public void TestSkewDigitalEuroCallPut()
{
DateTime today = new DateTime(2010, 06, 22);
DateTime expiry = new DateTime(2011, 03, 24);
double tau = System.Convert.ToDouble(expiry.Subtract(today).Days) / 365.0;
double spot = 42.00;
double upperbarrier = 44.00;
double lowerbarrier = 40.00;
int[] zerodays = { 0, 10 };
double[] zerorates = { 0.05, 0.05 };
int[] divdays = new int[] { 0, 51, 247 };
double[] divAmts = new double[] { 4, 1.350, 1.420 };
double fwd = EquityAnalytics.GetForwardCCLin365(spot, tau, divdays, divAmts, zerodays, zerorates);
OrcWingParameters owp = new OrcWingParameters()
{
AtmForward = fwd, CallCurve = 0.1250, CurrentVol = 0.26, DnCutoff = -0.25, Dsr = 0.9, PutCurve = 0.10, RefFwd = fwd, RefVol = 0.26, Scr = 0.0, SlopeRef = -0.1750, Ssr = 100, TimeToMaturity = tau, UpCutoff = 0.20, Usr = 0.50, Vcr = 0.0
};
List <OrcWingParameters> owpList = new List <OrcWingParameters>();
owpList.Add(owp);
double skew1 = EquityAnalytics.GetWingSkew(owpList, new LinearInterpolation(), tau, upperbarrier);
double vol = EquityAnalytics.GetWingValue(owpList, new LinearInterpolation(), tau, upperbarrier);
BinaryEuro op1 = new BinaryEuro(spot, upperbarrier, true, tau, vol, zerodays, zerorates, divdays, divAmts, skew1);
double pr1 = op1.GetPrice();
Assert.AreEqual(0.35784, pr1, 0.005);
double skew2 = EquityAnalytics.GetWingSkew(owpList, new LinearInterpolation(), tau, lowerbarrier);
BinaryEuro op2 = new BinaryEuro(spot, lowerbarrier, false, tau, vol, zerodays, zerorates, divdays, divAmts, skew2);
double pr2 = op2.GetPrice();
Assert.AreEqual(0.43275, pr2, 0.005);
}
/// <summary>
/// We can roughly proxy the alpha-th percentile of the simulation
/// by calculating the upper alpha percentile confidence point for a given time slice
/// of the ln-OU process and pricing the collar with this asset value for a given profile point
/// This should roughly compare to the calculated (1-alpha)-worst PCE point.
/// </summary>
/// <param name="spot"></param>
/// <param name="kappa"></param>
/// <param name="theta"></param>
/// <param name="histvol"></param>
/// <param name="time0"></param>
/// <param name="maturity"></param>
/// <param name="callstrike"></param>
/// <param name="putstrike"></param>
/// <param name="zerodays"></param>
/// <param name="zerorates"></param>
/// <param name="divdays"></param>
/// <param name="divamts"></param>
/// <param name="volSurface"></param>
/// <returns></returns>
public static double PCEProxyCalc(double spot,
double kappa,
double theta,
double histvol,
double time0,
double maturity,
double callstrike,
double putstrike,
int[] zerodays,
double[] zerorates,
int[] divdays,
double[] divamts,
List <OrcWingParameters> volSurface)
{
double tau = maturity - time0;
double upperBound = EquityPCEAnalytics.LNOUUpperBound(spot, 0.95, kappa, theta, histvol, time0);
double fStar = EquityAnalytics.GetForwardCCLin365(upperBound, time0, maturity, divdays, divamts, zerodays, zerorates);
double callvol = EquityAnalytics.GetWingValue(volSurface, new LinearInterpolation(), tau, callstrike);
double putvol = EquityAnalytics.GetWingValue(volSurface, new LinearInterpolation(), tau, putstrike);
double r0 = EquityAnalytics.GetRateCCLin365(time0, maturity, zerodays, zerorates);
BlackScholes bs = new BlackScholes();
