// Longstaff-Schwartz for American puts and calls
// Also returns Euro price
public double[] HestonLSM(double[,] S, double K, double r, double q, double T, int NT, int NS, string PutCall)
{
Regression RE = new Regression();
// Time increment
double dt = T / Convert.ToDouble(NT);
// Initialize the Cash Flows.
double[,] CF = new double[NS, NT];
// Set the last cash flows to the intrinsic value.
for (int s = 0; s <= NS - 1; s++)
{
if (PutCall == "P")
{
CF[s, NT - 1] = Math.Max(K - S[s, NT - 1], 0.0);
}
else if (PutCall == "C")
{
CF[s, NT - 1] = Math.Max(S[s, NT - 1] - K, 0.0);
}
}
// European price
double EuroPrice = 0.0;
for (int s = 0; s <= NS - 1; s++)
{
EuroPrice += Math.Exp(-r * T) * CF[s, NT - 1] / Convert.ToDouble(NS);
}
// Work backwards through the stock prices until time t=2.
// We could work through to time t=1 but the regression will not be
// of full rank at time 1, so this is cleaner.
for (int t = NT - 2; t >= 1; t--)
{
// Indices for stock paths in-the-money at time t
int[] I = new int[NS];
for (int s = 0; s <= NS - 1; s++)
{
I[s] = 0;
if (((PutCall == "P") & (S[s, t] < K)) | ((PutCall == "C") & (S[s, t] > K)))
{
I[s] = 1;
}
}
// Stock paths in-the-money at time t
int NI = 0;
List <double> X = new List <double>();
List <int> Xi = new List <int>();
for (int s = 0; s <= NS - 1; s++)
{
if (I[s] == 1)
{
X.Add(S[s, t]);
Xi.Add(s);
NI += 1;
}
}
// Cash flows at time t+1, discounted one period
double[] Y = new double[NI];
for (int s = 0; s <= NI - 1; s++)
{
Y[s] = CF[Xi[s], t + 1] * Math.Exp(-r * dt);
}
// Design matrix for regression to predict cash flows
double[,] Z = new double[NI, 3];
for (int s = 0; s <= NI - 1; s++)
{
Z[s, 0] = 1.0;
Z[s, 1] = (1.0 - X[s]);
Z[s, 2] = (2.0 - 4.0 * X[s] - X[s] * X[s]) / 2.0;
}
// Regression parameters and predicted cash flows
double[] beta = RE.Beta(Z, Y);
double[] PredCF = RE.MVMult(Z, beta);
// Indices for stock paths where immediate exercise is optimal
// J[s] contains the path number
// E[s] contains the
List <int> E = new List <int>();
List <int> J = new List <int>();
int NE = 0;
for (int s = 0; s <= NI - 1; s++)
{
if ((PutCall == "P" & (K - X[s] > 0 & K - X[s] > PredCF[s])) |
(PutCall == "C" & (X[s] - K > 0 & X[s] - K > PredCF[s])))
{
J.Add(s);
E.Add(Xi[s]);
NE += 1;
}
}
// All other paths --> Continuation is optimal
int[] All = new int[NS];
for (int k = 0; k <= NS - 1; k++)
{
All[k] = k;
}
// C contains indices for continuation paths
IEnumerable <int> C = All.Except(E);
int[] Co = C.ToArray();
int NC = Co.Length;
// Replace cash flows with exercise value where exercise is optimal
for (int s = 0; s <= NE - 1; s++)
{
if (PutCall == "P")
{
CF[E[s], t] = Math.Max(K - X[J[s]], 0.0);
}
else if (PutCall == "C")
{
CF[E[s], t] = Math.Max(X[J[s]] - K, 0.0);
}
}
for (int s = 0; s <= NC - 1; s++)
{
CF[Co[s], t] = Math.Exp(-r * dt) * CF[Co[s], t + 1];
}
}
// Calculate the cash flow at time 2
double[] Op = new double[NS];
for (int s = 0; s <= NS - 1; s++)
{
Op[s] = CF[s, 1];
}
// Calculate the American price
double AmerPrice = Math.Exp(-r * dt) * RE.VMean(Op);
// Return the European and American prices
double[] output = new double[2] {
EuroPrice, AmerPrice
};
return(output);
}
public DHSim DHTransVolSim(string scheme, DHParam param, double S0, double Strike, double Mat, double r, double q, int T, int N, string PutCall)
{
// Heston parameters
double kappa1 = param.kappa1;
double theta1 = param.theta1;
double sigma1 = param.sigma1;
double v01 = param.v01;
double rho1 = param.rho1;
double kappa2 = param.kappa2;
double theta2 = param.theta2;
double sigma2 = param.sigma2;
double v02 = param.v02;
