public double MSPriceBS(double S, double K, double T, double sigma, double r, double q, int MaxIter, double Tol, double b, double dt)
{
BisectionAlgo BA = new BisectionAlgo();
MSset mssettings = new MSset();
mssettings.theta = Math.Log(K / S) / sigma / Math.Sqrt(T);
mssettings.K = K;
mssettings.sigma = sigma;
mssettings.r = r;
mssettings.q = q;
mssettings.T = T;
double a = 0.5 * mssettings.theta;
double y = BA.Bisection(mssettings, a, b, Tol, MaxIter, dt);
if (y < mssettings.theta)
{
y = mssettings.theta;
}
return(MSPutBS(y, mssettings));
}
// Bisection Algorithm for Black Scholes pricing
public double Bisection(MSset mssettings, double a, double b, double Tol, int MaxIter, double dy)
{
double lowCdif = MSPutBSdiff(a, mssettings, dy);
double highCdif = MSPutBSdiff(b, mssettings, dy);
double y = 0.0;
double midP;
if (lowCdif * highCdif > 0.0)
{
y = -999.0;
}
else
{
for (int x = 0; x <= MaxIter; x++)
{
midP = (a + b) / 2.0;
double midCdif = MSPutBSdiff(midP, mssettings, dy);
if (Math.Abs(midCdif) < Tol)
{
break;
}
else
{
if (midCdif > 0.0)
{
a = midP;
}
else
{
b = midP;
}
}
y = midP;
}
}
return(y);
}
public double MSPutBS(double y, MSset mssettings)
{
// The settings and parameters
double theta = mssettings.theta;
double K = mssettings.K;
double sigma = mssettings.sigma;
double r = mssettings.r;
double q = mssettings.q;
double T = mssettings.T;
// The drift
double mu = r - q;
// The "C" coefficients evaluated at theta = y
BlackScholesPrice BS = new BlackScholesPrice();
double cdf = BS.NormCDF(y);
double pdf = BS.NormPDF(y);
double C1 = sigma * y * K / (y * cdf + pdf);
double C2 = -1.0 / 2.0 * (C1 * cdf * Math.Pow(sigma, 2.0) - 2.0 * C1 * cdf * mu + Math.Pow(sigma, 3.0) * Math.Pow(y, 2.0) * K) / sigma / (cdf * Math.Pow(y, 2.0) + cdf + y * pdf);
double C3 = 1.0 / 24.0 * (-24.0 * y * cdf * Math.Pow(sigma, 3.0) * C2 + 48.0 * y * cdf * sigma * C2 * mu + 24.0 * y * cdf * Math.Pow(sigma, 2.0) * r * C1 - 24.0 * pdf * C2 * Math.Pow(sigma, 3.0) + 48.0 * pdf * C2 * mu * sigma + 24.0 * pdf * r * C1 * Math.Pow(sigma, 2.0) - 3.0 * pdf * C1 * Math.Pow(sigma, 4.0) + 12.0 * pdf * C1 * Math.Pow(sigma, 2.0) * mu - 12.0 * pdf * C1 * Math.Pow(mu, 2.0) + 4.0 * Math.Pow(sigma, 5.0) * Math.Pow(y, 3.0) * K) / Math.Pow(sigma, 2.0) / (cdf * Math.Pow(y, 3.0) + 3.0 * y * cdf + pdf * Math.Pow(y, 2.0) + 2.0 * pdf);
