/// <summary>Computes the 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(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 CdfNorm(int n, double[] a, double[] y, int startIndexA, int startIndexY)
{
for (int j = 0; j < n; j++)
{
y[j + startIndexY] = StandardNormalDistribution.GetCdfValue(a[j + startIndexA]);
}
}
/// <summary>Gets the implied strike for a specific european straddle.
/// </summary>
/// <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="undiscountedValue">The value of the option at the time of expiry, thus the price but <b>not</b> discounted to time 0.</param>
/// <param name="blackVolatility">The Black volatility.</param>
/// <param name="initialStrike">The initial strike used as initial guess for the Newton method.</param>
/// <param name="impliedStrike">The implied strike (output).</param>
/// <returns>
/// Returns a value indicating whether <paramref name="impliedStrike"/> contains valid data.
/// </returns>
/// <remarks>This method checks some academic conditions and is based on the Newton approach.
/// <para>The implied strike is not unique.</para>
/// </remarks>
protected static ImpliedCalculationResultState TryGetImpliedStrike(double forward, double timeToExpiry, double undiscountedValue, double blackVolatility, double initialStrike, out double impliedStrike)
{
/* the value of a straddle is given by K-F + 2 * CallValue(K) thus we do it in a simular way as in the case of a call option */
impliedStrike = initialStrike;
double optionValueForGivenStrike = undiscountedValue;
double absUndiscountedValue = Math.Abs(undiscountedValue);
double sqrtOfTime = Math.Sqrt(timeToExpiry);
double dplus, CDFAtdMinus;
for (int i = 1; i <= MaxNumberIterationImpliedStrike; i++)
{
if (impliedStrike < MachineConsts.SuperTinyEpsilon)
{
optionValueForGivenStrike = Math.Abs(forward - impliedStrike);
impliedStrike = forward - undiscountedValue;
}
else
{
dplus = (Math.Log(forward / impliedStrike) + 0.5 * blackVolatility * blackVolatility * timeToExpiry) / (blackVolatility * sqrtOfTime);
CDFAtdMinus = StandardNormalDistribution.GetCdfValue(dplus - blackVolatility * sqrtOfTime);
optionValueForGivenStrike = impliedStrike - forward + 2 * (forward * StandardNormalDistribution.GetCdfValue(dplus) - impliedStrike * CDFAtdMinus);
impliedStrike += optionValueForGivenStrike * Math.Log(optionValueForGivenStrike / undiscountedValue) / (1 + 2.0 * CDFAtdMinus);
}
if (Math.Abs(undiscountedValue - optionValueForGivenStrike) < absUndiscountedValue * MachineConsts.TinyEpsilon)
{
return(ImpliedCalculationResultState.ProperResult);
}
}
return(ImpliedCalculationResultState.NoProperResult);
}
/// <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 cumulative normal distribution function values of vector elements [in-place].
/// </summary>
/// <param name="n">The number of elements to be calculated.</param>
/// <param name="a">The input vector a; overwritten by the result of the operation, i.e. N(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 CdfNorm(int n, double[] a)
{
for (int j = 0; j < n; j++)
{
a[j] = StandardNormalDistribution.GetCdfValue(a[j]);
}
}
/// <summary>Gets the normalized value (price) of a call or put option.
/// </summary>
/// <param name="theta">Indicates if a call (=1) or put (=-1) is given.</param>
/// <param name="x">The logarithm of 'forward / strike'.</param>
/// <param name="expOfXDivTwo"><paramref name="x"/> divided by two and applied to the exponential function. This
/// argument will be computed once only (performance reason).</param>
/// <param name="normalisedVolatility">The normalized volatility, i.e. the volatility divided by the
/// square root of the time span between valuation and expiry date.</param>
/// <returns>The normalized value of the call/put option, thus multiplying the result
/// by the specific discount factor and the square root of forward * strike gives the value of the option.</returns>
/// <remarks>
/// The normalized price (value) of the option is given by
/// <para>
/// value / (discount factor * Sqrt(forward * strike))
/// </para>
/// thus multiply the result by the discount factor * Sqrt(forward * strike)
/// to get the correct price.
