VG_pricer.py

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Mon Aug 12 18:47:05 2019

@author: cantaro86
"""

from scipy import sparse
from scipy.sparse.linalg import splu
from time import time
import numpy as np
import scipy as scp
from scipy import signal
from scipy.integrate import quad
import scipy.stats as ss
import scipy.special as scps

import matplotlib.pyplot as plt
from matplotlib import cm
from FMNM.CF import cf_VG
from FMNM.probabilities import Q1, Q2
from functools import partial
from FMNM.FFT import fft_Lewis, IV_from_Lewis


class VG_pricer:
    """
    Closed Formula.
    Monte Carlo.
    Finite-difference PIDE: Explicit-implicit scheme, with Brownian approximation

        0 = dV/dt + (r -(1/2)sig^2 -w) dV/dx + (1/2)sig^2 d^V/dx^2
                 + \int[ V(x+y) nu(dy) ] -(r+lam)V
    """

    def __init__(self, Option_info, Process_info):
        """
        Process_info:  of type VG_process.
        It contains the interest rate r and the VG parameters (sigma, theta, kappa)

        Option_info:  of type Option_param.
        It contains (S0,K,T) i.e. current price, strike, maturity in years
        """
        self.r = Process_info.r  # interest rate
        self.sigma = Process_info.sigma  # VG parameter
        self.theta = Process_info.theta  # VG parameter
        self.kappa = Process_info.kappa  # VG parameter
        self.exp_RV = Process_info.exp_RV  # function to generate exponential VG Random Variables
        self.w = -np.log(1 - self.theta * self.kappa - self.kappa / 2 * self.sigma**2) / self.kappa  # coefficient w

        self.S0 = Option_info.S0  # current price
        self.K = Option_info.K  # strike
        self.T = Option_info.T  # maturity in years

        self.price = 0
        self.S_vec = None
        self.price_vec = None
        self.mesh = None
        self.exercise = Option_info.exercise
        self.payoff = Option_info.payoff

    def payoff_f(self, S):
        if self.payoff == "call":
            Payoff = np.maximum(S - self.K, 0)
        elif self.payoff == "put":
            Payoff = np.maximum(self.K - S, 0)
        return Payoff

    def closed_formula(self):
        """
        VG closed formula.  Put is obtained by put/call parity.
        """

        def Psy(a, b, g):
            f = lambda u: ss.norm.cdf(a / np.sqrt(u) + b * np.sqrt(u)) * u ** (g - 1) * np.exp(-u) / scps.gamma(g)
            result = quad(f, 0, np.inf)
            return result[0]

        # Ugly parameters
        xi = -self.theta / self.sigma**2
        s = self.sigma / np.sqrt(1 + ((self.theta / self.sigma) ** 2) * (self.kappa / 2))
        alpha = xi * s

        c1 = self.kappa / 2 * (alpha + s) ** 2
        c2 = self.kappa / 2 * alpha**2
        d = 1 / s * (np.log(self.S0 / self.K) + self.r * self.T + self.T / self.kappa * np.log((1 - c1) / (1 - c2)))

        # Closed formula
        call = self.S0 * Psy(
            d * np.sqrt((1 - c1) / self.kappa),
            (alpha + s) * np.sqrt(self.kappa / (1 - c1)),
            self.T / self.kappa,
        ) - self.K * np.exp(-self.r * self.T) * Psy(
            d * np.sqrt((1 - c2) / self.kappa),
            (alpha) * np.sqrt(self.kappa / (1 - c2)),
            self.T / self.kappa,
        )

        if self.payoff == "call":
            return call
        elif self.payoff == "put":
            return call - self.S0 + self.K * np.exp(-self.r * self.T)
        else:
            raise ValueError("invalid type. Set 'call' or 'put'")

    def Fourier_inversion(self):
        """
        Price obtained by inversion of the characteristic function
        """
        k = np.log(self.K / self.S0)  # log moneyness
        cf_VG_b = partial(
            cf_VG,
            t=self.T,
            mu=(self.r - self.w),
            theta=self.theta,
            sigma=self.sigma,
            kappa=self.kappa,
        )

