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import math
import numpy as np


def vol_risk_price(price_unit=1):
    S0 = 100.  # initial price
    T = 1.0  # time
    sigma = 2  # volatility
    M = 200  # number of time steps
    dt = T / M  # time step
    Sim = 1000  # number of simulations
    I = 1

    std = list()
    for i in range(1, Sim + 1):
        ###reservation price:
        ### r(s,t) = s - q * gamma * sigma**2 * (T-t)

        S = np.zeros((M + 1, I))
        S[0] = S0
        for t in range(1, M + 1):
            z = np.random.standard_normal(I)
            S[t] = S[t - 1] + sigma * math.sqrt(dt) * z

        std.append(np.diff(S[:, 0] * price_unit).std(ddof=1) / np.sqrt(dt))

    average_risk_percent_in_s0 = np.mean(std) ** 2 / (S0 * price_unit)
    return average_risk_percent_in_s0


var = vol_risk_price()
# 0.04 = 2**2 / 100
np.testing.assert_allclose(var, 0.04, rtol=0.1)

var = vol_risk_price(10)
# 0.04 = 20**2 / 1000
np.testing.assert_allclose(var, 0.4, rtol=0.1)