/// <para>The implementation is based on 'By implication', Peter Jäckel, formula (2.2)-(2.3).</para>
/// </remarks>
private static double NormalizedCallPutValue(int theta, double x, double expOfXDivTwo, double normalisedVolatility)
{
if (Math.Abs(x) < MachineConsts.Epsilon)
{
return(1 - 2 * StandardNormalDistribution.GetCdfValue(-normalisedVolatility / 2.0));
}
return(theta * (expOfXDivTwo * StandardNormalDistribution.GetCdfValue(theta * (x / normalisedVolatility + normalisedVolatility / 2.0))
- 1.0 / expOfXDivTwo * StandardNormalDistribution.GetCdfValue(theta * (x / normalisedVolatility - normalisedVolatility / 2.0))));
}
/// <summary>Gets the price of the option <c>at time of expiry</c>, i.e. not discounted.
/// </summary>
/// <param name="volatility">The volatility.</param>
/// <returns>The value of the option <c>at the time of expiry</c>, thus not discounted. To get the price just multiply the return value with the discount factor.</returns>
public double GetNoneDiscountedValue(double volatility)
{
if ((m_Strike < MachineConsts.Epsilon) || (volatility * volatility * m_TimeToExpiration < MachineConsts.Epsilon))
{
return(m_Forward >= m_Strike ? 0.0 : 1.0);
}
double d = (m_Strike - m_Forward) / (volatility * SqrtOfTimeToExpiration);
return(StandardNormalDistribution.GetCdfValue(d));
}
/// <summary>Gets the forward-delta of the option, i.e. the partial derivative of the option value formula with respect to the forward.
/// </summary>
/// <param name="volatility">The volatility.</param>
/// <returns>The forward-delta of the option, i.e. the partial derivative of the option value formula with respect to the forward.
/// </returns>
public double GetForwardDelta(double volatility)
{
if ((m_Strike < MachineConsts.Epsilon) || (volatility * volatility * m_TimeToExpiration < MachineConsts.Epsilon))
{
return((m_Strike > m_Forward) ? 1 : 0);
}
double d = (m_Strike - m_Forward) / (volatility * SqrtOfTimeToExpiration);
return(-m_DiscountFactor *StandardNormalDistribution.GetCdfValue(d));
}
/// <summary>Gets the price of the option <c>at time of expiry</c>, i.e. not discounted.
/// </summary>
/// <param name="volatility">The volatility.</param>
/// <returns>The value of the option <c>at the time of expiry</c>, thus not discounted. To get the price just multiply the return value with the discount factor.</returns>
public double GetNoneDiscountedValue(double volatility)
{
if (volatility * volatility * m_TimeToExpiration < MachineConsts.Epsilon)
{
return(Math.Max(0, m_Strike - m_Forward));
}
double d = (m_Strike - m_Forward) / (volatility * SqrtOfTimeToExpiration);
return((m_Strike - m_Forward) * StandardNormalDistribution.GetCdfValue(d) + volatility * SqrtOfTimeToExpiration * StandardNormalDistribution.GetPdfValue(d));
}
/// <summary>Gets the strike-delta of the option, i.e. the partial derivative of the option value formula with respect to the strike.
/// </summary>
/// <param name="volatility">The volatility.</param>
/// <returns>The strike-delta of the option, i.e. the partial derivative of the option value formula with respect to the strike.</returns>
public double GetStrikeDelta(double volatility)
{
if (volatility * volatility * m_TimeToExpiration < MachineConsts.Epsilon)
{
return(0.0);
}
double d = (m_Forward - m_Strike) / (volatility * SqrtOfTimeToExpiration);
return(m_DiscountFactor * (1.0 - 2.0 * StandardNormalDistribution.GetCdfValue(d)));
}
/// <summary>Gets the forward-delta of the option, i.e. the partial derivative of the option value formula with respect to the forward.
