public static double bachelierBlackFormula( Option.Type optionType,
double strike,
double forward,
double stdDev,
double discount)
{
if ( stdDev < 0.0 )
throw new ArgumentException( "stdDev (" + stdDev + ") must be non-negative" );
if ( discount <= 0.0 )
throw new ArgumentException( "discount (" + discount + ") must be positive" );
double d = ( forward - strike ) * (int)optionType;
double h = d / stdDev;
if ( stdDev == 0.0 )
return discount * Math.Max( d, 0.0 );
CumulativeNormalDistribution phi = new CumulativeNormalDistribution();
double result = discount * ( stdDev * phi.derivative( h ) + d * phi.value( h ) );
if ( !( result >= 0.0 ) )
throw new ApplicationException( "negative value (" + result + ") for " +
stdDev + " stdDev, " +
optionType + " option, " +
strike + " strike , " +
forward + " forward" );
return result;
}
public static double blackFormulaStdDevDerivative(double strike,
double forward,
double stdDev,
double discount,
double displacement)
{
checkParameters(strike, forward, displacement);
if (stdDev < 0.0)
{
throw new ArgumentException("stdDev (" + stdDev + ") must be non-negative");
}
if (discount <= 0.0)
{
throw new ArgumentException("discount (" + discount + ") must be positive");
}
forward = forward + displacement;
strike = strike + displacement;
if (stdDev == 0.0 || strike == 0)
{
return(0.0);
}
double d1 = Math.Log(forward / strike) / stdDev + .5 * stdDev;
CumulativeNormalDistribution phi = new CumulativeNormalDistribution();
return(discount * forward * phi.derivative(d1));
}
public static double blackFormula( Option.Type optionType, double strike, double forward, double stdDev,
double discount, double displacement )
{
checkParameters( strike, forward, displacement );
if ( !( stdDev >= 0.0 ) )
throw new ApplicationException( "stdDev (" + stdDev + ") must be non-negative" );
if ( !( discount > 0.0 ) )
throw new ApplicationException( "discount (" + discount + ") must be positive" );
if ( stdDev == 0.0 )
return Math.Max( ( forward - strike ) * (int)optionType, 0.0 ) * discount;
forward = forward + displacement;
strike = strike + displacement;
// since displacement is non-negative strike==0 iff displacement==0
// so returning forward*discount is OK
if ( strike == 0.0 )
return ( optionType == Option.Type.Call ? forward * discount : 0.0 );
double d1 = Math.Log( forward / strike ) / stdDev + 0.5 * stdDev;
double d2 = d1 - stdDev;
CumulativeNormalDistribution phi = new CumulativeNormalDistribution();
double nd1 = phi.value( (int)optionType * d1 );
double nd2 = phi.value( (int)optionType * d2 );
double result = discount * (int)optionType * ( forward * nd1 - strike * nd2 );
if ( !( result >= 0.0 ) )
throw new ApplicationException( "negative value (" + result + ") for " +
stdDev + " stdDev, " +
optionType + " option, " +
strike + " strike , " +
forward + " forward" );
return result;
}
public GaussianKernel(double average, double sigma)
{
nd_ = new NormalDistribution(average,sigma);
cnd_ = new CumulativeNormalDistribution(average,sigma);
// normFact is \sqrt{2*\pi}.
normFact_ = Const.M_SQRT2*Const.M_SQRTPI;
}
protected double digitalPriceWithoutSmile(double strike, double initialValue, double expiry, double deflator)
{
double lambdaS = smilesOnExpiry_.volatility(strike);
double lambdaT = smilesOnPayment_.volatility(strike);
List <double> lambdaU = lambdasOverPeriod(expiry, lambdaS, lambdaT);
double variance = startTime_ * lambdaU[0] * lambdaU[0] + (expiry - startTime_) * lambdaU[1] * lambdaU[1];
double lambdaSATM = smilesOnExpiry_.volatility(initialValue);
double lambdaTATM = smilesOnPayment_.volatility(initialValue);
//drift of Lognormal process (of Libor) "a_U()" nel paper
List <double> muU = driftsOverPeriod(expiry, lambdaSATM, lambdaTATM, correlation_);
double adjustment = (startTime_ * muU[0] + (expiry - startTime_) * muU[1]);
double d2 = (Math.Log(initialValue / strike) + adjustment - 0.5 * variance) / Math.Sqrt(variance);
CumulativeNormalDistribution phi = new CumulativeNormalDistribution();
double result = deflator * phi.value(d2);
Utils.QL_REQUIRE(result > 0.0, () =>
"RangeAccrualPricerByBgm::digitalPriceWithoutSmile: result< 0. Result:" + result);
Utils.QL_REQUIRE(result / deflator <= 1.0, () =>
"RangeAccrualPricerByBgm::digitalPriceWithoutSmile: result/deflator > 1. Ratio: "
+ result / deflator + " result: " + result + " deflator: " + deflator);
return(result);
}
public static double bachelierBlackFormula(Option.Type optionType, double strike, double forward, double stdDev, double discount)
{
if (stdDev < 0.0)
{
throw new ArgumentException("stdDev (" + stdDev + ") must be non-negative");
}
if (discount <= 0.0)
{
throw new ArgumentException("discount (" + discount + ") must be positive");
}
double d = (forward - strike) * (int)optionType;
double h = d / stdDev;
if (stdDev == 0.0)
{
return(discount * Math.Max(d, 0.0));
}
CumulativeNormalDistribution phi = new CumulativeNormalDistribution();
double result = discount * (stdDev * phi.derivative(h) + d * phi.value(h));
if (!(result >= 0.0))
{
throw new ApplicationException("negative value (" + result + ") for " +
stdDev + " stdDev, " +
optionType + " option, " +
strike + " strike , " +
forward + " forward");
}
return(result);
}
/*! Black style formula when forward is normal rather than
* log-normal. This is essentially the model of Bachelier.
*
* \warning Bachelier model needs absolute volatility, not
* percentage volatility. Standard deviation is
* absoluteVolatility*sqrt(timeToMaturity)
*/
public static double bachelierBlackFormula(Option.Type optionType,
double strike,
double forward,
double stdDev,
double discount = 1.0)
{
QL_REQUIRE(stdDev >= 0.0, () =>
"stdDev (" + stdDev + ") must be non-negative");
QL_REQUIRE(discount > 0.0, () =>
"discount (" + discount + ") must be positive");
double d = (forward - strike) * (int)optionType, h = d / stdDev;
if (stdDev.IsEqual(0.0))
{
return(discount * Math.Max(d, 0.0));
}
CumulativeNormalDistribution phi = new CumulativeNormalDistribution();
double result = discount * (stdDev * phi.derivative(h) + d * phi.value(h));
QL_REQUIRE(result >= 0.0, () =>
"negative value (" + result + ") for " +
stdDev + " stdDev, " +
optionType + " option, " +
strike + " strike , " +
forward + " forward");
return(result);
}
public freeArbSVI(List <double> strikes,
List <double> times,
double spot,
Handle <YieldTermStructure> riskFreeTS,
Handle <YieldTermStructure> dividendTS,
Matrix crudeVolSurface,
double lambda)
{
// sanity check to add : check date + strikes vs vol surface
strikes_ = strikes;
times_ = times;
spot_ = spot;
riskFreeTS_ = riskFreeTS;
dividendTS_ = dividendTS;
fwMoneynessGrid_ = new Matrix(times_.Count, strikes_.Count, 0.0);
totalVariance_ = new Matrix(times_.Count, strikes_.Count, 0.0);
X_ = new Matrix(times_.Count, 2 * strikes_.Count + 2, 0.0);
crudeVolSurface_ = new Matrix(crudeVolSurface);
cumulNormdiv_ = new CumulativeNormalDistribution();
lambda_ = lambda;
g_i_ = new List <CubicInterpolation>();
A_ = new Matrix(2 * strikes_.Count + 2, strikes_.Count, 0.0);;
B_ = new Matrix(2 * strikes_.Count + 2, 2 * strikes_.Count + 1, 0.0);;
strikesSpline_ = new InitializedList <List <double> >();
}
public AnalyticDoubleBarrierEngine(GeneralizedBlackScholesProcess process, int series = 5)
{
process_ = process;
series_ = series;
f_ = new CumulativeNormalDistribution();
process_.registerWith(update);
}
//! gaussian-assumption Average Shortfall (averaged shortfallness)
public double gaussianAverageShortfall(double target)
{
double m = this.mean();
double std = this.standardDeviation();
CumulativeNormalDistribution gIntegral = new CumulativeNormalDistribution(m, std);
NormalDistribution g = new NormalDistribution(m, std);
return((target - m) + std * std * g.value(target) / gIntegral.value(target));
}
protected double smileCorrection(double strike,
double forward,
double expiry,
double deflator)
{
double previousStrike = strike - eps_ / 2;
double nextStrike = strike + eps_ / 2;
double derSmileS = (smilesOnExpiry_.volatility(nextStrike) -
smilesOnExpiry_.volatility(previousStrike)) / eps_;
double derSmileT = (smilesOnPayment_.volatility(nextStrike) -
smilesOnPayment_.volatility(previousStrike)) / eps_;
double lambdaS = smilesOnExpiry_.volatility(strike);
double lambdaT = smilesOnPayment_.volatility(strike);
double derLambdaDerK = derLambdaDerLambdaS(expiry) * derSmileS +
derLambdaDerLambdaT(expiry) * derSmileT;
double lambdaSATM = smilesOnExpiry_.volatility(forward);
double lambdaTATM = smilesOnPayment_.volatility(forward);
List <double> lambdasOverPeriodU = lambdasOverPeriod(expiry, lambdaS, lambdaT);
//drift of Lognormal process (of Libor) "a_U()" nel paper
List <double> muU = driftsOverPeriod(expiry, lambdaSATM, lambdaTATM, correlation_);
double variance = Math.Max(startTime_, 0.0) * lambdasOverPeriodU[0] * lambdasOverPeriodU[0] +
Math.Min(expiry - startTime_, expiry) * lambdasOverPeriodU[1] * lambdasOverPeriodU[1];
double forwardAdjustment = Math.Exp(Math.Max(startTime_, 0.0) * muU[0] +
Math.Min(expiry - startTime_, expiry) * muU[1]);
double forwardAdjusted = forward * forwardAdjustment;
double d1 = (Math.Log(forwardAdjusted / strike) + 0.5 * variance) / Math.Sqrt(variance);
double sqrtOfTimeToExpiry = (Math.Max(startTime_, 0.0) * lambdasOverPeriodU[0] +
Math.Min(expiry - startTime_, expiry) * lambdasOverPeriodU[1]) * (1.0 / Math.Sqrt(variance));
CumulativeNormalDistribution phi = new CumulativeNormalDistribution();
NormalDistribution psi = new NormalDistribution();
double result = -forwardAdjusted *psi.value(d1) * sqrtOfTimeToExpiry * derLambdaDerK;
result *= deflator;
Utils.QL_REQUIRE(Math.Abs(result / deflator) <= 1.0 + Math.Pow(eps_, .2), () =>
"RangeAccrualPricerByBgm::smileCorrection: abs(result/deflator) > 1. Ratio: "
+ result / deflator + " result: " + result + " deflator: " + deflator);
return(result);
}
/*! returns the variance of observations below target
* \f[ \frac{\sum w_i (min(0, x_i-target))^2 }{\sum w_i}. \f]
*
* See Dembo, Freeman "The Rules Of Risk", Wiley (2001)
*/
public double gaussianRegret(double target)
{
double m = this.mean();
double std = this.standardDeviation();
double variance = std * std;
CumulativeNormalDistribution gIntegral = new CumulativeNormalDistribution(m, std);
NormalDistribution g = new NormalDistribution(m, std);
double firstTerm = variance + m * m - 2.0 * target * m + target * target;
double alfa = gIntegral.value(target);
double secondTerm = m - target;
double beta = variance * g.value(target);
double result = alfa * firstTerm - beta * secondTerm;
return(result / alfa);
}
public static double blackFormula(Option.Type optionType, double strike, double forward, double stdDev, double discount, double displacement)
{
checkParameters(strike, forward, displacement);
if (!(stdDev >= 0.0))
{
throw new ApplicationException("stdDev (" + stdDev + ") must be non-negative");
}
if (!(discount > 0.0))
{
throw new ApplicationException("discount (" + discount + ") must be positive");
}
if (stdDev == 0.0)
{
return(Math.Max((forward - strike) * (int)optionType, 0.0) * discount);
}
forward = forward + displacement;
strike = strike + displacement;
// since displacement is non-negative strike==0 iff displacement==0
// so returning forward*discount is OK
if (strike == 0.0)
{
return(optionType == Option.Type.Call ? forward * discount : 0.0);
}
double d1 = Math.Log(forward / strike) / stdDev + 0.5 * stdDev;
double d2 = d1 - stdDev;
CumulativeNormalDistribution phi = new CumulativeNormalDistribution();
double nd1 = phi.value((int)optionType * d1);
double nd2 = phi.value((int)optionType * d2);
double result = discount * (int)optionType * (forward * nd1 - strike * nd2);
if (!(result >= 0.0))
{
throw new ApplicationException("negative value (" + result + ") for " +
stdDev + " stdDev, " +
optionType + " option, " +
strike + " strike , " +
forward + " forward");
}
return(result);
}
public double nD2(double strike) // n(d2)
{
double d2_ = 0.0;
double n_d2_ = 0.0; // n(d2)
if (stdDev_ >= Const.QL_EPSILON)
{
if (strike > 0)
{
d2_ = Math.Log(forward_ / strike) / stdDev_ - 0.5 * stdDev_;
CumulativeNormalDistribution f = new CumulativeNormalDistribution();
n_d2_ = f.derivative(d2_);
}
}
return(n_d2_);
}
public BlackImpliedStdDevHelper(Option.Type optionType,
double strike,
double forward,
double undiscountedBlackPrice,
double displacement = 0.0)
{
halfOptionType_ = 0.5 * (int)optionType;
signedStrike_ = (int)optionType * (strike + displacement);
signedForward_ = (int)optionType * (forward + displacement);
undiscountedBlackPrice_ = undiscountedBlackPrice;
N_ = new CumulativeNormalDistribution();
checkParameters(strike, forward, displacement);
Utils.QL_REQUIRE(undiscountedBlackPrice >= 0.0, () =>
"undiscounted Black price (" +
undiscountedBlackPrice + ") must be non-negative");
signedMoneyness_ = (int)optionType * Math.Log((forward + displacement) / (strike + displacement));
}
public double value(double x)
{
CumulativeNormalDistribution phi = new CumulativeNormalDistribution();
double temp = (x - mux_) / sigmax_;
double txy = Math.Sqrt(1.0 - rhoxy_ * rhoxy_);
Vector lambda = new Vector(size_);
int i;
for (i = 0; i < size_; i++)
{
double tau = (i == 0 ? t_[0] - T_ : t_[i] - t_[i - 1]);
double c = (i == size_ - 1 ? (1.0 + rate_ * tau) : rate_ * tau);
lambda[i] = c * A_[i] * Math.Exp(-Ba_[i] * x);
}
//SolvingFunction function = new SolvingFunction(lambda, Bb_);
//verifier le polymorphisme
SolvingFunction function = new SolvingFunction(lambda, Bb_);
Brent s1d = new Brent();
s1d.setMaxEvaluations(1000);
double yb = s1d.solve(function, 1e-6, 0.00, -100.0, 100.0);
double h1 = (yb - muy_) / (sigmay_ * txy) -
rhoxy_ * (x - mux_) / (sigmax_ * txy);
double value = phi.value(-w_ * h1);
for (i = 0; i < size_; i++)
{
double h2 = h1 +
Bb_[i] * sigmay_ *Math.Sqrt(1.0 - rhoxy_ *rhoxy_);
double kappa = -Bb_[i] *
(muy_ - 0.5 * txy * txy * sigmay_ * sigmay_ * Bb_[i] +
rhoxy_ * sigmay_ * (x - mux_) / sigmax_);
value -= lambda[i] * Math.Exp(kappa) * phi.value(-w_ * h2);
}
return(Math.Exp(-0.5 * temp * temp) * value /
(sigmax_ * Math.Sqrt(2.0 * QLNet.Const.M_PI)));
}
public double cumD2(double strike) // N(d2) or N(-d2)
{
double d2_ = 0.0;
double cum_d2_pos_ = 1.0; // N(d2)
double cum_d2_neg_ = 0.0; // N(-d2)
CumulativeNormalDistribution f = new CumulativeNormalDistribution();
if (stdDev_ >= Const.QL_EPSILON)
{
if (strike > 0)
{
d2_ = Math.Log(forward_ / strike) / stdDev_ - 0.5 * stdDev_;
return(f.value(phi_ * d2_));
}
}
else
{
if (forward_ < strike)
{
cum_d2_pos_ = 0.0;
cum_d2_neg_ = 1.0;
}
else if (forward_ == strike)
{
d2_ = -0.5 * stdDev_;
return(f.value(phi_ * d2_));
}
}
if (phi_ > 0)
{
// if Call
return(cum_d2_pos_);
}
else
{
return(cum_d2_neg_);
}
}
public double cumD1(double strike) // N(d1) or N(-d1)
{
double d1_ = 0.0;
double cum_d1_pos_ = 1.0; // N(d1)
double cum_d1_neg_ = 0.0; // N(-d1)
CumulativeNormalDistribution f = new CumulativeNormalDistribution();
if (stdDev_ >= Const.QL_EPSILON)
{
if (strike > 0)
{
d1_ = Math.Log(forward_ / strike) / stdDev_ + 0.5 * stdDev_;
return(f.value(phi_ * d1_));
}
}
else
{
if (forward_ < strike)
{
cum_d1_pos_ = 0.0;
cum_d1_neg_ = 1.0;
}
else if (forward_.IsEqual(strike))
{
d1_ = 0.5 * stdDev_;
return(f.value(phi_ * d1_));
}
}
if (phi_ > 0)
{
// if Call
return(cum_d1_pos_);
}
else
{
return(cum_d1_neg_);
}
}
// calculate the value of euro min basket call
private double euroTwoAssetMinBasketCall(double forward1, double forward2, double strike, double riskFreeDiscount,
double variance1, double variance2, double rho)
{
double stdDev1 = Math.Sqrt(variance1);
double stdDev2 = Math.Sqrt(variance2);
double variance = variance1 + variance2 - 2 * rho * stdDev1 * stdDev2;
double stdDev = Math.Sqrt(variance);
double modRho1 = (rho * stdDev2 - stdDev1) / stdDev;
double modRho2 = (rho * stdDev1 - stdDev2) / stdDev;
double D1 = (Math.Log(forward1 / forward2) + 0.5 * variance) / stdDev;
double alfa, beta, gamma;
if (strike != 0.0)
{
BivariateCumulativeNormalDistribution bivCNorm = new BivariateCumulativeNormalDistribution(rho);
BivariateCumulativeNormalDistribution bivCNormMod2 = new BivariateCumulativeNormalDistribution(modRho2);
BivariateCumulativeNormalDistribution bivCNormMod1 = new BivariateCumulativeNormalDistribution(modRho1);
double D1_1 = (Math.Log(forward1 / strike) + 0.5 * variance1) / stdDev1;
double D1_2 = (Math.Log(forward2 / strike) + 0.5 * variance2) / stdDev2;
alfa = bivCNormMod1.value(D1_1, -D1);
beta = bivCNormMod2.value(D1_2, D1 - stdDev);
gamma = bivCNorm.value(D1_1 - stdDev1, D1_2 - stdDev2);
}
else
{
CumulativeNormalDistribution cum = new CumulativeNormalDistribution();
alfa = cum.value(-D1);
beta = cum.value(D1 - stdDev);
gamma = 1.0;
}
return(riskFreeDiscount * (forward1 * alfa + forward2 * beta - strike * gamma));
}
/*! Black 1976 probability of being in the money (in the bond martingale measure), i.e. N(d2).
