public Uu HestonExplicitPDENonUniformGrid(HParam param, double K, double r, double q, double[] S, double[] V, double[] T)
{
// Finite differences for the Heston PDE for a European Call
// Uses uneven grid sizes
// In 'T Hout and Foulon "ADI Finite Difference Schemes for Option Pricing
// in the Heston Modelo with Correlation" Int J of Num Analysis and Modeling, 2010.
// INPUTS
// params = 6x1 vector of Heston parameters
// K = Strike price
// r = risk free rate
// q = Dividend
// S = vector for stock price grid
// V = vector for volatility grid
// T = vector for maturity grid
// OUTPUTS
// U = U(S,v) 2-D array of size (nS+1)x(nV+1)x(nT+1) for the call price
// Heston parameters
double kappa = param.kappa;
double theta = param.theta;
double sigma = param.sigma;
double v0 = param.v0;
double rho = param.rho;
double lambda = param.lambda;
// Grid measurements
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];
double dt = (Tmax - Tmin) / Convert.ToDouble(NT - 1);
// Initialize the 2-D grid with zeros
double[,] U = new double[NS, NV];
// Temporary grid for previous time steps
double[,] u = new double[NS, NV];
//// Solve the PDE
// Round each value of U(S,v) at each step
// Boundary condition for t=maturity
for (int s = 0; s <= NS - 1; s++)
{
for (int v = 0; v <= NV - 1; v++)
{
U[s, v] = Math.Max(S[s] - K, 0.0);
}
}
double LHS, derV, derS, derSS, derVV, derSV, L;
for (int t = 0; t <= NT - 2; t++)
{
// Boundary condition for Smin and Smax
for (int v = 0; v <= NV - 2; v++)
{
U[0, v] = 0.0;
U[NS - 1, v] = Math.Max(0.0, Smax - K);
}
// Boundary condition for Vmax
for (int s = 0; s <= NS - 1; s++)
{
U[s, NV - 1] = Math.Max(0.0, S[s] - K);
}
// Update the temporary grid u(s,t) with the boundary conditions
for (int s = 0; s <= NS - 1; s++)
{
for (int v = 0; v <= NV - 1; v++)
{
u[s, v] = U[s, v];
}
}
// Boundary condition for Vmin.
// Previous time step values are in the temporary grid u(s,t)
for (int s = 1; s <= NS - 2; s++)
{
derV = (u[s, 1] - u[s, 0]) / (V[1] - V[0]); // Forward difference
derS = (u[s + 1, 0] - u[s - 1, 0]) / (S[s + 1] - S[s - 1]); // Central difference
LHS = -r * u[s, 0] + (r - q) * S[s] * derS + kappa * theta * derV;
U[s, 0] = LHS * dt + u[s, 0];
}
// Update the temporary grid u(s,t) with the boundary conditions
for (int s = 0; s <= NS - 1; s++)
{
for (int v = 0; v <= NV - 1; v++)
{
u[s, v] = U[s, v];
}
}
// Interior points of the grid (non boundary).
