/// <summary>
/// Computes the price of a one-touch/no-touch option.
/// </summary>
/// <param name="spot"> the spot </param>
/// <param name="timeToExpiry"> the time to expiry </param>
/// <param name="costOfCarry"> the cost of carry </param>
/// <param name="rate"> the interest rate </param>
/// <param name="lognormalVol"> the lognormal volatility </param>
/// <param name="barrier"> the barrier </param>
/// <returns> the price </returns>
public virtual double price(double spot, double timeToExpiry, double costOfCarry, double rate, double lognormalVol, SimpleConstantContinuousBarrier barrier)
{
ArgChecker.notNull(barrier, "barrier");
bool isKnockIn = barrier.KnockType.KnockIn;
bool isDown = barrier.BarrierType.Down;
double h = barrier.BarrierLevel;
ArgChecker.isFalse(isDown && spot <= barrier.BarrierLevel, "The Data is not consistent with an alive barrier (DOWN and spot<=barrier).");
ArgChecker.isFalse(!isDown && spot >= barrier.BarrierLevel, "The Data is not consistent with an alive barrier (UP and spot>=barrier).");
double eta = isDown ? 1 : -1;
double df2 = Math.Exp(-rate * timeToExpiry);
double lognormalVolSq = lognormalVol * lognormalVol;
double lognormalVolT = lognormalVol * Math.Sqrt(timeToExpiry);
if (DoubleMath.fuzzyEquals(Math.Min(timeToExpiry, lognormalVolSq), 0d, SMALL))
{
return(isKnockIn ? 0d : df2);
}
double mu = (costOfCarry - 0.5 * lognormalVolSq) / lognormalVolSq;
double lambda = Math.Sqrt(mu * mu + 2 * rate / lognormalVolSq);
double m1 = lognormalVolT * (1 + mu);
double x2 = Math.Log(spot / h) / lognormalVolT + m1;
double y2 = Math.Log(h / spot) / lognormalVolT + m1;
double z = Math.Log(h / spot) / lognormalVolT + lambda * lognormalVolT;
double xE = isKnockIn ? getF(spot, z, lognormalVolT, h, mu, lambda, eta) : getE(spot, df2, x2, y2, lognormalVolT, h, mu, eta);
return(xE);
}
/// <summary>
/// {@inheritDoc} </summary>
/// <exception cref="MathException"> If there are no real roots; if the Commons method could not evaluate the function; if the Commons method could not converge. </exception>
public virtual double?[] getRoots(RealPolynomialFunction1D function)
{
ArgChecker.notNull(function, "function");
try
{
Complex[] roots = ROOT_FINDER.solveAllComplex(function.Coefficients, 0);
IList <double> realRoots = new List <double>();
foreach (Complex c in roots)
{
if (DoubleMath.fuzzyEquals(c.Imaginary, 0d, EPS))
{
realRoots.Add(c.Real);
}
}
if (realRoots.Count == 0)
{
throw new MathException("Could not find any real roots");
}
return(realRoots.ToArray());
}
catch (TooManyEvaluationsException e)
{
throw new MathException(e);
}
}
public virtual void test_currencyExposureBetweenFixingAndPayment()
{
double eps = 1.0e-14;
LocalDate valuationDate = date(2014, 6, 30);
LocalDate paymentDate = date(2014, 7, 1);
LocalDate fixingDate = date(2014, 6, 27);
FxResetNotionalExchange resetNotionalUSD = FxResetNotionalExchange.of(CurrencyAmount.of(USD, NOTIONAL), paymentDate, FxIndexObservation.of(GBP_USD_WM, fixingDate, REF_DATA));
FxResetNotionalExchange resetNotionalGBP = FxResetNotionalExchange.of(CurrencyAmount.of(GBP, -NOTIONAL), paymentDate, FxIndexObservation.of(GBP_USD_WM, fixingDate, REF_DATA));
LocalDateDoubleTimeSeries ts = LocalDateDoubleTimeSeries.of(LocalDate.of(2014, 6, 27), 1.65);
ImmutableRatesProvider prov = ImmutableRatesProvider.builder(valuationDate).fxRateProvider(FX_MATRIX).discountCurve(GBP, DISCOUNT_CURVE_GBP).discountCurve(USD, DISCOUNT_CURVE_USD).timeSeries(FxIndices.GBP_USD_WM, ts).build();
DiscountingFxResetNotionalExchangePricer test = new DiscountingFxResetNotionalExchangePricer();
// USD
MultiCurrencyAmount computedUSD = test.currencyExposure(resetNotionalUSD, prov);
PointSensitivities pointUSD = test.presentValueSensitivity(resetNotionalUSD, prov).build();
MultiCurrencyAmount expectedUSD = prov.currencyExposure(pointUSD.convertedTo(USD, prov)).plus(CurrencyAmount.of(resetNotionalUSD.Currency, test.presentValue(resetNotionalUSD, prov)));
assertFalse(computedUSD.contains(USD)); // 0 USD
assertEquals(computedUSD.getAmount(GBP).Amount, expectedUSD.getAmount(GBP).Amount, eps * NOTIONAL);
// GBP
MultiCurrencyAmount computedGBP = test.currencyExposure(resetNotionalGBP, prov);
PointSensitivities pointGBP = test.presentValueSensitivity(resetNotionalGBP, prov).build();
MultiCurrencyAmount expectedGBP = prov.currencyExposure(pointGBP.convertedTo(GBP, prov)).plus(CurrencyAmount.of(resetNotionalGBP.Currency, test.presentValue(resetNotionalGBP, prov)));
assertFalse(computedGBP.contains(GBP)); // 0 GBP
assertEquals(computedGBP.getAmount(USD).Amount, expectedGBP.getAmount(USD).Amount, eps * NOTIONAL);
// FD approximation
FxMatrix fxMatrixUp = FxMatrix.of(GBP, USD, FX_RATE + EPS_FD);
ImmutableRatesProvider provUp = ImmutableRatesProvider.builder(valuationDate).fxRateProvider(fxMatrixUp).discountCurve(GBP, DISCOUNT_CURVE_GBP).discountCurve(USD, DISCOUNT_CURVE_USD).timeSeries(FxIndices.GBP_USD_WM, ts).build();
double expectedFdUSD = -(test.presentValue(resetNotionalUSD, provUp) - test.presentValue(resetNotionalUSD, prov)) * FX_RATE * FX_RATE / EPS_FD;
assertTrue(!computedUSD.contains(USD) && DoubleMath.fuzzyEquals(expectedFdUSD, 0d, eps));
double expectedFdGBP = (test.presentValue(resetNotionalGBP, provUp) - test.presentValue(resetNotionalGBP, prov)) / EPS_FD;
assertTrue(!computedGBP.contains(GBP) && DoubleMath.fuzzyEquals(expectedFdGBP, 0d, eps));
}
/// <summary>
/// Creates an instance.