* It is a risk-neutral probability, not the real world one.
* \warning instead of volatility it uses standard deviation, i.e. volatility*sqrt(timeToMaturity)
*/
public static double blackFormulaCashItmProbability(Option.Type optionType,
double strike,
double forward,
double stdDev,
double displacement = 0.0)
{
checkParameters(strike, forward, displacement);
if (stdDev.IsEqual(0.0))
{
return(forward * (int)optionType > strike * (int)optionType ? 1.0 : 0.0);
}
forward = forward + displacement;
strike = strike + displacement;
if (strike.IsEqual(0.0))
{
return(optionType == Option.Type.Call ? 1.0 : 0.0);
}
double d2 = Math.Log(forward / strike) / stdDev - 0.5 * stdDev;
CumulativeNormalDistribution phi = new CumulativeNormalDistribution();
return(phi.value((int)optionType * d2));
}
//Hagan, 3.5b, 3.5c
protected override double optionletPrice(Option.Type optionType, double strike)
{
double variance = swaptionVolatility().link.blackVariance(fixingDate_, swapTenor_, swapRateValue_);
double firstDerivativeOfGAtForwardValue = gFunction_.firstDerivative(swapRateValue_);
double price = 0;
double CK = vanillaOptionPricer_.value(strike, optionType, annuity_);
price += (discount_ / annuity_) * CK;
double sqrtSigma2T = Math.Sqrt(variance);
double lnRoverK = Math.Log(swapRateValue_ / strike);
double d32 = (lnRoverK + 1.5 * variance) / sqrtSigma2T;
double d12 = (lnRoverK + .5 * variance) / sqrtSigma2T;
double dminus12 = (lnRoverK - .5 * variance) / sqrtSigma2T;
CumulativeNormalDistribution cumulativeOfNormal = new CumulativeNormalDistribution();
double N32 = cumulativeOfNormal.value(((int)optionType) * d32);
double N12 = cumulativeOfNormal.value(((int)optionType) * d12);
double Nminus12 = cumulativeOfNormal.value(((int)optionType) * dminus12);
price += ((int)optionType) * firstDerivativeOfGAtForwardValue * annuity_ * swapRateValue_ * (swapRateValue_ * Math.Exp(variance) * N32 - (swapRateValue_ + strike) * N12 + strike * Nminus12);
price *= coupon_.accrualPeriod();
return price;
}
//Hagan, 3.5b, 3.5c
protected override double optionletPrice(Option.Type optionType, double strike)
{
double variance = swaptionVolatility().link.blackVariance(fixingDate_, swapTenor_, swapRateValue_);
double firstDerivativeOfGAtForwardValue = gFunction_.firstDerivative(swapRateValue_);
double price = 0;
double CK = vanillaOptionPricer_.value(strike, optionType, annuity_);
price += (discount_ / annuity_) * CK;
double sqrtSigma2T = Math.Sqrt(variance);
double lnRoverK = Math.Log(swapRateValue_ / strike);
double d32 = (lnRoverK + 1.5 * variance) / sqrtSigma2T;
double d12 = (lnRoverK + .5 * variance) / sqrtSigma2T;
double dminus12 = (lnRoverK - .5 * variance) / sqrtSigma2T;
CumulativeNormalDistribution cumulativeOfNormal = new CumulativeNormalDistribution();
double N32 = cumulativeOfNormal.value(((int)optionType) * d32);
double N12 = cumulativeOfNormal.value(((int)optionType) * d12);
double Nminus12 = cumulativeOfNormal.value(((int)optionType) * dminus12);
price += ((int)optionType) * firstDerivativeOfGAtForwardValue * annuity_ * swapRateValue_ * (swapRateValue_ * Math.Exp(variance) * N32 - (swapRateValue_ + strike) * N12 + strike * Nminus12);
price *= coupon_.accrualPeriod();
return(price);
}
// public BlackCalculator(StrikedTypePayoff payoff, double forward, double stdDev, double discount = 1.0) {
public BlackCalculator(StrikedTypePayoff payoff, double forward, double stdDev, double discount) {
strike_ = payoff.strike();
forward_ = forward;
stdDev_ = stdDev;
discount_ = discount;
variance_ = stdDev*stdDev;
if (!(forward>0.0))
throw new ApplicationException("positive forward value required: " + forward + " not allowed");
if (!(stdDev>=0.0))
throw new ApplicationException("non-negative standard deviation required: " + stdDev + " not allowed");
if (!(discount>0.0))
throw new ApplicationException("positive discount required: " + discount + " not allowed");
if (stdDev_>=Const.QL_EPSILON) {
if (strike_==0.0) {
n_d1_ = 0.0;
n_d2_ = 0.0;
cum_d1_ = 1.0;
cum_d2_= 1.0;
} else {
D1_ = Math.Log(forward/strike_)/stdDev_ + 0.5*stdDev_;
D2_ = D1_-stdDev_;
CumulativeNormalDistribution f = new CumulativeNormalDistribution();
cum_d1_ = f.value(D1_);
cum_d2_= f.value(D2_);
n_d1_ = f.derivative(D1_);
n_d2_ = f.derivative(D2_);
}
} else {
if (forward>strike_) {
cum_d1_ = 1.0;
cum_d2_= 1.0;
} else {
cum_d1_ = 0.0;
cum_d2_= 0.0;
}
n_d1_ = 0.0;
n_d2_ = 0.0;
}
X_ = strike_;
DXDstrike_ = 1.0;
// the following one will probably disappear as soon as
// super-share will be properly handled
DXDs_ = 0.0;
// this part is always executed.
// in case of plain-vanilla payoffs, it is also the only part
// which is executed.
switch (payoff.optionType()) {
case Option.Type.Call:
alpha_ = cum_d1_;// N(d1)
DalphaDd1_ = n_d1_;// n(d1)
beta_ = -cum_d2_;// -N(d2)
DbetaDd2_ = - n_d2_;// -n(d2)
break;
case Option.Type.Put:
alpha_ = -1.0+cum_d1_;// -N(-d1)
DalphaDd1_ = n_d1_;// n( d1)
beta_ = 1.0-cum_d2_;// N(-d2)
DbetaDd2_ = - n_d2_;// -n( d2)
break;
default:
throw new ArgumentException("invalid option type");
}
// now dispatch on type.
Calculator calc = new Calculator(this);
payoff.accept(calc);
}
public override void calculate()
{
Utils.QL_REQUIRE(arguments_.exercise.type() == Exercise.Type.American, () => "not an American Option");
AmericanExercise ex = arguments_.exercise as AmericanExercise;
Utils.QL_REQUIRE(ex != null, () => "non-American exercise given");
Utils.QL_REQUIRE(!ex.payoffAtExpiry(), () => "payoff at expiry not handled");
StrikedTypePayoff payoff = arguments_.payoff as StrikedTypePayoff;
Utils.QL_REQUIRE(payoff != null, () => "non-striked payoff given");
double variance = process_.blackVolatility().link.blackVariance(ex.lastDate(), payoff.strike());
double dividendDiscount = process_.dividendYield().link.discount(ex.lastDate());
double riskFreeDiscount = process_.riskFreeRate().link.discount(ex.lastDate());
double spot = process_.stateVariable().link.value();
Utils.QL_REQUIRE(spot > 0.0, () => "negative or null underlying given");
double forwardPrice = spot * dividendDiscount / riskFreeDiscount;
BlackCalculator black = new BlackCalculator(payoff, forwardPrice, Math.Sqrt(variance), riskFreeDiscount);
if (dividendDiscount >= 1.0 && payoff.optionType() == Option.Type.Call)
{
// early exercise never optimal
results_.value = black.value();
results_.delta = black.delta(spot);
results_.deltaForward = black.deltaForward();
results_.elasticity = black.elasticity(spot);
results_.gamma = black.gamma(spot);
DayCounter rfdc = process_.riskFreeRate().link.dayCounter();
DayCounter divdc = process_.dividendYield().link.dayCounter();
DayCounter voldc = process_.blackVolatility().link.dayCounter();
double t = rfdc.yearFraction(process_.riskFreeRate().link.referenceDate(), arguments_.exercise.lastDate());
results_.rho = black.rho(t);
t = divdc.yearFraction(process_.dividendYield().link.referenceDate(), arguments_.exercise.lastDate());
results_.dividendRho = black.dividendRho(t);
t = voldc.yearFraction(process_.blackVolatility().link.referenceDate(), arguments_.exercise.lastDate());
results_.vega = black.vega(t);
results_.theta = black.theta(spot, t);
results_.thetaPerDay = black.thetaPerDay(spot, t);
results_.strikeSensitivity = black.strikeSensitivity();
results_.itmCashProbability = black.itmCashProbability();
}
else
{
// early exercise can be optimal
CumulativeNormalDistribution cumNormalDist = new CumulativeNormalDistribution();
NormalDistribution normalDist = new NormalDistribution();
double tolerance = 1e-6;
double Sk = BaroneAdesiWhaleyApproximationEngine.criticalPrice(payoff, riskFreeDiscount, dividendDiscount,
variance, tolerance);
double forwardSk = Sk * dividendDiscount / riskFreeDiscount;
double alpha = -2.0 * Math.Log(riskFreeDiscount) / (variance);
double beta = 2.0 * Math.Log(dividendDiscount / riskFreeDiscount) / (variance);
double h = 1 - riskFreeDiscount;
double phi = 0;
switch (payoff.optionType())
{
case Option.Type.Call:
phi = 1;
break;
case Option.Type.Put:
phi = -1;
break;
default:
Utils.QL_FAIL("invalid option type");
break;
}
//it can throw: to be fixed
// FLOATING_POINT_EXCEPTION
double temp_root = Math.Sqrt((beta - 1) * (beta - 1) + (4 * alpha) / h);
double lambda = (-(beta - 1) + phi * temp_root) / 2;
double lambda_prime = -phi * alpha / (h * h * temp_root);
double black_Sk = Utils.blackFormula(payoff.optionType(), payoff.strike(), forwardSk, Math.Sqrt(variance)) *
riskFreeDiscount;
double hA = phi * (Sk - payoff.strike()) - black_Sk;
double d1_Sk = (Math.Log(forwardSk / payoff.strike()) + 0.5 * variance) / Math.Sqrt(variance);
double d2_Sk = d1_Sk - Math.Sqrt(variance);
double part1 = forwardSk * normalDist.value(d1_Sk) / (alpha * Math.Sqrt(variance));
double part2 = -phi *forwardSk *cumNormalDist.value(phi *d1_Sk) * Math.Log(dividendDiscount) /
Math.Log(riskFreeDiscount);
double part3 = +phi *payoff.strike() * cumNormalDist.value(phi * d2_Sk);
double V_E_h = part1 + part2 + part3;
double b = (1 - h) * alpha * lambda_prime / (2 * (2 * lambda + beta - 1));
double c = -((1 - h) * alpha / (2 * lambda + beta - 1)) *
(V_E_h / (hA) + 1 / h + lambda_prime / (2 * lambda + beta - 1));
double temp_spot_ratio = Math.Log(spot / Sk);
double chi = temp_spot_ratio * (b * temp_spot_ratio + c);
if (phi * (Sk - spot) > 0)
{
results_.value = black.value() + hA * Math.Pow((spot / Sk), lambda) / (1 - chi);
}
else
{
results_.value = phi * (spot - payoff.strike());
}
double temp_chi_prime = (2 * b / spot) * Math.Log(spot / Sk);
double chi_prime = temp_chi_prime + c / spot;
double chi_double_prime = 2 * b / (spot * spot) - temp_chi_prime / spot - c / (spot * spot);
results_.delta = phi * dividendDiscount * cumNormalDist.value(phi * d1_Sk) +
(lambda / (spot * (1 - chi)) + chi_prime / ((1 - chi) * (1 - chi))) *
(phi * (Sk - payoff.strike()) - black_Sk) * Math.Pow((spot / Sk), lambda);
results_.gamma = phi * dividendDiscount * normalDist.value(phi * d1_Sk) / (spot * Math.Sqrt(variance)) +
(2 * lambda * chi_prime / (spot * (1 - chi) * (1 - chi)) +
2 * chi_prime * chi_prime / ((1 - chi) * (1 - chi) * (1 - chi)) +
chi_double_prime / ((1 - chi) * (1 - chi)) +
lambda * (1 - lambda) / (spot * spot * (1 - chi))) *
(phi * (Sk - payoff.strike()) - black_Sk) * Math.Pow((spot / Sk), lambda);
} // end of "early exercise can be optimal"
}
public double payoffAtExpiry(double spot, double variance, double discount)
{
double dividendDiscount = process_.dividendYield().link.discount(exercise_.lastDate());
Utils.QL_REQUIRE(spot > 0.0, () => "positive spot value required");
Utils.QL_REQUIRE(discount > 0.0, () => "positive discount required");
Utils.QL_REQUIRE(dividendDiscount > 0.0, () => "positive dividend discount required");
Utils.QL_REQUIRE(variance >= 0.0, () => "negative variance not allowed");
Option.Type type = payoff_.optionType();
double strike = payoff_.strike();
double? barrier = arguments_.barrier;
Utils.QL_REQUIRE(barrier > 0.0, () => "positive barrier value required");
Barrier.Type barrierType = arguments_.barrierType;
double stdDev = Math.Sqrt(variance);
double mu = Math.Log(dividendDiscount / discount) / variance - 0.5;
double K = 0;
// binary cash-or-nothing payoff?
