public double WeightedPrice(double thet, double[,] L, double S0, double V0, double K, double r, double q, double Mat, double[] S, double[] V, double[] T, double[,] A, double[,] invA, double[,] B)
{
// Heston Call price using the Weighted Method
// Requires a uniform grid for the stock price, volatility, and maturity
// INPUTS
// thet = theta parameter for the Weighted method
// L = operator matrix for the Heston model
// params = vector of Heston parameters
// S0 = Spot price at which to price the call
// V0 = variance at which to price the call
// K = Strike price
// r = risk free rate
// q = dividend yield
// Mat = maturity
// S = uniform grid for the stock price
// V = uniform grid for the volatility
// T = uniform grid for the maturity
// A = "A" matrix for weighted method
// invA = A inverse
// B = "B" matrix for weighted method
// OUTPUT
// y = 2-D interpolated European Call price
MatrixOps MO = new MatrixOps();
Interpolation IP = new Interpolation();
// Required vector lengths and time increment
int NS = S.Length;
int NV = V.Length;
int NT = T.Length;
double dt = T[1] - T[0];
// Identity matrix
int N = NS * NV;
double[,] I = MO.CreateI(N);
// Initialize the U and u vectors
double[] U = new double[N];
double[] u = new double[N];
// U(0) vector - value of U(T) at maturity
double[] Si = new double[N];
int k = 0;
for (int v = 0; v <= NV - 1; v++)
{
for (int s = 0; s <= NS - 1; s++)
{
Si[k] = S[s];
U[k] = Math.Max(Si[k] - K, 0.0);
k += 1;
}
}
// Loop through the time increments, updating U(t) to U(t+1) at each step
for (int t = 2; t <= NT; t++)
{
for (k = 0; k <= N - 1; k++)
{
u[k] = U[k];
}
if (thet == 0.0)
{
U = MO.MVMult(B, u); // Explicit Method
}
else if (thet == 1.0)
{
U = MO.MVMult(invA, u); // Implicit Method
}
else
{
U = MO.MVMult(invA, MO.MVMult(B, u)); // Weighted Methods
}
}
// Restack the U vector to output a matrix
double[,] UU = new double[NS, NV];
k = 0;
for (int v = 0; v <= NV - 1; v++)
{
for (int s = 0; s <= NS - 1; s++)
{
UU[s, v] = U[k];
k += 1;
}
}
// Interpolate to get the price at S0 and v0
return(IP.interp2(V, S, UU, V0, S0));
}
public LMatrices BuildDerivatives(double[] S, double[] V, double[] T)
{
// Build the first- and second-order derivatives for the "L" operator matrix
// for the Weighted method
// INPUTS
// S = vector for uniform stock price grid
// V = vector for uniform volatility grid
// T = vector for uniform maturity grid
// thet = parameter for the weighted scheme
// OUTPUTS
// Matrices of dimnension N x N (N=NS+NV)
// derS = Matrix for first-order derivative dU/dS
// derSS = Matrix for second-order derivative dU2/dS2
// derV1 = Matrix for first-order derivative dU/dV, kappa*theta portion
// derV2 = Matrix for first-order derivative dU/dV, -kappa*V portion
// derVV = Matrix for first-order derivative dU2/dV2
// derSV = Matrix for first-order derivative dU2/dSdV
// R = Matrix for r*S(v,t) portion of the PDE
Interpolation IP = new Interpolation();
// Length of stock price, volatility, and maturity