/// </summary>
/// <param name="mu"> The location parameter </param>
/// <param name="sigma"> The scale parameter, not negative or zero </param>
/// <param name="ksi"> The shape parameter </param>
public GeneralizedExtremeValueDistribution(double mu, double sigma, double ksi)
{
ArgChecker.isTrue(sigma >= 0, "sigma must be >= 0");
_mu = mu;
_sigma = sigma;
_ksi = ksi;
_ksiIsZero = DoubleMath.fuzzyEquals(ksi, 0d, 1e-13);
}
/// <summary>
/// Creates an instance.
/// </summary>
/// <param name="mu"> The location parameter </param>
/// <param name="sigma"> The scale parameter </param>
/// <param name="ksi"> The shape parameter </param>
/// <param name="engine"> A uniform random number generator, not null </param>
public GeneralizedParetoDistribution(double mu, double sigma, double ksi, RandomEngine engine)
{
ArgChecker.isTrue(sigma > 0, "sigma must be > 0");
ArgChecker.isTrue(!DoubleMath.fuzzyEquals(ksi, 0d, 1e-15), "ksi cannot be zero");
ArgChecker.notNull(engine, "engine");
_mu = mu;
_sigma = sigma;
_ksi = ksi;
_engine = engine;
}
//-------------------------------------------------------------------------
/// <summary>
/// Compares each element in the array to zero within a tolerance.
/// <para>
/// An empty array returns true;
/// </para>
/// <para>
/// The input array is not mutated.
///
/// </para>
/// </summary>
/// <param name="array"> the array to check </param>
/// <param name="tolerance"> the tolerance to use </param>
/// <returns> true if the array is effectively equal to zero </returns>
public static bool fuzzyEqualsZero(double[] array, double tolerance)
{
for (int i = 0; i < array.Length; i++)
{
if (!DoubleMath.fuzzyEquals(array[i], 0, tolerance))
{
return(false);
}
}
return(true);
}
//-------------------------------------------------------------------------
/// <summary>
/// Converts this amount to an equivalent amount the specified currency.
/// <para>
/// The result will be expressed in terms of the given currency, converting
/// using the specified FX rate.
/// </para>
/// <para>
/// For example, if this represents 'GBP 100' and this method is called with
/// arguments {@code (USD, 1.6)} then the result will be 'USD 160'.
///
/// </para>
/// </summary>
/// <param name="resultCurrency"> the currency of the result </param>
/// <param name="fxRate"> the FX rate from this currency to the result currency </param>
/// <returns> the converted instance, which should be expressed in the specified currency </returns>
/// <exception cref="IllegalArgumentException"> if the FX is not 1 when no conversion is required </exception>
public virtual Money convertedTo(Currency resultCurrency, decimal fxRate)
{
if (currency.Equals(resultCurrency))
{
if (DoubleMath.fuzzyEquals(fxRate.doubleValue(), 1d, 1e-8))
{
return(this);
}
throw new System.ArgumentException("FX rate must be 1 when no conversion required");
}
return(Money.of(resultCurrency, amount * fxRate));
}
/// <summary>
/// Converts this amount to an equivalent amount the specified currency.
/// <para>
/// The result will be expressed in terms of the given currency, converting
/// using the specified FX rate.
/// </para>
/// <para>
/// For example, if this represents 'GBP 100' and this method is called with
/// arguments {@code (USD, 1.6)} then the result will be 'USD 160'.
///
/// </para>
/// </summary>
/// <param name="resultCurrency"> the currency of the result </param>
/// <param name="fxRate"> the FX rate from this currency to the result currency </param>
/// <returns> the converted instance, which should be expressed in the specified currency </returns>
/// <exception cref="IllegalArgumentException"> if the FX is not 1 when no conversion is required </exception>
public CurrencyAmount convertedTo(Currency resultCurrency, double fxRate)
{
if (currency.Equals(resultCurrency))
{
if (DoubleMath.fuzzyEquals(fxRate, 1d, 1e-8))
{
return(this);
}
throw new System.ArgumentException("FX rate must be 1 when no conversion required");
}
return(CurrencyAmount.of(resultCurrency, amount * fxRate));
}
/// <summary>
/// Compares each element in the first array to the matching index in the second array within a tolerance.
/// <para>
/// If the arrays differ in length, false is returned.
/// </para>
/// <para>
/// The input arrays are not mutated.