// Previous time step values are in the temporary grid u(s,t)
for (int s = 1; s <= NS - 2; s++)
{
for (int v = 1; v <= NV - 2; v++)
{
derS = (u[s + 1, v] - u[s - 1, v]) / (S[s + 1] - S[s - 1]); // Central difference for dU/dS
derV = (u[s, v + 1] - u[s, v - 1]) / (V[v + 1] - V[v - 1]); // Central difference for dU/dV
derSS = ((u[s + 1, v] - u[s, v]) / (S[s + 1] - S[s]) - (u[s, v] - u[s - 1, v]) / (S[s] - S[s - 1])) / (S[s + 1] - S[s]); // d2U/dS2
derVV = ((u[s, v + 1] - u[s, v]) / (V[v + 1] - V[v]) - (u[s, v] - u[s, v - 1]) / (V[v] - V[v - 1])) / (V[v + 1] - V[v]); // d2U/dV2
derSV = (u[s + 1, v + 1] - u[s - 1, v + 1] - U[s + 1, v - 1] + U[s - 1, v - 1]) / (S[s + 1] - S[s - 1]) / (V[v + 1] - V[v - 1]); // d2U/dSdV
L = 0.5 * V[v] * S[s] * S[s] * derSS + rho * sigma * V[v] * S[s] * derSV
+ 0.5 * sigma * sigma * V[v] * derVV - r * u[s, v]
+ (r - q) * S[s] * derS + kappa * (theta - V[v]) * derV;
U[s, v] = L * dt + u[s, v];
}
}
}
Uu output = new Uu();
output.bigU = U;
output.smallU = u;
return(output);
}
static void Main(string[] args)
{
// Classes
Interpolation IP = new Interpolation();
ExplicitPDE EP = new ExplicitPDE();
// Number of grid points for the stock, volatility, and maturity
int nS = 79; // Stock price
int nV = 39; // Volatility
int nT = 3000; // Maturity
// Option flavor
string PutCall = "P";
string EuroAmer = "A";
// Strike price, risk free rate, dividend yield, and maturity
// True prices from Clarke and Parrott (1999)
double K = 10.0;
double r = 0.1;
double q = 0.00;
double Mat = 0.25;
double[] S0 = new double[5] {
8.0, 9.0, 10.0, 11.0, 12.0
};
double[] TruePrice = new double[5] {
2.0000, 1.107641, 0.520030, 0.213668, 0.082036
};
// Heston parameters
HParam param;
param.kappa = 5;
param.theta = 0.16;
param.sigma = 0.9;
param.rho = 0.1;
param.v0 = 0.0625;
param.lambda = 0;
// Settings for the European option price calculation
// 32-point Gauss-Laguerre Abscissas and weights
double[] X = new Double[32];
double[] W = new Double[32];
using (TextReader reader = File.OpenText("../../GaussLaguerre32.txt"))
for (int k = 0; k <= 31; k++)
{
string text = reader.ReadLine();
string[] bits = text.Split(' ');
X[k] = double.Parse(bits[0]);
W[k] = double.Parse(bits[1]);
}
// Minimum and maximum values for the Stock Price, Volatility, and Maturity
double Smin = 0.0; double Smax = 2.0 * K;
double Vmin = 0.0; double Vmax = 0.5;
double Tmin = 0.0; double Tmax = Mat;
// The maturity time increment and grid
double dt = (Tmax - Tmin) / Convert.ToDouble(nT);
double[] T = new double[nT + 1];
for (int i = 0; i <= nT; i++)
{
T[i] = Convert.ToDouble(i) * dt;
}
// Obtain the prices by 2-D interpolation and the errors compared to Clarke and Parrott
// Also obtain the Greeks
double V0 = param.v0;
int N = S0.Length;
double[] PDEPrice = new double[N];
double D1, D2, V1, V2, dCdv0, C1, C2, C3, C4, dC2, T1, T2;
double[] DeltaPDE = new double[N];
