/// <summary>Gets the implied Black volatility of a specific european call option.
/// </summary>
/// <param name="strike">The strike.</param>
/// <param name="forward">The forward.</param>
/// <param name="timeToExpiry">The time span between valuation date and expiry date in its <see cref="System.Double"/> representation.</param>
/// <param name="c0">The dimensionless option price, i.e. option price divided by forward and discount factor.</param>
/// <param name="value">The implied Black volatility (output).</param>
/// <returns>A value indicating whether <paramref name="value"/> contains valid data.</returns>
private ImpliedCalculationResultState TryGetImpliedCallVolatility(double strike, double forward, double timeToExpiry, double c0, out double value)
{
var x = Math.Log(forward / strike);
var expOfMinusx = strike / forward;
if (x > 0) // "in-out" duality
{
x = -x;
expOfMinusx = forward / strike;
c0 = expOfMinusx * c0 + 1 - expOfMinusx; // expOfMinusx = exp( [original] x )
}
var v0 = GetInitialCallOptionTotalVolatility(x, c0);
var oneOverOnePlusW = 1.0 / (1 + m_RelaxationParameter);
var v = v0;
for (int j = 0; j < m_MaxNumberOfIterations; j++)
{
var nPlus = StandardNormalDistribution.GetCdfValue(x / v + 0.5 * v);
var nMinus = expOfMinusx * StandardNormalDistribution.GetCdfValue(x / v - 0.5 * v);
double c = nPlus - nMinus; // undiscounted normalized option value
var temp = StandardNormalDistribution.GetInverseCdfValue((c0 + nMinus + m_RelaxationParameter * nPlus) * oneOverOnePlusW);
v = temp + Math.Sqrt(temp * temp + 2.0 * Math.Abs(x));
if (Math.Abs(c - c0) < m_Tolerance)
{
value = v / Math.Sqrt(timeToExpiry); // here, we assume that the 'new' total volatility estimation is better than the one used for the calculation of 'c'
return(ImpliedCalculationResultState.ProperResult);
}
}
value = v / Math.Sqrt(timeToExpiry);
return(ImpliedCalculationResultState.NoProperResult);
}
/// <summary>Computes the inverse cumulative normal distribution function values of vector elements.
/// </summary>
/// <param name="n">The number of elements to be calculated.</param>
/// <param name="a">The input vector a.</param>
/// <param name="y">The output vector N^{-1}(a[j]), j=0,...,<paramref name="n"/>-1, where N(x) =\int_{-\infty}^x 1/\sqrt(2*PI) * exp(-1/2 *t^2) dt.</param>
public void CdfNormInv(int n, double[] a, double[] y)
{
for (int j = 0; j < n; j++)
{
y[j] = StandardNormalDistribution.GetInverseCdfValue(a[j]);
}
}
/// <summary>Computes the inverse cumulative normal distribution function values of vector elements.
/// </summary>
/// <param name="n">The number of elements to be calculated.</param>
/// <param name="a">The input vector a.</param>
/// <param name="y">The output vector y with y[<paramref name="startIndexY"/> + j] := N^{-1}(a[<paramref name="startIndexA"/> + j]) for j = 0,...,<paramref name="n"/>-1, where N(x) =\int_{-\infty}^x 1/\sqrt(2*PI) * exp(-1/2 *t^2) dt.</param>
/// <param name="startIndexA">The null-based start index of <paramref name="a"/>.</param>
/// <param name="startIndexY">The null-based start index of <paramref name="y"/>.</param>
public void CdfNormInv(int n, double[] a, double[] y, int startIndexA, int startIndexY)
{
for (int j = 0; j < n; j++)
{
y[j + startIndexY] = StandardNormalDistribution.GetInverseCdfValue(a[j + startIndexA]);
}
}
/// <summary>Gets the implied strike for a specific option price.
/// </summary>
/// <param name="optionValue">The value of the option.</param>
/// <param name="volatility">The volatility.</param>
/// <param name="impliedStrike">The implied strike (output).</param>
/// <returns>A value indicating whether <paramref name="impliedStrike"/> contains valid data.</returns>
public ImpliedCalculationResultState TryGetImpliedStrike(double optionValue, double volatility, out double impliedStrike)
{
if (m_TimeToExpiration < MachineConsts.Epsilon)
{
impliedStrike = m_Forward;
return(ImpliedCalculationResultState.ProperResult);
}
impliedStrike = m_Forward * Math.Exp(StandardNormalDistribution.GetInverseCdfValue(optionValue / m_DiscountFactor) * volatility * Math.Sqrt(m_TimeToExpiration) - 0.5 * volatility * volatility * m_TimeToExpiration);
return((Double.IsNaN(impliedStrike) == false) ? ImpliedCalculationResultState.ProperResult : ImpliedCalculationResultState.NoProperResult);
}
/// <summary>Gets the value of the copula at some point.