///
/// </para>
/// </summary>
/// <param name="array1"> the first array to check </param>
/// <param name="array2"> the second array to check </param>
/// <param name="tolerance"> the tolerance to use </param>
/// <returns> true if the arrays are effectively equal </returns>
public static bool fuzzyEquals(double[] array1, double[] array2, double tolerance)
{
if (array1.Length != array2.Length)
{
return(false);
}
for (int i = 0; i < array1.Length; i++)
{
if (!DoubleMath.fuzzyEquals(array1[i], array2[i], tolerance))
{
return(false);
}
}
return(true);
}
private static CurveNode curveFxSwapCurveNode(string conventionStr, string timeStr, string label, QuoteId quoteId, double spread, CurveNodeDate date, CurveNodeDateOrder order)
{
if (!DoubleMath.fuzzyEquals(spread, 0d, 1e-10d))
{
throw new System.ArgumentException("Additional spread must be zero for FX swaps");
}
Matcher matcher = SIMPLE_YMD_TIME_REGEX.matcher(timeStr.ToUpper(Locale.ENGLISH));
if (!matcher.matches())
{
throw new System.ArgumentException(Messages.format("Invalid time format for FX swap: {}", timeStr));
}
Period periodToEnd = Period.parse("P" + matcher.group(1));
FxSwapConvention convention = FxSwapConvention.of(conventionStr);
FxSwapTemplate template = FxSwapTemplate.of(periodToEnd, convention);
return(FxSwapCurveNode.builder().template(template).farForwardPointsId(quoteId).label(label).date(date).dateOrder(order).build());
}
public virtual double?[] getRoots(RealPolynomialFunction1D function)
{
ArgChecker.notNull(function, "function");
double[] coefficients = function.Coefficients;
if (coefficients.Length != 4)
{
throw new System.ArgumentException("Function is not a cubic");
}
ComplexNumber[] result = ROOT_FINDER.getRoots(function);
IList <double> reals = new List <double>();
foreach (ComplexNumber c in result)
{
if (DoubleMath.fuzzyEquals(c.Imaginary, 0d, 1e-16))
{
reals.Add(c.Real);
}
}
ArgChecker.isTrue(reals.Count > 0, "Could not find any real roots");
return(reals.toArray(EMPTY_ARRAY));
}
/// <summary>
/// {@inheritDoc} </summary>
/// <exception cref="IllegalArgumentException"> If the function is not cubic </exception>
public virtual ComplexNumber[] getRoots(RealPolynomialFunction1D function)
{
ArgChecker.notNull(function, "function");
double[] coefficients = function.Coefficients;
ArgChecker.isTrue(coefficients.Length == 4, "Function is not a cubic");
double divisor = coefficients[3];
double a = coefficients[2] / divisor;
double b = coefficients[1] / divisor;
double c = coefficients[0] / divisor;
double aSq = a * a;
double q = (aSq - 3 * b) / 9;
double r = (2 * a * aSq - 9 * a * b + 27 * c) / 54;
double rSq = r * r;
double qCb = q * q * q;
double constant = a / 3;
if (rSq < qCb)
{
double mult = -2 * Math.Sqrt(q);
double theta = Math.Acos(r / Math.Sqrt(qCb));
return(new ComplexNumber[]
{
new ComplexNumber(mult * Math.Cos(theta / 3) - constant, 0),
new ComplexNumber(mult * Math.Cos((theta + TWO_PI) / 3) - constant, 0),
new ComplexNumber(mult * Math.Cos((theta - TWO_PI) / 3) - constant, 0)
});
}
double s = -Math.Sign(r) * Math.cbrt(Math.Abs(r) + Math.Sqrt(rSq - qCb));
double t = DoubleMath.fuzzyEquals(s, 0d, 1e-16) ? 0 : q / s;
double sum = s + t;
double real = -0.5 * sum - constant;
double imaginary = Math.Sqrt(3) * (s - t) / 2;
return(new ComplexNumber[]
{
new ComplexNumber(sum - constant, 0),
new ComplexNumber(real, imaginary),
new ComplexNumber(real, -imaginary)
});
}
public virtual void test_pv01_quote()
{
FixedCouponBondTradeCalculationFunction <FixedCouponBondTrade> function = FixedCouponBondTradeCalculationFunction.TRADE;
ScenarioMarketData md = marketData();
LegalEntityDiscountingProvider provider = LOOKUP.marketDataView(md.scenario(0)).discountingProvider();
DiscountingFixedCouponBondTradePricer pricer = DiscountingFixedCouponBondTradePricer.DEFAULT;
PointSensitivities pvPointSens = pricer.presentValueSensitivity(RTRADE, provider);
CurrencyParameterSensitivities pvParamSens = provider.parameterSensitivity(pvPointSens);
CurrencyParameterSensitivities expectedPv01CalBucketed = MQ_CALC.sensitivity(pvParamSens, provider).multipliedBy(1e-4);
MultiCurrencyAmount expectedPv01Cal = expectedPv01CalBucketed.total();
ISet <Measure> measures = ImmutableSet.of(Measures.PV01_MARKET_QUOTE_SUM, Measures.PV01_MARKET_QUOTE_BUCKETED);
//JAVA TO C# CONVERTER WARNING: Java wildcard generics have no direct equivalent in .NET:
//ORIGINAL LINE: java.util.Map<com.opengamma.strata.calc.Measure, com.opengamma.strata.collect.result.Result<?>> computed = function.calculate(TRADE, measures, PARAMS, md, REF_DATA);
IDictionary <Measure, Result <object> > computed = function.calculate(TRADE, measures, PARAMS, md, REF_DATA);
MultiCurrencyScenarioArray sumComputed = (MultiCurrencyScenarioArray)computed[Measures.PV01_MARKET_QUOTE_SUM].Value;
ScenarioArray <CurrencyParameterSensitivities> bucketedComputed = (ScenarioArray <CurrencyParameterSensitivities>)computed[Measures.PV01_MARKET_QUOTE_BUCKETED].Value;
assertEquals(sumComputed.ScenarioCount, 1);
assertEquals(sumComputed.get(0).Currencies, ImmutableSet.of(GBP));
assertTrue(DoubleMath.fuzzyEquals(sumComputed.get(0).getAmount(GBP).Amount, expectedPv01Cal.getAmount(GBP).Amount, 1.0e-10));
assertEquals(bucketedComputed.ScenarioCount, 1);
assertTrue(bucketedComputed.get(0).equalWithTolerance(expectedPv01CalBucketed, 1.0e-10));
}
private double getZOverChi(double rho, double z)
{
// Implementation comment: To avoid numerical instability (0/0) around ATM the first order approximation is used.
if (DoubleMath.fuzzyEquals(z, 0.0, SMALL_Z))
{
return(1.0 - rho * z / 2.0);
}
double rhoStar = 1 - rho;
if (DoubleMath.fuzzyEquals(rhoStar, 0.0, RHO_EPS))
{
if (z < 1.0)
{
return(-z / Math.Log(1.0d - z));
}
else
{
throw new System.ArgumentException("can't handle z>=1, rho=1");
}
}
double rhoHat = 1 + rho;
if (DoubleMath.fuzzyEquals(rhoHat, 0.0, RHO_EPS_NEGATIVE))
{
if (z > -1)
{
return(z / Math.Log(1 + z));
}
else if (z < -1)
{
if (rhoHat == 0)
{
return(0.0);
}
double chi = Math.Log(rhoHat) - Math.Log(-(1 + z) / rhoStar);
return(z / chi);
}
else
{
return(0.0);
}
}
double arg;
if (z < LARGE_NEG_Z)
{
arg = (rho * rho - 1) / 2 / z; //get rounding errors due to fine balanced cancellation for very large negative z
}
else if (z > LARGE_POS_Z)
{
arg = 2 * (z - rho);
}
else
{
arg = (Math.Sqrt(1 - 2 * rho * z + z * z) + z - rho);
//Mathematically this cannot be less than zero, but you know what computers are like.
if (arg <= 0.0)
{
return(0.0);
}
}
double chi = Math.Log(arg) - Math.Log(rhoStar);
return(z / chi);
}
/// <summary>
/// Computes the implied volatility in the SABR model and its derivatives.
