/**
* This static method calculated the gamma of a call option under a Black-Scholes model
*
* @param initialStockValue The initial value of the underlying, i.e., the spot.
* @param riskFreeRate The risk free rate of the bank account numerarie.
* @param volatility The Black-Scholes volatility.
* @param optionMaturity The option maturity T.
* @param optionStrike The option strike.
* @return The gamma of the option
*/
public static RandomVariableInterface blackScholesOptionGamma(
RandomVariableInterface initialStockValue,
RandomVariableInterface riskFreeRate,
RandomVariableInterface volatility,
double optionMaturity,
double optionStrike) {
if (optionStrike <= 0.0 || optionMaturity <= 0.0) {
// The Black-Scholes model does not consider it being an option
return initialStockValue.mult(0.0);
} else {
// Calculate gamma
RandomVariableInterface dPlus =
initialStockValue
.div(optionStrike)
.log()
.add(volatility.squared().mult(0.5).add(riskFreeRate).mult(optionMaturity))
.div(volatility)
.div(Math.sqrt(optionMaturity));
RandomVariableInterface gamma =
dPlus
.squared()
.mult(-0.5)
.exp()
.div(
initialStockValue
.mult(volatility)
.mult(Math.sqrt(2.0 * Math.PI * optionMaturity)));
return gamma;
}
}
/**
* Calculates the delta of a call option under a Black-Scholes model
*
* <p>The method also handles cases where the forward and/or option strike is negative and some
* limit cases where the forward or the option strike is zero. In the case forward = option strike
* = 0 the method returns 1.0.
*
* @param initialStockValue The initial value of the underlying, i.e., the spot.
* @param riskFreeRate The risk free rate of the bank account numerarie.
* @param volatility The Black-Scholes volatility.
* @param optionMaturity The option maturity T.
* @param optionStrike The option strike.
* @return The delta of the option
*/
public static RandomVariableInterface blackScholesOptionDelta(
RandomVariableInterface initialStockValue,
RandomVariableInterface riskFreeRate,
RandomVariableInterface volatility,
double optionMaturity,
RandomVariableInterface optionStrike) {
if (optionMaturity < 0) {
return initialStockValue.mult(0.0);
} else {
// Calculate delta
RandomVariableInterface dPlus =
initialStockValue
.div(optionStrike)
.log()
.add(volatility.squared().mult(0.5).add(riskFreeRate).mult(optionMaturity))
.div(volatility)
.div(Math.sqrt(optionMaturity));
UnivariateFunction cummulativeNormal =
new UnivariateFunction() {
public double value(double x) {
return NormalDistribution.cumulativeDistribution(x);
}
};
RandomVariableInterface delta = dPlus.apply(cummulativeNormal);
return delta;
}
}
/**
* Calculates the Black-Scholes option value of a call, i.e., the payoff max(S(T)-K,0) P, where S
* follows a log-normal process with constant log-volatility.
*
* <p>The model specific quantities are considered to be random variable, i.e., the function may
* calculate an per-path valuation in a single call.
*
* @param forward The forward of the underlying.
* @param volatility The Black-Scholes volatility.
* @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)
* @return Returns the value of a European call option under the Black-Scholes model.
*/
public static RandomVariableInterface blackScholesGeneralizedOptionValue(
RandomVariableInterface forward,
RandomVariableInterface volatility,
double optionMaturity,
double optionStrike,
RandomVariableInterface payoffUnit) {
if (optionMaturity < 0) {
return forward.mult(0.0);
} else {
RandomVariableInterface dPlus =
forward
.div(optionStrike)
.log()
.add(volatility.squared().mult(0.5 * optionMaturity))
.div(volatility)
.div(Math.sqrt(optionMaturity));
RandomVariableInterface dMinus = dPlus.sub(volatility.mult(Math.sqrt(optionMaturity)));
UnivariateFunction cumulativeNormal =
new UnivariateFunction() {
public double value(double x) {
return NormalDistribution.cumulativeDistribution(x);
}
};
RandomVariableInterface valueAnalytic =
dPlus
.apply(cumulativeNormal)
.mult(forward)
.sub(dMinus.apply(cumulativeNormal).mult(optionStrike))
.mult(payoffUnit);
return valueAnalytic;
}
}