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 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;
}
/*! 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
)
{
return bachelierBlackFormula( optionType, strike, forward, stdDev, 1 );
}
protected override double optionletPriceImp(Option.Type optionType, double effStrike,
double forward, double stdDev)
{
return Utils.blackFormula(optionType,
effStrike,
forward,
stdDev);
}
public override double discountBondOption(Option.Type type,
double strike,
double maturity,
double bondMaturity)
{
if (!(strike > 0.0))
throw new ApplicationException("strike must be positive");
double discountT = discountBond(0.0, maturity, x0());
double discountS = discountBond(0.0, bondMaturity, x0());
if (maturity < Const.QL_EPSILON)
{
switch (type){
case Option.Type.Call:
return Math.Max(discountS - strike, 0.0);
case Option.Type.Put:
return Math.Max(strike - discountS, 0.0);
default:
throw new ApplicationException("unsupported option type");
}
}
double sigma2 = sigma()*sigma();
double h = Math.Sqrt(k()*k() + 2.0*sigma2);
double b = B(maturity,bondMaturity);
double rho = 2.0*h/(sigma2*(Math.Exp(h*maturity) - 1.0));
double psi = (k() + h)/sigma2;
double df = 4.0*k()*theta()/sigma2;
double ncps = 2.0*rho*rho*x0()*Math.Exp(h*maturity)/(rho+psi+b);
double ncpt = 2.0*rho*rho*x0()*Math.Exp(h*maturity)/(rho+psi);
NonCentralChiSquareDistribution chis = new NonCentralChiSquareDistribution(df, ncps);
NonCentralChiSquareDistribution chit = new NonCentralChiSquareDistribution(df, ncpt);
double z = Math.Log(A(maturity,bondMaturity)/strike)/b;
double call = discountS*chis.value(2.0*z*(rho+psi+b)) -
strike*discountT*chit.value(2.0*z*(rho+psi));
if (type == Option.Type.Call)
return call;
else
return call - discountS + strike*discountT;
}
public override double discountBondOption(Option.Type type,
double strike,
double maturity,
double bondMaturity) {
double v;
double _a = a();
if (Math.Abs(maturity) < Const.QL_EPSILON){
v = 0.0;
}
else if (_a < Math.Sqrt(Const.QL_EPSILON)){
v = sigma()*B(maturity, bondMaturity)* Math.Sqrt(maturity);
} else {
v = sigma()*B(maturity, bondMaturity)*
Math.Sqrt(0.5*(1.0 - Math.Exp(-2.0*_a*maturity))/_a);
}
double f = discountBond(0.0, bondMaturity, r0_);
double k = discountBond(0.0, maturity, r0_)*strike;
return Utils.blackFormula(type, k, f, v);
}
//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 DiscreteAverageData(Option.Type Type,
double Underlying,
double Strike,
double DividendYield,
double RiskFreeRate,
double First,
double Length,
int Fixings,
double Volatility,
bool ControlVariate,
double Result)
{
type = Type;
underlying = Underlying;
strike = Strike;
dividendYield = DividendYield;
riskFreeRate = RiskFreeRate;
first = First;
length = Length;
fixings = Fixings;
volatility = Volatility;
controlVariate = ControlVariate;
result = Result;
}
public abstract double discountBondOption(Option.Type type, double strike, double maturity, double bondMaturity);
public CashOrNothingPayoff(Option.Type type, double strike, double cashPayoff)
: base(type, strike)
{
_cashPayoff = cashPayoff;
}
//! usually only need implement this (of course they may need
//! to re-implement initialize too ...)