/// <para>
/// The derivatives are stored in an array with:
/// <ul>
/// <li>[0] derivative with respect to the forward
/// <li>[1] derivative with respect to the strike
/// <li>[2] derivative with respect to the alpha
/// <li>[3] derivative with respect to the beta
/// <li>[4] derivative with respect to the rho
/// <li>[5] derivative with respect to the nu
/// </ul>
///
/// </para>
/// </summary>
/// <param name="forward"> the forward value of the underlying </param>
/// <param name="strike"> the strike value of the option </param>
/// <param name="timeToExpiry"> the time to expiry of the option </param>
/// <param name="alpha"> the SABR alpha value </param>
/// <param name="beta"> the SABR beta value </param>
/// <param name="rho"> the SABR rho value </param>
/// <param name="nu"> the SABR nu value </param>
/// <returns> the volatility and associated derivatives </returns>
public ValueDerivatives volatilityAdjoint(double forward, double strike, double timeToExpiry, double alpha, double beta, double rho, double nu)
{
ArgChecker.isTrue(forward > 0.0, "forward must be greater than zero");
ArgChecker.isTrue(strike >= 0.0, "strike must be greater than zero");
ArgChecker.isTrue(timeToExpiry >= 0.0, "timeToExpiry must be greater than zero");
double cutoff = forward * CUTOFF_MONEYNESS;
double k = strike;
if (k < cutoff)
{
log.info("Given strike of {} is less than cutoff at {}, therefore the strike is taken as {}", new object[] { k, cutoff, cutoff });
k = cutoff;
}
double betaStar = 1 - beta;
double rhoStar = 1.0 - rho;
if (alpha == 0.0)
{
double alphaBar;
if (DoubleMath.fuzzyEquals(forward, k, ATM_EPS))
{ //TODO should this is relative
alphaBar = (1 + (2 - 3 * rho * rho) * nu * nu / 24 * timeToExpiry) / Math.Pow(forward, betaStar);
}
else
{
//for non-atm options the alpha sensitivity at alpha = 0 is infinite. Returning this will most likely break calibrations,
// so we return an arbitrary large number
alphaBar = 1e7;
}
return(ValueDerivatives.of(0d, DoubleArray.of(0, 0, alphaBar, 0, 0, 0)));
}
// Implementation note: Forward sweep.
double sfK = Math.Pow(forward * k, betaStar / 2);
double lnrfK = Math.Log(forward / k);
double z = nu / alpha * sfK * lnrfK;
double rzxz;
double xz = 0;
if (DoubleMath.fuzzyEquals(z, 0.0, SMALL_Z))
{
rzxz = 1.0 - 0.5 * z * rho; //small z expansion to z^2 terms
}
else
{
if (DoubleMath.fuzzyEquals(rhoStar, 0.0, RHO_EPS))
{
if (z < 1.0)
{
xz = -Math.Log(1.0d - z);
rzxz = z / xz;
}
else
{
throw new System.ArgumentException("can't handle z>=1, rho=1");
}
}
else
{
double arg;
if (z < LARGE_NEG_Z)
{
arg = (rho * rho - 1) / 2 / z; //get rounding errors due to fine balanced cancellation for very large negative z
}
else if (z > LARGE_POS_Z)
{
arg = 2 * (z - rho);
}
else
{
arg = (Math.Sqrt(1 - 2 * rho * z + z * z) + z - rho);
}
if (arg <= 0.0)
{ //Mathematically this cannot be less than zero, but you know what computers are like.
rzxz = 0.0;
}
else
{
xz = Math.Log(arg / (1 - rho));
rzxz = z / xz;
}
}
}
double sf1 = sfK * (1 + betaStar * betaStar / 24 * (lnrfK * lnrfK) + Math.Pow(betaStar, 4) / 1920 * Math.Pow(lnrfK, 4));
double sf2 = (1 + (Math.Pow(betaStar * alpha / sfK, 2) / 24 + (rho * beta * nu * alpha) / (4 * sfK) + (2 - 3 * rho * rho) * nu * nu / 24) * timeToExpiry);
double volatility = Math.Max(MIN_VOL, alpha / sf1 * rzxz * sf2);
// Implementation note: Backward sweep.