double C4 = -1.0 / 48.0 * (-48.0 * cdf * Math.Pow(sigma, 3.0) * Math.Pow(y, 2.0) * r * C2 + 72.0 * cdf * Math.Pow(sigma, 4.0) * Math.Pow(y, 2.0) * C3 - 144.0 * cdf * Math.Pow(sigma, 2.0) * Math.Pow(y, 2.0) * C3 * mu - 48.0 * cdf * Math.Pow(sigma, 3.0) * r * C2 + 72.0 * cdf * Math.Pow(sigma, 4.0) * C3 - 144.0 * cdf * Math.Pow(sigma, 2.0) * C3 * mu + 12.0 * cdf * Math.Pow(sigma, 5.0) * C2 - 48.0 * cdf * Math.Pow(sigma, 3.0) * C2 * mu - 24.0 * cdf * Math.Pow(sigma, 4.0) * r * C1 + 48.0 * cdf * sigma * C2 * Math.Pow(mu, 2.0) + 48.0 * cdf * Math.Pow(sigma, 2.0) * mu * r * C1 - 48.0 * y * pdf * r * C2 * Math.Pow(sigma, 3.0) + 72.0 * y * pdf * C3 * Math.Pow(sigma, 4.0) - 144.0 * y * pdf * C3 * mu * Math.Pow(sigma, 2.0) - y * pdf * Math.Pow(sigma, 6.0) * C1 + 6.0 * y * pdf * Math.Pow(sigma, 4.0) * C1 * mu - 12.0 * y * pdf * Math.Pow(sigma, 2.0) * C1 * Math.Pow(mu, 2.0) + 8.0 * y * pdf * C1 * Math.Pow(mu, 3.0) + 2.0 * Math.Pow(sigma, 7.0) * Math.Pow(y, 4.0) * K) / Math.Pow(sigma, 3.0) / (cdf * Math.Pow(y, 4.0) + 6.0 * cdf * Math.Pow(y, 2.0) + 3.0 * cdf + pdf * Math.Pow(y, 3.0) + 5.0 * y * pdf);
double C5 = 1.0 / 1920.0 * (16.0 * Math.Pow(sigma, 9.0) * Math.Pow(y, 5.0) * K - 80.0 * pdf * Math.Pow(sigma, 7.0) * C2 + 5.0 * pdf * Math.Pow(sigma, 8.0) * C1 + 80.0 * pdf * C1 * Math.Pow(mu, 4.0) - 640.0 * pdf * Math.Pow(r, 2.0) * C3 * Math.Pow(sigma, 4.0) * C1 + 320.0 * pdf * r * C3 * Math.Pow(sigma, 2.0) * C1 * Math.Pow(mu, 2.0) + 640.0 * pdf * Math.Pow(mu, 2.0) * r * C1 * Math.Pow(sigma, 2.0) + 640.0 * pdf * r * C3 * Math.Pow(sigma, 5.0) * C2 + 640.0 * pdf * sigma * C2 * Math.Pow(mu, 3.0) + 480.0 * pdf * Math.Pow(sigma, 5.0) * C2 * mu - 960.0 * pdf * Math.Pow(sigma, 3.0) * C2 * Math.Pow(mu, 2.0) + 160.0 * pdf * Math.Pow(sigma, 6.0) * r * C1 + 120.0 * pdf * Math.Pow(sigma, 4.0) * C1 * Math.Pow(mu, 2.0) - 40.0 * pdf * Math.Pow(sigma, 6.0) * C1 * mu - 160.0 * pdf * Math.Pow(sigma, 2.0) * C1 * Math.Pow(mu, 3.0) + 3840.0 * pdf * r * Math.Pow(sigma, 4.0) * C3 + 1280.0 * pdf * r * Math.Pow(sigma, 5.0) * C2 - 320.0 * pdf * Math.Pow(r, 2.0) * Math.Pow(sigma, 4.0) * C1 + 15360.0 * pdf * Math.Pow(sigma, 3.0) * C4 * mu + 5760.0 * pdf * Math.Pow(sigma, 4.0) * C3 * mu - 5760.0 * pdf * Math.Pow(sigma, 2.0) * C3 * Math.Pow(mu, 2.0) + 23040.0 * y * cdf * Math.Pow(sigma, 3.0) * C4 * mu + 1920.0 * y * cdf * Math.Pow(sigma, 5.0) * r * C2 - 3840.0 * y * cdf * Math.Pow(sigma, 3.0) * r * C2 * mu - 3840.0 * pdf * Math.Pow(y, 2.0) * Math.Pow(sigma, 5.0) * C4 - 5.0 * pdf * Math.Pow(y, 2.0) * Math.Pow(sigma, 