/// </summary>
/// <param name="x">The argument x.</param>
/// <param name="y">The argument y.</param>
/// <returns>
/// The value of the copula at (<paramref name="x"/>,<paramref name="y"/>).
/// </returns>
public double GetValue(double x, double y)
{
if ((x < MachineConsts.SuperTinyEpsilon) || (y < MachineConsts.SuperTinyEpsilon))
{
return(0.0);
}
if (1.0 - x < MachineConsts.SuperTinyEpsilon)
{
if (1.0 - y < MachineConsts.SuperTinyEpsilon)
{
return(x);
}
return(y);
}
if (1.0 - y < MachineConsts.SuperTinyEpsilon)
{
return(x);
}
return(BivariateNormalDistribution.StandardCDFValue(StandardNormalDistribution.GetInverseCdfValue(x), StandardNormalDistribution.GetInverseCdfValue(y), m_Correlation));
}
/// <summary>Gets the implied volatility for a specific non-discounted option price.
/// </summary>
/// <param name="noneDiscountedValue">The value of the option at the time of expiry, thus the price but <b>not</b> discounted to time 0.</param>
/// <param name="impliedVolatility">The implied volatility (output).</param>
/// <returns>A value indicating whether <paramref name="impliedVolatility" /> contains valid data.</returns>
/// <remarks>This method is the inverse function of <see cref="IConstantVolatilityStandardEuropeanOption.GetNoneDiscountedValue(double)" />.</remarks>
public ImpliedCalculationResultState TryGetImpliedVolatilityOfNonDiscountedValue(double noneDiscountedValue, out double impliedVolatility)
{
double x = Math.Log(m_Forward / m_Strike);
double NInverse = StandardNormalDistribution.GetInverseCdfValue(noneDiscountedValue);
if (NInverse * NInverse + 2 * x >= 0)
{
double root = Math.Sqrt(NInverse * NInverse + 2 * x);
impliedVolatility = NInverse + root;
if (NInverse - root >= 0)
{
impliedVolatility = NInverse - root;
}
impliedVolatility /= Math.Sqrt(m_TimeToExpiration);
return(ImpliedCalculationResultState.ProperResult);
}
impliedVolatility = 0.0;
return(ImpliedCalculationResultState.NoProperResult);
}
/// <summary>Gets the implied strike for a specific option price.
/// </summary>
/// <param name="optionValue">The value of the option.</param>
/// <param name="volatility">The volatility.</param>
/// <param name="impliedStrike">The implied strike (output).</param>
/// <returns>A value indicating whether <paramref name="impliedStrike"/> contains valid data.</returns>
public ImpliedCalculationResultState TryGetImpliedStrike(double optionValue, double volatility, out double impliedStrike)
{
impliedStrike = m_Forward + volatility * SqrtOfTimeToExpiration * StandardNormalDistribution.GetInverseCdfValue(optionValue / m_DiscountFactor);
return((Double.IsInfinity(impliedStrike) || Double.IsNaN(impliedStrike)) ? ImpliedCalculationResultState.InputError : ImpliedCalculationResultState.ProperResult);
}
/// <summary>Gets the implied volatility for a specific non-discounted option price.
/// </summary>
/// <param name="noneDiscountedValue">The value of the option at the time of expiry, thus the price but <b>not</b> discounted to time 0.</param>
/// <param name="impliedVolatility">The implied volatility (output).</param>
/// <returns>A value indicating whether <paramref name="impliedVolatility" /> contains valid data.</returns>
/// <remarks>This method is the inverse function of <see cref="IConstantVolatilityStandardEuropeanOption.GetNoneDiscountedValue(double)" />.</remarks>
public ImpliedCalculationResultState TryGetImpliedVolatilityOfNonDiscountedValue(double noneDiscountedValue, out double impliedVolatility)
{
impliedVolatility = (m_Strike - m_Forward) / (SqrtOfTimeToExpiration * StandardNormalDistribution.GetInverseCdfValue(noneDiscountedValue));
return((Double.IsInfinity(impliedVolatility) || Double.IsNaN(impliedVolatility)) ? ImpliedCalculationResultState.InputError : ImpliedCalculationResultState.ProperResult);
}
/// <summary>Gets the implied Black-Scholes volatility for a given value (price) of a specific call or put option.