double vBar = 1;
double sf2Bar = alpha / sf1 * rzxz * vBar;
double sf1Bar = -alpha / (sf1 * sf1) * rzxz * sf2 * vBar;
double rzxzBar = alpha / sf1 * sf2 * vBar;
double zBar;
double xzBar = 0.0;
if (DoubleMath.fuzzyEquals(z, 0.0, SMALL_Z))
{
zBar = -rho / 2 * rzxzBar;
}
else
{
if (DoubleMath.fuzzyEquals(rhoStar, 0.0, RHO_EPS))
{
if (z < 1.0)
{
xzBar = -z / (xz * xz) * rzxzBar;
zBar = 1.0d / xz * rzxzBar + 1.0d / (1.0d - z) * xzBar;
}
else
{
throw new System.ArgumentException("can't handle z>=1, rho=1");
}
}
else
{
if (z < LARGE_NEG_Z)
{
zBar = 1 / xz * rzxzBar + xzBar / (xz * xz) * rzxzBar;
}
else if (z > LARGE_POS_Z)
{
zBar = 1 / xz * rzxzBar - xzBar / (xz * xz) * rzxzBar;
}
else
{
xzBar = -z / (xz * xz) * rzxzBar;
zBar = 1 / xz * rzxzBar + 1 / ((Math.Sqrt(1 - 2 * rho * z + z * z) + z - rho)) * (0.5 * Math.Pow(1 - 2 * rho * z + z * z, -0.5) * (-2 * rho + 2 * z) + 1) * xzBar;
}
}
}
double lnrfKBar = sfK * (betaStar * betaStar / 12 * lnrfK + Math.Pow(betaStar, 4) / 1920 * 4 * Math.Pow(lnrfK, 3)) * sf1Bar + nu / alpha * sfK * zBar;
double sfKBar = nu / alpha * lnrfK * zBar + sf1 / sfK * sf1Bar - (Math.Pow(betaStar * alpha, 2) / Math.Pow(sfK, 3) / 12 + (rho * beta * nu * alpha) / 4 / (sfK * sfK)) * timeToExpiry * sf2Bar;
double strikeBar = -1 / k * lnrfKBar + betaStar * sfK / (2 * k) * sfKBar;
double forwardBar = 1 / forward * lnrfKBar + betaStar * sfK / (2 * forward) * sfKBar;
double nuBar = 1 / alpha * sfK * lnrfK * zBar + ((rho * beta * alpha) / (4 * sfK) + (2 - 3 * rho * rho) * nu / 12) * timeToExpiry * sf2Bar;
double rhoBar;
if (Math.Abs(forward - k) < ATM_EPS)
{
rhoBar = -z / 2 * rzxzBar;
}
else
{
if (DoubleMath.fuzzyEquals(rhoStar, 0.0, RHO_EPS))
{
if (z >= 1)
{
if (rhoStar == 0.0)
{
rhoBar = double.NegativeInfinity; //the derivative at rho = 1 is infinite - this sets it to some arbitrary large number
}
else
{
rhoBar = xzBar * (1.0 / rhoStar + (0.5 - z) / (z - 1.0) / (z - 1.0));
}
}
else
{
rhoBar = (0.5 * Math.Pow(z / (1 - z), 2) + 0.25 * (z - 4.0) * Math.Pow(z / (1.0 - z), 3) / (1.0 - z) * rhoStar) * xzBar;
}
}
else
{
rhoBar = (1 / (Math.Sqrt(1 - 2 * rho * z + z * z) + z - rho) * (-Math.Pow(1 - 2 * rho * z + z * z, -0.5) * z - 1) + 1 / rhoStar) * xzBar;
}
}
rhoBar += ((beta * nu * alpha) / (4 * sfK) - rho * nu * nu / 4) * timeToExpiry * sf2Bar;
double alphaBar = -nu / (alpha * alpha) * sfK * lnrfK * zBar + ((betaStar * alpha / sfK) * (betaStar / sfK) / 12 + (rho * beta * nu) / (4 * sfK)) * timeToExpiry * sf2Bar + 1 / sf1 * rzxz * sf2 * vBar;
double betaBar = -0.5 * Math.Log(forward * k) * sfK * sfKBar - sfK * (betaStar / 12 * (lnrfK * lnrfK) + Math.Pow(betaStar, 3) / 480 * Math.Pow(lnrfK, 4)) * sf1Bar + (-betaStar * alpha * alpha / sfK / sfK / 12 + rho * nu * alpha / 4 / sfK) * timeToExpiry * sf2Bar;
return(ValueDerivatives.of(volatility, DoubleArray.of(forwardBar, strikeBar, alphaBar, betaBar, rhoBar, nuBar)));
}
//-------------------------------------------------------------------------
/// <summary>
/// Gets the amount as a string.
/// <para>
/// The format is the currency code, followed by a space, followed by the
/// amount: '${currency} ${amount}'.
///
/// </para>
/// </summary>
/// <returns> the currency amount </returns>
//JAVA TO C# CONVERTER TODO TASK: Most Java annotations will not have direct .NET equivalent attributes:
//ORIGINAL LINE: @Override @ToString public String toString()
public override string ToString()
{
return(currency + " " + (DoubleMath.isMathematicalInteger(amount) ? Convert.ToString((long)amount) : Convert.ToString(amount)));
}
public double volatility(double forward, double strike, double timeToExpiry, double alpha, double beta, double rho, double nu)
{
ArgChecker.isTrue(forward > 0.0, "forward must be greater than zero");
ArgChecker.isTrue(strike >= 0.0, "strike must be greater than zero");
ArgChecker.isTrue(timeToExpiry >= 0.0, "timeToExpiry must be greater than zero");
if (alpha == 0.0)
{
return(0.0);
}
double cutoff = forward * CUTOFF_MONEYNESS;
double k;
if (strike < cutoff)
{
log.info("Given strike of {} is less than cutoff at {}, therefore the strike is taken as {}", new object[] { strike, cutoff, cutoff });
k = cutoff;
}
else
{
k = strike;
}
double vol, z, zOverChi;
double beta1 = 1 - beta;
if (DoubleMath.fuzzyEquals(forward, k, ATM_EPS))
{
double f1 = Math.Pow(forward, beta1);