int NS = S.Length;
int NV = V.Length;
int NT = T.Length;
double Smin = S[0]; double Smax = S[NS - 1];
double Vmin = V[0]; double Vmax = V[NV - 1];
double Tmin = T[0]; double Tmax = T[NT - 1];
// Increment for Stock Price, Volatility, and Maturity
double ds = (Smax - Smin) / Convert.ToDouble(NS - 1);
double dv = (Vmax - Vmin) / Convert.ToDouble(NV - 1);
double dt = (Tmax - Tmin) / Convert.ToDouble(NT - 1);
// Preliminary quantities
// Size of the U(t) vector and L matrix
int N = NS * NV;
// The vectors for S and V, stacked
double[] Si = new double[N];
double[] Vi = new double[N];
int k = 0;
for (int v = 0; v <= NV - 1; v++)
{
for (int s = 0; s <= NS - 1; s++)
{
Si[k] = S[s];
Vi[k] = V[v];
k += 1;
}
}
// Identification of the boundary points
int[] VminB = new int[N];
int[] VmaxB = new int[N];
int[] SminB = new int[N];
int[] SmaxB = new int[N];
// Vmin and Vmax
for (int v = 0; v <= NS - 2; v++)
{
VminB[v] = 1;
}
for (int v = N - NS + 1; v <= N - 2; v++)
{
VmaxB[v] = 1;
}
VmaxB[N - 1] = 1;
// Smin and Smax
k = 0;
for (int v = 0; v <= NV - 1; v++)
{
for (int s = 0; s <= NS - 1; s++)
{
if (s == 0)
{
SminB[k] = 1;
}
else if (s == NS - 1)
{
SmaxB[k] = 1;
}
k += 1;
}
}
SminB[0] = 0;
SmaxB[NS - 1] = 0;
SmaxB[N - 1] = 0;
// Identification of the non-boundary points
int[] NB = new int[N];
for (k = 0; k <= N - 1; k++)
{
if (SminB[k] == 0 & SmaxB[k] == 0 & VminB[k] == 0 & VmaxB[k] == 0)
{
NB[k] = 1;
}
}
// Forward, backward and central differences for S
int[] Cs = new int[N];
int[] Fs = new int[N];
int[] Bs = new int[N];
k = 0;
for (int v = 0; v <= NV - 2; v++)
{
for (int s = 0; s <= NS - 1; s++)
{
if (s == 1)
{
Fs[k] = 1;
}
else if (s == NS - 2)
{
Bs[k] = 1;
}
else if (s >= 2 & s <= NS - 3)
{
Cs[k] = 1;
}
k += 1;
}
}
Fs[1] = 0;
Bs[NS - 2] = 0;
for (k = 2; k <= NS - 3; k++)
{
Cs[k] = 0;
}
// Forward, backward and central differences for V
int[] Cv = new int[N];
int[] Fv = new int[N];
int[] Bv = new int[N];
for (k = NS + 1; k <= 2 * NS - 2; k++)
{
Fv[k] = 1;
}
for (k = (NV - 2) * NS + 1; k <= (NV - 1) * NS - 2; k++)
{
Bv[k] = 1;
}
k = 0;
for (int v = 0; v <= NV - 3; v++)
{
for (int s = 0; s <= NS - 1; s++)
{
if (s >= 1 & s <= NS - 2)
{
Cv[k] = 1;
}
k += 1;
}
}
for (k = 0; k <= 2 * NS - 1; k++)
{
Cv[k] = 0;
}
// Central difference for SV-derivatives
int[] Csv = new int[N];
k = 0;
for (int v = 0; v <= NV - 2; v++)
{
for (int s = 0; s <= NS - 1; s++)
{
if (s >= 1 & s <= NS - 2)
{
Csv[k] = 1;
}
k += 1;
}
}
for (k = 0; k <= NS - 1; k++)
{
Csv[k] = 0;
}
// Create the matrices for the derivatives
double[,] derS = new double[N, N];
double[,] derSS = new double[N, N];
double[,] derV1 = new double[N, N];
double[,] derV2 = new double[N, N];
double[,] derVV = new double[N, N];
double[,] derSV = new double[N, N];
double[,] R = new double[N, N];
for (int i = 0; i <= N - 1; i++)
{
for (int j = 0; j <= N - 1; j++)
{
if (i == j)
{
R[i, j] = 1.0;
}
}
}
int M;
int[] I;
// Create the matrices for the derivatives
// NON BOUNDARY POINTS ----------------------------------