double lhs = Math.Max(bs.BSprice(fStar, tau, callstrike, r0, callvol, true) - bs.BSprice(fStar, tau, putstrike, r0, putvol, false), 0);
return(lhs);
}
///<summary>
///</summary>
///<param name="vols"></param>
///<param name="days"></param>
///<param name="spot"></param>
///<param name="ratedays"></param>
///<param name="rateamts"></param>
///<param name="divdays"></param>
///<param name="divamts"></param>
///<returns></returns>
private List <OrcWingParameters> UnpackWing(double[,] vols, double[] days, double spot, int[] ratedays, double[] rateamts, int[] divdays, double[] divamts)
{
const double daybasis = 365.0;
int rows = vols.GetLength(0);
var opList = new List <OrcWingParameters>();
for (int idx = 0; idx < rows; idx++)
{
double fwd = EquityAnalytics.GetForwardCCLin365(spot, days[idx] / daybasis, divdays, divamts, ratedays, rateamts);
var op = new OrcWingParameters
{
AtmForward = fwd,
CurrentVol = vols[idx, 0],
SlopeRef = vols[idx, 1],
PutCurve = vols[idx, 2],
CallCurve = vols[idx, 3],
DnCutoff = vols[idx, 4],
UpCutoff = vols[idx, 5],
Vcr = vols[idx, 6],
Scr = vols[idx, 7],
Ssr = 100 * vols[idx, 8],
Dsr = vols[idx, 9],
Usr = vols[idx, 10],
RefVol = vols[idx, 11],
RefFwd = fwd,
TimeToMaturity = days[idx] / daybasis
};
opList.Add(op);
}
return(opList);
}
public void TestPCEProfile()
{
DateTime today = new DateTime(2010, 10, 22);
DateTime expiry = new DateTime(2011, 10, 22);
double maturity = Convert.ToDouble(expiry.Subtract(today).Days) / 365.0;
double spot = 4200.00;
int simuls = 10000;
double callstrike = 1.3 * spot;
double putstrike = 1.0 * spot;
int[] zerodays = { 0, 365 };
double r0 = 0.065;
double confidence = 0.95;
double[] zerorates = { r0, r0 };
int[] divdays = { 1, 51, 247 };
double[] divamts = { 22.5, 50, 50 };
double fwd = EquityAnalytics.GetForwardCCLin365(spot, maturity, divdays, divamts, zerodays, zerorates);
double histvol = 0.60;
double theta = Math.Log(spot);
double kappa = 0.9;
double[] times = { 7.0 / 365.0, 128.0 / 365.0, 250.0 / 365.0, 365.0 / 365.0, 730.0 / 365.0 };
OrcWingParameters owp = new OrcWingParameters {
AtmForward = fwd, CallCurve = 0.1250, CurrentVol = 0.26, DnCutoff = -0.25, Dsr = 0.9, PutCurve = 0.10, RefFwd = fwd, RefVol = 0.26, Scr = 0.0, SlopeRef = -0.1750, Ssr = 100, TimeToMaturity = maturity, UpCutoff = 0.20, Usr = 0.50, Vcr = 0.0
};
List <OrcWingParameters> owpList = new List <OrcWingParameters> {
owp
};
double[,] results = EquityPCEAnalytics.GetCollarPCE("CollarPCE",
zerodays,
zerorates,
divdays,
divamts,
owpList,
spot,
callstrike,
putstrike,
maturity,
kappa,
theta,
histvol,
times,
confidence,
0.25 / 365.0,
simuls,
3151);
double lhs1 = PCEProxyCalc(spot, kappa, theta, histvol, times[0], maturity, callstrike, putstrike, zerodays, zerorates, divdays, divamts, owpList);
double lhs2 = PCEProxyCalc(spot, kappa, theta, histvol, times[1], maturity, callstrike, putstrike, zerodays, zerorates, divdays, divamts, owpList);
double lhs3 = PCEProxyCalc(spot, kappa, theta, histvol, times[2], maturity, callstrike, putstrike, zerodays, zerorates, divdays, divamts, owpList);
Assert.AreEqual(results[0, 1], lhs1, Math.Abs(0.10 * results[0, 1]));
Assert.AreEqual(results[1, 1], lhs2, Math.Abs(0.10 * results[1, 1]));
Assert.AreEqual(results[2, 1], lhs3, Math.Abs(0.10 * results[2, 1]));
}
public void TestCrankNicolson()
{
