def generate_gbm(lambda_0: float, T: int, paths: int = 1, mu: float = 0.5, sigma: float = 1, rng: np.random.Generator = rng):
t = np.repeat(np.array([np.arange(1, T+1)]), paths, axis = 0)
b = rng.normal(size=[paths, T])
w = b.cumsum(axis = 1)
drift = (mu - 0.5 * sigma**2) * t
diffusion = sigma * w
S = lambda_0 * np.exp(drift + diffusion)
return S
def generate_abm(lambda_0: float,
T: int,
paths: int = 1,
mu: float = 0.5,
sigma: float = 1,
rng: np.random.Generator = rng):
t = np.repeat(np.array([np.arange(1, T + 1)]), paths, axis=0)
b = rng.normal(size=[paths, T])
w = b.cumsum(axis=1)
S = lambda_0 + mu * t + sigma * w
return S