CashOrNothingPayoff coo = payoff_ as CashOrNothingPayoff;
if (coo != null)
{
K = coo.cashPayoff();
}
// binary asset-or-nothing payoff?
AssetOrNothingPayoff aoo = payoff_ as AssetOrNothingPayoff;
if (aoo != null)
{
mu += 1.0;
K = spot * dividendDiscount / discount; // forward
}
double log_S_X = Math.Log(spot / strike);
double log_S_H = Math.Log(spot / barrier.GetValueOrDefault());
double log_H_S = Math.Log(barrier.GetValueOrDefault() / spot);
double log_H2_SX = Math.Log(barrier.GetValueOrDefault() * barrier.GetValueOrDefault() / (spot * strike));
double H_S_2mu = Math.Pow(barrier.GetValueOrDefault() / spot, 2 * mu);
double eta = (barrierType == Barrier.Type.DownIn ||
barrierType == Barrier.Type.DownOut ? 1.0 : -1.0);
double phi = (type == Option.Type.Call ? 1.0 : -1.0);
double x1, x2, y1, y2;
double cum_x1, cum_x2, cum_y1, cum_y2;
if (variance >= Const.QL_EPSILON)
{
// we calculate using mu*stddev instead of (mu+1)*stddev
// because cash-or-nothing don't need it. asset-or-nothing
// mu is really mu+1
x1 = phi * (log_S_X / stdDev + mu * stdDev);
x2 = phi * (log_S_H / stdDev + mu * stdDev);
y1 = eta * (log_H2_SX / stdDev + mu * stdDev);
y2 = eta * (log_H_S / stdDev + mu * stdDev);
CumulativeNormalDistribution f = new CumulativeNormalDistribution();
cum_x1 = f.value(x1);
cum_x2 = f.value(x2);
cum_y1 = f.value(y1);
cum_y2 = f.value(y2);
}
else
{
if (log_S_X > 0)
{
cum_x1 = 1.0;
}
else
{
cum_x1 = 0.0;
}
if (log_S_H > 0)
{
cum_x2 = 1.0;
}
else
{
cum_x2 = 0.0;
}
if (log_H2_SX > 0)
{
cum_y1 = 1.0;
}
else
{
cum_y1 = 0.0;
}
if (log_H_S > 0)
{
cum_y2 = 1.0;
}
else
{
cum_y2 = 0.0;
}
}
double alpha = 0;
switch (barrierType)
{
case Barrier.Type.DownIn:
if (type == Option.Type.Call)
{
// down-in and call
if (strike >= barrier)
{
// B3 (eta=1, phi=1)
alpha = H_S_2mu * cum_y1;
}
else
{
// B1-B2+B4 (eta=1, phi=1)
alpha = cum_x1 - cum_x2 + H_S_2mu * cum_y2;
}
}
else
{
// down-in and put
if (strike >= barrier)
{
// B2-B3+B4 (eta=1, phi=-1)
alpha = cum_x2 + H_S_2mu * (-cum_y1 + cum_y2);
}
else
{
// B1 (eta=1, phi=-1)
alpha = cum_x1;
}
}
break;
case Barrier.Type.UpIn:
if (type == Option.Type.Call)
{
// up-in and call
if (strike >= barrier)
{
// B1 (eta=-1, phi=1)
alpha = cum_x1;
}
else
{
// B2-B3+B4 (eta=-1, phi=1)
alpha = cum_x2 + H_S_2mu * (-cum_y1 + cum_y2);
}
}
else
{
// up-in and put
if (strike >= barrier)
{
// B1-B2+B4 (eta=-1, phi=-1)
alpha = cum_x1 - cum_x2 + H_S_2mu * cum_y2;
}
else
{
// B3 (eta=-1, phi=-1)
alpha = H_S_2mu * cum_y1;
}
}
break;
case Barrier.Type.DownOut:
if (type == Option.Type.Call)
{
// down-out and call
if (strike >= barrier)
{
// B1-B3 (eta=1, phi=1)
alpha = cum_x1 - H_S_2mu * cum_y1;
}
else
{
// B2-B4 (eta=1, phi=1)
alpha = cum_x2 - H_S_2mu * cum_y2;
}
}
else
{
// down-out and put
if (strike >= barrier)
{
// B1-B2+B3-B4 (eta=1, phi=-1)
alpha = cum_x1 - cum_x2 + H_S_2mu * (cum_y1 - cum_y2);
}
else
{
// always 0
alpha = 0;
}
}
break;
case Barrier.Type.UpOut:
if (type == Option.Type.Call)
{
// up-out and call
if (strike >= barrier)
{
// always 0
alpha = 0;
}
else
{
// B1-B2+B3-B4 (eta=-1, phi=1)
alpha = cum_x1 - cum_x2 + H_S_2mu * (cum_y1 - cum_y2);
}
}
else
{
// up-out and put
if (strike >= barrier)
{
// B2-B4 (eta=-1, phi=-1)
alpha = cum_x2 - H_S_2mu * cum_y2;
}
else
{
// B1-B3 (eta=-1, phi=-1)
alpha = cum_x1 - H_S_2mu * cum_y1;
}
}
break;
default:
Utils.QL_FAIL("invalid barrier type");
break;
}
return(discount * K * alpha);
}
// critical commodity price
//public static double criticalPrice(StrikedTypePayoff payoff, double riskFreeDiscount, double dividendDiscount,
// double variance, double tolerance = 1e-6);
public static double criticalPrice(StrikedTypePayoff payoff, double riskFreeDiscount, double dividendDiscount,
double variance, double tolerance)
{
// Calculation of seed value, Si
double n = 2.0 * Math.Log(dividendDiscount / riskFreeDiscount) / (variance);
double m = -2.0 * Math.Log(riskFreeDiscount) / (variance);
double bT = Math.Log(dividendDiscount / riskFreeDiscount);
double qu, Su, h, Si;
switch (payoff.optionType())
{
case Option.Type.Call:
qu = (-(n - 1.0) + Math.Sqrt(((n - 1.0) * (n - 1.0)) + 4.0 * m)) / 2.0;
Su = payoff.strike() / (1.0 - 1.0 / qu);
h = -(bT + 2.0 * Math.Sqrt(variance)) * payoff.strike() /
(Su - payoff.strike());
Si = payoff.strike() + (Su - payoff.strike()) *
(1.0 - Math.Exp(h));
break;
case Option.Type.Put:
qu = (-(n - 1.0) - Math.Sqrt(((n - 1.0) * (n - 1.0)) + 4.0 * m)) / 2.0;
Su = payoff.strike() / (1.0 - 1.0 / qu);
h = (bT - 2.0 * Math.Sqrt(variance)) * payoff.strike() /
(payoff.strike() - Su);
Si = Su + (payoff.strike() - Su) * Math.Exp(h);
break;
default:
throw new ArgumentException("unknown option type");
}
// Newton Raphson algorithm for finding critical price Si
double Q, LHS, RHS, bi;
double forwardSi = Si * dividendDiscount / riskFreeDiscount;
double d1 = (Math.Log(forwardSi / payoff.strike()) + 0.5 * variance) /
Math.Sqrt(variance);
CumulativeNormalDistribution cumNormalDist = new CumulativeNormalDistribution();
double K = (riskFreeDiscount != 1.0 ? -2.0 * Math.Log(riskFreeDiscount) /
(variance * (1.0 - riskFreeDiscount)) : 0.0);
double temp = Utils.blackFormula(payoff.optionType(), payoff.strike(),
forwardSi, Math.Sqrt(variance)) * riskFreeDiscount;
switch (payoff.optionType())
{
case Option.Type.Call:
Q = (-(n - 1.0) + Math.Sqrt(((n - 1.0) * (n - 1.0)) + 4 * K)) / 2;
LHS = Si - payoff.strike();
RHS = temp + (1 - dividendDiscount * cumNormalDist.value(d1)) * Si / Q;
bi = dividendDiscount * cumNormalDist.value(d1) * (1 - 1 / Q) +
(1 - dividendDiscount *
cumNormalDist.derivative(d1) / Math.Sqrt(variance)) / Q;
while (Math.Abs(LHS - RHS) / payoff.strike() > tolerance)
{
Si = (payoff.strike() + RHS - bi * Si) / (1 - bi);
forwardSi = Si * dividendDiscount / riskFreeDiscount;
d1 = (Math.Log(forwardSi / payoff.strike()) + 0.5 * variance)
/ Math.Sqrt(variance);
LHS = Si - payoff.strike();
double temp2 = Utils.blackFormula(payoff.optionType(), payoff.strike(),
forwardSi, Math.Sqrt(variance)) * riskFreeDiscount;
RHS = temp2 + (1 - dividendDiscount * cumNormalDist.value(d1)) * Si / Q;
bi = dividendDiscount * cumNormalDist.value(d1) * (1 - 1 / Q)
+ (1 - dividendDiscount *
cumNormalDist.derivative(d1) / Math.Sqrt(variance))
/ Q;
}
break;
case Option.Type.Put:
Q = (-(n - 1.0) - Math.Sqrt(((n - 1.0) * (n - 1.0)) + 4 * K)) / 2;
LHS = payoff.strike() - Si;
RHS = temp - (1 - dividendDiscount * cumNormalDist.value(-d1)) * Si / Q;
bi = -dividendDiscount *cumNormalDist.value(-d1) * (1 - 1 / Q)
- (1 + dividendDiscount * cumNormalDist.derivative(-d1)
/ Math.Sqrt(variance)) / Q;
while (Math.Abs(LHS - RHS) / payoff.strike() > tolerance)
{
Si = (payoff.strike() - RHS + bi * Si) / (1 + bi);
forwardSi = Si * dividendDiscount / riskFreeDiscount;
d1 = (Math.Log(forwardSi / payoff.strike()) + 0.5 * variance)
/ Math.Sqrt(variance);
LHS = payoff.strike() - Si;
double temp2 = Utils.blackFormula(payoff.optionType(), payoff.strike(),
forwardSi, Math.Sqrt(variance)) * riskFreeDiscount;
RHS = temp2 - (1 - dividendDiscount * cumNormalDist.value(-d1)) * Si / Q;
bi = -dividendDiscount *cumNormalDist.value(-d1) * (1 - 1 / Q)
- (1 + dividendDiscount * cumNormalDist.derivative(-d1)
/ Math.Sqrt(variance)) / Q;
}
break;
default:
throw new ArgumentException("unknown option type");
}
return(Si);
}
// critical commodity price
//public static double criticalPrice(StrikedTypePayoff payoff, double riskFreeDiscount, double dividendDiscount,
// double variance, double tolerance = 1e-6);
public static double criticalPrice(StrikedTypePayoff payoff, double riskFreeDiscount, double dividendDiscount,
double variance, double tolerance)
{
// Calculation of seed value, Si
double n= 2.0*Math.Log(dividendDiscount/riskFreeDiscount)/(variance);
double m=-2.0*Math.Log(riskFreeDiscount)/(variance);
double bT = Math.Log(dividendDiscount/riskFreeDiscount);
double qu, Su, h, Si;
switch (payoff.optionType()) {
case Option.Type.Call:
qu = (-(n-1.0) + Math.Sqrt(((n-1.0)*(n-1.0)) + 4.0*m))/2.0;
Su = payoff.strike() / (1.0 - 1.0/qu);
h = -(bT + 2.0*Math.Sqrt(variance)) * payoff.strike() /
(Su - payoff.strike());
Si = payoff.strike() + (Su - payoff.strike()) *
(1.0 - Math.Exp(h));
break;
case Option.Type.Put:
qu = (-(n-1.0) - Math.Sqrt(((n-1.0)*(n-1.0)) + 4.0*m))/2.0;
Su = payoff.strike() / (1.0 - 1.0/qu);
h = (bT - 2.0*Math.Sqrt(variance)) * payoff.strike() /
(payoff.strike() - Su);
Si = Su + (payoff.strike() - Su) * Math.Exp(h);
break;
default:
throw new ArgumentException("unknown option type");
}
// Newton Raphson algorithm for finding critical price Si
double Q, LHS, RHS, bi;
double forwardSi = Si * dividendDiscount / riskFreeDiscount;
double d1 = (Math.Log(forwardSi/payoff.strike()) + 0.5*variance) /
Math.Sqrt(variance);
CumulativeNormalDistribution cumNormalDist = new CumulativeNormalDistribution();
double K = (riskFreeDiscount!=1.0 ? -2.0*Math.Log(riskFreeDiscount)/
(variance*(1.0-riskFreeDiscount)) : 0.0);
double temp = Utils.blackFormula(payoff.optionType(), payoff.strike(),
forwardSi, Math.Sqrt(variance))*riskFreeDiscount;
switch (payoff.optionType()) {
case Option.Type.Call:
Q = (-(n-1.0) + Math.Sqrt(((n-1.0)*(n-1.0)) + 4 * K)) / 2;
LHS = Si - payoff.strike();
RHS = temp + (1 - dividendDiscount * cumNormalDist.value(d1)) * Si / Q;
bi = dividendDiscount * cumNormalDist.value(d1) * (1 - 1 / Q) +
(1 - dividendDiscount *
cumNormalDist.derivative(d1) / Math.Sqrt(variance)) / Q;
while (Math.Abs(LHS - RHS)/payoff.strike() > tolerance) {
Si = (payoff.strike() + RHS - bi * Si) / (1 - bi);
forwardSi = Si * dividendDiscount / riskFreeDiscount;
d1 = (Math.Log(forwardSi/payoff.strike())+0.5*variance)
/Math.Sqrt(variance);
LHS = Si - payoff.strike();
double temp2 = Utils.blackFormula(payoff.optionType(), payoff.strike(),
forwardSi, Math.Sqrt(variance))*riskFreeDiscount;
RHS = temp2 + (1 - dividendDiscount * cumNormalDist.value(d1)) * Si / Q;
bi = dividendDiscount * cumNormalDist.value(d1) * (1 - 1 / Q)
+ (1 - dividendDiscount *
cumNormalDist.derivative(d1) / Math.Sqrt(variance))
/ Q;
}
break;
case Option.Type.Put:
Q = (-(n-1.0) - Math.Sqrt(((n-1.0)*(n-1.0)) + 4 * K)) / 2;
LHS = payoff.strike() - Si;
RHS = temp - (1 - dividendDiscount * cumNormalDist.value(-d1)) * Si / Q;
bi = -dividendDiscount * cumNormalDist.value(-d1) * (1 - 1 / Q)
- (1 + dividendDiscount * cumNormalDist.derivative(-d1)
/ Math.Sqrt(variance)) / Q;
while (Math.Abs(LHS - RHS)/payoff.strike() > tolerance) {
Si = (payoff.strike() - RHS + bi * Si) / (1 + bi);
forwardSi = Si * dividendDiscount / riskFreeDiscount;
d1 = (Math.Log(forwardSi/payoff.strike())+0.5*variance)
/Math.Sqrt(variance);
LHS = payoff.strike() - Si;
double temp2 = Utils.blackFormula(payoff.optionType(), payoff.strike(),
forwardSi, Math.Sqrt(variance))*riskFreeDiscount;
RHS = temp2 - (1 - dividendDiscount * cumNormalDist.value(-d1)) * Si / Q;
bi = -dividendDiscount * cumNormalDist.value(-d1) * (1 - 1 / Q)
- (1 + dividendDiscount * cumNormalDist.derivative(-d1)
/ Math.Sqrt(variance)) / Q;
}
break;
default:
throw new ArgumentException("unknown option type");
}
return Si;
}
// public BlackCalculator(StrikedTypePayoff payoff, double forward, double stdDev, double discount = 1.0) {
public BlackCalculator(StrikedTypePayoff payoff, double forward, double stdDev, double discount)
{
strike_ = payoff.strike();
forward_ = forward;
stdDev_ = stdDev;
discount_ = discount;
variance_ = stdDev * stdDev;
if (!(forward > 0.0))
{
throw new ApplicationException("positive forward value required: " + forward + " not allowed");
}
if (!(stdDev >= 0.0))
{
throw new ApplicationException("non-negative standard deviation required: " + stdDev + " not allowed");
}
if (!(discount > 0.0))
{
throw new ApplicationException("positive discount required: " + discount + " not allowed");
}
if (stdDev_ >= Const.QL_Epsilon)
{
if (strike_ == 0.0)
{
n_d1_ = 0.0;
n_d2_ = 0.0;
cum_d1_ = 1.0;
cum_d2_ = 1.0;
}
else
{
D1_ = Math.Log(forward / strike_) / stdDev_ + 0.5 * stdDev_;
D2_ = D1_ - stdDev_;
CumulativeNormalDistribution f = new CumulativeNormalDistribution();
cum_d1_ = f.value(D1_);
cum_d2_ = f.value(D2_);
n_d1_ = f.derivative(D1_);
n_d2_ = f.derivative(D2_);
}
}
else
{
if (forward > strike_)
{
cum_d1_ = 1.0;
cum_d2_ = 1.0;
}
else
{
cum_d1_ = 0.0;
cum_d2_ = 0.0;
}
n_d1_ = 0.0;
n_d2_ = 0.0;
}
X_ = strike_;
DXDstrike_ = 1.0;
// the following one will probably disappear as soon as
// super-share will be properly handled
DXDs_ = 0.0;
// this part is always executed.
