += a1_b / betaMinusOne;
var t2_b = -0.5 * e4_b * e4;
betaMinusOneSq_b += -one_over_two_betaMinusOneSq_b * 0.5 / betaMinusOneSq / betaMinusOneSq;
t_b += t2_b * 2.0 * t;
mu_b += -t_b / sigma;
sigma_b += t_b * mu / sigma / sigma;
betaMinusOne_b += betaMinusOneSq_b * 2.0 * betaMinusOne;
beta_b += betaMinusOne_b;
return(res);
}
public static double beta_black_cdf(double beta, double mu, double sigma, double x)
{
return(SpecialFunction.cdf_normal((inv_g(beta, x) - mu) / sigma));
}
public static double beta_black_cdfinv(double beta, double mu, double sigma, double p)
{
return(g(beta, SpecialFunction.cdfinv_normal(p) * sigma + mu));
}
public static double beta_black_trunc_b(double beta, double mu, double sigma, double x, ref double beta_b, ref double mu_b, ref double sigma_b, ref double x_b, double res_b)
{
var betaMinusOne = beta - 1.0;
var betaMinusOneSq = betaMinusOne * betaMinusOne;
var sigma2 = sigma * sigma;
var sqrt2 = Math.Sqrt(2.0);
var sqrtPi = Math.Sqrt(Math.PI);
var t = -mu / sigma;
var one_over_two_betaMinusOneSq = 0.5 / betaMinusOneSq;
var f0 = -1.0 / (2.0 * sqrtPi);
var res = 0.0;
var betaMinusOne_b = 0.0;
var betaMinusOneSq_b = 0.0;
var one_over_two_betaMinusOneSq_b = 0.0;
var x2_b = 0.0;
var t_b = 0.0;
if (x < t)
{
var x2 = x * x;
var ar2 = -0.5 * x2;
var e2 = Math.Exp(ar2);
var a4 = x + sigma / betaMinusOne;
var ar4 = -beta * mu / betaMinusOneSq + (sigma2 + 2.0 * mu) * one_over_two_betaMinusOneSq;
var dgroup4_dar4 = 0.0;
var dgroup4_da4 = 0.0;
var group4 = SpecialFunction.exp_x_N_y_b(ar4, a4, ref dgroup4_dar4, ref dgroup4_da4, 1.0);
var dNx_dx = 0.0;
var Nx = SpecialFunction.cdf_normal_b(x, ref dNx_dx, 1.0);
res = f0 * sqrt2 * e2 * beta * sigma
+ betaMinusOneSq * group4
+ beta * Nx * (mu - beta + 2.0);
var e2_b = res_b * f0 * sqrt2 * beta * sigma;
beta_b += res_b * (f0 * sqrt2 * e2 * sigma + Nx * (mu - 2.0 * beta + 2.0));
sigma_b += res_b * f0 * sqrt2 * e2 * beta;
betaMinusOneSq_b += res_b * group4;
var group4_b = res_b * betaMinusOneSq;
var Nx_b = res_b * beta * (mu - beta + 2.0);
mu_b += res_b * beta * Nx;
x_b += Nx_b * dNx_dx;
var ar4_b = group4_b * dgroup4_dar4;
var a4_b = group4_b * dgroup4_da4;
beta_b += ar4_b * (-mu / betaMinusOneSq);
mu_b += ar4_b * (-beta / betaMinusOneSq + 2.0 * one_over_two_betaMinusOneSq);
betaMinusOneSq_b += ar4_b * (beta * mu / betaMinusOneSq / betaMinusOneSq);
sigma_b += ar4_b * 2.0 * sigma * one_over_two_betaMinusOneSq;
one_over_two_betaMinusOneSq_b += ar4_b * (sigma2 + 2.0 * mu);
x_b += a4_b;
sigma_b += a4_b / betaMinusOne;
betaMinusOne_b += a4_b * (-sigma / betaMinusOne / betaMinusOne);
var ar2_b = e2_b * e2;
x2_b += -0.5 * ar2_b;
x_b += x2_b * 2.0 * x;
}
else
{
var t2 = t * t;
var ar4 = -0.5 * t2;
var e4 = Math.Exp(ar4);
var a1 = t + sigma / betaMinusOne;
var ar1 = -beta * mu / betaMinusOneSq + (sigma2 + 2.0 * mu) * one_over_two_betaMinusOneSq;
var dmgroup1_dar1 = 0.0;
var dmgroup1_da1 = 0.0;
var group1 = -SpecialFunction.exp_x_N_y_b(ar1, a1, ref dmgroup1_dar1, ref dmgroup1_da1, 1.0);
var a2 = t - sigma;
var a3 = x - sigma;
var ar2 = mu + 0.5 * sigma2;
var dmgroup2_dar2 = 0.0;
var dmgroup2_da2 = 0.0;
var group2 = -SpecialFunction.exp_x_N_y_b(ar2, a2, ref dmgroup2_dar2, ref dmgroup2_da2, 1.0);
var dgroup3_dar2 = 0.0;
var dgroup3_da3 = 0.0;
var group3 = SpecialFunction.exp_x_N_y_b(ar2, a3, ref dgroup3_dar2, ref dgroup3_da3, 1.0);
var dnmt_dmt = 0.0;
var nmt = SpecialFunction.cdf_normal_b(-t, ref dnmt_dmt, 1.0);
