public void Compute(OptionPosition option)
{
var t = option.TimeToExpiry.TotalDays / 365.0;
if (Math.Abs(t) < Double.Epsilon)
{
return;
}
var d1 = Math.Log(option.UnderlyingPrice / option.Strike) + (option.InterestRate - option.DividendRate + Math.Pow(option.Volatility, 2)) / t;
var d2 = d1 - option.Volatility * Math.Sqrt(t);
// when calculating option greeks keep in mind that we calculate greek for position,
// and that's why BS formulas for greeks must be multiplied by -1 in case we're selling options
// so formulas for greeks are [sign * BSFormula]
if (option.Type == OptionType.Call)
{
option.OptionPrice = option.UnderlyingPrice * NormalDistribution.Phi(d1) -
option.Strike * Math.Exp(-option.InterestRate * t) * NormalDistribution.Phi(d2);
option.Delta = Math.Exp(-option.DividendRate * t) * NormalDistribution.Phi(d1);
option.Theta = -Math.Exp(-option.DividendRate * t) * option.UnderlyingPrice * NormalDistribution.Density(d1) *
option.Volatility / (2 * Math.Sqrt(t))
+
option.DividendRate * Math.Exp(-option.DividendRate * t) * option.UnderlyingPrice *
NormalDistribution.Phi(d1)
-
option.InterestRate * option.Strike * Math.Exp(-option.InterestRate * t) *
NormalDistribution.Phi(d2);
option.Rho = t * option.Strike * Math.Exp(-option.InterestRate * t) * NormalDistribution.Phi(d2);
}
else
{
option.OptionPrice = -option.UnderlyingPrice * NormalDistribution.Phi(-d1) +
option.Strike * Math.Exp(-option.InterestRate * t) * NormalDistribution.Phi(-d2);
option.Delta = Math.Exp(-option.DividendRate * t) * (NormalDistribution.Phi(d1) - 1);
option.Theta = -Math.Exp(-option.DividendRate * t) * option.UnderlyingPrice * NormalDistribution.Density(d1) *
option.Volatility / (2 * Math.Sqrt(t))
-
option.DividendRate * Math.Exp(-option.DividendRate * t) * option.UnderlyingPrice *
NormalDistribution.Phi(-d1)
+
option.InterestRate * option.Strike * Math.Exp(-option.InterestRate * t) *
NormalDistribution.Phi(-d2);
option.Rho = -t *option.Strike *Math.Exp(-option.InterestRate *t) * NormalDistribution.Phi(-d2);
}
option.Gamma = Math.Exp(-option.DividendRate * t) * NormalDistribution.Density(d1) /
(option.UnderlyingPrice * option.Volatility * Math.Sqrt(t));
option.Vega = Math.Exp(-option.DividendRate * t) * option.UnderlyingPrice * NormalDistribution.Density(d1) *
Math.Sqrt(t);
SideHelper.FixGreeksAccordingToSide(option);
}
private static double CdfProxy(double x)
{//c.f. D.Lamberton, B.Lapeyre "Introduction au calcul stochastique applique a la finance" p158.
if (x < 0.0)
{
throw new Exception("positive value expected !");
}
double t = 1 / (1 + 0.2316419 * x);
return(1.0 - NormalDistribution.Density(x)
* t * (0.319381530 + t * (-0.356563782 + t * (1.781477937 + t * (-1.821255978 + t * 1.330274429)))));
}
public void CallNaiveImplemXCheck()
{
var maturities = GridUtils.RegularGrid(0.1, 5.0, 100);
var vols = GridUtils.RegularGrid(0.001, 0.015, 10);
var volMoneynesses = GridUtils.RegularGrid(-3, 3, 100);
foreach (var mat in maturities)
{
foreach (var vol in vols)
{
var sigma = Math.Sqrt(mat) * vol;
foreach (var vm in volMoneynesses)
{
var strike = vm * sigma;
var call = BachelierOption.Price(0.0, strike, vol, mat, 1);
//Naive implementation of bachelier formulae
var d = strike / sigma;
var call_xcheck = -strike *NormalDistribution.Cumulative(-d) + sigma * NormalDistribution.Density(d);
if (DoubleUtils.EqualZero(call))
{
Assert.IsTrue(DoubleUtils.EqualZero(call_xcheck));
}
else
{
var errRelative = (call - call_xcheck) / call_xcheck;
Assert.IsTrue(Math.Abs(errRelative) < 1.0e-13);
}
}
}
}
}