// in case of plain-vanilla payoffs, it is also the only part
// which is executed.
switch (payoff.optionType())
{
case Option.Type.Call:
alpha_ = cum_d1_; // N(d1)
DalphaDd1_ = n_d1_; // n(d1)
beta_ = -cum_d2_; // -N(d2)
DbetaDd2_ = -n_d2_; // -n(d2)
break;
case Option.Type.Put:
alpha_ = -1.0 + cum_d1_; // -N(-d1)
DalphaDd1_ = n_d1_; // n( d1)
beta_ = 1.0 - cum_d2_; // N(-d2)
DbetaDd2_ = -n_d2_; // -n( d2)
break;
default:
throw new ArgumentException("invalid option type");
}
// now dispatch on type.
Calculator calc = new Calculator(this);
payoff.accept(calc);
}
public override void calculate()
{
/* this engine cannot really check for the averageType==Geometric
* since it can be used as control variate for the Arithmetic version
* QL_REQUIRE(arguments_.averageType == Average::Geometric,"not a geometric average option")
*/
Utils.QL_REQUIRE(arguments_.exercise.type() == Exercise.Type.European, () => "not an European Option");
double runningLog;
int pastFixings;
if (arguments_.averageType == Average.Type.Geometric)
{
Utils.QL_REQUIRE(arguments_.runningAccumulator > 0.0, () =>
"positive running product required: " + arguments_.runningAccumulator + " not allowed");
runningLog = Math.Log(arguments_.runningAccumulator.GetValueOrDefault());
pastFixings = arguments_.pastFixings.GetValueOrDefault();
}
else
{ // it is being used as control variate
runningLog = 1.0;
pastFixings = 0;
}
PlainVanillaPayoff payoff = arguments_.payoff as PlainVanillaPayoff;
Utils.QL_REQUIRE(payoff != null, () => "non-plain payoff given");
Date referenceDate = process_.riskFreeRate().link.referenceDate();
DayCounter rfdc = process_.riskFreeRate().link.dayCounter();
DayCounter divdc = process_.dividendYield().link.dayCounter();
DayCounter voldc = process_.blackVolatility().link.dayCounter();
List <double> fixingTimes = new List <double>();
int i;
for (i = 0; i < arguments_.fixingDates.Count; i++)
{
if (arguments_.fixingDates[i] >= referenceDate)
{
double t = voldc.yearFraction(referenceDate, arguments_.fixingDates[i]);
fixingTimes.Add(t);
}
}
int remainingFixings = fixingTimes.Count;
int numberOfFixings = pastFixings + remainingFixings;
double N = numberOfFixings;
double pastWeight = pastFixings / N;
double futureWeight = 1.0 - pastWeight;
double timeSum = 0;
fixingTimes.ForEach((ii, vv) => timeSum += fixingTimes[ii]);
double vola = process_.blackVolatility().link.blackVol(arguments_.exercise.lastDate(), payoff.strike());
double temp = 0.0;
for (i = pastFixings + 1; i < numberOfFixings; i++)
{
temp += fixingTimes[i - pastFixings - 1] * (N - i);
}
double variance = vola * vola / N / N * (timeSum + 2.0 * temp);
double dsigG_dsig = Math.Sqrt((timeSum + 2.0 * temp)) / N;
double sigG = vola * dsigG_dsig;
double dmuG_dsig = -(vola * timeSum) / N;
Date exDate = arguments_.exercise.lastDate();
double dividendRate = process_.dividendYield().link.
zeroRate(exDate, divdc, Compounding.Continuous, Frequency.NoFrequency).rate();
double riskFreeRate = process_.riskFreeRate().link.
zeroRate(exDate, rfdc, Compounding.Continuous, Frequency.NoFrequency).rate();
double nu = riskFreeRate - dividendRate - 0.5 * vola * vola;
double s = process_.stateVariable().link.value();
Utils.QL_REQUIRE(s > 0.0, () => "positive underlying value required");
int M = (pastFixings == 0 ? 1 : pastFixings);
double muG = pastWeight * runningLog / M + futureWeight * Math.Log(s) + nu * timeSum / N;
double forwardPrice = Math.Exp(muG + variance / 2.0);
double riskFreeDiscount = process_.riskFreeRate().link.discount(arguments_.exercise.lastDate());
BlackCalculator black = new BlackCalculator(payoff, forwardPrice, Math.Sqrt(variance), riskFreeDiscount);
results_.value = black.value();
results_.delta = futureWeight * black.delta(forwardPrice) * forwardPrice / s;
results_.gamma = forwardPrice * futureWeight / (s * s)
* (black.gamma(forwardPrice) * futureWeight * forwardPrice
- pastWeight * black.delta(forwardPrice));
double Nx_1, nx_1;
CumulativeNormalDistribution CND = new CumulativeNormalDistribution();
NormalDistribution ND = new NormalDistribution();
if (sigG > Const.QL_EPSILON)
{
double x_1 = (muG - Math.Log(payoff.strike()) + variance) / sigG;
Nx_1 = CND.value(x_1);
nx_1 = ND.value(x_1);
}
else
{
Nx_1 = (muG > Math.Log(payoff.strike()) ? 1.0 : 0.0);
nx_1 = 0.0;
}
results_.vega = forwardPrice * riskFreeDiscount *
((dmuG_dsig + sigG * dsigG_dsig) * Nx_1 + nx_1 * dsigG_dsig);
if (payoff.optionType() == Option.Type.Put)
{
results_.vega -= riskFreeDiscount * forwardPrice *
(dmuG_dsig + sigG * dsigG_dsig);
}
double tRho = rfdc.yearFraction(process_.riskFreeRate().link.referenceDate(),
arguments_.exercise.lastDate());
results_.rho = black.rho(tRho) * timeSum / (N * tRho)
- (tRho - timeSum / N) * results_.value;
double tDiv = divdc.yearFraction(
process_.dividendYield().link.referenceDate(),
arguments_.exercise.lastDate());
results_.dividendRho = black.dividendRho(tDiv) * timeSum / (N * tDiv);
results_.strikeSensitivity = black.strikeSensitivity();
results_.theta = Utils.blackScholesTheta(process_,
results_.value.GetValueOrDefault(),
results_.delta.GetValueOrDefault(),
results_.gamma.GetValueOrDefault());
}
//! gaussian-assumption Shortfall (observations below target)
public double gaussianShortfall(double target)
{
CumulativeNormalDistribution gIntegral = new CumulativeNormalDistribution(this.mean(), this.standardDeviation());
return(gIntegral.value(target));
}
public double value(double x)
{
CumulativeNormalDistribution phi = new CumulativeNormalDistribution();
double temp = (x - mux_)/sigmax_;
double txy = Math.Sqrt(1.0 - rhoxy_*rhoxy_);
Vector lambda = new Vector(size_);
int i;
for (i=0; i<size_; i++) {
double tau = (i==0 ? t_[0] - T_ : t_[i] - t_[i-1]);
double c = (i==size_-1 ? (1.0+rate_*tau) : rate_*tau);
lambda[i] = c*A_[i]*Math.Exp(-Ba_[i]*x);
}
//SolvingFunction function = new SolvingFunction(lambda, Bb_);
//verifier le polymorphisme
SolvingFunction function = new SolvingFunction(lambda, Bb_);
Brent s1d = new Brent();
s1d.setMaxEvaluations(1000);
double yb = s1d.solve(function, 1e-6, 0.00, -100.0, 100.0);
double h1 = (yb - muy_)/(sigmay_*txy) -
rhoxy_*(x - mux_)/(sigmax_*txy);
double value = phi.value(-w_*h1);
for (i=0; i<size_; i++) {
double h2 = h1 +
Bb_[i]*sigmay_*Math.Sqrt(1.0-rhoxy_*rhoxy_);
double kappa = - Bb_[i] *
(muy_ - 0.5*txy*txy*sigmay_*sigmay_*Bb_[i] +
rhoxy_*sigmay_*(x-mux_)/sigmax_);
value -= lambda[i] *Math.Exp(kappa)*phi.value(-w_*h2);
}
return Math.Exp(-0.5*temp*temp)*value/
(sigmax_*Math.Sqrt(2.0*QLNet.Const.M_PI));
}
public override void calculate()
{
/* this engine cannot really check for the averageType==Geometric
since it can be used as control variate for the Arithmetic version
QL_REQUIRE(arguments_.averageType == Average::Geometric,
"not a geometric average option");
*/
if(!(arguments_.exercise.type() == Exercise.Type.European))
throw new ApplicationException("not an European Option");
double runningLog;
int pastFixings;
if (arguments_.averageType == Average.Type.Geometric) {
if(!(arguments_.runningAccumulator>0.0))
throw new ApplicationException("positive running product required: "
+ arguments_.runningAccumulator + " not allowed");
runningLog =
Math.Log(arguments_.runningAccumulator.GetValueOrDefault());
pastFixings = arguments_.pastFixings.GetValueOrDefault();
} else { // it is being used as control variate
runningLog = 1.0;
pastFixings = 0;
}
PlainVanillaPayoff payoff = (PlainVanillaPayoff)(arguments_.payoff);
if (payoff == null)
throw new ApplicationException("non-plain payoff given");
Date referenceDate = process_.riskFreeRate().link.referenceDate();
DayCounter rfdc = process_.riskFreeRate().link.dayCounter();
DayCounter divdc = process_.dividendYield().link.dayCounter();
DayCounter voldc = process_.blackVolatility().link.dayCounter();
List<double> fixingTimes = new InitializedList<double>(arguments_.fixingDates.Count());
int i;
for (i=0; i<arguments_.fixingDates.Count(); i++) {
if (arguments_.fixingDates[i]>=referenceDate) {
double t = voldc.yearFraction(referenceDate,
arguments_.fixingDates[i]);
fixingTimes.Add(t);
}
}
int remainingFixings = fixingTimes.Count();
int numberOfFixings = pastFixings + remainingFixings;
double N = numberOfFixings;
double pastWeight = pastFixings/N;
double futureWeight = 1.0-pastWeight;
/*double timeSum = std::accumulate(fixingTimes.begin(),
fixingTimes.end(), 0.0);*/
double timeSum = 0;
fixingTimes.ForEach((ii, vv) => timeSum += fixingTimes[ii]);
double vola = process_.blackVolatility().link.blackVol(
arguments_.exercise.lastDate(),
payoff.strike());
double temp = 0.0;
for (i=pastFixings+1; i<numberOfFixings; i++)
temp += fixingTimes[i-pastFixings-1]*(N-i);
double variance = vola*vola /N/N * (timeSum+ 2.0*temp);
double dsigG_dsig = Math.Sqrt((timeSum + 2.0*temp))/N;
double sigG = vola * dsigG_dsig;
double dmuG_dsig = -(vola * timeSum)/N;
Date exDate = arguments_.exercise.lastDate();
double dividendRate = process_.dividendYield().link.
zeroRate(exDate, divdc, Compounding.Continuous, Frequency.NoFrequency).rate();
double riskFreeRate = process_.riskFreeRate().link.