8.0) * C1 - 80.0 * pdf * Math.Pow(y, 2.0) * C1 * Math.Pow(mu, 4.0) - 960.0 * y * cdf * Math.Pow(sigma, 4.0) * Math.Pow(r, 2.0) * C1 + 5760.0 * y * cdf * Math.Pow(sigma, 4.0) * C3 * mu - 5760.0 * y * cdf * Math.Pow(sigma, 2.0) * C3 * Math.Pow(mu, 2.0) + 40.0 * pdf * Math.Pow(y, 2.0) * Math.Pow(sigma, 6.0) * C1 * mu - 120.0 * pdf * Math.Pow(y, 2.0) * Math.Pow(sigma, 4.0) * C1 * Math.Pow(mu, 2.0) + 160.0 * pdf * Math.Pow(y, 2.0) * Math.Pow(sigma, 2.0) * C1 * Math.Pow(mu, 3.0) + 7680.0 * pdf * Math.Pow(y, 2.0) * Math.Pow(sigma, 3.0) * C4 * mu + 1920.0 * pdf * Math.Pow(y, 2.0) * r * Math.Pow(sigma, 4.0) * C3 - 3840.0 * Math.Pow(y, 3.0) * cdf * Math.Pow(sigma, 5.0) * C4 - 11520.0 * y * cdf * Math.Pow(sigma, 5.0) * C4 - 1440.0 * y * cdf * Math.Pow(sigma, 6.0) * C3 - 320.0 * pdf * r * C3 * Math.Pow(sigma, 4.0) * C1 * mu - 1280.0 * pdf * r * C3 * Math.Pow(sigma, 3.0) * C2 * mu - 640.0 * pdf * Math.Pow(sigma, 4.0) * mu * r * C1 + 80.0 * pdf * r * C3 * Math.Pow(sigma, 6.0) * C1 - 7680.0 * pdf * Math.Pow(sigma, 5.0) * C4 - 1440.0 * pdf * Math.Pow(sigma, 6.0) * C3 + 1920.0 * Math.Pow(y, 3.0) * cdf * Math.Pow(sigma, 4.0) * r * C3 + 7680.0 * Math.Pow(y, 3.0) * cdf * Math.Pow(sigma, 3.0) * C4 * mu + 5760.0 * y * cdf * Math.Pow(sigma, 4.0) * r * C3 - 2560.0 * pdf * r * Math.Pow(sigma, 3.0) * C2 * mu) / Math.Pow(sigma, 4.0) / (cdf * Math.Pow(y, 5.0) + 10.0 * cdf * Math.Pow(y, 3.0) + 15.0 * y * cdf + pdf * Math.Pow(y, 4.0) + 9.0 * pdf * Math.Pow(y, 2.0) + 8.0 * pdf);
//Set 1 polynomials
double P01 = theta;
double P11 = 0.0;
double Q01 = 1.0;
double Q11 = 0.0;
//Set 2 polynomials
double P02 = Math.Pow(theta, 2.0) + 1.0;
double P12 = -1.0 / 2.0 * C1 * (-Math.Pow(sigma, 2.0) + 2.0 * mu) / sigma;
double Q02 = theta;
double Q12 = 0.0;
//Set 3 polynomials
double P03 = Math.Pow(theta, 3.0) + 3.0 * theta;
double P13 = -theta * (-C2 * Math.Pow(sigma, 2.0) + 2.0 * C2 * mu + r * C1 * sigma) / sigma;
double Q03 = Math.Pow(theta, 2.0) + 2.0;
double Q13 = -1.0 / 8.0 * (-8.0 * C2 * Math.Pow(sigma, 3.0) + 16.0 * C2 * mu * sigma + 8.0 * r * C1 * Math.Pow(sigma, 2.0) - C1 * Math.Pow(sigma, 4.0) + 4.0 * C1 * Math.Pow(sigma, 2.0) * mu - 4.0 * C1 * Math.Pow(mu, 2.0)) / Math.Pow(sigma, 2.0);
//Set 4 polynomials
double P04 = Math.Pow(theta, 4.0) + 6.0 * Math.Pow(theta, 2.0) + 3.0;