/// </summary>
/// <param name="volatility">The volatility.</param>
/// <returns>The forward-delta of the option, i.e. the partial derivative of the option value formula with respect to the forward.</returns>
public double GetForwardDelta(double volatility)
{
if ((m_Strike < MachineConsts.Epsilon) || (volatility * volatility * m_TimeToExpiration < MachineConsts.Epsilon))
{
return((m_Forward >= m_Strike) ? 1 : -1);
}
double dplus = (LogOfMoneyness + 0.5 * volatility * volatility * m_TimeToExpiration) / (volatility * SqrtOfTimeToExpiration);
return(m_DiscountFactor * (2.0 * StandardNormalDistribution.GetCdfValue(dplus) - 1.0));
}
/// <summary>Gets the strike-delta of the option, i.e. the partial derivative of the option value formula with
/// respect to the strike.
/// </summary>
/// <param name="volatility">The volatility.</param>
/// <returns>The strike-delta of the option, i.e. the partial derivative of the option value formula with respect to the strike.</returns>
public double GetStrikeDelta(double volatility)
{
if ((m_Strike < MachineConsts.Epsilon) || (volatility * volatility * m_TimeToExpiration < MachineConsts.Epsilon))
{
return(0.0);
}
double dminus = (LogOfMoneyness - 0.5 * volatility * volatility * m_TimeToExpiration) / (volatility * SqrtOfTimeToExpiration);
return(-m_DiscountFactor *StandardNormalDistribution.GetCdfValue(dminus));
}
/// <summary>Gets the price of the option <c>at time of expiry</c>, i.e. not discounted.
/// </summary>
/// <param name="volatility">The volatility.</param>
/// <returns>The value of the option <c>at the time of expiry</c>, thus not discounted. To get the price just multiply the return value with the discount factor.
/// </returns>
public double GetNoneDiscountedValue(double volatility)
{
/* mind numerical traps: */
if ((m_Strike < MachineConsts.Epsilon) || (volatility * volatility * m_TimeToExpiration < MachineConsts.Epsilon))
{
return(m_Forward <= m_Strike ? 1.0 : 0.0);
}
double dminus = (LogOfMoneyness - 0.5 * volatility * volatility * m_TimeToExpiration) / (volatility * SqrtOfTimeToExpiration);
return(StandardNormalDistribution.GetCdfValue(-dminus));
}
/// <summary>Gets the price of the option <c>at time of expiry</c>, i.e. not discounted.
/// </summary>
/// <param name="volatility">The volatility.</param>
/// <returns>The value of the option <c>at the time of expiry</c>, thus not discounted. To get the price just multiply the return value with the discount factor.</returns>
public double GetNoneDiscountedValue(double volatility)
{
if ((m_Strike < MachineConsts.Epsilon) || (volatility * volatility * m_TimeToExpiration < MachineConsts.Epsilon))
{
return(Math.Max(0, m_Strike - m_Forward));
}
double dplus = (LogOfMoneyness + 0.5 * volatility * volatility * m_TimeToExpiration) / (volatility * SqrtOfTimeToExpiration);
double dminus = dplus - volatility * SqrtOfTimeToExpiration;
return(m_Strike * StandardNormalDistribution.GetCdfValue(-dminus) - m_Forward * StandardNormalDistribution.GetCdfValue(-dplus));
}
/// <summary>Gets the price of the option <c>at time of expiry</c>, i.e. not discounted.
/// </summary>
/// <param name="volatility">The volatility.</param>
/// <returns>The value of the option <c>at the time of expiry</c>, thus not discounted. To get the price just multiply the return value with the discount factor.</returns>
public double GetNoneDiscountedValue(double volatility)
{
/* mind numerical traps: */
if ((m_Strike < MachineConsts.Epsilon) || (volatility * volatility * m_TimeToExpiration < MachineConsts.Epsilon))
{
return(Math.Max(0, m_Forward - m_Strike));
}
double dplus = LogOfMoneyness / (volatility * SqrtOfTimeToExpiration) + 0.5 * volatility * SqrtOfTimeToExpiration;
double dminus = dplus - volatility * SqrtOfTimeToExpiration;
return(m_Forward * StandardNormalDistribution.GetCdfValue(dplus) - m_Strike * StandardNormalDistribution.GetCdfValue(dminus));
}
/// <summary>Gets the price of the option <c>at time of expiry</c>, i.e. not discounted.