zeroRate(exDate, rfdc, Compounding.Continuous, Frequency.NoFrequency).rate();
double nu = riskFreeRate - dividendRate - 0.5*vola*vola;
double s = process_.stateVariable().link.value();
if(!(s > 0.0))
throw new ApplicationException("positive underlying value required");
int M = (pastFixings == 0 ? 1 : pastFixings);
double muG = pastWeight * runningLog/M +
futureWeight * Math.Log(s) + nu*timeSum/N;
double forwardPrice = Math.Exp(muG + variance / 2.0);
double riskFreeDiscount = process_.riskFreeRate().link.discount(
arguments_.exercise.lastDate());
BlackCalculator black = new BlackCalculator(payoff, forwardPrice, Math.Sqrt(variance),
riskFreeDiscount);
results_.value = black.value();
results_.delta = futureWeight*black.delta(forwardPrice)*forwardPrice/s;
results_.gamma = forwardPrice*futureWeight/(s*s)
*( black.gamma(forwardPrice)*futureWeight*forwardPrice
- pastWeight*black.delta(forwardPrice) );
double Nx_1, nx_1;
CumulativeNormalDistribution CND = new CumulativeNormalDistribution();
NormalDistribution ND = new NormalDistribution();
if (sigG > Const.QL_Epsilon) {
double x_1 = (muG-Math.Log(payoff.strike())+variance)/sigG;
Nx_1 = CND.value(x_1);
nx_1 = ND.value( x_1);
} else {
Nx_1 = (muG > Math.Log(payoff.strike()) ? 1.0 : 0.0);
nx_1 = 0.0;
}
results_.vega = forwardPrice * riskFreeDiscount *
( (dmuG_dsig + sigG * dsigG_dsig)*Nx_1 + nx_1*dsigG_dsig );
if (payoff.optionType() == Option.Type.Put)
results_.vega -= riskFreeDiscount * forwardPrice *
(dmuG_dsig + sigG * dsigG_dsig);
double tRho = rfdc.yearFraction(process_.riskFreeRate().link.referenceDate(),
arguments_.exercise.lastDate());
results_.rho = black.rho(tRho)*timeSum/(N*tRho)
- (tRho-timeSum/N)*results_.value;
double tDiv = divdc.yearFraction(
process_.dividendYield().link.referenceDate(),
arguments_.exercise.lastDate());
results_.dividendRho = black.dividendRho(tDiv)*timeSum /(N*tDiv);
results_.strikeSensitivity = black.strikeSensitivity();
results_.theta = Utils.blackScholesTheta(process_,
results_.value.GetValueOrDefault(),
results_.delta.GetValueOrDefault(),
results_.gamma.GetValueOrDefault());
}
public override void calculate()
{
if (!(arguments_.exercise.type() == Exercise.Type.American))
throw new ApplicationException("not an American Option");
AmericanExercise ex = arguments_.exercise as AmericanExercise;
if (ex == null)
throw new ApplicationException("non-American exercise given");
if (ex.payoffAtExpiry())
throw new ApplicationException("payoff at expiry not handled");
StrikedTypePayoff payoff = arguments_.payoff as StrikedTypePayoff;
if (payoff == null)
throw new ApplicationException("non-striked payoff given");
double variance = process_.blackVolatility().link.blackVariance(ex.lastDate(), payoff.strike());
double dividendDiscount = process_.dividendYield().link.discount(ex.lastDate());
double riskFreeDiscount = process_.riskFreeRate().link.discount(ex.lastDate());
double spot = process_.stateVariable().link.value();
if (!(spot > 0.0))
throw new ApplicationException("negative or null underlying given");
double forwardPrice = spot * dividendDiscount / riskFreeDiscount;
BlackCalculator black = new BlackCalculator(payoff, forwardPrice, Math.Sqrt(variance), riskFreeDiscount);
if (dividendDiscount>=1.0 && payoff.optionType()==Option.Type.Call)
{
// early exercise never optimal
results_.value = black.value();
results_.delta = black.delta(spot);
results_.deltaForward = black.deltaForward();
results_.elasticity = black.elasticity(spot);
results_.gamma = black.gamma(spot);
DayCounter rfdc = process_.riskFreeRate().link.dayCounter();
DayCounter divdc = process_.dividendYield().link.dayCounter();
DayCounter voldc = process_.blackVolatility().link.dayCounter();
double t = rfdc.yearFraction(process_.riskFreeRate().link.referenceDate(), arguments_.exercise.lastDate());
results_.rho = black.rho(t);
t = divdc.yearFraction(process_.dividendYield().link.referenceDate(), arguments_.exercise.lastDate());
results_.dividendRho = black.dividendRho(t);
t = voldc.yearFraction(process_.blackVolatility().link.referenceDate(), arguments_.exercise.lastDate());
results_.vega = black.vega(t);
results_.theta = black.theta(spot, t);
results_.thetaPerDay = black.thetaPerDay(spot, t);
results_.strikeSensitivity = black.strikeSensitivity();
results_.itmCashProbability = black.itmCashProbability();
}
else
{
// early exercise can be optimal
CumulativeNormalDistribution cumNormalDist = new CumulativeNormalDistribution();
NormalDistribution normalDist = new NormalDistribution();
double tolerance = 1e-6;
double Sk = BaroneAdesiWhaleyApproximationEngine.criticalPrice(payoff, riskFreeDiscount, dividendDiscount, variance, tolerance);
double forwardSk = Sk * dividendDiscount / riskFreeDiscount;
double alpha = -2.0 *Math.Log(riskFreeDiscount)/(variance);
double beta = 2.0 *Math.Log(dividendDiscount/riskFreeDiscount)/ (variance);
double h = 1 - riskFreeDiscount;
double phi;
switch (payoff.optionType())
{
case Option.Type.Call:
phi = 1;
break;
case Option.Type.Put:
phi = -1;
break;
default:
throw new ArgumentException("invalid option type");
}
//it can throw: to be fixed
// FLOATING_POINT_EXCEPTION
double temp_root = Math.Sqrt ((beta-1)*(beta-1) + (4 *alpha)/h);
double lambda = (-(beta-1) + phi * temp_root) / 2;
double lambda_prime = - phi * alpha / (h *h * temp_root);
double black_Sk = Utils.blackFormula(payoff.optionType(), payoff.strike(), forwardSk, Math.Sqrt(variance)) * riskFreeDiscount;
double hA = phi * (Sk - payoff.strike()) - black_Sk;
double d1_Sk = (Math.Log(forwardSk/payoff.strike()) + 0.5 *variance) /Math.Sqrt(variance);
double d2_Sk = d1_Sk - Math.Sqrt(variance);
double part1 = forwardSk * normalDist.value(d1_Sk) / (alpha * Math.Sqrt(variance));
double part2 = - phi * forwardSk * cumNormalDist.value(phi * d1_Sk) * Math.Log(dividendDiscount) / Math.Log(riskFreeDiscount);
double part3 = + phi * payoff.strike() * cumNormalDist.value(phi * d2_Sk);
double V_E_h = part1 + part2 + part3;
double b = (1-h) * alpha * lambda_prime / (2*(2 *lambda + beta - 1));
double c = - ((1 - h) * alpha / (2 * lambda + beta - 1)) * (V_E_h / (hA) + 1 / h + lambda_prime / (2 *lambda + beta - 1));
double temp_spot_ratio = Math.Log(spot / Sk);
double chi = temp_spot_ratio * (b * temp_spot_ratio + c);
if (phi*(Sk-spot) > 0)
{
results_.value = black.value() + hA * Math.Pow((spot/Sk), lambda) / (1 - chi);
}
else
{
results_.value = phi * (spot - payoff.strike());
}
double temp_chi_prime = (2 * b / spot) * Math.Log(spot/Sk);
double chi_prime = temp_chi_prime + c / spot;
double chi_double_prime = 2 *b/(spot *spot) - temp_chi_prime / spot - c / (spot *spot);
results_.delta = phi * dividendDiscount * cumNormalDist.value (phi * d1_Sk) + (lambda / (spot * (1 - chi)) + chi_prime / ((1 - chi)*(1 - chi))) * (phi * (Sk - payoff.strike()) - black_Sk) * Math.Pow((spot/Sk), lambda);
results_.gamma = phi * dividendDiscount * normalDist.value (phi *d1_Sk) / (spot * Math.Sqrt(variance)) + (2 * lambda * chi_prime / (spot * (1 - chi) * (1 - chi)) + 2 * chi_prime * chi_prime / ((1 - chi) * (1 - chi) * (1 - chi)) + chi_double_prime / ((1 - chi) * (1 - chi)) + lambda * (1 - lambda) / (spot * spot * (1 - chi))) * (phi * (Sk - payoff.strike()) - black_Sk) * Math.Pow((spot/Sk), lambda);
} // end of "early exercise can be optimal"
}
public override void calculate()
{
// First: tests on types
Utils.QL_REQUIRE(arguments_.exercise.type() == Exercise.Type.European, () =>
"not an European Option");
PlainVanillaPayoff payoff = arguments_.payoff as PlainVanillaPayoff;
Utils.QL_REQUIRE(payoff != null, () => "not a plain-vanilla payoff");
// forward values - futures, so b=0
double forward1 = process1_.stateVariable().link.value();
double forward2 = process2_.stateVariable().link.value();
Date exerciseDate = arguments_.exercise.lastDate();
// Volatilities
double sigma1 = process1_.blackVolatility().link.blackVol(exerciseDate,
forward1);
double sigma2 = process2_.blackVolatility().link.blackVol(exerciseDate,
forward2);
double riskFreeDiscount = process1_.riskFreeRate().link.discount(exerciseDate);
double strike = payoff.strike();
// Unique F (forward) value for pricing
double F = forward1 / (forward2 + strike);
// Its volatility
double sigma =
Math.Sqrt(Math.Pow(sigma1, 2)
+ Math.Pow((sigma2 * (forward2 / (forward2 + strike))), 2)
- 2 * rho_.link.value() * sigma1 * sigma2 * (forward2 / (forward2 + strike)));
// Day counter and Dates handling variables
DayCounter rfdc = process1_.riskFreeRate().link.dayCounter();
double t = rfdc.yearFraction(process1_.riskFreeRate().link.referenceDate(),
arguments_.exercise.lastDate());
// Black-Scholes solution values
double d1 = (Math.Log(F) + 0.5 * Math.Pow(sigma,
2) * t) / (sigma * Math.Sqrt(t));
double d2 = d1 - sigma * Math.Sqrt(t);
NormalDistribution pdf = new NormalDistribution();
CumulativeNormalDistribution cum = new CumulativeNormalDistribution();
double Nd1 = cum.value(d1);
double Nd2 = cum.value(d2);
double NMd1 = cum.value(-d1);
double NMd2 = cum.value(-d2);
Option.Type optionType = payoff.optionType();
if (optionType == Option.Type.Call)
{
results_.value = riskFreeDiscount * (F * Nd1 - Nd2) * (forward2 + strike);
}
else
{
results_.value = riskFreeDiscount * (NMd2 - F * NMd1) * (forward2 + strike);
}
double?callValue = optionType == Option.Type.Call ? results_.value :
riskFreeDiscount * (F * Nd1 - Nd2) * (forward2 + strike);
results_.theta = (Math.Log(riskFreeDiscount) / t) * callValue +
riskFreeDiscount * (forward1 * sigma) / (2 * Math.Sqrt(t)) * pdf.value(d1);
}
public override void calculate()
{
if (!(arguments_.exercise.type() == Exercise.Type.American))
throw new ApplicationException("not an American Option");
AmericanExercise ex = arguments_.exercise as AmericanExercise;
if (ex == null) throw new ApplicationException("non-American exercise given");
if(ex.payoffAtExpiry()) throw new ApplicationException("payoff at expiry not handled");
StrikedTypePayoff payoff = arguments_.payoff as StrikedTypePayoff;
if (payoff == null) throw new ApplicationException("non-striked payoff given");
double variance = process_.blackVolatility().link.blackVariance(ex.lastDate(), payoff.strike());
double dividendDiscount = process_.dividendYield().link.discount(ex.lastDate());
double riskFreeDiscount = process_.riskFreeRate().link.discount(ex.lastDate());
double spot = process_.stateVariable().link.value();
if (!(spot > 0.0)) throw new ApplicationException("negative or null underlying given");
double forwardPrice = spot * dividendDiscount / riskFreeDiscount;
BlackCalculator black = new BlackCalculator(payoff, forwardPrice, Math.Sqrt(variance), riskFreeDiscount);
if (dividendDiscount>=1.0 && payoff.optionType()==Option.Type.Call) {
// early exercise never optimal
results_.value = black.value();
results_.delta = black.delta(spot);
results_.deltaForward = black.deltaForward();
results_.elasticity = black.elasticity(spot);
results_.gamma = black.gamma(spot);
DayCounter rfdc = process_.riskFreeRate().link.dayCounter();
DayCounter divdc = process_.dividendYield().link.dayCounter();
DayCounter voldc = process_.blackVolatility().link.dayCounter();
double t = rfdc.yearFraction(process_.riskFreeRate().link.referenceDate(), arguments_.exercise.lastDate());
results_.rho = black.rho(t);
t = divdc.yearFraction(process_.dividendYield().link.referenceDate(), arguments_.exercise.lastDate());
results_.dividendRho = black.dividendRho(t);
t = voldc.yearFraction(process_.blackVolatility().link.referenceDate(), arguments_.exercise.lastDate());
results_.vega = black.vega(t);
results_.theta = black.theta(spot, t);
results_.thetaPerDay = black.thetaPerDay(spot, t);
results_.strikeSensitivity = black.strikeSensitivity();
results_.itmCashProbability = black.itmCashProbability();
} else {
// early exercise can be optimal
CumulativeNormalDistribution cumNormalDist = new CumulativeNormalDistribution();
double tolerance = 1e-6;
double Sk = criticalPrice(payoff, riskFreeDiscount,
dividendDiscount, variance, tolerance);
double forwardSk = Sk * dividendDiscount / riskFreeDiscount;
double d1 = (Math.Log(forwardSk/payoff.strike()) + 0.5*variance)
/Math.Sqrt(variance);
double n = 2.0*Math.Log(dividendDiscount/riskFreeDiscount)/variance;
double K = -2.0*Math.Log(riskFreeDiscount)/
(variance*(1.0-riskFreeDiscount));
double Q, a;
switch (payoff.optionType()) {
case Option.Type.Call:
Q = (-(n-1.0) + Math.Sqrt(((n-1.0)*(n-1.0))+4.0*K))/2.0;
a = (Sk/Q) * (1.0 - dividendDiscount * cumNormalDist.value(d1));
if (spot<Sk) {
results_.value = black.value() +
a * Math.Pow((spot/Sk), Q);
} else {
results_.value = spot - payoff.strike();
}
break;
case Option.Type.Put:
Q = (-(n-1.0) - Math.Sqrt(((n-1.0)*(n-1.0))+4.0*K))/2.0;
a = -(Sk/Q) *
(1.0 - dividendDiscount * cumNormalDist.value(-d1));
if (spot>Sk) {
results_.value = black.value() +
a * Math.Pow((spot / Sk), Q);
} else {
results_.value = payoff.strike() - spot;
}
break;
default:
throw new ApplicationException("unknown option type");
}
} // end of "early exercise can be optimal"
}
public override void calculate()
{
if (!(arguments_.exercise.type() == Exercise.Type.American))
{
throw new ApplicationException("not an American Option");
}
AmericanExercise ex = arguments_.exercise as AmericanExercise;
if (ex == null)
{
throw new ApplicationException("non-American exercise given");
}
if (ex.payoffAtExpiry())
{
throw new ApplicationException("payoff at expiry not handled");
}
StrikedTypePayoff payoff = arguments_.payoff as StrikedTypePayoff;
if (payoff == null)
{
throw new ApplicationException("non-striked payoff given");
}
double variance = process_.blackVolatility().link.blackVariance(ex.lastDate(), payoff.strike());
double dividendDiscount = process_.dividendYield().link.discount(ex.lastDate());
double riskFreeDiscount = process_.riskFreeRate().link.discount(ex.lastDate());
double spot = process_.stateVariable().link.value();
if (!(spot > 0.0))
{
throw new ApplicationException("negative or null underlying given");
}
double forwardPrice = spot * dividendDiscount / riskFreeDiscount;
BlackCalculator black = new BlackCalculator(payoff, forwardPrice, Math.Sqrt(variance), riskFreeDiscount);
if (dividendDiscount >= 1.0 && payoff.optionType() == Option.Type.Call)
{
// early exercise never optimal
results_.value = black.value();
results_.delta = black.delta(spot);
results_.deltaForward = black.deltaForward();
results_.elasticity = black.elasticity(spot);
results_.gamma = black.gamma(spot);
DayCounter rfdc = process_.riskFreeRate().link.dayCounter();
DayCounter divdc = process_.dividendYield().link.dayCounter();
DayCounter voldc = process_.blackVolatility().link.dayCounter();
double t = rfdc.yearFraction(process_.riskFreeRate().link.referenceDate(), arguments_.exercise.lastDate());
results_.rho = black.rho(t);
t = divdc.yearFraction(process_.dividendYield().link.referenceDate(), arguments_.exercise.lastDate());
results_.dividendRho = black.dividendRho(t);
t = voldc.yearFraction(process_.blackVolatility().link.referenceDate(), arguments_.exercise.lastDate());
results_.vega = black.vega(t);
results_.theta = black.theta(spot, t);
results_.thetaPerDay = black.thetaPerDay(spot, t);
results_.strikeSensitivity = black.strikeSensitivity();
results_.itmCashProbability = black.itmCashProbability();
}
else
{
// early exercise can be optimal
CumulativeNormalDistribution cumNormalDist = new CumulativeNormalDistribution();
double tolerance = 1e-6;
double Sk = criticalPrice(payoff, riskFreeDiscount,
dividendDiscount, variance, tolerance);
double forwardSk = Sk * dividendDiscount / riskFreeDiscount;
double d1 = (Math.Log(forwardSk / payoff.strike()) + 0.5 * variance)
/ Math.Sqrt(variance);
double n = 2.0 * Math.Log(dividendDiscount / riskFreeDiscount) / variance;
double K = -2.0 * Math.Log(riskFreeDiscount) /
(variance * (1.0 - riskFreeDiscount));
double Q, a;
switch (payoff.optionType())
{
case Option.Type.Call:
Q = (-(n - 1.0) + Math.Sqrt(((n - 1.0) * (n - 1.0)) + 4.0 * K)) / 2.0;
a = (Sk / Q) * (1.0 - dividendDiscount * cumNormalDist.value(d1));
if (spot < Sk)
{
results_.value = black.value() +
a * Math.Pow((spot / Sk), Q);
}
else
{
results_.value = spot - payoff.strike();
}
break;
case Option.Type.Put:
Q = (-(n - 1.0) - Math.Sqrt(((n - 1.0) * (n - 1.0)) + 4.0 * K)) / 2.0;
a = -(Sk / Q) *
(1.0 - dividendDiscount * cumNormalDist.value(-d1));
if (spot > Sk)
{
results_.value = black.value() +
a * Math.Pow((spot / Sk), Q);
}
else
{
results_.value = payoff.strike() - spot;
}
break;
default:
throw new ApplicationException("unknown option type");
}
} // end of "early exercise can be optimal"
}
//private double DXDstrike_;
public AmericanPayoffAtExpiry(double spot, double discount, double dividendDiscount, double variance, StrikedTypePayoff payoff)
{
spot_ = spot;
discount_ = discount;
dividendDiscount_ = dividendDiscount;
variance_ = variance;
if (!(spot_ > 0.0))
{
throw new ApplicationException("positive spot_ value required");
}
forward_ = spot_ * dividendDiscount_ / discount_;
if (!(discount_ > 0.0))
{
throw new ApplicationException("positive discount required");
}
if (!(dividendDiscount_ > 0.0))
{
throw new ApplicationException("positive dividend discount_ required");
}
if (!(variance_ >= 0.0))
{
throw new ApplicationException("negative variance_ not allowed");
}
stdDev_ = Math.Sqrt(variance_);
Option.Type type = payoff.optionType();
strike_ = payoff.strike();
mu_ = Math.Log(dividendDiscount_ / discount_) / variance_ - 0.5;
// binary cash-or-nothing payoff?
