/**
* Calculates the Bachelier option implied volatility of a call, i.e., the payoff
*
* <p><i>max(S(T)-K,0)</i>, where <i>S</i> follows a normal process with constant volatility.
*
* @param forward The forward of the underlying.
* @param optionMaturity The option maturity T.
* @param optionStrike The option strike. If the option strike is ≤ 0.0 the method returns the
* value of the forward contract paying S(T)-K in T.
* @param payoffUnit The payoff unit (e.g., the discount factor)
* @param optionValue The option value.
* @return Returns the implied volatility of a European call option under the Bachelier model.
*/
public static double bachelierOptionImpliedVolatility(
double forward,
double optionMaturity,
double optionStrike,
double payoffUnit,
double optionValue) {
// Limit the maximum number of iterations, to ensure this calculation returns fast, e.g. in
// cases when there is no such thing as an implied vol
// TODO: An exception should be thrown, when there is no implied volatility for the given value.
int maxIterations = 100;
double maxAccuracy = 0.0;
if (optionStrike <= 0.0) {
// Actually it is not an option
return 0.0;
} else {
// Calculate an lower and upper bound for the volatility
double volatilityLowerBound = 0.0;
double volatilityUpperBound =
(optionValue + Math.abs(forward - optionStrike)) / Math.sqrt(optionMaturity) / payoffUnit;
volatilityUpperBound /=
Math.min(
1.0,
NormalDistribution.density(
(forward - optionStrike) / (volatilityUpperBound * Math.sqrt(optionMaturity))));
// Solve for implied volatility
GoldenSectionSearch solver =
new GoldenSectionSearch(volatilityLowerBound, volatilityUpperBound);
while (solver.getAccuracy() > maxAccuracy
&& !solver.isDone()
&& solver.getNumberOfIterations() < maxIterations) {
double volatility = solver.getNextPoint();
double valueAnalytic =
bachelierOptionValue(forward, volatility, optionMaturity, optionStrike, payoffUnit);
double error = valueAnalytic - optionValue;
solver.setValue(error * error);
}
return solver.getBestPoint();
}
}
/**
* Calculates the option value of a call, i.e., the payoff max(S(T)-K,0) P, where S follows a
* normal process with constant volatility, i.e., a Bachelier model.
*
* @param forward The forward of the underlying.
* @param volatility The Bachelier volatility.
* @param optionMaturity The option maturity T.
* @param optionStrike The option strike.
* @param payoffUnit The payoff unit (e.g., the discount factor)
* @return Returns the value of a European call option under the Bachelier model.
*/
public static double bachelierOptionValue(
double forward,
double volatility,
double optionMaturity,
double optionStrike,
double payoffUnit) {
if (optionMaturity < 0) {
return 0;
} else {
// Calculate analytic value
double dPlus = (forward - optionStrike) / (volatility * Math.sqrt(optionMaturity));
double valueAnalytic =
((forward - optionStrike) * NormalDistribution.cumulativeDistribution(dPlus)
+ (volatility * Math.sqrt(optionMaturity)) * NormalDistribution.density(dPlus))
* payoffUnit;
return valueAnalytic;
}
}