I = IP.Find(Cs);
M = I.Length;
for (k = 0; k <= M - 1; k++)
{ // Central differences
derS[I[k], I[k] - 1] = -0.5 / ds * Si[I[k]]; // U(s-1,v)
derS[I[k], I[k]] = 0.0; // U(s,v)
derS[I[k], I[k] + 1] = 0.5 / ds * Si[I[k]]; // U(s+1,v)
derSS[I[k], I[k] - 1] = 1.0 / ds / ds * Vi[I[k]] * Si[I[k]] * Si[I[k]]; // U(s-1,v)
derSS[I[k], I[k]] = -2.0 / ds / ds * Vi[I[k]] * Si[I[k]] * Si[I[k]]; // U(s,v)
derSS[I[k], I[k] + 1] = 1.0 / ds / ds * Vi[I[k]] * Si[I[k]] * Si[I[k]]; // U(s+1,v)
}
I = IP.Find(Fs);
M = I.Length;
for (k = 0; k <= M - 1; k++)
{ // Forward differences
derS[I[k], I[k]] = -1.5 / ds * Si[I[k]]; // U(s,v)
derS[I[k], I[k] + 1] = 2.0 / ds * Si[I[k]]; // U(s+1,v)
derS[I[k], I[k] + 2] = -1.0 / ds * Si[I[k]]; // U(s+2,v)
derSS[I[k], I[k]] = 1.0 / ds / ds * Vi[I[k]] * Si[I[k]] * Si[I[k]]; // U(s,v)
derSS[I[k], I[k] + 1] = -2.0 / ds / ds * Vi[I[k]] * Si[I[k]] * Si[I[k]]; // U(s+1,v)
derSS[I[k], I[k] + 2] = 1.0 / ds / ds * Vi[I[k]] * Si[I[k]] * Si[I[k]]; // U(s+2,v)
}
I = IP.Find(Bs);
M = I.Length;
for (k = 0; k <= M - 1; k++)
{ // Backward differences
derS[I[k], I[k] - 2] = -0.5 / ds * Si[I[k]]; // U(s-2,v)
derS[I[k], I[k] - 1] = -2.0 / ds * Si[I[k]]; // U(s-1,v)
derS[I[k], I[k]] = 1.5 / ds * Si[I[k]]; // U(s,v)
derSS[I[k], I[k] - 2] = 1.0 / ds / ds * Vi[I[k]] * Si[I[k]] * Si[I[k]]; // U(s-2,v)
derSS[I[k], I[k] - 1] = -2.0 / ds / ds * Vi[I[k]] * Si[I[k]] * Si[I[k]]; // U(s-1,v)
derSS[I[k], I[k]] = 1.0 / ds / ds * Vi[I[k]] * Si[I[k]] * Si[I[k]]; // U(s,v)
}
// Create the matrix for V-derivatives
I = IP.Find(Cv);
M = I.Length;
for (k = 0; k <= M - 1; k++)
{ // Central differences
derV1[I[k], I[k] - NS] = -0.5 / dv; // U(s,v-1)
derV1[I[k], I[k]] = 0.0; // U(s,v)
derV1[I[k], I[k] + NS] = 0.5 / dv; // U(s,v+1)
derV2[I[k], I[k] - NS] = -0.5 / dv * Vi[I[k]]; // U(s,v-1)
derV2[I[k], I[k]] = 0.0; // U(s,v)
derV2[I[k], I[k] + NS] = 0.5 / dv * Vi[I[k]]; // U(s,v+1)
derVV[I[k], I[k] - NS] = 1.0 / dv / dv * Vi[I[k]]; // U(s,v-1)
derVV[I[k], I[k]] = -2.0 / dv / dv * Vi[I[k]]; // U(s,v)
derVV[I[k], I[k] + NS] = 1.0 / dv / dv * Vi[I[k]]; // U(s,v+1)
}
I = IP.Find(Fv);
M = I.Length;
for (k = 0; k <= M - 1; k++)
{ // Forward differences
derV1[I[k], I[k]] = -1.5 / dv; // U(s,v)
derV1[I[k], I[k] + NS] = 2.0 / dv; // U(s,v+1)
derV1[I[k], I[k] + 2 * NS] = -1.0 / dv; // U(s,v+2)
derV2[I[k], I[k]] = -1.5 / dv * Vi[I[k]]; // U(s,v)
derV2[I[k], I[k] + NS] = 2.0 / dv * Vi[I[k]]; // U(s,v+1)
derV2[I[k], I[k] + 2 * NS] = -1.0 / dv * Vi[I[k]]; // U(s,v+2)
derVV[I[k], I[k]] = 1.0 / dv / dv * Vi[I[k]]; // U(s,v)
derVV[I[k], I[k] + NS] = -2.0 / dv / dv * Vi[I[k]]; // U(s,v+1)
derVV[I[k], I[k] + 2 * NS] = 1.0 / dv / dv * Vi[I[k]]; // U(s,v+2)
}
I = IP.Find(Bv);
M = I.Length;
for (k = 0; k <= M - 1; k++)
{ // Backward differences
derV1[I[k], I[k] - 2 * NS] = -0.5 / dv; // U(s,v-2)
derV1[I[k], I[k] - NS] = -2.0 / dv; // U(s,v-1)
derV1[I[k], I[k]] = 1.5 / dv; // U(s,v)
derV2[I[k], I[k] - 2 * NS] = -0.5 / dv * Vi[I[k]]; // U(s,v-2)
derV2[I[k], I[k] - NS] = -2.0 / dv * Vi[I[k]]; // U(s,v-1)