CashOrNothingPayoff coo = payoff as CashOrNothingPayoff;
if (coo != null)
{
K_ = coo.cashPayoff();
//DKDstrike_ = 0.0;
}
// binary asset-or-nothing payoff?
AssetOrNothingPayoff aoo = payoff as AssetOrNothingPayoff;
if (aoo != null)
{
K_ = forward_;
//DKDstrike_ = 0.0;
mu_ += 1.0;
}
log_H_S_ = Math.Log(strike_ / spot_);
double n_d1;
double n_d2;
double cum_d1_;
double cum_d2_;
if (variance_ >= Const.QL_Epsilon)
{
D1_ = log_H_S_ / stdDev_ + mu_ * stdDev_;
D2_ = D1_ - 2.0 * mu_ * stdDev_;
CumulativeNormalDistribution f = new CumulativeNormalDistribution();
cum_d1_ = f.value(D1_);
cum_d2_ = f.value(D2_);
n_d1 = f.derivative(D1_);
n_d2 = f.derivative(D2_);
}
else
{
if (log_H_S_ > 0)
{
cum_d1_ = 1.0;
cum_d2_ = 1.0;
}
else
{
cum_d1_ = 0.0;
cum_d2_ = 0.0;
}
n_d1 = 0.0;
n_d2 = 0.0;
}
switch (type)
{
// up-and-in cash-(at-hit)-or-nothing option
// a.k.a. american call with cash-or-nothing payoff
case Option.Type.Call:
if (strike_ > spot_)
{
alpha_ = 1.0 - cum_d2_; // N(-d2)
DalphaDd1_ = -n_d2; // -n( d2)
beta_ = 1.0 - cum_d1_; // N(-d1)
DbetaDd2_ = -n_d1; // -n( d1)
}
else
{
alpha_ = 0.5;
DalphaDd1_ = 0.0;
beta_ = 0.5;
DbetaDd2_ = 0.0;
}
break;
// down-and-in cash-(at-hit)-or-nothing option
// a.k.a. american put with cash-or-nothing payoff
case Option.Type.Put:
if (strike_ < spot_)
{
alpha_ = cum_d2_; // N(d2)
DalphaDd1_ = n_d2; // n(d2)
beta_ = cum_d1_; // N(d1)
DbetaDd2_ = n_d1; // n(d1)
}
else
{
alpha_ = 0.5;
DalphaDd1_ = 0.0;
beta_ = 0.5;
DbetaDd2_ = 0.0;
}
break;
default:
throw new ApplicationException("invalid option type");
}
inTheMoney_ = (type == Option.Type.Call && strike_ < spot_) || (type == Option.Type.Put && strike_ > spot_);
if (inTheMoney_)
{
Y_ = 1.0;
X_ = 1.0;
//DYDstrike_ = 0.0;
//DXDstrike_ = 0.0;
}
else
{
Y_ = 1.0;
X_ = Math.Pow((double)(strike_ / spot_), (double)(2.0 * mu_));
// DXDstrike_ = ......;
}
}
public void testOperatorConsistency()
{
//("Testing differential operators...");
NormalDistribution normal = new NormalDistribution(average, sigma);
CumulativeNormalDistribution cum = new CumulativeNormalDistribution(average, sigma);
double xMin = average - 4 * sigma,
xMax = average + 4 * sigma;
int N = 10001;
double h = (xMax - xMin) / (N - 1);
Vector x = new Vector(N),
y = new Vector(N),
yi = new Vector(N),
yd = new Vector(N),
temp = new Vector(N),
diff = new Vector(N);
for (int i = 0; i < N; i++)
x[i] = xMin + h * i;
for (int i = 0; i < x.Count; i++)
y[i] = normal.value(x[i]);
for (int i = 0; i < x.Count; i++)
yi[i] = cum.value(x[i]);
for (int i = 0; i < x.size(); i++)
yd[i] = normal.derivative(x[i]);
// define the differential operators
DZero D = new DZero(N, h);
DPlusDMinus D2 = new DPlusDMinus(N, h);
// check that the derivative of cum is Gaussian
temp = D.applyTo(yi);
for (int i = 0; i < y.Count; i++)
diff[i] = y[i] - temp[i];
double e = Utilities.norm(diff, diff.size(), h);
if (e > 1.0e-6)
{
Assert.Fail("norm of 1st derivative of cum minus Gaussian: " + e + "\ntolerance exceeded");
}
// check that the second derivative of cum is normal.derivative
temp = D2.applyTo(yi);
for (int i = 0; i < yd.Count; i++)
diff[i] = yd[i] - temp[i];
e = Utilities.norm(diff, diff.size(), h);
if (e > 1.0e-4)
{
Assert.Fail("norm of 2nd derivative of cum minus Gaussian derivative: " + e + "\ntolerance exceeded");
}
}
public AmericanPayoffAtHit(double spot, double discount, double dividendDiscount, double variance, StrikedTypePayoff payoff) {
spot_ = spot;
discount_ = discount;
dividendDiscount_ = dividendDiscount;
variance_ = variance;
if (!(spot_ > 0.0))
throw new ApplicationException("positive spot value required");
if (!(discount_ > 0.0))
throw new ApplicationException("positive discount required");
if (!(dividendDiscount_ > 0.0))
throw new ApplicationException("positive dividend discount required");
if (!(variance_ >= 0.0))
throw new ApplicationException("negative variance not allowed");
stdDev_ = Math.Sqrt(variance_);
Option.Type type = payoff.optionType();
strike_ = payoff.strike();
log_H_S_ = Math.Log(strike_ / spot_);
double n_d1;
double n_d2;
double cum_d1_;
double cum_d2_;
if (variance_ >= Const.QL_Epsilon) {
if (discount_ == 0.0 && dividendDiscount_ == 0.0) {
mu_ = -0.5;
lambda_ = 0.5;
} else if (discount_ == 0.0) {
throw new ApplicationException("null discount not handled yet");
} else {
mu_ = Math.Log(dividendDiscount_ / discount_) / variance_ - 0.5;
lambda_ = Math.Sqrt(mu_ * mu_ - 2.0 * Math.Log(discount_) / variance_);
}
D1_ = log_H_S_ / stdDev_ + lambda_ * stdDev_;
D2_ = D1_ - 2.0 * lambda_ * stdDev_;
CumulativeNormalDistribution f = new CumulativeNormalDistribution();
cum_d1_ = f.value(D1_);
cum_d2_ = f.value(D2_);
n_d1 = f.derivative(D1_);
n_d2 = f.derivative(D2_);
} else {
// not tested yet
mu_ = Math.Log(dividendDiscount_ / discount_) / variance_ - 0.5;
lambda_ = Math.Sqrt(mu_ * mu_ - 2.0 * Math.Log(discount_) / variance_);
if (log_H_S_ > 0) {
cum_d1_ = 1.0;
cum_d2_ = 1.0;
} else {
cum_d1_ = 0.0;
cum_d2_ = 0.0;
}
n_d1 = 0.0;
n_d2 = 0.0;
}
switch (type) {
// up-and-in cash-(at-hit)-or-nothing option
// a.k.a. american call with cash-or-nothing payoff
case Option.Type.Call:
if (strike_ > spot_) {
alpha_ = 1.0 - cum_d1_; // N(-d1)
DalphaDd1_ = -n_d1; // -n( d1)
beta_ = 1.0 - cum_d2_; // N(-d2)
DbetaDd2_ = -n_d2; // -n( d2)
} else {
alpha_ = 0.5;
DalphaDd1_ = 0.0;
beta_ = 0.5;
DbetaDd2_ = 0.0;
}
break;
// down-and-in cash-(at-hit)-or-nothing option
// a.k.a. american put with cash-or-nothing payoff
case Option.Type.Put:
if (strike_ < spot_) {
alpha_ = cum_d1_; // N(d1)
DalphaDd1_ = n_d1; // n(d1)
beta_ = cum_d2_; // N(d2)
DbetaDd2_ = n_d2; // n(d2)
} else {
alpha_ = 0.5;
DalphaDd1_ = 0.0;
beta_ = 0.5;
DbetaDd2_ = 0.0;
}
break;
default:
throw new ApplicationException("invalid option type");
}
muPlusLambda_ = mu_ + lambda_;
muMinusLambda_ = mu_ - lambda_;
inTheMoney_ = (type == Option.Type.Call && strike_ < spot_) || (type == Option.Type.Put && strike_ > spot_);
if (inTheMoney_) {
forward_ = 1.0;
X_ = 1.0;
DXDstrike_ = 0.0;
} else {
forward_ = Math.Pow(strike_ / spot_, muPlusLambda_);
X_ = Math.Pow(strike_ / spot_, muMinusLambda_);
// DXDstrike_ = ......;
}
// Binary Cash-Or-Nothing payoff?
CashOrNothingPayoff coo = payoff as CashOrNothingPayoff;
if (coo != null) {
K_ = coo.cashPayoff();
DKDstrike_ = 0.0;
}
// Binary Asset-Or-Nothing payoff?
AssetOrNothingPayoff aoo = payoff as AssetOrNothingPayoff;
if (aoo != null) {
if (inTheMoney_) {
K_ = spot_;
DKDstrike_ = 0.0;
} else {
K_ = aoo.strike();
DKDstrike_ = 1.0;
}
}
}
//private double DXDstrike_;
public AmericanPayoffAtExpiry(double spot, double discount, double dividendDiscount, double variance, StrikedTypePayoff payoff) {
spot_ = spot;
discount_ = discount;
dividendDiscount_ = dividendDiscount;
variance_ = variance;
if (!(spot_ > 0.0))
throw new ApplicationException("positive spot_ value required");
forward_ = spot_ * dividendDiscount_ / discount_;
if (!(discount_ > 0.0))
throw new ApplicationException("positive discount required");
if (!(dividendDiscount_ > 0.0))
throw new ApplicationException("positive dividend discount_ required");
if (!(variance_ >= 0.0))
throw new ApplicationException("negative variance_ not allowed");
stdDev_ = Math.Sqrt(variance_);
Option.Type type = payoff.optionType();
strike_ = payoff.strike();
mu_ = Math.Log(dividendDiscount_ / discount_) / variance_ - 0.5;
// binary cash-or-nothing payoff?
CashOrNothingPayoff coo = payoff as CashOrNothingPayoff;
if (coo != null) {
K_ = coo.cashPayoff();
//DKDstrike_ = 0.0;
}
// binary asset-or-nothing payoff?