vol = alpha * (1 + timeToExpiry * (beta1 * beta1 * alpha * alpha / 24 / f1 / f1 + rho * alpha * beta * nu / 4 / f1 + nu * nu * (2 - 3 * rho * rho) / 24)) / f1;
}
else
{
if (DoubleMath.fuzzyEquals(beta, 0, BETA_EPS))
{
double ln = Math.Log(forward / k);
z = nu * Math.Sqrt(forward * k) * ln / alpha;
zOverChi = getZOverChi(rho, z);
vol = alpha * ln * zOverChi * (1 + timeToExpiry * (alpha * alpha / forward / k + nu * nu * (2 - 3 * rho * rho)) / 24) / (forward - k);
}
else if (DoubleMath.fuzzyEquals(beta, 1, BETA_EPS))
{
double ln = Math.Log(forward / k);
z = nu * ln / alpha;
zOverChi = getZOverChi(rho, z);
vol = alpha * zOverChi * (1 + timeToExpiry * (rho * alpha * nu / 4 + nu * nu * (2 - 3 * rho * rho) / 24));
}
else
{
double ln = Math.Log(forward / k);
double f1 = Math.Pow(forward * k, beta1);
double f1Sqrt = Math.Sqrt(f1);
double lnBetaSq = Math.Pow(beta1 * ln, 2);
z = nu * f1Sqrt * ln / alpha;
zOverChi = getZOverChi(rho, z);
double first = alpha / (f1Sqrt * (1 + lnBetaSq / 24 + lnBetaSq * lnBetaSq / 1920));
double second = zOverChi;
double third = 1 + timeToExpiry * (beta1 * beta1 * alpha * alpha / 24 / f1 + rho * nu * beta * alpha / 4 / f1Sqrt + nu * nu * (2 - 3 * rho * rho) / 24);
vol = first * second * third;
}
}
//There is nothing to prevent the nu * nu * (2 - 3 * rho * rho) / 24 to be large negative, and hence the volatility negative
return(Math.Max(MIN_VOL, vol));
}
//-------------------------------------------------------------------------
public override double?apply(double?x)
{
ArgChecker.inRangeInclusive(x, 0d, 1d, "x");
double pp, p, t, h, w, lnA, lnB, u, a1 = _a - 1;
double b1 = _b - 1;
if (_a >= 1 && _b >= 1)
{
pp = x < 0.5 ? x.Value : 1 - x;
t = Math.Sqrt(-2 * Math.Log(pp));
p = (2.30753 + t * 0.27061) / (1 + t * (0.99229 + t * 0.04481)) - t;
if (p < 0.5)
{
p *= -1;
}
a1 = (Math.Sqrt(p) - 3.0) / 6.0;
double tempA = 1.0 / (2 * _a - 1);
double tempB = 1.0 / (2 * _b - 1);
h = 2.0 / (tempA + tempB);
w = p * Math.Sqrt(a1 + h) / h - (tempB - tempA) * (a1 + 5.0 / 6 - 2.0 / (3 * h));
p = _a / (_a + _b + Math.Exp(2 * w));
}
else
{
lnA = Math.Log(_a / (_a + _b));
lnB = Math.Log(_b / (_a + _b));
t = Math.Exp(_a * lnA) / _a;
u = Math.Exp(_b * lnB) / _b;
w = t + u;
if (x.Value < t / w)
{
p = Math.Pow(_a * w * x, 1.0 / _a);
}
else
{
p = 1 - Math.Pow(_b * w * (1 - x), 1.0 / _b);
}
}
double afac = -_lnGamma.apply(_a) - _lnGamma.apply(_b) + _lnGamma.apply(_a + _b);
double error;
for (int j = 0; j < 10; j++)
{
if (DoubleMath.fuzzyEquals(p, 0d, 1e-16) || DoubleMath.fuzzyEquals(p, (double)1, 1e-16))
{
throw new MathException("a or b too small for accurate evaluation");
}
error = _beta.apply(p) - x;
t = Math.Exp(a1 * Math.Log(p) + b1 * Math.Log(1 - p) + afac);
u = error / t;
t = u / (1 - 0.5 * Math.Min(1, u * (a1 / p - b1 / (1 - p))));
p -= t;
if (p <= 0)
{
p = 0.5 * (p + t);
}
if (p >= 1)
{
p = 0.5 * (p + t + 1);
}
if (Math.Abs(t) < EPS * p && j > 0)
{
break;
}
}
return(p);
}
/// <summary>
/// Computes the first and second order derivatives of the Black implied volatility in the SABR model.
/// <para>
/// The first derivative values will be stored in the input array {@code volatilityD}
/// The array contains, [0] Derivative w.r.t the forward, [1] the derivative w.r.t the strike, [2] the derivative w.r.t. to alpha,
/// [3] the derivative w.r.t. to beta, [4] the derivative w.r.t. to rho, and [5] the derivative w.r.t. to nu.
/// Thus the length of the array should be 6.
/// </para>
/// <para>
/// The second derivative values will be stored in the input array {@code volatilityD2}.
/// Only the second order derivative with respect to the forward and strike are implemented.
/// The array contains [0][0] forward-forward; [0][1] forward-strike; [1][1] strike-strike.
/// Thus the size should be 2 x 2.
/// </para>
/// <para>
/// Around ATM, a first order expansion is used to due to some 0/0-type indetermination.
/// The second order derivative produced is poor around ATM.
///
/// </para>
/// </summary>
/// <param name="forward"> the forward value of the underlying </param>
/// <param name="strike"> the strike value of the option </param>
/// <param name="timeToExpiry"> the time to expiry of the option </param>
/// <param name="data"> the SABR data. </param>
/// <param name="volatilityD"> the array used to return the first order derivative </param>