double[] GammaPDE = new double[N];
double[] Vega1PDE = new double[N];
double[] VannaPDE = new double[N];
double[] VolgaPDE = new double[N];
double[] ThetaPDE = new double[N];
double dS, dv, c, dz, d, dn;
double[] z = new double[nS + 1];
double[] S = new double[nS + 1];
double[] n = new double[nV + 1];
double[] V = new double[nV + 1];
for (int k = 0; k <= N - 1; k++)
{
//// Pricing Using a Non-Uniform Grid
// The stock price grid
c = S0[k] / 5.0;
dz = 1.0 / nS * (IP.aSinh((Smax - S0[k]) / c) - IP.aSinh(-S0[k] / c));
for (int i = 0; i <= nS; i++)
{
z[i] = IP.aSinh(-S0[k] / c) + Convert.ToDouble(i) * dz;
S[i] = S0[k] + c * Math.Sinh(z[i]);
}
S[0] = 0;
// The volatility grid
d = Vmax / 500.0;
dn = IP.aSinh(Vmax / d) / nV;
for (int j = 0; j <= nV; j++)
{
n[j] = Convert.ToDouble(j) * dn;
V[j] = d * Math.Sinh(n[j]);
}
// Solve the PDE and return U(S,v,T) and U(S,v,T-dt);
Uu output = EP.HestonExplicitPDENonUniformGrid(param, K, r, q, S, V, T, PutCall, EuroAmer);
double[,] U = output.bigU;
double[,] u = output.smallU;
// Price, Delta, Gamma
dS = 0.01 * S0[k];
dv = 0.01 * V0;
PDEPrice[k] = IP.interp2(V, S, U, V0, S0[k]);
D1 = IP.interp2(V, S, U, V0, S0[k] + dS);
D2 = IP.interp2(V, S, U, V0, S0[k] - dS);
DeltaPDE[k] = (D1 - D2) / 2.0 / dS;
GammaPDE[k] = (D1 - 2.0 * PDEPrice[k] + D2) / dS / dS;
// Vega #1
V1 = IP.interp2(V, S, U, V0 + dv, S0[k]);
V2 = IP.interp2(V, S, U, V0 - dv, S0[k]);
dCdv0 = (V1 - V2) / 2.0 / dv;
Vega1PDE[k] = dCdv0 * 2.0 * Math.Sqrt(V0);
// Vanna and volga
C1 = IP.interp2(V, S, U, V0 + dv, S0[k] + dS);
C2 = IP.interp2(V, S, U, V0 - dv, S0[k] + dS);
C3 = IP.interp2(V, S, U, V0 + dv, S0[k] - dS);
C4 = IP.interp2(V, S, U, V0 - dv, S0[k] - dS);
VannaPDE[k] = (C1 - C2 - C3 + C4) / 4.0 / dv / dS * 2.0 * Math.Sqrt(V0);
dC2 = (V1 - 2 * PDEPrice[k] + V2) / dv / dv;
VolgaPDE[k] = 4.0 * Math.Sqrt(V0) * (dC2 * Math.Sqrt(V0) + Vega1PDE[k] / 4.0 / V0);
// Theta
T1 = IP.interp2(V, S, U, V0, S0[k]); // U(S,v,T)
T2 = IP.interp2(V, S, u, V0, S0[k]); // U(s,v,T-dt)
ThetaPDE[k] = -(T1 - T2) / dt;
}
// Output the results
Console.WriteLine("Stock price grid size {0}", nS + 1);
Console.WriteLine("Volatility grid size {0}", nV + 1);
Console.WriteLine("Number of time steps {0}", nT);
Console.WriteLine("--------------------------------------------------------------------------");
Console.WriteLine("Spot Price Delta Gamma Vega1 Vanna Volga Theta");
Console.WriteLine("--------------------------------------------------------------------------");
for (int k = 0; k <= N - 1; k++)
{
Console.WriteLine(" {0,2:F0} {1,10:F4} {2,10:F4} {3,8:F4} {4,8:F4} {5,8:F4} {6,8:F4} {7,8:F4}",
S0[k], PDEPrice[k], DeltaPDE[k], GammaPDE[k], Vega1PDE[k], VannaPDE[k], VolgaPDE[k], ThetaPDE[k]);
}
Console.WriteLine("--------------------------------------------------------------------------");
}
static void Main(string[] args)
{
// Classes
