fat_tails_and_their_impact_on_option_prices2

773041c7-3e60-455a-a52b-b37cb6cfe183.py

# define our gaussian look-a-like distribution
class cust_dist(stats.rv_continuous):
    
    # define init with sigma deviation param e
    def __init__(self, e, *args, **kwargs):
        super().__init__(*args, **kwargs)
        # init our variance divergence
        self.e = e
        # init our cdf and ppf functions
        self.cdf_func, self.ppf_func = self.create_cdf_ppf()
        
    # function to return normal distribution pdf
    def norm_p(self, x, loc=0, scale=1):
        return np.exp(-0.5 * ((x - loc )/ scale)**2) / (scale * 2.5066282746310002)
        
    # function to normalise the pdf over chosen domain
    def normalisation(self, x):
        return simps(self.pdf(x), x)
    
    # function to
    def create_cdf_ppf(self):
        # define domain as +/-25 sigma
        xs = np.linspace(-25, 25, 10000001)
        # normalise our pdf to sum to 1 so it satisfies a distribution
        norm_constant = self.normalisation(xs)
        # compute pdfs to be summed to form cdf
        my_pdfs = self.pdf(xs) / norm_constant
        # cumsum to form cdf
        my_cdf = np.cumsum(my_pdfs)
        # make sure cdf bounded on [0,1]
        my_cdf = my_cdf / my_cdf[-1]
        # create cdf and ppf
        func_cdf = interp1d(xs, my_cdf)
        func_ppf = interp1d(my_cdf, xs, fill_value='extrapolate')
        return func_cdf, func_ppf
        
    # pdf function for averaged normals
    def _pdf(self, x):
        # define lower var distribution prob
        low = self.norm_p(x, scale=(1 - self.e)**0.5)
        # define higher var distribution prob
        high = self.norm_p(x, scale=(1 + self.e)**0.5)
        return 0.5 * (low + high)
    
    # cdf function
    def _cdf(self, x):
        return self.cdf_func(x)
    
    # inverse cdf function
    def _ppf(self, x):
        return self.ppf_func(x)