/// </summary>
/// <param name="volatility">The volatility.</param>
/// <returns>The value of the option <c>at the time of expiry</c>, thus not discounted. To get the price just multiply the return value with the discount factor.</returns>
public double GetNoneDiscountedValue(double volatility)
{
double forwardMinusStrike = m_Forward - m_Strike;
if (volatility * volatility * m_TimeToExpiration < MachineConsts.Epsilon)
{
return(Math.Max(0, forwardMinusStrike));
}
double d = forwardMinusStrike / (volatility * SqrtOfTimeToExpiration);
return(forwardMinusStrike * StandardNormalDistribution.GetCdfValue(d) + volatility * SqrtOfTimeToExpiration * StandardNormalDistribution.GetPdfValue(d));
}
/// <summary>Gets the price of the option <c>at time of expiry</c>, i.e. not discounted.
/// </summary>
/// <param name="volatility">The volatility.</param>
/// <returns>The value of the option <c>at the time of expiry</c>, thus not discounted. To get
/// the price just multiply the return value with the discount factor.
/// </returns>
public double GetNoneDiscountedValue(double volatility)
{
if ((m_Strike < MachineConsts.Epsilon) || (volatility * volatility * m_TimeToExpiration < MachineConsts.Epsilon))
{
return(Math.Abs(m_Strike - m_Forward));
}
double dplus = (LogOfMoneyness + 0.5 * volatility * volatility * m_TimeToExpiration) / (volatility * SqrtOfTimeToExpiration);
double dminus = dplus - volatility * SqrtOfTimeToExpiration;
/* the value is given by: (K: strike, f: forward), use N(-t) = 1-N(t)
* discfactor * [ f * N(d+) - K * N(d-) + K * N(-d-) - f * N(-d+) ], i.e. call + put value
* = discfactor * [ f * N(d+) - K * N(d-) + K* (1-N(d-)) - f * (1-N(d+))
* = discfactor * [2 * f * N(d+) - 2 * K * N(d-) + K -f] */
return(m_Strike - m_Forward + 2.0 * m_Forward * StandardNormalDistribution.GetCdfValue(dplus) - 2.0 * m_Strike * StandardNormalDistribution.GetCdfValue(dminus));
}
/// <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)
{
/* Initial value is taken from rational approximation of a straddle option. The algorithm converges with quaratic order of convergence, therefore
* for an arbitrary point for which the algorithm is well-defined. Because of the good initial point, often it is one iteration required only. */
double v0;
double sqrtOfTime = Math.Sqrt(timeToExpiry);
if (RationalApproximation.TryGetStraddleImpliedVolatility(strike - forward, sqrtOfTime, 2.0 * c0 * forward + strike - forward, out value) == ImpliedCalculationResultState.ProperResult)
{
v0 = value * sqrtOfTime / forward;
}
else
{
v0 = 0.1 * sqrtOfTime / forward; // default if approximation fails.
}
var x = 1 - strike / forward;
if (x > 0) // "in-out" duality
{
c0 = c0 - x;
x = -x;
}
var oneOverOnePlusW = 1.0 / (1 + m_RelaxationParameter);
var v = v0;
for (int j = 0; j < m_MaxNumberOfIterations; j++)
{
var n = StandardNormalDistribution.GetPdfValue(x / v);
var N = StandardNormalDistribution.GetCdfValue(x / v);
double c = v * n + x * N; // undiscounted normalized option value
v = (c0 - x * N + m_RelaxationParameter * v * n) * oneOverOnePlusW / n;
if (Math.Abs(c - c0) < m_Tolerance)
{
value = forward * v / sqrtOfTime; // 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 / sqrtOfTime;
return(ImpliedCalculationResultState.NoProperResult);
}
/// <summary>Gets the implied strike for a given value (price) and Black-Scholes volatility of a specific european call or put option.