AssetOrNothingPayoff aoo = payoff as AssetOrNothingPayoff;
if (aoo != null) {
K_ = forward_;
//DKDstrike_ = 0.0;
mu_ += 1.0;
}
log_H_S_ = Math.Log(strike_ / spot_);
double n_d1;
double n_d2;
double cum_d1_;
double cum_d2_;
if (variance_ >= Const.QL_EPSILON) {
D1_ = log_H_S_ / stdDev_ + mu_ * stdDev_;
D2_ = D1_ - 2.0 * mu_ * stdDev_;
CumulativeNormalDistribution f = new CumulativeNormalDistribution();
cum_d1_ = f.value(D1_);
cum_d2_ = f.value(D2_);
n_d1 = f.derivative(D1_);
n_d2 = f.derivative(D2_);
} else {
if (log_H_S_ > 0) {
cum_d1_ = 1.0;
cum_d2_ = 1.0;
} else {
cum_d1_ = 0.0;
cum_d2_ = 0.0;
}
n_d1 = 0.0;
n_d2 = 0.0;
}
switch (type) {
// up-and-in cash-(at-hit)-or-nothing option
// a.k.a. american call with cash-or-nothing payoff
case Option.Type.Call:
if (strike_ > spot_) {
alpha_ = 1.0 - cum_d2_; // N(-d2)
DalphaDd1_ = -n_d2; // -n( d2)
beta_ = 1.0 - cum_d1_; // N(-d1)
DbetaDd2_ = -n_d1; // -n( d1)
} else {
alpha_ = 0.5;
DalphaDd1_ = 0.0;
beta_ = 0.5;
DbetaDd2_ = 0.0;
}
break;
// down-and-in cash-(at-hit)-or-nothing option
// a.k.a. american put with cash-or-nothing payoff
case Option.Type.Put:
if (strike_ < spot_) {
alpha_ = cum_d2_; // N(d2)
DalphaDd1_ = n_d2; // n(d2)
beta_ = cum_d1_; // N(d1)
DbetaDd2_ = n_d1; // n(d1)
} else {
alpha_ = 0.5;
DalphaDd1_ = 0.0;
beta_ = 0.5;
DbetaDd2_ = 0.0;
}
break;
default:
throw new ApplicationException("invalid option type");
}
inTheMoney_ = (type == Option.Type.Call && strike_ < spot_) || (type == Option.Type.Put && strike_ > spot_);
if (inTheMoney_) {
Y_ = 1.0;
X_ = 1.0;
//DYDstrike_ = 0.0;
//DXDstrike_ = 0.0;
} else {
Y_ = 1.0;
X_ = Math.Pow((double)(strike_ / spot_), (double)(2.0 * mu_));
// DXDstrike_ = ......;
}
}
public override Vector evolve(double t0, Vector x0, double dt, Vector dw)
{
Vector retVal = new Vector(2);
double vol, vol2, mu, nu, dy;
double sdt = Math.Sqrt(dt);
double sqrhov = Math.Sqrt(1.0 - rho_ * rho_);
switch (discretization_)
{
// For the definition of PartialTruncation, FullTruncation
// and Reflection see Lord, R., R. Koekkoek and D. van Dijk (2006),
// "A Comparison of biased simulation schemes for
// stochastic volatility models",
// Working Paper, Tinbergen Institute
case Discretization.PartialTruncation:
vol = (x0[1] > 0.0) ? Math.Sqrt(x0[1]) : 0.0;
vol2 = sigma_ * vol;
mu = riskFreeRate_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()
- dividendYield_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()
- 0.5 * vol * vol;
nu = kappa_ * (theta_ - x0[1]);
retVal[0] = x0[0] * Math.Exp(mu * dt + vol * dw[0] * sdt);
retVal[1] = x0[1] + nu * dt + vol2 * sdt * (rho_ * dw[0] + sqrhov * dw[1]);
break;
case Discretization.FullTruncation:
vol = (x0[1] > 0.0) ? Math.Sqrt(x0[1]) : 0.0;
vol2 = sigma_ * vol;
mu = riskFreeRate_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()
- dividendYield_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()
- 0.5 * vol * vol;
nu = kappa_ * (theta_ - vol * vol);
retVal[0] = x0[0] * Math.Exp(mu * dt + vol * dw[0] * sdt);
retVal[1] = x0[1] + nu * dt + vol2 * sdt * (rho_ * dw[0] + sqrhov * dw[1]);
break;
case Discretization.Reflection:
vol = Math.Sqrt(Math.Abs(x0[1]));
vol2 = sigma_ * vol;
mu = riskFreeRate_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()
- dividendYield_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()
- 0.5 * vol * vol;
nu = kappa_ * (theta_ - vol * vol);
retVal[0] = x0[0] * Math.Exp(mu * dt + vol * dw[0] * sdt);
retVal[1] = vol * vol
+ nu * dt + vol2 * sdt * (rho_ * dw[0] + sqrhov * dw[1]);
break;
case Discretization.NonCentralChiSquareVariance:
// use Alan Lewis trick to decorrelate the equity and the variance
// process by using y(t)=x(t)-\frac{rho}{sigma}\nu(t)
// and Ito's Lemma. Then use exact sampling for the variance
// process. For further details please read the Wilmott thread
// "QuantLib code is very high quality"
vol = (x0[1] > 0.0) ? Math.Sqrt(x0[1]) : 0.0;
mu = riskFreeRate_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()
- dividendYield_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()
- 0.5 * vol * vol;
retVal[1] = varianceDistribution(x0[1], dw[1], dt);
dy = (mu - rho_ / sigma_ * kappa_
* (theta_ - vol * vol)) * dt + vol * sqrhov * dw[0] * sdt;
retVal[0] = x0[0] * Math.Exp(dy + rho_ / sigma_ * (retVal[1] - x0[1]));
break;
case Discretization.QuadraticExponential:
case Discretization.QuadraticExponentialMartingale:
{
// for details of the quadratic exponential discretization scheme
// see Leif Andersen,
// Efficient Simulation of the Heston Stochastic Volatility Model
double ex = Math.Exp(-kappa_ * dt);
double m = theta_ + (x0[1] - theta_) * ex;
double s2 = x0[1] * sigma_ * sigma_ * ex / kappa_ * (1 - ex)
+ theta_ * sigma_ * sigma_ / (2 * kappa_) * (1 - ex) * (1 - ex);
double psi = s2 / (m * m);
double g1 = 0.5;
double g2 = 0.5;
double k0 = -rho_ * kappa_ * theta_ * dt / sigma_;
double k1 = g1 * dt * (kappa_ * rho_ / sigma_ - 0.5) - rho_ / sigma_;
double k2 = g2 * dt * (kappa_ * rho_ / sigma_ - 0.5) + rho_ / sigma_;
double k3 = g1 * dt * (1 - rho_ * rho_);
double k4 = g2 * dt * (1 - rho_ * rho_);
double A = k2 + 0.5 * k4;
if (psi < 1.5)
{
double b2 = 2 / psi - 1 + Math.Sqrt(2 / psi * (2 / psi - 1));
double b = Math.Sqrt(b2);
double a = m / (1 + b2);
if (discretization_ == Discretization.QuadraticExponentialMartingale)
{
// martingale correction
Utils.QL_REQUIRE(A < 1 / (2 * a), () => "illegal value");
k0 = -A * b2 * a / (1 - 2 * A * a) + 0.5 * Math.Log(1 - 2 * A * a)
- (k1 + 0.5 * k3) * x0[1];
}
retVal[1] = a * (b + dw[1]) * (b + dw[1]);
}
else
{
double p = (psi - 1) / (psi + 1);
double beta = (1 - p) / m;
double u = new CumulativeNormalDistribution().value(dw[1]);
if (discretization_ == Discretization.QuadraticExponentialMartingale)
{
// martingale correction
Utils.QL_REQUIRE(A < beta, () => "illegal value");
k0 = -Math.Log(p + beta * (1 - p) / (beta - A)) - (k1 + 0.5 * k3) * x0[1];
}
retVal[1] = ((u <= p) ? 0.0 : Math.Log((1 - p) / (1 - u)) / beta);
}
mu = riskFreeRate_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()
- dividendYield_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value();
retVal[0] = x0[0] * Math.Exp(mu * dt + k0 + k1 * x0[1] + k2 * retVal[1]
+ Math.Sqrt(k3 * x0[1] + k4 * retVal[1]) * dw[0]);
}
break;
case Discretization.BroadieKayaExactSchemeLobatto:
case Discretization.BroadieKayaExactSchemeLaguerre:
case Discretization.BroadieKayaExactSchemeTrapezoidal:
{
double nu_0 = x0[1];
double nu_t = varianceDistribution(nu_0, dw[1], dt);
double x = Math.Min(1.0 - Const.QL_EPSILON,
Math.Max(0.0, new CumulativeNormalDistribution().value(dw[2])));
cdf_nu_ds f = new cdf_nu_ds(this, nu_0, nu_t, dt, discretization_);
double vds = new Brent().solve(f, 1e-5, -x, theta_ * dt, 0.1 * theta_ * dt);
//double vds = new Brent().solve( boost::lambda::bind(&cdf_nu_ds, *this, boost::lambda::_1,
// nu_0, nu_t, dt, discretization_)-x,
// 1e-5, theta_*dt, 0.1*theta_*dt);
double vdw = (nu_t - nu_0 - kappa_ * theta_ * dt + kappa_ * vds) / sigma_;
mu = (riskFreeRate_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()
- dividendYield_.link.forwardRate(t0, t0 + dt, Compounding.Continuous).value()) * dt
- 0.5 * vds + rho_ * vdw;
double sig = Math.Sqrt((1 - rho_ * rho_) * vds);
double s = x0[0] * Math.Exp(mu + sig * dw[0]);
retVal[0] = s;
retVal[1] = nu_t;
}
break;
default:
Utils.QL_FAIL("unknown discretization schema");
break;
}
return(retVal);
}
public static double blackFormulaStdDevDerivative( double strike,
double forward,
double stdDev,
double discount,
double displacement)
{
checkParameters( strike, forward, displacement );
if ( stdDev < 0.0 )
throw new ArgumentException( "stdDev (" + stdDev + ") must be non-negative" );
if ( discount <= 0.0 )
throw new ArgumentException( "discount (" + discount + ") must be positive" );
forward = forward + displacement;
strike = strike + displacement;
if ( stdDev == 0.0 || strike == 0)
return 0.0;
double d1 = Math.Log( forward / strike ) / stdDev + .5 * stdDev;
CumulativeNormalDistribution phi = new CumulativeNormalDistribution();
return discount * forward * phi.derivative( d1 );
}
public override void calculate()
{
double sigmaShift_vega = 0.0001;
double sigmaShift_volga = 0.0001;
double spotShift_delta = 0.0001 * spotFX_.link.value();
double sigmaShift_vanna = 0.0001;
Handle <Quote> x0Quote = new Handle <Quote>(new SimpleQuote(spotFX_.link.value())); //used for shift
Handle <Quote> atmVolQuote = new Handle <Quote>(new SimpleQuote(atmVol_.link.value())); //used for shift
BlackVolTermStructure blackVolTS = new BlackConstantVol(Settings.evaluationDate(),
new NullCalendar(), atmVolQuote, new Actual365Fixed());
BlackScholesMertonProcess stochProcess = new BlackScholesMertonProcess(x0Quote, foreignTS_, domesticTS_,
new Handle <BlackVolTermStructure>(blackVolTS));
IPricingEngine engineBS = new AnalyticBarrierEngine(stochProcess);
BlackDeltaCalculator blackDeltaCalculatorAtm = new BlackDeltaCalculator(
Option.Type.Call, atmVol_.link.deltaType(), x0Quote.link.value(),
domesticTS_.link.discount(T_), foreignTS_.link.discount(T_),
atmVol_.link.value() * Math.Sqrt(T_));
double atmStrike = blackDeltaCalculatorAtm.atmStrike(atmVol_.link.atmType());
double call25Vol = vol25Call_.link.value();
double put25Vol = vol25Put_.link.value();
BlackDeltaCalculator blackDeltaCalculatorPut25 = new BlackDeltaCalculator(
Option.Type.Put, vol25Put_.link.deltaType(), x0Quote.link.value(),
domesticTS_.link.discount(T_), foreignTS_.link.discount(T_),
put25Vol * Math.Sqrt(T_));
double put25Strike = blackDeltaCalculatorPut25.strikeFromDelta(-0.25);
BlackDeltaCalculator blackDeltaCalculatorCall25 = new BlackDeltaCalculator(
Option.Type.Call, vol25Call_.link.deltaType(), x0Quote.link.value(),
domesticTS_.link.discount(T_), foreignTS_.link.discount(T_),
call25Vol * Math.Sqrt(T_));
double call25Strike = blackDeltaCalculatorCall25.strikeFromDelta(0.25);
//here use vanna volga interpolated smile to price vanilla
List <double> strikes = new List <double>();
List <double> vols = new List <double>();
strikes.Add(put25Strike);
vols.Add(put25Vol);
strikes.Add(atmStrike);
vols.Add(atmVol_.link.value());
strikes.Add(call25Strike);
vols.Add(call25Vol);
VannaVolga vannaVolga = new VannaVolga(x0Quote.link.value(), domesticTS_.link.discount(T_),
foreignTS_.link.discount(T_), T_);
Interpolation interpolation = vannaVolga.interpolate(strikes, strikes.Count, vols);
interpolation.enableExtrapolation();
StrikedTypePayoff payoff = arguments_.payoff as StrikedTypePayoff;
double strikeVol = interpolation.value(payoff.strike());
//vannila option price
double vanillaOption = Utils.blackFormula(payoff.optionType(), payoff.strike(),
x0Quote.link.value() * foreignTS_.link.discount(T_) / domesticTS_.link.discount(T_),
strikeVol * Math.Sqrt(T_),
domesticTS_.link.discount(T_));
//spot > barrier up&out 0
if (x0Quote.link.value() >= arguments_.barrier && arguments_.barrierType == Barrier.Type.UpOut)
{
results_.value = 0.0;
results_.additionalResults["VanillaPrice"] = adaptVanDelta_? bsPriceWithSmile_ : vanillaOption;
results_.additionalResults["BarrierInPrice"] = adaptVanDelta_? bsPriceWithSmile_ : vanillaOption;
results_.additionalResults["BarrierOutPrice"] = 0.0;
}
//spot > barrier up&in vanilla
else if (x0Quote.link.value() >= arguments_.barrier && arguments_.barrierType == Barrier.Type.UpIn)
{
results_.value = adaptVanDelta_? bsPriceWithSmile_ : vanillaOption;
results_.additionalResults["VanillaPrice"] = adaptVanDelta_? bsPriceWithSmile_ : vanillaOption;
results_.additionalResults["BarrierInPrice"] = adaptVanDelta_? bsPriceWithSmile_ : vanillaOption;
results_.additionalResults["BarrierOutPrice"] = 0.0;
}
//spot < barrier down&out 0
else if (x0Quote.link.value() <= arguments_.barrier && arguments_.barrierType == Barrier.Type.DownOut)
{
results_.value = 0.0;
results_.additionalResults["VanillaPrice"] = adaptVanDelta_? bsPriceWithSmile_ : vanillaOption;
results_.additionalResults["BarrierInPrice"] = adaptVanDelta_? bsPriceWithSmile_ : vanillaOption;
results_.additionalResults["BarrierOutPrice"] = 0.0;
}
//spot < barrier down&in vanilla
else if (x0Quote.link.value() <= arguments_.barrier && arguments_.barrierType == Barrier.Type.DownIn)
{
results_.value = adaptVanDelta_? bsPriceWithSmile_ : vanillaOption;
results_.additionalResults["VanillaPrice"] = adaptVanDelta_? bsPriceWithSmile_ : vanillaOption;
results_.additionalResults["BarrierInPrice"] = adaptVanDelta_? bsPriceWithSmile_ : vanillaOption;