/// <param name="volatilityD2"> the array of array used to return the second order derivative </param>
/// <returns> the Black implied volatility </returns>
public override double volatilityAdjoint2(double forward, double strike, double timeToExpiry, SabrFormulaData data, double[] volatilityD, double[][] volatilityD2)
{
double k = Math.Max(strike, 0.000001);
double alpha = data.Alpha;
double beta = data.Beta;
double rho = data.Rho;
double nu = data.Nu;
// Forward
double h0 = (1 - beta) / 2;
double h1 = forward * k;
double h1h0 = Math.Pow(h1, h0);
double h12 = h1h0 * h1h0;
double h2 = Math.Log(forward / k);
double h22 = h2 * h2;
double h23 = h22 * h2;
double h24 = h23 * h2;
double f1 = h1h0 * (1 + h0 * h0 / 6.0 * (h22 + h0 * h0 / 20.0 * h24));
double f2 = nu / alpha * h1h0 * h2;
double f3 = h0 * h0 / 6.0 * alpha * alpha / h12 + rho * beta * nu * alpha / 4.0 / h1h0 + (2 - 3 * rho * rho) / 24.0 * nu * nu;
double sqrtf2 = Math.Sqrt(1 - 2 * rho * f2 + f2 * f2);
double f2x = 0.0;
double x = 0.0, xp = 0, xpp = 0;
if (DoubleMath.fuzzyEquals(f2, 0.0, SMALL_Z))
{
f2x = 1.0 - 0.5 * f2 * rho; //small f2 expansion to f2^2 terms
}
else
{
if (DoubleMath.fuzzyEquals(rho, 1.0, RHO_EPS))
{
x = f2 < 1.0 ? -Math.Log(1.0 - f2) - 0.5 * Math.Pow(f2 / (f2 - 1.0), 2) * (1.0 - rho) : Math.Log(2.0 * f2 - 2.0) - Math.Log(1.0 - rho);
}
else
{
x = Math.Log((sqrtf2 + f2 - rho) / (1 - rho));
}
xp = 1.0 / sqrtf2;
xpp = (rho - f2) / Math.Pow(sqrtf2, 3.0);
f2x = f2 / x;
}
double sigma = Math.Max(MIN_VOL, alpha / f1 * f2x * (1 + f3 * timeToExpiry));
// First level
double h0Dbeta = -0.5;
double sigmaDf1 = -sigma / f1;
double sigmaDf2 = 0;
if (DoubleMath.fuzzyEquals(f2, 0.0, SMALL_Z))
{
sigmaDf2 = alpha / f1 * (1 + f3 * timeToExpiry) * -0.5 * rho;
}
else
{
sigmaDf2 = alpha / f1 * (1 + f3 * timeToExpiry) * (1.0 / x - f2 * xp / (x * x));
}
double sigmaDf3 = alpha / f1 * f2x * timeToExpiry;
double sigmaDf4 = f2x / f1 * (1 + f3 * timeToExpiry);
double sigmaDx = -alpha / f1 * f2 / (x * x) * (1 + f3 * timeToExpiry);
//JAVA TO C# CONVERTER NOTE: The following call to the 'RectangularArrays' helper class reproduces the rectangular array initialization that is automatic in Java:
//ORIGINAL LINE: double[][] sigmaD2ff = new double[3][3];
double[][] sigmaD2ff = RectangularArrays.ReturnRectangularDoubleArray(3, 3);
sigmaD2ff[0][0] = -sigmaDf1 / f1 + sigma / (f1 * f1); //OK
sigmaD2ff[0][1] = -sigmaDf2 / f1;
sigmaD2ff[0][2] = -sigmaDf3 / f1;
if (DoubleMath.fuzzyEquals(f2, 0.0, SMALL_Z))
{
sigmaD2ff[1][2] = alpha / f1 * -0.5 * rho * timeToExpiry;
}
else
{
sigmaD2ff[1][1] = alpha / f1 * (1 + f3 * timeToExpiry) * (-2 * xp / (x * x) - f2 * xpp / (x * x) + 2 * f2 * xp * xp / (x * x * x));
sigmaD2ff[1][2] = alpha / f1 * timeToExpiry * (1.0 / x - f2 * xp / (x * x));
}
sigmaD2ff[2][2] = 0.0;
// double sigma = alpha / f1 * f2x * (1 + f3 * theta);
// Second level
double[] f1Dh = new double[3];
double[] f2Dh = new double[3];
double[] f3Dh = new double[3];
f1Dh[0] = h1h0 * (h0 * (h22 / 3.0 + h0 * h0 / 40.0 * h24)) + Math.Log(h1) * f1;
f1Dh[1] = h0 * f1 / h1;
f1Dh[2] = h1h0 * (h0 * h0 / 6.0 * (2.0 * h2 + h0 * h0 / 5.0 * h23));
f2Dh[0] = Math.Log(h1) * f2;
f2Dh[1] = h0 * f2 / h1;
f2Dh[2] = nu / alpha * h1h0;
f3Dh[0] = h0 / 3.0 * alpha * alpha / h12 - 2 * h0 * h0 / 6.0 * alpha * alpha / h12 * Math.Log(h1) - rho * beta * nu * alpha / 4.0 / h1h0 * Math.Log(h1);
f3Dh[1] = -2 * h0 * h0 / 6.0 * alpha * alpha / h12 * h0 / h1 - rho * beta * nu * alpha / 4.0 / h1h0 * h0 / h1;
f3Dh[2] = 0.0;
double[] f1Dp = new double[4]; // Derivative to sabr parameters
double[] f2Dp = new double[4];
double[] f3Dp = new double[4];
double[] f4Dp = new double[4];
f1Dp[0] = 0.0;
f1Dp[1] = f1Dh[0] * h0Dbeta;
f1Dp[2] = 0.0;
f1Dp[3] = 0.0;
f2Dp[0] = -f2 / alpha;
f2Dp[1] = f2Dh[0] * h0Dbeta;
f2Dp[2] = 0.0;
f2Dp[3] = h1h0 * h2 / alpha;
f3Dp[0] = h0 * h0 / 3.0 * alpha / h12 + rho * beta * nu / 4.0 / h1h0;
f3Dp[1] = rho * nu * alpha / 4.0 / h1h0 + f3Dh[0] * h0Dbeta;
f3Dp[2] = beta * nu * alpha / 4.0 / h1h0 - rho / 4.0 * nu * nu;
f3Dp[3] = rho * beta * alpha / 4.0 / h1h0 + (2 - 3 * rho * rho) / 12.0 * nu;
f4Dp[0] = 1.0;
f4Dp[1] = 0.0;
f4Dp[2] = 0.0;
f4Dp[3] = 0.0;
double sigmaDh1 = sigmaDf1 * f1Dh[1] + sigmaDf2 * f2Dh[1] + sigmaDf3 * f3Dh[1];
double sigmaDh2 = sigmaDf1 * f1Dh[2] + sigmaDf2 * f2Dh[2] + sigmaDf3 * f3Dh[2];