double spot = 385.5;
double epsilon = 0.0001;
double strike = 400;
double[] rt = { 0.0425,
0.0452,
0.0457,
0.0462 };
int[] tt = { 1, 32, 63, 93 };
int[] divdays = { 0, 48 };
double[] divs = { 20, 10 };
double vol = 0.2046;
int steps = 80;
double tStepSize = 0.01;
DateTime today = new DateTime(2010, 05, 07);
DateTime expiry = new DateTime(2010, 06, 24);
//string style = "European";
double t = expiry.Subtract(today).Days / 365.0 + epsilon; // For CN backwards propagation div time needs to be strictly less than expiry time
double fwd = EquityAnalytics.GetForwardCCLin365(spot, t, divdays, divs, tt, rt);
double df = EquityAnalytics.GetDFCCLin365(0, t, tt, rt);
BlackScholes bs = new BlackScholes(spot, strike, false, t, vol, tt, rt, divdays, divs);
CrankNicolson lhs = new CrankNicolson(false, false, spot, strike, t, vol, steps, tStepSize, 8.0, divdays, divs, tt, rt);
double[,] res0 = OptionAnalytics.BlackScholesWithGreeks(false, fwd, strike, vol, t);
double[] res_cn = lhs.GetPriceAndGreeks();
double pr_bs = bs.GetPrice();
double delta_bs = bs.GetDelta();
double gamma_bs = bs.GetGamma();
double theta_bs = bs.GetTheta();
double pr_cn = res_cn[0];
double delta_cn = res_cn[1];
double gamma_cn = res_cn[2];
double theta_cn = res_cn[3];
Assert.AreEqual(pr_cn, pr_bs, 0.50);
Assert.AreEqual(delta_cn, delta_bs, 0.03);
Assert.AreEqual(gamma_cn, 0.012931145370580023, 0.005);
Assert.AreEqual(bs.GetTheta(), theta_cn, 0.01);
}
/// <summary>
/// Evaluate the PCE for a collar given the parameters passed below
/// </summary>
/// <param name="ratedays"></param>
/// <param name="rateamts"></param>
/// <param name="divdays"></param>
/// <param name="divamts"></param>
/// <param name="volSurface"></param>
/// <param name="spot"></param>
/// <param name="s"></param>
/// <param name="callstrike"></param>
/// <param name="putstrike"></param>
/// <param name="t0"></param>
/// <param name="maturity"></param>
/// <returns></returns>
private static double CollarPCEFunction(int[] ratedays,
double[] rateamts,
int[] divdays,
double[] divamts,
List <OrcWingParameters> volSurface,
double spot,
double s,
double callstrike,
double putstrike,
double t0,
double maturity)
{
t0 = Math.Min(t0, maturity); // cap profile point to maturity;
double tau = maturity - t0;
double r = EquityAnalytics.GetRateCCLin365(t0, maturity, ratedays, rateamts);
double q = EquityAnalytics.GetYieldCCLin365(spot, t0, maturity, divdays, divamts, ratedays, rateamts);
double callvol = EquityAnalytics.GetWingValue(volSurface, new LinearInterpolation(), tau, callstrike);
double putvol = EquityAnalytics.GetWingValue(volSurface, new LinearInterpolation(), tau, putstrike);
double fwd = s * Math.Exp((r - q) * tau);
BlackScholes bs = new BlackScholes();
double lhs = Math.Max(bs.BSprice(fwd, tau, callstrike, r, callvol, true) - bs.BSprice(fwd, tau, putstrike, r, putvol, false), 0);
return(lhs);
}
/// <summary>
/// Before calculating any PCE numbers on the simulation set, first check the simulated asset values
/// at the time slices and compare to the expected values of the theoretical distribution.
/// Check we're within 3 standard errors, reason for 3 being that there is a discretisation error introduced
/// by the time scheme approximation as well as sample error introduced by the simulation sample mean.