results_.additionalResults["BarrierOutPrice"] = 0.0;
}
else
{
//set up BS barrier option pricing
//only calculate out barrier option price
// in barrier price = vanilla - out barrier
Barrier.Type barrierTyp;
if (arguments_.barrierType == Barrier.Type.UpOut)
{
barrierTyp = arguments_.barrierType;
}
else if (arguments_.barrierType == Barrier.Type.UpIn)
{
barrierTyp = Barrier.Type.UpOut;
}
else if (arguments_.barrierType == Barrier.Type.DownOut)
{
barrierTyp = arguments_.barrierType;
}
else
{
barrierTyp = Barrier.Type.DownOut;
}
BarrierOption barrierOption = new BarrierOption(barrierTyp,
arguments_.barrier.GetValueOrDefault(),
arguments_.rebate.GetValueOrDefault(),
(StrikedTypePayoff)arguments_.payoff,
arguments_.exercise);
barrierOption.setPricingEngine(engineBS);
//BS price with atm vol
double priceBS = barrierOption.NPV();
double priceAtmCallBS = Utils.blackFormula(Option.Type.Call, atmStrike,
x0Quote.link.value() * foreignTS_.link.discount(T_) / domesticTS_.link.discount(T_),
atmVol_.link.value() * Math.Sqrt(T_),
domesticTS_.link.discount(T_));
double price25CallBS = Utils.blackFormula(Option.Type.Call, call25Strike,
x0Quote.link.value() * foreignTS_.link.discount(T_) / domesticTS_.link.discount(T_),
atmVol_.link.value() * Math.Sqrt(T_),
domesticTS_.link.discount(T_));
double price25PutBS = Utils.blackFormula(Option.Type.Put, put25Strike,
x0Quote.link.value() * foreignTS_.link.discount(T_) / domesticTS_.link.discount(T_),
atmVol_.link.value() * Math.Sqrt(T_),
domesticTS_.link.discount(T_));
//market price
double priceAtmCallMkt = Utils.blackFormula(Option.Type.Call, atmStrike,
x0Quote.link.value() * foreignTS_.link.discount(T_) / domesticTS_.link.discount(T_),
atmVol_.link.value() * Math.Sqrt(T_),
domesticTS_.link.discount(T_));
double price25CallMkt = Utils.blackFormula(Option.Type.Call, call25Strike,
x0Quote.link.value() * foreignTS_.link.discount(T_) / domesticTS_.link.discount(T_),
call25Vol * Math.Sqrt(T_),
domesticTS_.link.discount(T_));
double price25PutMkt = Utils.blackFormula(Option.Type.Put, put25Strike,
x0Quote.link.value() * foreignTS_.link.discount(T_) / domesticTS_.link.discount(T_),
put25Vol * Math.Sqrt(T_),
domesticTS_.link.discount(T_));
//Analytical Black Scholes formula for vanilla option
NormalDistribution norm = new NormalDistribution();
double d1atm = (Math.Log(x0Quote.link.value() * foreignTS_.link.discount(T_) /
domesticTS_.link.discount(T_) / atmStrike)
+ 0.5 * Math.Pow(atmVolQuote.link.value(), 2.0) * T_) /
(atmVolQuote.link.value() * Math.Sqrt(T_));
double vegaAtm_Analytical = x0Quote.link.value() * norm.value(d1atm) * Math.Sqrt(T_) *
foreignTS_.link.discount(T_);
double vannaAtm_Analytical = vegaAtm_Analytical / x0Quote.link.value() *
(1.0 - d1atm / (atmVolQuote.link.value() * Math.Sqrt(T_)));
double volgaAtm_Analytical = vegaAtm_Analytical * d1atm * (d1atm - atmVolQuote.link.value() * Math.Sqrt(T_)) /
atmVolQuote.link.value();
double d125call = (Math.Log(x0Quote.link.value() * foreignTS_.link.discount(T_) /
domesticTS_.link.discount(T_) / call25Strike)
+ 0.5 * Math.Pow(atmVolQuote.link.value(), 2.0) * T_) / (atmVolQuote.link.value() * Math.Sqrt(T_));
double vega25Call_Analytical = x0Quote.link.value() * norm.value(d125call) * Math.Sqrt(T_) *
foreignTS_.link.discount(T_);
double vanna25Call_Analytical = vega25Call_Analytical / x0Quote.link.value() *
(1.0 - d125call / (atmVolQuote.link.value() * Math.Sqrt(T_)));
double volga25Call_Analytical = vega25Call_Analytical * d125call *
(d125call - atmVolQuote.link.value() * Math.Sqrt(T_)) / atmVolQuote.link.value();
double d125Put = (Math.Log(x0Quote.link.value() * foreignTS_.link.discount(T_) /
domesticTS_.link.discount(T_) / put25Strike)
+ 0.5 * Math.Pow(atmVolQuote.link.value(), 2.0) * T_) / (atmVolQuote.link.value() * Math.Sqrt(T_));
double vega25Put_Analytical = x0Quote.link.value() * norm.value(d125Put) * Math.Sqrt(T_) *
foreignTS_.link.discount(T_);
double vanna25Put_Analytical = vega25Put_Analytical / x0Quote.link.value() *
(1.0 - d125Put / (atmVolQuote.link.value() * Math.Sqrt(T_)));
double volga25Put_Analytical = vega25Put_Analytical * d125Put *
(d125Put - atmVolQuote.link.value() * Math.Sqrt(T_)) / atmVolQuote.link.value();
//BS vega
((SimpleQuote)atmVolQuote.currentLink()).setValue(atmVolQuote.link.value() + sigmaShift_vega);
barrierOption.recalculate();
double vegaBarBS = (barrierOption.NPV() - priceBS) / sigmaShift_vega;
((SimpleQuote)atmVolQuote.currentLink()).setValue(atmVolQuote.link.value() - sigmaShift_vega);//setback
//BS volga
//vegaBar2
//base NPV
((SimpleQuote)atmVolQuote.currentLink()).setValue(atmVolQuote.link.value() + sigmaShift_volga);
barrierOption.recalculate();
double priceBS2 = barrierOption.NPV();
//shifted npv
((SimpleQuote)atmVolQuote.currentLink()).setValue(atmVolQuote.link.value() + sigmaShift_vega);
barrierOption.recalculate();
double vegaBarBS2 = (barrierOption.NPV() - priceBS2) / sigmaShift_vega;
double volgaBarBS = (vegaBarBS2 - vegaBarBS) / sigmaShift_volga;
((SimpleQuote)atmVolQuote.currentLink()).setValue(atmVolQuote.link.value()
- sigmaShift_volga
- sigmaShift_vega); //setback
//BS Delta
//base delta
((SimpleQuote)x0Quote.currentLink()).setValue(x0Quote.link.value() + spotShift_delta);//shift forth
barrierOption.recalculate();
double priceBS_delta1 = barrierOption.NPV();
((SimpleQuote)x0Quote.currentLink()).setValue(x0Quote.link.value() - 2 * spotShift_delta);//shift back
barrierOption.recalculate();
double priceBS_delta2 = barrierOption.NPV();
((SimpleQuote)x0Quote.currentLink()).setValue(x0Quote.link.value() + spotShift_delta);//set back
double deltaBar1 = (priceBS_delta1 - priceBS_delta2) / (2.0 * spotShift_delta);
//shifted delta
((SimpleQuote)atmVolQuote.currentLink()).setValue(atmVolQuote.link.value() + sigmaShift_vanna); //shift sigma
((SimpleQuote)x0Quote.currentLink()).setValue(x0Quote.link.value() + spotShift_delta); //shift forth
barrierOption.recalculate();
priceBS_delta1 = barrierOption.NPV();
((SimpleQuote)x0Quote.currentLink()).setValue(x0Quote.link.value() - 2 * spotShift_delta);//shift back
barrierOption.recalculate();
priceBS_delta2 = barrierOption.NPV();
((SimpleQuote)x0Quote.currentLink()).setValue(x0Quote.link.value() + spotShift_delta);//set back
double deltaBar2 = (priceBS_delta1 - priceBS_delta2) / (2.0 * spotShift_delta);
double vannaBarBS = (deltaBar2 - deltaBar1) / sigmaShift_vanna;
((SimpleQuote)atmVolQuote.currentLink()).setValue(atmVolQuote.link.value() - sigmaShift_vanna);//set back
//Matrix
Matrix A = new Matrix(3, 3, 0.0);
//analytical
A[0, 0] = vegaAtm_Analytical;
A[0, 1] = vega25Call_Analytical;
A[0, 2] = vega25Put_Analytical;
A[1, 0] = vannaAtm_Analytical;
A[1, 1] = vanna25Call_Analytical;
A[1, 2] = vanna25Put_Analytical;
A[2, 0] = volgaAtm_Analytical;
A[2, 1] = volga25Call_Analytical;
A[2, 2] = volga25Put_Analytical;
Vector b = new Vector(3, 0.0);
b[0] = vegaBarBS;
b[1] = vannaBarBS;
b[2] = volgaBarBS;
//Vector q = inverse(A) * b; TODO implements transpose
Vector q = Matrix.inverse(A) * b;
//touch probability
CumulativeNormalDistribution cnd = new CumulativeNormalDistribution();
double mu = domesticTS_.link.zeroRate(T_, Compounding.Continuous).value() -
foreignTS_.link.zeroRate(T_, Compounding.Continuous).value() -
Math.Pow(atmVol_.link.value(), 2.0) / 2.0;
double h2 = (Math.Log(arguments_.barrier.GetValueOrDefault() / x0Quote.link.value()) + mu * T_) /
(atmVol_.link.value() * Math.Sqrt(T_));
double h2Prime = (Math.Log(x0Quote.link.value() / arguments_.barrier.GetValueOrDefault()) + mu * T_) /
(atmVol_.link.value() * Math.Sqrt(T_));
double probTouch = 0.0;
if (arguments_.barrierType == Barrier.Type.UpIn || arguments_.barrierType == Barrier.Type.UpOut)
{
probTouch = cnd.value(h2Prime) + Math.Pow(arguments_.barrier.GetValueOrDefault() / x0Quote.link.value(),
2.0 * mu / Math.Pow(atmVol_.link.value(), 2.0)) * cnd.value(-h2);
}
else
{
probTouch = cnd.value(-h2Prime) + Math.Pow(arguments_.barrier.GetValueOrDefault() / x0Quote.link.value(),
2.0 * mu / Math.Pow(atmVol_.link.value(), 2.0)) * cnd.value(h2);
}
double p_survival = 1.0 - probTouch;
double lambda = p_survival;
double adjust = q[0] * (priceAtmCallMkt - priceAtmCallBS)
+ q[1] * (price25CallMkt - price25CallBS)
+ q[2] * (price25PutMkt - price25PutBS);
double outPrice = priceBS + lambda * adjust;//
double inPrice;
//adapt Vanilla delta
if (adaptVanDelta_ == true)
{
outPrice += lambda * (bsPriceWithSmile_ - vanillaOption);
//capfloored by (0, vanilla)
outPrice = Math.Max(0.0, Math.Min(bsPriceWithSmile_, outPrice));
inPrice = bsPriceWithSmile_ - outPrice;
}
else
{
//capfloored by (0, vanilla)
outPrice = Math.Max(0.0, Math.Min(vanillaOption, outPrice));
inPrice = vanillaOption - outPrice;
}
if (arguments_.barrierType == Barrier.Type.DownOut || arguments_.barrierType == Barrier.Type.UpOut)
{
results_.value = outPrice;
}
else
{
results_.value = inPrice;
}
results_.additionalResults["VanillaPrice"] = vanillaOption;
results_.additionalResults["BarrierInPrice"] = inPrice;
results_.additionalResults["BarrierOutPrice"] = outPrice;
results_.additionalResults["lambda"] = lambda;
}
}
public AmericanPayoffAtExpiry(double spot, double discount, double dividendDiscount, double variance,
StrikedTypePayoff payoff, bool knock_in = true)
{
spot_ = spot;
discount_ = discount;
dividendDiscount_ = dividendDiscount;
variance_ = variance;
knock_in_ = knock_in;
Utils.QL_REQUIRE(spot_ > 0.0, () => "positive spot value required");
Utils.QL_REQUIRE(discount_ > 0.0, () => "positive discount required");
Utils.QL_REQUIRE(dividendDiscount_ > 0.0, () => "positive dividend discount required");
Utils.QL_REQUIRE(variance_ >= 0.0, () => "negative variance not allowed");
stdDev_ = Math.Sqrt(variance_);
Option.Type type = payoff.optionType();
strike_ = payoff.strike();
forward_ = spot_ * dividendDiscount_ / discount_;
mu_ = Math.Log(dividendDiscount_ / discount_) / variance_ - 0.5;
// binary cash-or-nothing payoff?
CashOrNothingPayoff coo = payoff as CashOrNothingPayoff;
if (coo != null)
{
K_ = coo.cashPayoff();
}
// binary asset-or-nothing payoff?
AssetOrNothingPayoff aoo = payoff as AssetOrNothingPayoff;
if (aoo != null)
{
K_ = forward_;
mu_ += 1.0;
}
log_H_S_ = Math.Log(strike_ / spot_);
double log_S_H_ = Math.Log(spot_ / strike_);
double eta = 0.0;
double phi = 0.0;
switch (type)
{
case Option.Type.Call:
if (knock_in_)
{
// up-and-in cash-(at-expiry)-or-nothing option
// a.k.a. american call with cash-or-nothing payoff
eta = -1.0;
phi = 1.0;
}
else
{
// up-and-out cash-(at-expiry)-or-nothing option
eta = -1.0;
phi = -1.0;
}
break;
case Option.Type.Put:
if (knock_in_)
{
// down-and-in cash-(at-expiry)-or-nothing option
// a.k.a. american put with cash-or-nothing payoff
eta = 1.0;
phi = -1.0;
}
else
{
// down-and-out cash-(at-expiry)-or-nothing option
eta = 1.0;
phi = 1.0;
}
break;
default:
Utils.QL_FAIL("invalid option type");
break;
}
if (variance_ >= Const.QL_EPSILON)
{
D1_ = phi * (log_S_H_ / stdDev_ + mu_ * stdDev_);
D2_ = eta * (log_H_S_ / stdDev_ + mu_ * stdDev_);
CumulativeNormalDistribution f = new CumulativeNormalDistribution();
cum_d1_ = f.value(D1_);
cum_d2_ = f.value(D2_);
n_d1_ = f.derivative(D1_);
n_d2_ = f.derivative(D2_);
}
else
{
if (log_S_H_ * phi > 0)
{
cum_d1_ = 1.0;
}
else
{
cum_d1_ = 0.0;
}
if (log_H_S_ * eta > 0)
{
cum_d2_ = 1.0;
}
else
{
cum_d2_ = 0.0;
}
n_d1_ = 0.0;
n_d2_ = 0.0;
}
switch (type)
{
case Option.Type.Call:
if (strike_ <= spot_)
{
if (knock_in_)
{
// up-and-in cash-(at-expiry)-or-nothing option
// a.k.a. american call with cash-or-nothing payoff
cum_d1_ = 0.5;
cum_d2_ = 0.5;
}
else
{
// up-and-out cash-(at-expiry)-or-nothing option
// already knocked out
cum_d1_ = 0.0;
cum_d2_ = 0.0;
}
n_d1_ = 0.0;
n_d2_ = 0.0;
}
break;
case Option.Type.Put:
if (strike_ >= spot_)
{
if (knock_in_)
{
// down-and-in cash-(at-expiry)-or-nothing option
// a.k.a. american put with cash-or-nothing payoff
cum_d1_ = 0.5;
cum_d2_ = 0.5;
}
else
{
// down-and-out cash-(at-expiry)-or-nothing option
// already knocked out
cum_d1_ = 0.0;
cum_d2_ = 0.0;
}
n_d1_ = 0.0;
n_d2_ = 0.0;
}
break;
default:
Utils.QL_FAIL("invalid option type");
break;
}
inTheMoney_ = (type == Option.Type.Call && strike_ < spot_) ||
(type == Option.Type.Put && strike_ > spot_);
if (inTheMoney_)
{
X_ = 1.0;
Y_ = 1.0;
}
else
{
X_ = 1.0;
if (cum_d2_ == 0.0)
{
Y_ = 0.0; // check needed on some extreme cases
}
else
{
Y_ = Math.Pow((strike_ / spot_), (2.0 * mu_));
}
}
if (!knock_in_)
{
Y_ *= -1.0;
}
}