//JAVA TO C# CONVERTER NOTE: The following call to the 'RectangularArrays' helper class reproduces the rectangular array initialization that is automatic in Java:
//ORIGINAL LINE: double[][] f1D2hh = new double[2][2]; // No h0
double[][] f1D2hh = RectangularArrays.ReturnRectangularDoubleArray(2, 2); // No h0
//JAVA TO C# CONVERTER NOTE: The following call to the 'RectangularArrays' helper class reproduces the rectangular array initialization that is automatic in Java:
//ORIGINAL LINE: double[][] f2D2hh = new double[2][2];
double[][] f2D2hh = RectangularArrays.ReturnRectangularDoubleArray(2, 2);
//JAVA TO C# CONVERTER NOTE: The following call to the 'RectangularArrays' helper class reproduces the rectangular array initialization that is automatic in Java:
//ORIGINAL LINE: double[][] f3D2hh = new double[2][2];
double[][] f3D2hh = RectangularArrays.ReturnRectangularDoubleArray(2, 2);
f1D2hh[0][0] = h0 * (h0 - 1) * f1 / (h1 * h1);
f1D2hh[0][1] = h0 * h1h0 / h1 * h0 * h0 / 6.0 * (2.0 * h2 + 4.0 * h0 * h0 / 20.0 * h23);
f1D2hh[1][1] = h1h0 * (h0 * h0 / 6.0 * (2.0 + 12.0 * h0 * h0 / 20.0 * h2));
f2D2hh[0][0] = h0 * (h0 - 1) * f2 / (h1 * h1);
f2D2hh[0][1] = nu / alpha * h0 * h1h0 / h1;
f2D2hh[1][1] = 0.0;
f3D2hh[0][0] = 2 * h0 * (2 * h0 + 1) * h0 * h0 / 6.0 * alpha * alpha / (h12 * h1 * h1) + h0 * (h0 + 1) * rho * beta * nu * alpha / 4.0 / (h1h0 * h1 * h1);
f3D2hh[0][1] = 0.0;
f3D2hh[1][1] = 0.0;
//JAVA TO C# CONVERTER NOTE: The following call to the 'RectangularArrays' helper class reproduces the rectangular array initialization that is automatic in Java:
//ORIGINAL LINE: double[][] sigmaD2hh = new double[2][2]; // No h0
double[][] sigmaD2hh = RectangularArrays.ReturnRectangularDoubleArray(2, 2); // No h0
for (int loopx = 0; loopx < 2; loopx++)
{
for (int loopy = loopx; loopy < 2; loopy++)
{
sigmaD2hh[loopx][loopy] = (sigmaD2ff[0][0] * f1Dh[loopy + 1] + sigmaD2ff[0][1] * f2Dh[loopy + 1] + sigmaD2ff[0][2] * f3Dh[loopy + 1]) * f1Dh[loopx + 1] + sigmaDf1 * f1D2hh[loopx][loopy] + (sigmaD2ff[0][1] * f1Dh[loopy + 1] + sigmaD2ff[1][1] * f2Dh[loopy + 1] + sigmaD2ff[1][2] * f3Dh[loopy + 1]) * f2Dh[loopx + 1] + sigmaDf2 * f2D2hh[loopx][loopy] + (sigmaD2ff[0][2] * f1Dh[loopy + 1] + sigmaD2ff[1][2] * f2Dh[loopy + 1] + sigmaD2ff[2][2] * f3Dh[loopy + 1]) * f3Dh[loopx + 1] + sigmaDf3 * f3D2hh[loopx][loopy];
}
}
// Third level
double h1Df = k;
double h1Dk = forward;
double h1D2ff = 0.0;
double h1D2kf = 1.0;
double h1D2kk = 0.0;
double h2Df = 1.0 / forward;
double h2Dk = -1.0 / k;
double h2D2ff = -1 / (forward * forward);
double h2D2fk = 0.0;
double h2D2kk = 1.0 / (k * k);
volatilityD[0] = sigmaDh1 * h1Df + sigmaDh2 * h2Df;
volatilityD[1] = sigmaDh1 * h1Dk + sigmaDh2 * h2Dk;
volatilityD[2] = sigmaDf1 * f1Dp[0] + sigmaDf2 * f2Dp[0] + sigmaDf3 * f3Dp[0] + sigmaDf4 * f4Dp[0];
volatilityD[3] = sigmaDf1 * f1Dp[1] + sigmaDf2 * f2Dp[1] + sigmaDf3 * f3Dp[1] + sigmaDf4 * f4Dp[1];
if (DoubleMath.fuzzyEquals(f2, 0.0, SMALL_Z))
{
volatilityD[4] = -0.5 * f2 + sigmaDf3 * f3Dp[2];
}
else
{
double xDr;
if (DoubleMath.fuzzyEquals(rho, 1.0, RHO_EPS))
{
xDr = f2 > 1.0 ? 1.0 / (1.0 - rho) + (0.5 - f2) / (f2 - 1.0) / (f2 - 1.0) : 0.5 * Math.Pow(f2 / (1.0 - f2), 2.0) + 0.25 * (f2 - 4.0) * Math.Pow(f2 / (f2 - 1.0), 3) / (f2 - 1.0) * (1.0 - rho);
if (Doubles.isFinite(xDr))
{
volatilityD[4] = sigmaDf1 * f1Dp[2] + sigmaDx * xDr + sigmaDf3 * f3Dp[2] + sigmaDf4 * f4Dp[2];
}
else
{
volatilityD[4] = double.NegativeInfinity;
}
}
else
{
xDr = (-f2 / sqrtf2 - 1 + (sqrtf2 + f2 - rho) / (1 - rho)) / (sqrtf2 + f2 - rho);
volatilityD[4] = sigmaDf1 * f1Dp[2] + sigmaDx * xDr + sigmaDf3 * f3Dp[2] + sigmaDf4 * f4Dp[2];
}
}
volatilityD[5] = sigmaDf1 * f1Dp[3] + sigmaDf2 * f2Dp[3] + sigmaDf3 * f3Dp[3] + sigmaDf4 * f4Dp[3];
volatilityD2[0][0] = (sigmaD2hh[0][0] * h1Df + sigmaD2hh[0][1] * h2Df) * h1Df + sigmaDh1 * h1D2ff + (sigmaD2hh[0][1] * h1Df + sigmaD2hh[1][1] * h2Df) * h2Df + sigmaDh2 * h2D2ff;
volatilityD2[0][1] = (sigmaD2hh[0][0] * h1Dk + sigmaD2hh[0][1] * h2Dk) * h1Df + sigmaDh1 * h1D2kf + (sigmaD2hh[0][1] * h1Dk + sigmaD2hh[1][1] * h2Dk) * h2Df + sigmaDh2 * h2D2fk;
volatilityD2[1][0] = volatilityD2[0][1];
volatilityD2[1][1] = (sigmaD2hh[0][0] * h1Dk + sigmaD2hh[0][1] * h2Dk) * h1Dk + sigmaDh1 * h1D2kk + (sigmaD2hh[0][1] * h1Dk + sigmaD2hh[1][1] * h2Dk) * h2Dk + sigmaDh2 * h2D2kk;
return(sigma);
}