/// </summary>
/// <param name="kappa"></param>
/// <param name="vol"></param>
/// <param name="theta"></param>
/// <param name="simuls"></param>
public void TestDistroConvergence(double kappa, double vol, double theta, int simuls)
{
DateTime today = new DateTime(2010, 10, 22);
DateTime expiry = new DateTime(2011, 10, 22);
double maturity = Convert.ToDouble(expiry.Subtract(today).Days) / 365.0;
double spot = 4200.00;
double callstrike = 1.3 * spot;
double putstrike = 1.0 * spot;
double confidence = 0.95;
int[] zerodays = { 0, 365 };
double[] zerorates = { 0.065, 0.065 };
int[] divdays = { 1, 51, 247 };
double[] divamts = { 22.5, 50, 50 };
double fwd = EquityAnalytics.GetForwardCCLin365(spot, maturity, divdays, divamts, zerodays, zerorates);
double histvol = vol;
double[] times = { 128.0 / 365.0, 250.0 / 365.0, 365.0 / 365.0 };
OrcWingParameters owp = new OrcWingParameters()
{
AtmForward = fwd, CallCurve = 0.1250, CurrentVol = 0.26, DnCutoff = -0.25, Dsr = 0.9, PutCurve = 0.10, RefFwd = fwd, RefVol = 0.26, Scr = 0.0, SlopeRef = -0.1750, Ssr = 100, TimeToMaturity = maturity, UpCutoff = 0.20, Usr = 0.50, Vcr = 0.0
};
List <OrcWingParameters> owpList = new List <OrcWingParameters> {
owp
};
double[,] results = EquityPCEAnalytics.GetCollarPCE("Asset",
zerodays,
zerorates,
divdays,
divamts,
owpList,
spot,
callstrike,
putstrike,
maturity,
kappa,
theta,
histvol,
times, confidence, 1.0 / 365.0, simuls, 3151);
// Check convergence to mean ln(S_t) ~ N( theta + exp(-kappa*t)*( ln(S_0) - theta , sigma^2/(2*kappa)*(1-exp(-2*kappa*t) )
// => E(S_t) = exp(theta + 0.5*sigma^2)
//
double LNmean1 = EquityPCEAnalytics.OUMean(kappa, theta, spot, times[0]);
double LNmean2 = EquityPCEAnalytics.OUMean(kappa, theta, spot, times[1]);
double LNmean3 = EquityPCEAnalytics.OUMean(kappa, theta, spot, times[2]);
double Var_1 = EquityPCEAnalytics.OUVar(histvol, kappa, times[0]);
double Var_2 = EquityPCEAnalytics.OUVar(histvol, kappa, times[1]);
double Var_3 = EquityPCEAnalytics.OUVar(histvol, kappa, times[2]);
//lognormal moments
double lhs1 = Math.Exp(LNmean1 + Var_1 / 2.0);
double lhs2 = Math.Exp(LNmean2 + Var_2 / 2.0);
double lhs3 = Math.Exp(LNmean3 + Var_3 / 2.0);
//double deltat = 1.0/365.0;
double logvar1 = EquityPCEAnalytics.LNOUVar(spot, histvol, kappa, theta, times[0]);
double logvar2 = EquityPCEAnalytics.LNOUVar(spot, histvol, kappa, theta, times[1]);
double logvar3 = EquityPCEAnalytics.LNOUVar(spot, histvol, kappa, theta, times[2]);
double stderr1 = Math.Sqrt(logvar1 / simuls);
double stderr2 = Math.Sqrt(logvar2 / simuls);
double stderr3 = Math.Sqrt(logvar3 / simuls);
//double accuracy = histvol*histvol*(1/(2*kappa) - deltat /(1-Math.Exp(-2*kappa*deltat)))*(1-Math.Exp(-2*kappa*times[0]));
Assert.AreEqual(lhs1, results[0, 0], 3.0 * stderr1);
Assert.AreEqual(lhs2, results[1, 0], 3.0 * stderr2);
Assert.AreEqual(lhs3, results[2, 0], 3.0 * stderr3);
}
