public static double Delta(double forward, double knownAverage, double sigma, double K, double tAvgStart, double tExpiry, double riskFree, OptionType callPut)
{
var tau = Max(0, tAvgStart);
var M = 2 * (Exp(sigma * sigma * tExpiry) - Exp(sigma * sigma * tau) * (1 + sigma * sigma * (tExpiry - tau)));
M /= Pow(sigma, 4.0) * (tExpiry - tau) * (tExpiry - tau);
var sigma_a = Sqrt(Log(M) / tExpiry);
K = AsianUtils.AdjustedStrike(K, knownAverage, tExpiry, tAvgStart);
if (K <= 0)
{
if (callPut == OptionType.P)
{
return(0);
}
var t2 = tExpiry - tAvgStart;
var expAvg = knownAverage * (t2 - tExpiry) / t2 + forward * tExpiry / t2;
var df = Exp(-riskFree * tExpiry);
return(df * (expAvg - K));
}
var delta = BlackFunctions.BlackDelta(forward, K, riskFree, tExpiry, sigma_a, callPut);
if (tAvgStart < 0)
{
delta *= tExpiry / (tExpiry - tAvgStart);
}
return(delta);
}
public static double PV(double forward, double knownAverage, double sigma, double K, double tAvgStart, double tExpiry, double riskFree, OptionType callPut)
{
if (tExpiry <= 0) //work out intrinsic
{
return(callPut == OptionType.Call ? Max(0, knownAverage - K) : Max(0, K - knownAverage));
}
var tau = Max(0, tAvgStart);
var M = 2 * (Exp(sigma * sigma * tExpiry) - Exp(sigma * sigma * tau) * (1 + sigma * sigma * (tExpiry - tau)));
M /= Pow(sigma, 4.0) * (tExpiry - tau) * (tExpiry - tau);
//hack to fix deep ITM options have imaginary vol
var sigma_a = tExpiry == 0 ? 0.0 : Sqrt(Log(M) / tExpiry);
K = AsianUtils.AdjustedStrike(K, knownAverage, tExpiry, tAvgStart);
if (K <= 0 && tAvgStart < 0)
{
if (callPut == OptionType.P)
{
return(0);
}
var t2 = tExpiry - tAvgStart;
var expAvg = knownAverage * (t2 - tExpiry) / t2 + forward * tExpiry / t2;
var df = Exp(-riskFree * tExpiry);
return(df * expAvg);
}
var pv = BlackFunctions.BlackPV(forward, K, riskFree, tExpiry, sigma_a, callPut);
if (tAvgStart < 0)
{
pv *= tExpiry / (tExpiry - tAvgStart);
}
return(pv);
}
public static double PV(double[] forwards, DateTime[] fixingDates, DateTime evalDate, DateTime payDate, double[] sigmas, double K, double riskFree, OptionType callPut, bool todayFixed = false)
{
if (payDate < evalDate)
{
return(0.0);
}
if (forwards.Length != fixingDates.Length || fixingDates.Length != sigmas.Length)
{
throw new DataMisalignedException();
}
var nFixed = evalDate < fixingDates.First() ? 0 :
fixingDates.Where(x => (todayFixed ? x <= evalDate : x < evalDate)).Count();
var nFloat = fixingDates.Length - nFixed;
var m1 = forwards.Skip(nFixed).Average();
var wholeAverage = forwards.Average();
var tExpiry = evalDate.CalculateYearFraction(fixingDates.Last(), DayCountBasis.Act365F, false);
var tPay = evalDate.CalculateYearFraction(payDate, DayCountBasis.Act365F, false);
var df = Exp(-riskFree * tPay);
if (tExpiry <= 0) //work out intrinsic
{
return(df * (callPut == OptionType.Call ? Max(0, wholeAverage - K) : Max(0, K - wholeAverage)));
}
var m2 = 0.0;
var ts = fixingDates.Select(x => Max(0, evalDate.CalculateYearFraction(x, DayCountBasis.Act365F, false))).ToArray();
for (var i = nFixed; i < fixingDates.Length; i++)
{
for (var j = nFixed; j < fixingDates.Length; j++)
{
m2 += forwards[i] * forwards[j] * Exp(sigmas[i] * sigmas[j] * ts[Min(i, j)]);
}
}
m2 /= nFloat * nFloat;
var sigma_a = Sqrt(1 / tExpiry * Log(m2 / (m1 * m1)));
var tAvgStart = evalDate.CalculateYearFraction(fixingDates.First(), DayCountBasis.Act365F, false);
var knownAverage = nFixed == 0 ? 0.0 : forwards.Take(nFixed).Average();
var k0 = K;
K = AsianUtils.AdjustedStrike(K, knownAverage, tExpiry, tAvgStart);
if (K <= 0)
{
return((callPut == OptionType.P) ? 0.0 : df *Max(wholeAverage - k0, 0));
}
var pv = BlackFunctions.BlackPV(m1, K, 0.0, tExpiry, sigma_a, callPut);
if (tAvgStart < 0)
{
pv *= tExpiry / (tExpiry - tAvgStart);
}
return(df * pv);
}
