Efficiency improvements to calculate risk params

[mikeyrf]

The csv scripts to calculate risk params around funding, market impact, and caps currently run unnecessary for loops over the time TS and possible k values. Removing the time dependence from calls to pystable and using np.array math should allow us to improve the efficiency of these scripts by removing the for loops.

[mikeyrf]

With a bit of math around CDF and inverse CDF, we can use a numpy array for time in VaR, ES, EV calculations. Basically I'm taking advantage of stable distribution properties: X = mu * t + sigma * (t/a)**(1/a) * Z where Z is a standard stable.

For example, we can change nvalue_at_risk to (my code might be wrong, but math should be correct):

def nvalue_at_risk(a: float, b: float, mu: float, sigma: float,
                   k_n: float, v: float, g_inv: float, alpha: float,
                   t: np.array) -> (np.array, np.array):
    z = pystable.create(alpha=a, beta=b, mu=0, sigma=1, parameterization=1)

    z_ginv_max = (g_inv - mu * t) / (sigma * (t/a) ** (1/a))  # this is an np.array
    cdf_x_ginv = np.array(pystable.cdf(z, list(z_ginv_max), len(z_ginv_max)))  # this is also now a np.array

    # var long
    p_long = cdf_x_ginv - alpha  # np.array
    q_long = mu * t + sigma * (t/a) ** (1/a) * np.array(pystable.q(z, list(p_long), len(p_long)))  # also a np.array
    nvar_long = ((1-2*k_n)**(np.floor(t/v))) * (np.exp(q_long) - 1)  # now also a np.array

    # var short
    p_short = np.array([alpha for i in range(len(t))])  # a np.array
    q_short =  mu * t + sigma * (t/a) ** (1/a) * np.array(pystable.q(z, p_short, len(p_short)))  # a np.array
    nvar_short = ((1-2*k_n)**(np.floor(t/v))) * (1 - np.exp(q_short))  # a np.array

    return nvar_long, nvar_short

Using math

Fx(g_inv) = Fz((g_inv - mu * t)/(sigma * (t/a)**(1/a))

p = Fx(g_inv) - alpha
q = mu * t + sigma * (t/a) ** (1/a) * F^{-1}z (p)