Esempio n. 1
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  /**
   * 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;
    }
  }
Esempio n. 2
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  /**
   * 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;
    }
  }
Esempio n. 3
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  /**
   * 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 &le; 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;
    }
  }