/// </summary>
/// <param name="theta">A value indicating whether a call (=1) or put (=-1) is given.</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="blackVolatility">The Black volatility.</param>
/// <param name="sqrtOfTime">The square root of <paramref name="timeToExpiry"/>.</param>
/// <param name="impliedStrike">The implied strike (output).</param>
/// <returns>Returns a value indicating whether <paramref name="impliedStrike"/> contains valid data.</returns>
/// <remarks>The implementation is based on the Newton approach.</remarks>
protected static ImpliedCalculationResultState TryGetPlainVanillaImpliedStrike(int theta, double forward, double timeToExpiry, double noneDiscountedValue, double blackVolatility, double sqrtOfTime, out double impliedStrike)
{
/* we apply Newtons method to the function
* H(K) = ln( f(K) / c),
* where f(K) is the (undiscounted) value of the option and c is the given undiscounted value
* of the option. It holds H'(K) = f'(K)/f(K) and Newtons method gives
*
* K_{n+1} = K_n - H(K_n)/H'(K_n) = K_n - ln(f(K)/c) * f(K) / f'(K).
*/
impliedStrike = forward;
double absOfValueAtExpiry = Math.Abs(noneDiscountedValue);
double optionValueForGivenStrike = noneDiscountedValue;
double dplus, CDFAtThetaTimesdMinus;
for (int i = 1; i <= MaxNumberIterationImpliedStrike; i++)
{
if (impliedStrike < MachineConsts.SuperTinyEpsilon)
{
optionValueForGivenStrike = forward + impliedStrike;
impliedStrike = forward - theta * noneDiscountedValue;
}
else
{
dplus = (Math.Log(forward / impliedStrike) + 0.5 * blackVolatility * blackVolatility * timeToExpiry) / (blackVolatility * sqrtOfTime);
CDFAtThetaTimesdMinus = StandardNormalDistribution.GetCdfValue(theta * (dplus - blackVolatility * sqrtOfTime));
optionValueForGivenStrike = (theta * forward * StandardNormalDistribution.GetCdfValue(theta * dplus) - theta * impliedStrike * CDFAtThetaTimesdMinus);
impliedStrike += theta * optionValueForGivenStrike * Math.Log(optionValueForGivenStrike / noneDiscountedValue) / CDFAtThetaTimesdMinus;
}
if (Math.Abs(noneDiscountedValue - optionValueForGivenStrike) < absOfValueAtExpiry * MachineConsts.TinyEpsilon)
{
return(ImpliedCalculationResultState.ProperResult);
}
}
return(ImpliedCalculationResultState.NoProperResult);
}
/// <summary>Gets the implied strike for a given value (price) and Bachelier (=Normal Black) volatility of a specific european call or put option.
/// </summary>
/// <param name="theta">A value indicating whether a call (=1) or put (=-1) is given.</param>
/// <param name="forward">The forward.</param>
/// <param name="undiscountedValue">The value of the option at the time of expiry, thus the price but <b>not</b> discounted to time 0.</param>
/// <param name="bachelierVolatility">The Bachelier (=Normal Black) volatility.</param>
/// <param name="sqrtOfTime">The square root of the time span between valuation date and expiry date.</param>
/// <param name="impliedStrike">The implied strike (output), input represents the initial value of the Newton-Method.</param>
/// <returns>Returns a value indicating whether <paramref name="impliedStrike"/> contains valid data.
/// </returns>
/// <remarks>The implementation is based on the Newton approach and the forward is used as initial guess.</remarks>
protected static ImpliedCalculationResultState TryGetPlainVanillaImpliedStrike(int theta, double forward, double undiscountedValue, double bachelierVolatility, double sqrtOfTime, ref double impliedStrike)
{
/* we apply Newtons method to the function
* H(K) = ln( f(K) / c),
* where f(K) is the (undiscounted) value of the option and c is the given undiscounted value
* of the option. It holds H'(K) = f'(K)/f(K) and Newtons method gives
*
* K_{n+1} = K_n - H(K_n)/H'(K_n) = K_n - ln(f(K)/c) * f(K) / f'(K).