derV2[I[k], I[k]] = 1.5 / dv * Vi[I[k]]; // U(s,v)
derVV[I[k], I[k] - 2 * NS] = 1.0 / dv / dv * Vi[I[k]]; // U(s,v-2)
derVV[I[k], I[k] - NS] = -2.0 / dv / dv * Vi[I[k]]; // U(s,v-1)
derVV[I[k], I[k]] = 1.0 / dv / dv * Vi[I[k]]; // U(s,v)
}
// Create the matrix for SV-derivatives
I = IP.Find(Csv);
M = I.Length;
for (k = 0; k <= M - 1; k++)
{
derSV[I[k], I[k] + NS + 1] = 1.0 / (4.0 * ds * dv) * Vi[I[k]] * Si[I[k]]; // U(s+1,v+1)
derSV[I[k], I[k] + NS - 1] = -1.0 / (4.0 * ds * dv) * Vi[I[k]] * Si[I[k]]; // U(s-1,v+1)
derSV[I[k], I[k] - NS - 1] = 1.0 / (4.0 * ds * dv) * Vi[I[k]] * Si[I[k]]; // U(s-1,v-1)
derSV[I[k], I[k] - NS + 1] = -1.0 / (4.0 * ds * dv) * Vi[I[k]] * Si[I[k]]; // U(s+1,v-1)
}
// BOUNDARY POINTS ----------------------------------
// Boundary for Smin
I = IP.Find(SminB);
M = I.Length;
for (k = 0; k <= M - 1; k++)
{
derS[I[k], I[k]] = 0.0;
derSS[I[k], I[k]] = 0.0;
derV1[I[k], I[k]] = 0.0;
derV2[I[k], I[k]] = 0.0;
derVV[I[k], I[k]] = 0.0;
derSV[I[k], I[k]] = 0.0;
R[I[k], I[k]] = 0.0;
}
// Boundary condition for Smax
I = IP.Find(SmaxB);
M = I.Length;
for (k = 0; k <= M - 1; k++)
{
derS[I[k], I[k]] = Si[I[k]];
derSS[I[k], I[k]] = 0.0;
derSV[I[k], I[k]] = 0.0; // Central difference
derV1[I[k], I[k] - NS] = -0.5 / dv; // U(s,v-1)
derV1[I[k], I[k]] = 0; // U(s,v)
derV1[I[k], I[k] + NS] = 0.5 / dv; // U(s,v+1)
derV2[I[k], I[k] - NS] = -0.5 / dv * Vi[I[k]]; // U(s,v-1)
derV2[I[k], I[k]] = 0.0; // U(s,v)
derV2[I[k], I[k] + NS] = 0.5 / dv * Vi[I[k]]; // U(s,v+1)
derVV[I[k], I[k] - NS] = 1.0 / dv / dv * Vi[I[k]]; // U(s,v-1)
derVV[I[k], I[k]] = -2.0 / dv / dv * Vi[I[k]]; // U(s,v)
derVV[I[k], I[k] + NS] = 1.0 / dv / dv * Vi[I[k]]; // U(s,v+1)
}
// Boundary condition for Vmax. Only the LS submatrix is non-zero.
I = IP.Find(VmaxB);
M = I.Length;
for (k = 0; k <= M - 1; k++)
{
derS[I[k], I[k]] = Si[I[k]];
}
// Boundary condition for Vmin
// First row
derS[0, 0] = -1.5 / ds * Si[0]; // U(s,v)
derS[0, 1] = 2.0 / ds * Si[0]; // U(s+1,v)
derS[0, 2] = -1.0 / ds * Si[0]; // U(s+2,v)
derV1[0, 0] = -1.5 / dv; // U(s,v)
derV1[0, NS] = 2.0 / dv; // U(s,v+1)
derV1[0, 2 * NS] = -1.0 / dv; // U(s,v+2)
// Last row
derS[N - 2, N - 4] = -0.5 / ds * Si[N - 2]; // U(s-2,v)
derS[N - 2, N - 3] = -2.0 / ds * Si[N - 2]; // U(s-1,v)
derS[N - 2, N - 2] = 1.5 / ds * Si[N - 2]; // U(s,v)
derV1[NS - 2, NS - 2 + 2 * NS] = -0.5 / dv * Vi[NS - 2]; // U(s,v+2)
derV1[NS - 2, NS - 2 + NS] = 2.0 / dv * Vi[NS - 2]; // U(s,v+1)
derV1[NS - 2, NS - 2] = -1.5 / dv * Vi[NS - 2]; // U(s,v)
// Other rows
for (int i = 1; i <= NS - 3; i++)
{
derS[i, i - 1] = -0.5 / ds * Si[i]; // U(s-1,v)
derS[i, i] = 0.0; // U(s,v)
derS[i, i + 1] = 0.5 / ds * Si[i]; // U(s+1,v)
derV1[i, i] = -1.5 / dv; // U(s,v)
derV1[i, i + NS] = 2.0 / dv; // U(s,v+1)
derV1[i, i + 2 * NS] = -1.0 / dv; // U(s,v+2)
}
// Output the sub-matrices
LMatrices LMat;
LMat.derS = derS;
LMat.derSS = derSS;
LMat.derV1 = derV1;
LMat.derV2 = derV2;
LMat.derVV = derVV;
LMat.derSV = derSV;
LMat.R = R;
return(LMat);
}
