How to deal with the negative value appearing in the np.log() and np.sqrt() function when calculating density

[yangysc]

Hi, Justin

Thanks for your great codes. However, I am wondering how to deal with the negative value appearing in the np.log() and np.sqrt() function when calculating the density.

For example, for the OzakiDensity, Kt can be negative if 1 + self._model.drift(x0, t) * (np.exp(self._model.drift_x(x0, t) * t) - 1) / ( x0 * self._model.drift_x(x0, t)) < -1, and Vt can be negative when performing Vt=np.sqrt(Vt)

Kt = (1 / t) * np.log(1 + self._model.drift(x0, t) * (np.exp(self._model.drift_x(x0, t) * t) - 1) / (
                    x0 * self._model.drift_x(x0, t)))
Vt = sig ** 2 * (np.exp(2 * Kt * t) - 1) / (2 * Kt)
Vt = np.sqrt(Vt)

Thanks in advance for any suggestion! I am not sure if the strategy you did for other densities is optimal, e.g.,

pymle/pymle/TransitionDensity.py

Lines 182 to 183 in 6344c2e

	if z <= 0:
	return 0

Could you kindly share some references for handling this negative values like returning zero?
Best,
Shanchao

[jkirkby3]

Thanks for your great question! I have rewritten the Ozaki density function to make clear that each of these terms is indeed well-defined (this form is also more efficient :). You can see that (exp(x*t) -1)/x is always positive for positive t ):

    sig = self._model.diffusion(x0, t)
    mu = self._model.drift(x0, t)
    mu_x = self._model.drift_x(x0, t)
    temp = mu * (np.exp(mu_x * t) - 1) / mu_x

    Mt = x0 + temp
    Kt = (2 / t) * np.log(1 + temp / x0)
    Vt = sig * np.sqrt((np.exp(Kt * t) - 1) / Kt)

    return np.exp(-0.5 * ((xt - Mt) / Vt) ** 2) / (np.sqrt(2 * np.pi) * Vt)

Also, I removed the if z <=0 expression from above, as you mention this is not going to be optimal, and its better that if something like this happens we should return a nan value, allowing the optimizer to adapt the parameters accordingly