/// </summary>
/// <param name="theta">A value indicating whether a call (=1) or put (=-1) is given.</param>
/// <param name="strike">The strike.</param>
/// <param name="forward">The forward.</param>
/// <param name="timeToExpiry">The time span between valuation date and expiry date in its <see cref="System.Double"/> representation.</param>
/// <param name="noneDiscountedValue">The value of the option at the time of expiry, thus the price but <b>not</b> discounted to time 0.</param>
/// <param name="impliedBlackVolatility">The implied Black volatility (output).</param>
/// <returns>A value indicating whether <paramref name="impliedBlackVolatility"/> contains valid data.</returns>
private ImpliedCalculationResultState TryGetPlainVanillaImpliedVolatility(int theta, double strike, double forward, double timeToExpiry, double noneDiscountedValue, out double impliedBlackVolatility)
{
impliedBlackVolatility = 0.0;
/* calculate some constants only once: */
double x = Math.Log(forward / strike);
if (Double.IsNaN(x)) // strike <= 0?
{
return(ImpliedCalculationResultState.InputError);
}
double beta = noneDiscountedValue / Math.Sqrt(forward * strike); // normalized value: the given value is already discounted
/* 1. case: at-the-money option:
* Compute the implied vola via inverse commulative distribution function */
if (Math.Abs(x) < MachineConsts.Epsilon) // at-the-money option
{
impliedBlackVolatility = -2.0 * StandardNormalDistribution.GetInverseCdfValue(0.5 - 0.5 * beta);
return(ImpliedCalculationResultState.ProperResult);
}
/* 2. case: in-the-money option:
* subtracting the normalized intrinsic value from the normalized value if necessary */
double expOfXDivTwo = Math.Sqrt(forward / strike); // = Math.Exp(x / 2.0);
double iota = theta * DoMath.HeavisideFunction(theta * x) * (expOfXDivTwo - 1 / expOfXDivTwo);
/* the normalized value has to be an element of the interval
* [iota, exp( \theta * x/2)],
* otherwise no implied value can be calculated!
*/
if (beta < iota)
{
return(ImpliedCalculationResultState.InputError); // one may sets 'beta = iota;'
}
if (beta > ((theta == 1) ? expOfXDivTwo : 1 / expOfXDivTwo))
{
return(ImpliedCalculationResultState.InputError); // one may sets 'beta = ((theta == 1) ? expOfXDivTwo : 1 / expOfXDivTwo);'
}
bool isInTheMoneyOption = false; // indicates if it is a in-the-money option
if (theta == -1)
{
if (x < 0)
{
isInTheMoneyOption = true;
}
}
else
{
if (x > 0) // strike < forward
{
isInTheMoneyOption = true;
}
}
if (isInTheMoneyOption == true)
{
beta -= iota;
theta = 1 - 2 * DoMath.HeavisideFunction(x);
/* now consider some out-of-the money option and recalculate the intrinsic value */
iota = theta * DoMath.HeavisideFunction(theta * x) * (expOfXDivTwo - 1 / expOfXDivTwo);
}
/* 3.) Set initual guess: */
double bc = NormalizedCallPutValue(theta, x, expOfXDivTwo, Math.Sqrt(2 * Math.Abs(x)));
/* 3a.) sigmaLow and sigmaHigh are given by formula (3.6)-(3.7) */
double tempForSigmaHighExp = (theta == 1) ? expOfXDivTwo : 1 / expOfXDivTwo; // = Math.Exp(theta * x / 2.0)
double tempForSigmaHighCDF = StandardNormalDistribution.GetCdfValue(-Math.Sqrt(Math.Abs(x) / 2.0));
double sigmaLow = Math.Sqrt(2 * x * x / (Math.Abs(x) - 4 * Math.Log((beta - iota) / (bc - iota))));
double sigmaHigh = -2 * StandardNormalDistribution.GetInverseCdfValue((tempForSigmaHighExp - beta) / (tempForSigmaHighExp - bc) * tempForSigmaHighCDF);
