/// <summary>Gets the Forward-Vanna of the option, i.e. the derivative \partial^2/ {\partial \sigma \partial F} of the option value.
/// </summary>
/// <param name="optionParameters">The parameters of the option.</param>
/// <param name="getOptionValue">A delegate that calculates the value of the option for specified parameters.</param>
/// <returns>A approximation of the gamma.</returns>
public static double GetOptionForwardVanna(ConstantVolatilityStandardEuropeanOptionConfiguration optionParameters, GetOptionValue getOptionValue)
{
double h = 0.00005; // should be not too small, otherwise one get numerical problems
double k = 0.00005;
double value = getOptionValue(optionParameters.Strike, optionParameters.Forward, optionParameters.TimeToExpiry, optionParameters.DiscountFactor, optionParameters.Volatility);
double valuePlusHPlusK = getOptionValue(optionParameters.Strike, optionParameters.Forward + k, optionParameters.TimeToExpiry, optionParameters.DiscountFactor, optionParameters.Volatility + h);
double valuePlusHMinusK = getOptionValue(optionParameters.Strike, optionParameters.Forward - k, optionParameters.TimeToExpiry, optionParameters.DiscountFactor, optionParameters.Volatility + h);
double valueMinusHPlusK = getOptionValue(optionParameters.Strike, optionParameters.Forward + k, optionParameters.TimeToExpiry, optionParameters.DiscountFactor, optionParameters.Volatility - h);
double valueMinusHMinusK = getOptionValue(optionParameters.Strike, optionParameters.Forward - k, optionParameters.TimeToExpiry, optionParameters.DiscountFactor, optionParameters.Volatility - h);
return((valuePlusHPlusK - valuePlusHMinusK - valueMinusHPlusK + valueMinusHMinusK) / (4.0 * h * k));
}
/// <summary>Gets the option kappa, i.e. strike-delta.
/// </summary>
/// <param name="optionParameters">The parameters of the option.</param>
/// <param name="getOptionValue">A delegate that calculates the value of the option for specified parameters.</param>
/// <returns>A approximation of the kappa.</returns>
public static double GetOptionKappa(ConstantVolatilityStandardEuropeanOptionConfiguration optionParameters, GetOptionValue getOptionValue)
{
double h = 0.0000001;
double value1 = getOptionValue(optionParameters.Strike + h, optionParameters.Forward, optionParameters.TimeToExpiry, optionParameters.DiscountFactor, optionParameters.Volatility);
double value2 = getOptionValue(optionParameters.Strike, optionParameters.Forward, optionParameters.TimeToExpiry, optionParameters.DiscountFactor, optionParameters.Volatility);
return((value1 - value2) / h);
}
/// <summary>Gets the Volga of the option, i.e. the second derivative with respect to the volatilty.
/// </summary>
/// <param name="optionParameters">The parameters of the option.</param>
/// <param name="getOptionValue">A delegate that calculates the value of the option for specified parameters.</param>
/// <returns>A approximation of the volga.</returns>
public static double GetOptionVolga(ConstantVolatilityStandardEuropeanOptionConfiguration optionParameters, GetOptionValue getOptionValue)
{
double h = 0.00005; // should be not too small, otherwise one get numerical problems
double value = getOptionValue(optionParameters.Strike, optionParameters.Forward, optionParameters.TimeToExpiry, optionParameters.DiscountFactor, optionParameters.Volatility);
double valueMinusH = getOptionValue(optionParameters.Strike, optionParameters.Forward, optionParameters.TimeToExpiry, optionParameters.DiscountFactor, optionParameters.Volatility - h);
double valueMinus2H = getOptionValue(optionParameters.Strike, optionParameters.Forward, optionParameters.TimeToExpiry, optionParameters.DiscountFactor, optionParameters.Volatility - 2.0 * h);
double valuePlusH = getOptionValue(optionParameters.Strike, optionParameters.Forward, optionParameters.TimeToExpiry, optionParameters.DiscountFactor, optionParameters.Volatility + h);
double valuePlus2H = getOptionValue(optionParameters.Strike, optionParameters.Forward, optionParameters.TimeToExpiry, optionParameters.DiscountFactor, optionParameters.Volatility + 2.0 * h);
/* approximation of the second derivative: (Richardson's extrapolation)
* f''(x) \approx [-f(x-2h) + 16 * f(x-h) - 30 * f(x) + 16 * f(x+h) - f(x+2h)] / (12 * h^2)
*/
return((-valueMinus2H + 16.0 * valueMinusH - 30.0 * value + 16.0 * valuePlusH - valuePlus2H) / (12 * h * h));
}