double P14 = -1.0 / 2.0 * (2.0 * r * C2 * sigma - 3.0 * C3 * Math.Pow(sigma, 2.0) + 6.0 * C3 * mu) / sigma * Math.Pow(theta, 2.0) + 1.0 / 4.0 * (-4.0 * r * C2 * Math.Pow(sigma, 2.0) + 6.0 * C3 * Math.Pow(sigma, 3.0) - 12.0 * C3 * sigma * mu + Math.Pow(sigma, 4.0) * C2 - 4.0 * Math.Pow(sigma, 2.0) * C2 * mu - 2.0 * Math.Pow(sigma, 3.0) * r * C1 + 4.0 * C2 * Math.Pow(mu, 2.0) + 4.0 * mu * r * C1 * sigma) / Math.Pow(sigma, 2.0);
double Q04 = Math.Pow(theta, 3.0) + 5.0 * theta;
double Q14 = -1.0 / 48.0 * (-72.0 * C3 * Math.Pow(sigma, 4.0) + 144.0 * C3 * mu * Math.Pow(sigma, 2.0) + 48.0 * r * C2 * Math.Pow(sigma, 3.0) + Math.Pow(sigma, 6.0) * C1 - 6.0 * Math.Pow(sigma, 4.0) * C1 * mu + 12.0 * Math.Pow(sigma, 2.0) * C1 * Math.Pow(mu, 2.0) - 8.0 * C1 * Math.Pow(mu, 3.0)) / Math.Pow(sigma, 3.0) * theta;
//Set 5 polynomials
double P05 = Math.Pow(theta, 5.0) + 10.0 * Math.Pow(theta, 3.0) + 15.0 * theta;
double P15 = -(r * C3 * sigma - 2.0 * C4 * Math.Pow(sigma, 2.0) + 4.0 * C4 * mu) / sigma * Math.Pow(theta, 3.0) + 1.0 / 4.0 * (-12.0 * r * C3 * Math.Pow(sigma, 2.0) + 24.0 * C4 * Math.Pow(sigma, 3.0) - 48.0 * C4 * sigma * mu - 4.0 * r * C2 * Math.Pow(sigma, 3.0) + 8.0 * r * sigma * C2 * mu + 2.0 * Math.Pow(r, 2.0) * Math.Pow(sigma, 2.0) * C1 + 3.0 * C3 * Math.Pow(sigma, 4.0) - 12.0 * C3 * mu * Math.Pow(sigma, 2.0) + 12.0 * C3 * Math.Pow(mu, 2.0)) / Math.Pow(sigma, 2.0) * theta;
double Q05 = Math.Pow(theta, 4.0) + 9.0 * Math.Pow(theta, 2.0) + 8;
double Q15 = -1.0 / 384.0 * (-Math.Pow(sigma, 8.0) * C1 + 8.0 * Math.Pow(sigma, 6.0) * C1 * mu - 24.0 * Math.Pow(sigma, 4.0) * C1 * Math.Pow(mu, 2.0) + 32.0 * Math.Pow(sigma, 2.0) * C1 * Math.Pow(mu, 3.0) - 16.0 * C1 * Math.Pow(mu, 4.0) - 768.0 * C4 * Math.Pow(sigma, 5.0) + 1536.0 * C4 * Math.Pow(sigma, 3.0) * mu + 384.0 * r * C3 * Math.Pow(sigma, 4.0)) / Math.Pow(sigma, 4.0) * Math.Pow(theta, 2.0) + 1.0 / 384.0 * (-128.0 * r * C3 * Math.Pow(sigma, 5.0) * C2 - 128.0 * Math.Pow(mu, 2.0) * r * C1 * Math.Pow(sigma, 2.0) + 1152.0 * C3 * Math.Pow(mu, 2.0) * Math.Pow(sigma, 2.0) + 32.0 * Math.Pow(sigma, 2.0) * C1 * Math.Pow(mu, 3.0) - 24.0 * Math.Pow(sigma, 4.0) * C1 * Math.Pow(mu, 2.0) + 8.0 * Math.Pow(sigma, 6.0) * C1 * mu - 256.0 * Math.Pow(sigma, 5.0) * r * C2 - 1152.0 * Math.Pow(sigma, 4.0) * C3 * mu - 768.0 * r * C3 * Math.Pow(sigma, 4.0) + 128.0 * Math.Pow(r, 2.0) * C3 * Math.Pow(sigma, 4.0) * C1 - 3072.0 * C4 * Math.Pow(sigma, 3.0) * mu + 256.0 * r * C3 * Math.Pow(sigma, 3.0) * C2 * mu - 16.0 * r * C3 * Math.Pow(sigma, 6.0) * C1 + 64.0 * r * C3 * Math.Pow(sigma, 4.0) * C1 * mu - 64.0 * r * C3 * Math.Pow(sigma, 2.0) * C1 * Math.Pow(mu, 2.0) + 64.0 * Math.Pow(sigma, 4.0) * Math.Pow(r, 2.0) * C1 - 128.0 * sigma * C2 * Math.Pow(mu, 3.0) + 192.0 * Math.Pow(sigma, 3.0) * C2 * Math.Pow(mu, 2.0) - 32.0 * Math.Pow(sigma, 6.0) * r * C1 - 96.0 * Math.Pow(sigma, 5.0) * C2 * mu + 1536.0 * C4 * Math.Pow(sigma, 5.0) + 512.0 * mu * r * C2 * Math.Pow(sigma, 3.0) - 16.0 * C1 * Math.Pow(mu, 4.0) + 288.0 * Math.Pow(sigma, 6.0) * C3 - Math.Pow(sigma, 8.0) * C1 + 16.0 * Math.Pow(sigma, 7.0) * C2 + 128.0 * Math.Pow(sigma, 4.0) * mu * r * C1) / Math.Pow(sigma, 4.0);
// The Black-Scholes American put approximation
cdf = BS.NormCDF(theta);
pdf = BS.NormPDF(theta);
double Price = (C1 * (P01 * cdf + Q01 * pdf) + P11 * cdf + Q11 * pdf) * Math.Pow(T, 0.5)
+ (C2 * (P02 * cdf + Q02 * pdf) + P12 * cdf + Q12 * pdf) * T
+ (C3 * (P03 * cdf + Q03 * pdf) + P13 * cdf + Q13 * pdf) * Math.Pow(T, 1.5)
+ (C4 * (P04 * cdf + Q04 * pdf) + P14 * cdf + Q14 * pdf) * Math.Pow(T, 2.0)
+ (C5 * (P05 * cdf + Q05 * pdf) + P15 * cdf + Q15 * pdf) * Math.Pow(T, 2.5);
return(Price);
}
static void Main(string[] args)
{
// Reproduces Table 2 in Medvedev and Scaillet (2010) for American puts
// under the Black Scholes model
// Option settings
double S = 40.0;
double r = 0.0488;
double q = 0.00;
// Medvedev-Scaillet put option settings
MSset mssettings = new MSset();
mssettings.r = r;
mssettings.q = q;
// Trinomial tree settings
int N = 500;
string PutCall = "P";
string EuroAmer = "A";
// Table 2 settings
double[] sigma = new double[9] {
0.2, 0.2, 0.2, 0.3, 0.3, 0.3, 0.4, 0.4, 0.4
};
double[] T = new double[9] {
1.0 / 12.0, 1.0 / 3.0, 7.0 / 12.0, 1.0 / 12.0, 1.0 / 3.0, 7.0 / 12.0, 1.0 / 12.0, 1.0 / 3.0, 7.0 / 12.0
};
double[,] BSPut = new double[3, 9]; // Black-Scholes
double[,] MSPut = new double[3, 9]; // Medvedev-Scaillet
double[,] BTPut = new double[3, 9]; // Trinomial tree
double SumError = 0.0;
// Settings for the Bisection Method
int MaxIter = 20000;
double tol = 1e-20;
double dt = 1e-10;
double b = 3.0;
// Find the tree and M-S prices
BlackScholesPrice BS = new BlackScholesPrice();
MSPrice MS = new MSPrice();
TrinomialPrice TP = new TrinomialPrice();
double[] K = new double[3] {
35.0, 40.0, 45.0
};
for (int k = 0; k <= 2; k++)
{
for (int i = 0; i <= 8; i++)
{
// Black Scholes European put
BSPut[k, i] = BS.BlackScholes(S, K[k], T[i], r, q, sigma[i], "P");
// Medvedev Scaillet American Put