double rho2 = param.rho2;
// Time increment
double dt = Mat / T;
// Required quantities
double K01 = -rho1 * kappa1 * theta1 * dt / sigma1;
double K11 = dt / 2.0 * (kappa1 * rho1 / sigma1 - 0.5) - rho1 / sigma1;
double K21 = dt / 2.0 * (kappa1 * rho1 / sigma1 - 0.5) + rho1 / sigma1;
double K31 = dt / 2.0 * (1.0 - rho1 * rho1);
double K02 = -rho2 * kappa2 * theta2 * dt / sigma2;
double K12 = dt / 2.0 * (kappa2 * rho2 / sigma2 - 0.5) - rho2 / sigma2;
double K22 = dt / 2.0 * (kappa2 * rho2 / sigma2 - 0.5) + rho2 / sigma2;
double K32 = dt / 2.0 * (1.0 - rho2 * rho2);
// Initialize the volatility and stock processes
double[,] w1 = new double[T, N];
double[,] w2 = new double[T, N];
double[,] S = new double[T, N];
// Starting values for the variance and stock processes
for (int k = 0; k <= N - 1; k++)
{
S[0, k] = S0; // Spot price
w1[0, k] = Math.Sqrt(v01); // Heston initial volatility
w2[0, k] = Math.Sqrt(v02);
}
// Generate the stock and volatility paths
RandomNumber RN = new RandomNumber();
double Zv1, m11, m12, beta1, thetav1;
double Zv2, m21, m22, beta2, thetav2;
double B1, B2, logS;
for (int k = 0; k <= N - 1; k++)
{
for (int t = 1; t <= T - 1; t++)
{
Zv1 = RN.RandomNorm();
Zv2 = RN.RandomNorm();
if (scheme == "ZhuEuler")
{
// Transformed variance scheme
w1[t, k] = w1[t - 1, k] + 0.5 * kappa1 * ((theta1 - sigma1 * sigma1 / 4.0 / kappa1) / w1[t - 1, k] - w1[t - 1, k]) * dt + 0.5 * sigma1 * Math.Sqrt(dt) * Zv1;
w2[t, k] = w2[t - 1, k] + 0.5 * kappa2 * ((theta2 - sigma2 * sigma2 / 4.0 / kappa2) / w2[t - 1, k] - w2[t - 1, k]) * dt + 0.5 * sigma2 * Math.Sqrt(dt) * Zv2;
}
else if (scheme == "ZhuTV")
{
// Zhu (2010) process for the transformed volatility
m11 = theta1 + (w1[t - 1, k] * w1[t - 1, k] - theta1) * Math.Exp(-kappa1 * dt);
m21 = theta2 + (w2[t - 1, k] * w2[t - 1, k] - theta2) * Math.Exp(-kappa2 * dt);
m12 = sigma1 * sigma1 / 4.0 / kappa1 * (1.0 - Math.Exp(-kappa1 * dt));
m22 = sigma2 * sigma2 / 4.0 / kappa2 * (1.0 - Math.Exp(-kappa2 * dt));
beta1 = Math.Sqrt(Math.Max(0, m11 - m12));
beta2 = Math.Sqrt(Math.Max(0, m21 - m22));
thetav1 = (beta1 - w1[t - 1, k] * Math.Exp(-kappa1 * dt / 2.0)) / (1.0 - Math.Exp(-kappa1 * dt / 2.0));
thetav2 = (beta2 - w2[t - 1, k] * Math.Exp(-kappa2 * dt / 2.0)) / (1.0 - Math.Exp(-kappa2 * dt / 2.0));
w1[t, k] = w1[t - 1, k] + 0.5 * kappa1 * (thetav1 - w1[t - 1, k]) * dt + 0.5 * sigma1 * Math.Sqrt(dt) * Zv1;
w2[t, k] = w2[t - 1, k] + 0.5 * kappa2 * (thetav2 - w2[t - 1, k]) * dt + 0.5 * sigma2 * Math.Sqrt(dt) * Zv2;
}
// Predictor-Corrector for the stock price
B1 = RN.RandomNorm();
B2 = RN.RandomNorm();
logS = Math.Log(Math.Exp(-r * Convert.ToDouble(t) * dt) * S[t - 1, k])
+ K01 + K11 * w1[t - 1, k] * w1[t - 1, k] + K21 * w1[t, k] * w1[t, k] + Math.Sqrt(K31 * (w1[t, k] * w1[t, k] + w1[t - 1, k] * w1[t - 1, k])) * B1
+ K02 + K12 * w2[t - 1, k] * w2[t - 1, k] + K22 * w2[t, k] * w2[t, k] + Math.Sqrt(K32 * (w2[t, k] * w2[t, k] + w2[t - 1, k] * w2[t - 1, k])) * B2;
S[t, k] = Math.Exp(logS) * Math.Exp(r * Convert.ToDouble(t + 1) * dt);
}
}
// Terminal stock prices
double[] ST = new double[N];
for (int k = 0; k <= N - 1; k++)
{
ST[k] = S[T - 1, k];
}
// Payoff vectors
double[] Payoff = new double[N];
for (int k = 0; k <= N - 1; k++)
{
if (PutCall == "C")
{
Payoff[k] = Math.Max(ST[k] - Strike, 0.0);
}
else if (PutCall == "P")
{
Payoff[k] = Math.Max(Strike - ST[k], 0.0);
}
}
// Simulated price
Regression RE = new Regression();
double SimPrice = Math.Exp(-r * Mat) * RE.VMean(Payoff);
// Output the results
DHSim output = new DHSim();
output.S = S;
output.EuroPrice = SimPrice;
return(output);
}