        right_lim = 5000  # using np.inf may create warnings
        if self.payoff == "call":
            call = self.S0 * Q1(k, cf_VG_b, right_lim) - self.K * np.exp(-self.r * self.T) * Q2(
                k, cf_VG_b, right_lim
            )  # pricing function
            return call
        elif self.payoff == "put":
            put = self.K * np.exp(-self.r * self.T) * (1 - Q2(k, cf_VG_b, right_lim)) - self.S0 * (
                1 - Q1(k, cf_VG_b, right_lim)
            )  # pricing function
            return put
        else:
            raise ValueError("invalid type. Set 'call' or 'put'")

    def MC(self, N, Err=False, Time=False):
        """
        Variance Gamma Monte Carlo
        Err = return Standard Error if True
        Time = return execution time if True
        """
        t_init = time()

        S_T = self.exp_RV(self.S0, self.T, N)
        V = scp.mean(np.exp(-self.r * self.T) * self.payoff_f(S_T), axis=0)

        if Err is True:
            if Time is True:
                elapsed = time() - t_init
                return V, ss.sem(np.exp(-self.r * self.T) * self.payoff_f(S_T)), elapsed
            else:
                return V, ss.sem(np.exp(-self.r * self.T) * self.payoff_f(S_T))
        else:
            if Time is True:
                elapsed = time() - t_init
                return V, elapsed
            else:
                return V

    def FFT(self, K):
        """
        FFT method. It returns a vector of prices.
        K is an array of strikes
        """
        K = np.array(K)
        cf_VG_b = partial(
            cf_VG,
            t=self.T,
            mu=(self.r - self.w),
            theta=self.theta,
            sigma=self.sigma,
            kappa=self.kappa,
        )

        if self.payoff == "call":
            return fft_Lewis(K, self.S0, self.r, self.T, cf_VG_b, interp="cubic")
        elif self.payoff == "put":  # put-call parity
            return (
                fft_Lewis(K, self.S0, self.r, self.T, cf_VG_b, interp="cubic") - self.S0 + K * np.exp(-self.r * self.T)
            )
        else:
            raise ValueError("invalid type. Set 'call' or 'put'")

    def IV_Lewis(self):
        """Implied Volatility from the Lewis formula"""

        cf_VG_b = partial(
            cf_VG,
            t=self.T,
            mu=(self.r - self.w),
            theta=self.theta,
            sigma=self.sigma,
            kappa=self.kappa,
        )

        if self.payoff == "call":
            return IV_from_Lewis(self.K, self.S0, self.T, self.r, cf_VG_b)
        elif self.payoff == "put":
            raise NotImplementedError
        else:
            raise ValueError("invalid type. Set 'call' or 'put'")

    def PIDE_price(self, steps, Time=False):
        """
        steps = tuple with number of space steps and time steps
        payoff = "call" or "put"
        exercise = "European" or "American"
        Time = Boolean. Execution time.
        """
        t_init = time()

        Nspace = steps[0]
        Ntime = steps[1]

        S_max = 6 * float(self.K)
        S_min = float(self.K) / 6
        x_max = np.log(S_max)
        x_min = np.log(S_min)

        dev_X = np.sqrt(self.sigma**2 + self.theta**2 * self.kappa)  # std dev VG process

        dx = (x_max - x_min) / (Nspace - 1)
        extraP = int(np.floor(5 * dev_X / dx))  # extra points beyond the B.C.
        x = np.linspace(x_min - extraP * dx, x_max + extraP * dx, Nspace + 2 * extraP)  # space discretization
        t, dt = np.linspace(0, self.T, Ntime, retstep=True)  # time discretization

        Payoff = self.payoff_f(np.exp(x))
        offset = np.zeros(Nspace - 2)
        V = np.zeros((Nspace + 2 * extraP, Ntime))  # grid initialization

        if self.payoff == "call":
            V[:, -1] = Payoff  # terminal conditions
            V[-extraP - 1 :, :] = np.exp(x[-extraP - 1 :]).reshape(extraP + 1, 1) * np.ones(
                (extraP + 1, Ntime)
            ) - self.K * np.exp(-self.r * t[::-1]) * np.ones(
                (extraP + 1, Ntime)
            )  # boundary condition
            V[: extraP + 1, :] = 0
        else:
            V[:, -1] = Payoff
            V[-extraP - 1 :, :] = 0
            V[: extraP + 1, :] = self.K * np.exp(-self.r * t[::-1]) * np.ones((extraP + 1, Ntime))