/* 2b.) calculate the gamma which is given in formula (4.7): */
double sigmaStar = -2 * StandardNormalDistribution.GetInverseCdfValue(tempForSigmaHighExp / (tempForSigmaHighExp - bc) * tempForSigmaHighCDF);
double bStar = NormalizedCallPutValue(theta, x, expOfXDivTwo, sigmaStar);
double sigmaLowStar = Math.Sqrt(2 * x * x / (Math.Abs(x) - 4 * Math.Log((bStar - iota) / (bc - iota))));
double sigmaHighStar = -2 * StandardNormalDistribution.GetInverseCdfValue((tempForSigmaHighExp - bStar) / (tempForSigmaHighExp - bc) * tempForSigmaHighCDF);
// the weight w^* given by formula (4.8)
double w = Math.Pow(Math.Min(Math.Max((sigmaStar - sigmaLowStar) / (sigmaHighStar - sigmaLowStar), 0), 1.0), Math.Log(bc / beta) / Math.Log(bc / bStar));
/* the code based on the 2006 edition of the paper, i.e. using formula (4.1) in "By implication", 24.11.2010:
* // double gamma = Math.Log((sigmaStar - sigmaLowStar) / (sigmaHighStar - sigmaLowStar)) / Math.Log(bStar / bc);
* // double w = Math.Min(1, Math.Pow(beta / bc, gamma)); // its the weight given by (4.2)
*
* /* 2c.) now the initial value is given; see formula (4.1): */
if (Double.IsNaN(sigmaLow) == false)
{
impliedBlackVolatility = sigmaLow * (1 - w) + sigmaHigh * w;
}
else
{
/* it is not hard to see that if \sigma_low is not defined then 'beta/bc' > 1
* which implies w = 1.0 */
impliedBlackVolatility = sigmaHigh;
}
/* 3.) Do the Halley's iteration method: */
double terminatingCondition = 1.0;
for (int i = 1; i <= m_MaxNumberOfIterations; i++)
{
/* A.) compute nu^_n(x,\sigma_n,\Theta) via formula (4.10) & (4.13): */
// first compute nu_n (without 'hat') via (3.10):
double b = NormalizedCallPutValue(theta, x, expOfXDivTwo, impliedBlackVolatility);
double impliedBSVolaSquare = impliedBlackVolatility * impliedBlackVolatility;
double bDash = MathConsts.OneOverSqrtTwoPi * Math.Exp(-0.5 * x * x / impliedBSVolaSquare - 0.125 * impliedBSVolaSquare);
double nuN = 0.0;
if (beta < bc)
{
nuN = Math.Log((beta - iota) / (b - iota)) * Math.Log(b - iota) / Math.Log(beta - iota) * (b - iota) / bDash;
}
else
{
nuN = (beta - b) / bDash;
}
double nuHat = Math.Max(nuN, -impliedBlackVolatility / 2.0);
/* B. compute eta^ via formula (4.14) & (4.15) & (4.11): */
double etaHat = x * x / (impliedBSVolaSquare * impliedBlackVolatility) - impliedBlackVolatility / 4.0;
if (beta < bc)
{
double logOfbMinusIota = Math.Log(b - iota);
etaHat -= (2 + logOfbMinusIota) / logOfbMinusIota * bDash / (b - iota);
}
etaHat = Math.Max(-0.75, etaHat / 2.0 * nuHat);
/* C. next step and store the terminating condition */
terminatingCondition = impliedBlackVolatility; // store the volatility which is used in the step before
impliedBlackVolatility = impliedBlackVolatility + Math.Max(nuHat / (1 + etaHat), -impliedBlackVolatility / 2.0);
terminatingCondition = Math.Abs(impliedBlackVolatility / terminatingCondition - 1);
if (Double.IsNaN(terminatingCondition))
{
impliedBlackVolatility = Double.NaN;
return(ImpliedCalculationResultState.NoProperResult);
}
if (terminatingCondition < m_Tolerance)
{
impliedBlackVolatility /= Math.Sqrt(timeToExpiry);
return(ImpliedCalculationResultState.ProperResult);
}
}
impliedBlackVolatility /= Math.Sqrt(timeToExpiry);
return(ImpliedCalculationResultState.NoProperResult);
}