protected virtual double optionletPriceImp(Option.Type optionType, double strike, double forward, double stdDev)
{
Utils.QL_FAIL("you must implement this to get a vol-dependent price");
return strike * forward * stdDev * (int)optionType;
}
//@}
//! can replace this if really required
protected virtual double optionletPrice(Option.Type optionType, double effStrike)
{
Date fixingDate = coupon_.fixingDate();
if (fixingDate <= Settings.evaluationDate())
{
// the amount is determined
double a, b;
if (optionType==Option.Type.Call)
{
a = coupon_.indexFixing();
b = effStrike;
}
else
{
a = effStrike;
b = coupon_.indexFixing();
}
return Math.Max(a - b, 0.0)* coupon_.accrualPeriod()*discount_;
}
else
{
// not yet determined, use Black/DD1/Bachelier/whatever from Impl
Utils.QL_REQUIRE( !capletVolatility().empty(), () => "missing optionlet volatility" );
double stdDev = Math.Sqrt(capletVolatility().link.totalVariance(fixingDate, effStrike));
double fixing = optionletPriceImp(optionType,
effStrike,
adjustedFixing(),
stdDev);
return fixing * coupon_.accrualPeriod() * discount_;
}
}
public PercentageStrikePayoff(Option.Type type, double moneyness)
: base(type, moneyness)
{
}
public PlainVanillaPayoff(Option.Type type, double strike) : base(type, strike) {}
protected override double optionletPrice(Option.Type optionType, double effStrike)
{
Date fixingDate = coupon_.fixingDate();
if (fixingDate <= Settings.evaluationDate()) {
// the amount is determined
double a;
double b;
if (optionType == Option.Type.Call) {
a = coupon_.indexFixing();
b = effStrike;
} else {
a = effStrike;
b = coupon_.indexFixing();
}
return Math.Max(a - b, 0.0) * coupon_.accrualPeriod() * discount_;
} else {
// not yet determined, use Black model
if (!(!capletVolatility().empty()))
throw new ApplicationException("missing optionlet volatility");
double stdDev = Math.Sqrt(capletVolatility().link.blackVariance(fixingDate, effStrike));
double fixing = Utils.blackFormula(optionType, effStrike, adjustedFixing(), stdDev);
return fixing * coupon_.accrualPeriod() * discount_;
}
}
public ConundrumIntegrand(VanillaOptionPricer o, YieldTermStructure curve, GFunction gFunction, Date fixingDate, Date paymentDate, double annuity, double forwardValue, double strike, Option.Type optionType) {
vanillaOptionPricer_ = o;
forwardValue_ = forwardValue;
annuity_ = annuity;
fixingDate_ = fixingDate;
paymentDate_ = paymentDate;
strike_ = strike;
optionType_ = optionType;
gFunction_ = gFunction;
}
public abstract double value(double strike, Option.Type optionType, double deflator);
// a.k.a. payoff strike
public GapPayoff(Option.Type type, double strike, double secondStrike)
: base(type, strike)
{
_secondStrike = secondStrike;
}
protected override double optionletPrice(Option.Type t, double d)
{ throw new ApplicationException("optionletPrice not available"); }
protected override double optionletImpl( Option.Type type, double strike,
double forward, double stdDev,
double d )
{
// could use displacement parameter in blackFormula but this is clearer
return Utils.blackFormula( type, strike + 1.0, forward + 1.0, stdDev, d );
}
public virtual double discountBondOption(Option.Type type,
double strike,
double maturity,
double bondMaturity)
{
double v = sigmaP(maturity, bondMaturity);
double f = termStructure().link.discount(bondMaturity);
double k = termStructure().link.discount(maturity) * strike;
return Utils.blackFormula(type, k, f, v);
}
protected override double optionletImpl( Option.Type type, double strike,
double forward, double stdDev,
double d )
{
return Utils.bachelierBlackFormula( type, strike, forward, stdDev, d );
}
public AssetOrNothingPayoff(Option.Type type, double strike)
: base(type, strike)
{
}
public GapPayoff(Option.Type type, double strike, double secondStrike) // a.k.a. payoff strike
: base(type, strike) {
secondStrike_ = secondStrike;
}
protected override double optionletPrice(Option.Type optionType, double strike) {
ConundrumIntegrand integrand = new ConundrumIntegrand(vanillaOptionPricer_, rateCurve_, gFunction_, fixingDate_, paymentDate_, annuity_, swapRateValue_, strike, optionType);
stdDeviationsForUpperLimit_ = requiredStdDeviations_;
double a;
double b;
double integralValue;
if (optionType == Option.Type.Call) {
upperLimit_ = resetUpperLimit(stdDeviationsForUpperLimit_);
// while(upperLimit_ <= strike){
// stdDeviationsForUpperLimit_ += 1.;
// upperLimit_ = resetUpperLimit(stdDeviationsForUpperLimit_);
// }
integralValue = integrate(strike, upperLimit_, integrand);
//refineIntegration(integralValue, *integrand);
} else {
a = Math.Min(strike, lowerLimit_);
b = strike;
integralValue = integrate(a, b, integrand);
}
double dFdK = integrand.firstDerivativeOfF(strike);
double swaptionPrice = vanillaOptionPricer_.value(strike, optionType, annuity_);
// v. HAGAN, Conundrums..., formule 2.17a, 2.18a
return coupon_.accrualPeriod() * (discount_ / annuity_) * ((1 + dFdK) * swaptionPrice + ((int)optionType) * integralValue);
}
public TypePayoff(Option.Type type) {
type_ = type;
}
public override double value(double strike, Option.Type optionType, double deflator)
{
double variance = _smile.variance(strike);
return deflator * Utils.blackFormula(optionType, strike, _forwardValue, Math.Sqrt(variance));
}
public FloatingTypePayoff(Option.Type type) : base(type) {}
protected abstract double optionletPrice(Option.Type optionType, double effStrike);
public StrikedTypePayoff(Option.Type type, double strike) : base(type) {
strike_ = strike;
}