        A = self.theta / (self.sigma**2)
        B = np.sqrt(self.theta**2 + 2 * self.sigma**2 / self.kappa) / self.sigma**2

        def levy_m(y):
            """Levy measure VG"""
            return np.exp(A * y - B * np.abs(y)) / (self.kappa * np.abs(y))

        eps = 1.5 * dx  # the cutoff near 0
        lam = (
            quad(levy_m, -(extraP + 1.5) * dx, -eps)[0] + quad(levy_m, eps, (extraP + 1.5) * dx)[0]
        )  # approximated intensity

        def int_w(y):
            """integrator"""
            return (np.exp(y) - 1) * levy_m(y)

        int_s = lambda y: np.abs(y) * np.exp(A * y - B * np.abs(y)) / self.kappa  # avoid division by zero

        w = (
            quad(int_w, -(extraP + 1.5) * dx, -eps)[0] + quad(int_w, eps, (extraP + 1.5) * dx)[0]
        )  # is the approx of omega

        sig2 = quad(int_s, -eps, eps)[0]  # the small jumps variance

        dxx = dx * dx
        a = (dt / 2) * ((self.r - w - 0.5 * sig2) / dx - sig2 / dxx)
        b = 1 + dt * (sig2 / dxx + self.r + lam)
        c = -(dt / 2) * ((self.r - w - 0.5 * sig2) / dx + sig2 / dxx)
        D = sparse.diags([a, b, c], [-1, 0, 1], shape=(Nspace - 2, Nspace - 2)).tocsc()
        DD = splu(D)

        nu = np.zeros(2 * extraP + 3)  # Lévy measure vector
        x_med = extraP + 1  # middle point in nu vector
        x_nu = np.linspace(-(extraP + 1 + 0.5) * dx, (extraP + 1 + 0.5) * dx, 2 * (extraP + 2))  # integration domain
        for i in range(len(nu)):
            if (i == x_med) or (i == x_med - 1) or (i == x_med + 1):
                continue
            nu[i] = quad(levy_m, x_nu[i], x_nu[i + 1])[0]

        if self.exercise == "European":
            # Backward iteration
            for i in range(Ntime - 2, -1, -1):
                offset[0] = a * V[extraP, i]
                offset[-1] = c * V[-1 - extraP, i]
                V_jump = V[extraP + 1 : -extraP - 1, i + 1] + dt * signal.convolve(
                    V[:, i + 1], nu[::-1], mode="valid", method="auto"
                )
                V[extraP + 1 : -extraP - 1, i] = DD.solve(V_jump - offset)
        elif self.exercise == "American":
            for i in range(Ntime - 2, -1, -1):
                offset[0] = a * V[extraP, i]
                offset[-1] = c * V[-1 - extraP, i]
                V_jump = V[extraP + 1 : -extraP - 1, i + 1] + dt * signal.convolve(
                    V[:, i + 1], nu[::-1], mode="valid", method="auto"
                )
                V[extraP + 1 : -extraP - 1, i] = np.maximum(DD.solve(V_jump - offset), Payoff[extraP + 1 : -extraP - 1])