MSPut[k, i] = MS.MSPriceBS(S, K[k], T[i], sigma[i], r, q, MaxIter, tol, b, dt);
// Trinomial Tree American put
BTPut[k, i] = TP.TrinomialTree(S, K[k], r, q, T[i], sigma[i], N, PutCall, EuroAmer);
SumError += Math.Abs(MSPut[k, i] - BTPut[k, i]);
}
}
// Output the results for K = 35, 40, 45
Console.WriteLine(" ");
Console.WriteLine(" Table 2 of Medvedev and Scaillet (2010)");
Console.WriteLine("----------------------------------------------------------------------------");
Console.WriteLine(" sigma = 0.2 sigma = 0.3 sigma = 0.4");
Console.WriteLine(" --------------------- --------------------- ---------------------");
Console.WriteLine("K = 35 T=1/12 T=1/3 T=7/12 T=1/12 T=1/3 T=7/12 T=1/12 T=1/3 T=7/12 ");
Console.WriteLine("----------------------------------------------------------------------------");
for (int k = 0; k <= 2; k++)
{
Console.WriteLine("EuroPut {0,6:F3} {1,6:F3} {2,6:F3} {3,8:F3} {4,6:F3} {5,6:F3} {6,8:F3} {7,6:F3} {8,7:F3}",
BSPut[k, 0], BSPut[k, 1], BSPut[k, 2], BSPut[k, 3], BSPut[k, 4], BSPut[k, 5], BSPut[k, 6], BSPut[k, 7], BSPut[k, 8]);
Console.WriteLine("MSPut {0,6:F3} {1,6:F3} {2,6:F3} {3,8:F3} {4,6:F3} {5,6:F3} {6,8:F3} {7,6:F3} {8,7:F3}",
MSPut[k, 0], MSPut[k, 1], MSPut[k, 2], MSPut[k, 3], MSPut[k, 4], MSPut[k, 5], MSPut[k, 6], MSPut[k, 7], MSPut[k, 8]);
Console.WriteLine("TreePut {0,6:F3} {1,6:F3} {2,6:F3} {3,8:F3} {4,6:F3} {5,6:F3} {6,8:F3} {7,6:F3} {8,7:F3}",
BTPut[k, 0], BTPut[k, 1], BTPut[k, 2], BTPut[k, 3], BTPut[k, 4], BTPut[k, 5], BTPut[k, 6], BTPut[k, 7], BTPut[k, 8]);
Console.WriteLine("----------------------------------------------------------------------------");
}
Console.WriteLine("EuroPut = Black-Scholes closed form European put price");
Console.WriteLine("MSPut = Fifth-order MS (2001) American put expansion");
Console.WriteLine("TreePut = Trinomial tree American put with {0} steps", N);
Console.WriteLine("Sum of absolute errors {0,8:F5}", SumError);
Console.WriteLine("----------------------------------------------------------------------------");
Console.WriteLine(" ");
}
private void import_bw_DoWork(object sender, DoWorkEventArgs e)
{
curMSset = new MSset();
curMSset.readFile(importDialog.FileName, RegexText.Text);
}
// Derivative of the MS approximation
public double MSPutBSdiff(double y, MSset mssettings, double dy)
{
MSPrice MS = new MSPrice();
return((MS.MSPutBS(y + dy, mssettings) - MS.MSPutBS(y - dy, mssettings)) / 2.0 / dy);
}