Interpolation IP = new Interpolation();
ExplicitPDE EP = new ExplicitPDE();
HestonPrice HP = new HestonPrice();
// Illustration of pricing using uniform and non-uniform grids
// Strike price, risk free rate, dividend yield, and maturity
double K = 100.0;
double r = 0.02;
double q = 0.0;
double Mat = 0.15;
// Heston parameters
HParam param;
param.kappa = 1.5;
param.theta = 0.04;
param.sigma = 0.3;
param.rho = -0.9;
param.v0 = 0.05;
param.lambda = 0.0;
// Minimum and maximum values for the Stock Price, Volatility, and Maturity
double Smin = 0.0; double Smax = 2.0 * K;
double Vmin = 0.0; double Vmax = 0.5;
double Tmin = 0.0; double Tmax = Mat;
// Number of grid points for the stock, volatility, and maturity
int nS = 79; // Stock price
int nV = 39; // Volatility
int nT = 3000; // Maturity
// The maturity time increment and grid
double dt = (Tmax - Tmin) / Convert.ToDouble(nT);
double[] T = new double[nT + 1];
for (int i = 0; i <= nT; i++)
{
T[i] = Convert.ToDouble(i) * dt;
}
//// Pricing Using a Non-Uniform Grid
// The stock price grid
double c = K / 5.0;
double dz = 1.0 / nS * (IP.aSinh((Smax - K) / c) - IP.aSinh(-K / c));
double[] z = new double[nS + 1];
double[] S = new double[nS + 1];
for (int i = 0; i <= nS; i++)
{
z[i] = IP.aSinh(-K / c) + Convert.ToDouble(i) * dz;
S[i] = K + c * Math.Sinh(z[i]);
}
S[0] = 0;
// The volatility grid
double d = Vmax / 500.0;
double dn = IP.aSinh(Vmax / d) / nV;
double[] n = new double[nV + 1];
double[] V = new double[nV + 1];
for (int j = 0; j <= nV; j++)
{
n[j] = Convert.ToDouble(j) * dn;
V[j] = d * Math.Sinh(n[j]);
}
// Solve the PDE and return U(S,v,T) and U(S,v,T-dt)
Uu output = EP.HestonExplicitPDENonUniformGrid(param, K, r, q, S, V, T);
double[,] U = output.bigU;
double[,] u = output.smallU;
// Settings for the option price calculation
// 32-point Gauss-Laguerre Abscissas and weights
double[] X = new Double[32];
double[] W = new Double[32];
using (TextReader reader = File.OpenText("../../GaussLaguerre32.txt"))
for (int k = 0; k <= 31; k++)
{
string text = reader.ReadLine();
string[] bits = text.Split(' ');
X[k] = double.Parse(bits[0]);
W[k] = double.Parse(bits[1]);
}
// Values
double S0 = 101.52;
double V0 = 0.05412;
int trap = 1;
string PutCall = "C";
// The closed form price and finite difference Greeks
param.v0 = V0;
double PriceClosed = HP.HestonPriceGaussLaguerre(param, S0, K, r, q, Mat, trap, PutCall, X, W);
double dS = 1.0;
double dV = 1e-2;
// Delta
double D1 = HP.HestonPriceGaussLaguerre(param, S0 + dS, K, r, q, Mat, trap, PutCall, X, W);
double D2 = HP.HestonPriceGaussLaguerre(param, S0 - dS, K, r, q, Mat, trap, PutCall, X, W);
double DeltaFD = (D1 - D2) / 2.0 / dS;
// Gamma
double GammaFD = (D1 - 2.0 * PriceClosed + D2) / dS / dS;
// Vega #1
param.v0 = V0 + dV;
double V1 = HP.HestonPriceGaussLaguerre(param, S0, K, r, q, Mat, trap, PutCall, X, W);
param.v0 = V0 - dV;