        X0 = np.log(self.S0)  # current log-price
        self.S_vec = np.exp(x[extraP + 1 : -extraP - 1])  # vector of S
        self.price = np.interp(X0, x, V[:, 0])
        self.price_vec = V[extraP + 1 : -extraP - 1, 0]
        self.mesh = V[extraP + 1 : -extraP - 1, :]

        if Time is True:
            elapsed = time() - t_init
            return self.price, elapsed
        else:
            return self.price

    def plot(self, axis=None):
        if type(self.S_vec) != np.ndarray or type(self.price_vec) != np.ndarray:
            self.PIDE_price((5000, 4000))

        plt.plot(self.S_vec, self.payoff_f(self.S_vec), color="blue", label="Payoff")
        plt.plot(self.S_vec, self.price_vec, color="red", label="VG curve")
        if type(axis) == list:
            plt.axis(axis)
        plt.xlabel("S")
        plt.ylabel("price")
        plt.title("VG price")
        plt.legend(loc="upper left")
        plt.show()

    def mesh_plt(self):
        if type(self.S_vec) != np.ndarray or type(self.mesh) != np.ndarray:
            self.PDE_price((7000, 5000))

        fig = plt.figure()
        ax = fig.add_subplot(111, projection="3d")

        X, Y = np.meshgrid(np.linspace(0, self.T, self.mesh.shape[1]), self.S_vec)
        ax.plot_surface(Y, X, self.mesh, cmap=cm.ocean)
        ax.set_title("VG price surface")
        ax.set_xlabel("S")
        ax.set_ylabel("t")
        ax.set_zlabel("V")
        ax.view_init(30, -100)  # this function rotates the 3d plot
        plt.show()

    def closed_formula_wrong(self):
        """
        VG closed formula. This implementation seems correct, BUT IT DOES NOT WORK!!
        Here I use the closed formula of Carr,Madan,Chang 1998.
        With scps.kv, a modified Bessel function of second kind.
        You can try to run it, but the output is slightly different from expected.
        """

        def Phi(alpha, beta, gamm, x, y):
            f = lambda u: u ** (alpha - 1) * (1 - u) ** (gamm - alpha - 1) * (1 - u * x) ** (-beta) * np.exp(u * y)
            result = quad(f, 0.00000001, 0.99999999)
            return (scps.gamma(gamm) / (scps.gamma(alpha) * scps.gamma(gamm - alpha))) * result[0]

        def Psy(a, b, g):
            c = np.abs(a) * np.sqrt(2 + b**2)
            u = b / np.sqrt(2 + b**2)

            value = (
                (c ** (g + 0.5) * np.exp(np.sign(a) * c) * (1 + u) ** g)
                / (np.sqrt(2 * np.pi) * g * scps.gamma(g))
                * scps.kv(g + 0.5, c)
                * Phi(g, 1 - g, 1 + g, (1 + u) / 2, -np.sign(a) * c * (1 + u))
                - np.sign(a)
                * (c ** (g + 0.5) * np.exp(np.sign(a) * c) * (1 + u) ** (1 + g))
                / (np.sqrt(2 * np.pi) * (g + 1) * scps.gamma(g))
                * scps.kv(g - 0.5, c)
                * Phi(g + 1, 1 - g, 2 + g, (1 + u) / 2, -np.sign(a) * c * (1 + u))
                + np.sign(a)
                * (c ** (g + 0.5) * np.exp(np.sign(a) * c) * (1 + u) ** (1 + g))
                / (np.sqrt(2 * np.pi) * (g + 1) * scps.gamma(g))
                * scps.kv(g - 0.5, c)
                * Phi(g, 1 - g, 1 + g, (1 + u) / 2, -np.sign(a) * c * (1 + u))
            )
            return value

        # Ugly parameters
        xi = -self.theta / self.sigma**2
        s = self.sigma / np.sqrt(1 + ((self.theta / self.sigma) ** 2) * (self.kappa / 2))
        alpha = xi * s

        c1 = self.kappa / 2 * (alpha + s) ** 2
        c2 = self.kappa / 2 * alpha**2
        d = 1 / s * (np.log(self.S0 / self.K) + self.r * self.T + self.T / self.kappa * np.log((1 - c1) / (1 - c2)))

        # Closed formula
        call = self.S0 * Psy(
            d * np.sqrt((1 - c1) / self.kappa),
            (alpha + s) * np.sqrt(self.kappa / (1 - c1)),
            self.T / self.kappa,
        ) - self.K * np.exp(-self.r * self.T) * Psy(
            d * np.sqrt((1 - c2) / self.kappa),
            (alpha) * np.sqrt(self.kappa / (1 - c2)),
            self.T / self.kappa,
        )

        return call