*/
double absOfValueAtExpiry = Math.Abs(undiscountedValue);
double optionValueForGivenStrike = undiscountedValue;
double d, CDF_Theta_d;
for (int i = 1; i <= MaxNumberIterationImpliedStrike; i++)
{
if (impliedStrike < MachineConsts.SuperTinyEpsilon)
{
optionValueForGivenStrike = forward + impliedStrike;
impliedStrike = forward - theta * undiscountedValue;
}
else
{
d = (forward - impliedStrike) / (bachelierVolatility * sqrtOfTime);
CDF_Theta_d = StandardNormalDistribution.GetCdfValue(theta * d);
optionValueForGivenStrike = theta * (forward - impliedStrike) * CDF_Theta_d + bachelierVolatility * sqrtOfTime * StandardNormalDistribution.GetPdfValue(theta * d);
impliedStrike += theta * optionValueForGivenStrike * Math.Log(optionValueForGivenStrike / undiscountedValue) / CDF_Theta_d;
}
if (Math.Abs(undiscountedValue - optionValueForGivenStrike) < absOfValueAtExpiry * MachineConsts.TinyEpsilon)
{
return(ImpliedCalculationResultState.ProperResult);
}
}
return(ImpliedCalculationResultState.NoProperResult);
}
/// <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);
}
/// <summary>Gets the price of the option <c>at time of expiry</c>, i.e. not discounted.
/// </summary>
/// <param name="volatility">The volatility.</param>
/// <returns>The value of the option <c>at the time of expiry</c>, thus not discounted. To get the price just multiply the return value with the discount factor.</returns>
public double GetNoneDiscountedValue(double volatility)
{
if (volatility * volatility * m_TimeToExpiration < MachineConsts.Epsilon)
{
return(Math.Abs(m_Strike - m_Forward));
}
// here, we use the fact N(-d) = 1 - N(d):
double d = (m_Forward - m_Strike) / (volatility * SqrtOfTimeToExpiration);
return(2.0 * (volatility * SqrtOfTimeToExpiration * StandardNormalDistribution.GetPdfValue(d) + (m_Strike - m_Forward) * (0.5 - StandardNormalDistribution.GetCdfValue(d))));
}
/// <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>
/// <remarks>This method is based on Newtons method where the initial guess is equal to the approximation given in
/// <para>How close are the option pricing formulas of Bachelier and Black-Merton-Scholes, W. Schachermayer, J. Teichmann.</para>
/// <para>The implied strike is not unique for straddle options.</para>
/// </remarks>
public ImpliedCalculationResultState TryGetImpliedStrike(double optionValue, double volatility, out double impliedStrike)
{
double undiscountedValue = optionValue / m_DiscountFactor;
if (m_TimeToExpiration < MachineConsts.Epsilon)
{
impliedStrike = m_Forward - undiscountedValue;
return(ImpliedCalculationResultState.ProperResult);
}
/* compute initial guess:
* Straddle(m) = C(m) + P(m) \approx 2 * a + m^2/( 2 pi * a),
* see eqn.(3.4)&(3.5), where the left hand side is already given.
*/
double a = volatility * SqrtOfTimeToExpiration / MathConsts.SqrtTwoPi;
double m = Math.Sqrt(Math.Abs((undiscountedValue - 2 * a) * MathConsts.TwoPi * a));
impliedStrike = m_Forward + m;
/* we apply Newtons method to the function
* H(K) = ln( f(K) / c),
* where f(K) is the (undiscounted) value of the option and c is the given undiscounted value
* of the option. It holds H'(K) = f'(K)/f(K) and Newtons method gives
*
* K_{n+1} = K_n - H(K_n)/H'(K_n) = K_n - ln(f(K)/c) * f(K) / f'(K),
*
* here f'(K) = 1 - 2 * N(d) which will be equal to 0, if K=forward.
*/
double absOfValueAtExpiry = Math.Abs(undiscountedValue);
double optionValueForGivenStrike = undiscountedValue;
double sqrtOfTime = SqrtOfTimeToExpiration;
double d, CDF_d;
for (int i = 1; i <= MaxNumberIterationImpliedStrike; i++)
{
d = (m_Forward - impliedStrike) / (volatility * sqrtOfTime);
CDF_d = StandardNormalDistribution.GetCdfValue(d);
optionValueForGivenStrike = 2.0 * (volatility * sqrtOfTime * StandardNormalDistribution.GetPdfValue(d) + (impliedStrike - m_Forward) * (0.5 - CDF_d));
if (Math.Abs(d) < MachineConsts.Epsilon) // forward = strike and f'(K) = 0, i.e. denominator = 0
{
if (Math.Abs(undiscountedValue - optionValueForGivenStrike) < absOfValueAtExpiry * MachineConsts.TinyEpsilon)
{
return(ImpliedCalculationResultState.ProperResult);
}
/* here, we do a better approximation for the initial guess of the implied strike,
* i.e. straddle = C(m) + P(m) = 2 a + m^2/(2Pi a) - m^4/(48 * Pi^2 a^3), we use
* a root-finding approach the these polynomials and we use the root 'm' (= strike - forward)
* where Straddle(m) - undiscountedValue is minimal with respect to each root.