res = f0 * sqrt2 * e4 * beta * sigma
- betaMinusOneSq * group1
+ group2 + group3
- beta * (mu - beta + 2) * (nmt - 1.0);
var e4_b = res_b * f0 * sqrt2 * beta * sigma;
beta_b += res_b * (f0 * sqrt2 * e4 * sigma - (mu - 2.0 * beta + 2.0) * (nmt - 1.0));
sigma_b += res_b * f0 * sqrt2 * e4 * beta;
betaMinusOneSq_b += -res_b * group1;
var group1_b = -res_b * betaMinusOneSq;
var group2_b = res_b;
var group3_b = res_b;
mu_b += -res_b * beta * (nmt - 1.0);
var nmt_b = res_b * -beta * (mu - beta + 2);
t_b += -nmt_b * dnmt_dmt;
var ar2_b = group3_b * dgroup3_dar2;
var a3_b = group3_b * dgroup3_da3;
var a2_b = -group2_b * dmgroup2_da2;
ar2_b += -group2_b * dmgroup2_dar2;
mu_b += ar2_b;
sigma_b += ar2_b * sigma;
x_b += a3_b;
sigma_b += -a3_b;
sigma_b += -a2_b;
t_b += a2_b;
var ar1_b = -group1_b * dmgroup1_dar1;
var a1_b = -group1_b * dmgroup1_da1;
beta_b += ar1_b * (-mu / betaMinusOneSq);
mu_b += ar1_b * (-beta / betaMinusOneSq + 2.0 * one_over_two_betaMinusOneSq);
betaMinusOneSq_b += ar1_b * (beta * mu / betaMinusOneSq / betaMinusOneSq);
sigma_b += ar1_b * 2.0 * sigma * one_over_two_betaMinusOneSq;
one_over_two_betaMinusOneSq_b += ar1_b * (sigma2 + 2.0 * mu);
t_b += a1_b;
sigma_b += a1_b / betaMinusOne;
betaMinusOne_b += -a1_b * sigma / betaMinusOne / betaMinusOne;
var ar4_b = e4_b * e4;
var t2_b = -0.5 * ar4_b;
t_b += t2_b * 2.0 * t;
}
betaMinusOneSq_b += -one_over_two_betaMinusOneSq_b * 0.5 / betaMinusOneSq / betaMinusOneSq;
mu_b += -t_b / sigma;
sigma_b += t_b * mu / sigma / sigma;
betaMinusOne_b += betaMinusOneSq_b * 2.0 * betaMinusOne;
beta_b += betaMinusOne_b;
return(res);
}
public static double beta_black_b(double beta, OptionType optType, double F, double K, double T, double sigma
, ref double beta_b, ref double F_b, ref double K_b, ref double T_b, ref double sigma_b, double res_b)
{
if (T < 0.0)
{
throw FS.FailWith($"T < 0 not allowed, {T}");
}
if (sigma < 0.0)
{
throw FS.FailWith($"sigma < 0 not allowed {sigma}");
}
var sqrtT = Math.Sqrt(T);
var stdev = sigma * sqrtT;
if (stdev == 0.0)
{
return(Math.Max(optType.ToEpsilon() * (F - K), 0.0));
}
var dmu_dbeta = 0.0;
var dmu_dF = 0.0;
var dmu_dstdev = 0.0;
var mu = inv_m1_beta_black_b(beta, F, stdev, ref dmu_dbeta, ref dmu_dF, ref dmu_dstdev, 1.0);
var digb_dbeta = 0.0;
var digb_dK = 0.0;
var igb = inv_g_b(beta, K, ref digb_dbeta, ref digb_dK, 1.0);
var d = (igb - mu) / stdev;
var dNd_dd = 0.0;
var Nd = SpecialFunction.cdf_normal_b(d, ref dNd_dd, 1.0);
var t1 = K * Nd;
var dt2_dbeta = 0.0;
var dt2_dmu = 0.0;
var dt2_dstdev = 0.0;
var dt2_dd = 0.0;
var t2 = beta_black_trunc_b(beta, mu, stdev, d, ref dt2_dbeta, ref dt2_dmu, ref dt2_dstdev, ref dt2_dd, 1.0);
var res = t1 - t2;
// now rewind
var t1_b = res_b;
var t2_b = -res_b;
beta_b += t2_b * dt2_dbeta;
var mu_b = t2_b * dt2_dmu;
var stdev_b = t2_b * dt2_dstdev;
var d_b = t2_b * dt2_dd;
K_b += t1_b * Nd;
var Nd_b = t1_b * K;
d_b += Nd_b * dNd_dd;
var igb_b = d_b / stdev;
mu_b += -d_b / stdev;
stdev_b += -d_b * (igb - mu) / stdev / stdev;
beta_b += igb_b * digb_dbeta;
K_b += igb_b * digb_dK;
beta_b += mu_b * dmu_dbeta;
F_b += mu_b * dmu_dF;
stdev_b += mu_b * dmu_dstdev;
sigma_b += stdev_b * sqrtT;
T_b += stdev_b * sigma / (2.0 * sqrtT);
if (optType == OptionType.Put)
{
return(res);
}
else if (optType == OptionType.Call)
{
F_b += res_b;
K_b += -res_b;
return(res + F - K);
}
else
{
throw FS.E_CASE(optType);
}
}