double V2 = HP.HestonPriceGaussLaguerre(param, S0, K, r, q, Mat, trap, PutCall, X, W);
double Vega1FD = (V1 - V2) / 2.0 / dV * 2.0 * Math.Sqrt(V0);
// Vanna
param.v0 = V0 + dV;
double C1 = HP.HestonPriceGaussLaguerre(param, S0 + dS, K, r, q, Mat, trap, PutCall, X, W);
double C3 = HP.HestonPriceGaussLaguerre(param, S0 - dS, K, r, q, Mat, trap, PutCall, X, W);
param.v0 = V0 - dV;
double C2 = HP.HestonPriceGaussLaguerre(param, S0 + dS, K, r, q, Mat, trap, PutCall, X, W);
double C4 = HP.HestonPriceGaussLaguerre(param, S0 - dS, K, r, q, Mat, trap, PutCall, X, W);
double VannaFD = (C1 - C2 - C3 + C4) / 4.0 / dV / dS * 2.0 * Math.Sqrt(V0);
param.v0 = V0;
// Volga
double dC2 = (V1 - 2.0 * PriceClosed + V2) / (dV * dV);
double VolgaFD = 4.0 * Math.Sqrt(V0) * (dC2 * Math.Sqrt(V0) + Vega1FD / 4.0 / V0);
// Theta
double T1 = HP.HestonPriceGaussLaguerre(param, S0, K, r, q, Mat, trap, PutCall, X, W);
double T2 = HP.HestonPriceGaussLaguerre(param, S0, K, r, q, Mat - dt, trap, PutCall, X, W);
double ThetaFD = -(T1 - T2) / dt;
// The PDE price and Greeks
double PricePDE = IP.interp2(V, S, U, V0, S0);
// Delta
D1 = IP.interp2(V, S, U, V0, S0 + dS);
D2 = IP.interp2(V, S, U, V0, S0 - dS);
double DeltaPDE = (D1 - D2) / 2.0 / dS;
// Gamma
double GammaPDE = (D1 - 2.0 * PricePDE + D2) / dS / dS;
// Vega #1
V1 = IP.interp2(V, S, U, V0 + dV, S0);
V2 = IP.interp2(V, S, U, V0 - dV, S0);
double Vega1PDE = (V1 - V2) / 2.0 / dV * 2.0 * Math.Sqrt(V0);
// Vanna
C1 = IP.interp2(V, S, U, V0 + dV, S0 + dS);
C2 = IP.interp2(V, S, U, V0 - dV, S0 + dS);
C3 = IP.interp2(V, S, U, V0 + dV, S0 - dS);
C4 = IP.interp2(V, S, U, V0 - dV, S0 - dS);
double VannaPDE = (C1 - C2 - C3 + C4) / 4.0 / dV / dS * 2.0 * Math.Sqrt(V0);
// Volga
dC2 = (V1 - 2.0 * PricePDE + V2) / (dV * dV);
double VolgaPDE = 4 * Math.Sqrt(V0) * (dC2 * Math.Sqrt(V0) + Vega1PDE / 4.0 / V0);
// Theta
T1 = IP.interp2(V, S, U, V0, S0);
T2 = IP.interp2(V, S, u, V0, S0);
double ThetaPDE = -(T1 - T2) / dt;
// Output the results
Console.WriteLine("Stock price grid size {0}", nS + 1);
Console.WriteLine("Volatility grid size {0}", nV + 1);
Console.WriteLine("Number of time steps {0}", nT);
Console.WriteLine("---------------------------------------");
Console.WriteLine("Greek PDE FiniteDiff");
Console.WriteLine("---------------------------------------");
Console.WriteLine("Price {0,10:F5} {1,12:F5} ", PricePDE, PriceClosed);
Console.WriteLine("Delta {0,10:F5} {1,12:F5} ", DeltaPDE, DeltaFD);
Console.WriteLine("Gamma {0,10:F5} {1,12:F5} ", GammaPDE, GammaFD);
Console.WriteLine("Vega #1 {0,10:F5} {1,12:F5} ", Vega1PDE, Vega1FD);
Console.WriteLine("Vanna {0,10:F5} {1,12:F5} ", VannaPDE, VannaFD);
Console.WriteLine("Volga {0,10:F5} {1,12:F5} ", VolgaPDE, VolgaFD);
Console.WriteLine("Theta {0,10:F5} {1,12:F5} ", ThetaPDE, ThetaFD);
Console.WriteLine("---------------------------------------");
}