*/
IRealPolynomial polynomial = Polynomial.Real.Create(2 * a, 0, 1.0 / (a * MathConsts.TwoPi), 0, -1.0 / (a * a * a * 48 * MathConsts.PiSquared));
List <double> roots = new List <double>();
polynomial.GetRealRoots(roots, Polynomial.RootFinder.LaguerreApproach.StandardPolishing);
if (roots.Count == 0)
{
throw new ArithmeticException("Straddle approximation polynomial of degree 4 has no real root.");
}
else
{
impliedStrike = m_Forward + roots[0];
double minDistance = Math.Abs(undiscountedValue - GetUndiscountedOptionValue(roots[0], volatility, sqrtOfTime));
for (int j = 1; j < roots.Count; j++)
{
double distance = Math.Abs(undiscountedValue - GetUndiscountedOptionValue(roots[j], volatility, sqrtOfTime));
if (distance < minDistance)
{
impliedStrike = m_Forward + roots[j];
minDistance = distance;
}
}
}
}
else
{
impliedStrike -= optionValueForGivenStrike * Math.Log(optionValueForGivenStrike / undiscountedValue) / (1 - 2 * CDF_d);
}
if (Math.Abs(undiscountedValue - optionValueForGivenStrike) < absOfValueAtExpiry * MachineConsts.TinyEpsilon)
{
return(ImpliedCalculationResultState.ProperResult);
}
}
return(ImpliedCalculationResultState.NoProperResult);
}
/// <summary>Gets the strike-delta of the option, i.e. the partial derivative of the option value formula with
/// respect to the strike.
/// </summary>
/// <param name="volatility">The volatility.</param>
/// <returns>The strike-delta of the option, i.e. the partial derivative of the option value formula with respect to the strike.</returns>
public double GetStrikeDelta(double volatility)
{
if ((m_Strike < MachineConsts.Epsilon) || (volatility * volatility * m_TimeToExpiration < MachineConsts.Epsilon))
{
return(0.0);
}
double dplus = (LogOfMoneyness + 0.5 * volatility * volatility * m_TimeToExpiration) / (volatility * SqrtOfTimeToExpiration);
double dminus = dplus - volatility * SqrtOfTimeToExpiration;
return(m_DiscountFactor * (-2.0 * m_Forward / (volatility * SqrtOfTimeToExpiration * m_Strike) * StandardNormalDistribution.GetPdfValue(dplus) - (2 * StandardNormalDistribution.GetCdfValue(dminus) - 1) + 2.0 * StandardNormalDistribution.GetPdfValue(dminus) / (volatility * SqrtOfTimeToExpiration)));
}
/// <summary>Gets the undiscounted option value using the Bachelier (=Normal Black) model thus the value <c>at expiry</c> of the option.
/// </summary>
/// <param name="strikeMinusForward">'Strike minus forward'.</param>
/// <param name="volatility">The Bachelier (=Normal Black) volatility.</param>
/// <param name="sqrtOfTime">The square root of the <see cref="System.Double"/> representation of the time span between valuation date and expiry date.</param>
/// <returns>The <b>undiscounted</b> Bachelier (=Normal Black) price of the call option, thus the value <c>at time of expiry</c>, i.e.
/// to get the price just multiply the return value with the discount factor.</returns>
private static double GetUndiscountedOptionValue(double strikeMinusForward, double volatility, double sqrtOfTime)
{
// here, we use the fact N(-d) = 1 - N(d):
double d = -strikeMinusForward / (volatility * sqrtOfTime);
return(2.0 * (volatility * sqrtOfTime * StandardNormalDistribution.GetPdfValue(d) + strikeMinusForward * (0.5 - StandardNormalDistribution.GetCdfValue(d))));
}
