BS, new mex factorization formulae.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Адаптируем новые формулы факторизации для модели Б-Ш, чтобы сравнить их впоследствии с имеющимися."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from numpy import log, sqrt, exp, fft, array, pi\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Параметры контракта down-and-out put"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# model parameters - we are going to evaluate a barrier put\n",
    "T = 1\n",
    "H_original = 90.0 # limit\n",
    "K_original = 100.0 # strike\n",
    "r_premia = 10 # annual interest rate\n",
    "r = log(r_premia/100 + 1)\n",
    "V0 = 0.316227766"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Определим функцию выплат:\n",
    "def G(y, barrier, strike):\n",
    "    \"\"\"payoff_function for a given option type (down-and-out put there)\"\"\"\n",
    "    if (barrier <= y) and (y <= strike):\n",
    "        return K_original - H_original*exp(y)\n",
    "    else:\n",
    "        return 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Характеристическая экспонента процесса Б-Ш\n",
    "$$\\psi(\\xi) = \\frac{{\\color{green}{\\sigma}}^2}{2} \\xi^2 - i{\\color{green}{\\gamma}}\\xi$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "sigma = V0\n",
    "gamma = r - 0.5 * sigma**2 # Black-Scholes parameter, from a no-arbitrage condition\n",
    "\n",
    "def psi(xi):\n",
    "    return (sigma**2/2) * xi * xi - 1j*gamma*xi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Подготавливаем дискретизацию, задаём шкалу по переменной, связанной с ценой, производим замены\n",
    "$$ y = ln(S)$$\n",
    "$$ K = ln(K), H = ln(H) $$\n",
    "\n",
    "Укладываем $M$ точек между $L \\cdot ln(1/2)$  и $ L \\cdot ln(2) $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "При этом заметим, что наши  действия с разбиением не будут безобидными - природа функций, интегралы от которых мы будем брать, требует мелкой сетки и по переменной $\\xi$ (частот), и по $y$ (или $x$) - (цен), а по переменной часот - ещё и довольно значительного (пока) разлёта.\n",
    "\n",
    "Схема преобразований Фурье будет переносить нас по сеткам в следующем порядке.\n",
    "\n",
    "1. $y \\rightarrow \\xi$ - FT для вычисления оператора\n",
    "2. $\\xi \\rightarrow y$ - iFT для вычисления 1-го слагаемого приближённого символа\n",
    "3. $y \\rightarrow \\xi$ - FT для вычисления 2-го слагаемого приближённого символа\n",
    "4. $\\xi \\rightarrow y$ - FT из оператора обратно\n",
    "\n",
    "Учитывая, что мы используем один и тот же вид FT, вид пространства после переноса определяется только количеством точек и шагом и известен с самого начала."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "В вычислениях будет использовано всего 65536 точек\n",
      "du =  0.0958737992428\n",
      "Частота меняется от  -3141.592653589793 до 3141.592653589793\n",
      "dx =  0.001\n",
      "Пространственная переменная меняется от  -32.768 до 32.767\n",
      "Отношение длин диапазонов пространство/частота: 0.0104302191955\n",
      "Отношение длин диапазонов частота/пространство: 95.8752621833\n",
      "L = 1\n",
      "Диапазон пространства в лог. шкале в терминах L был бы от  -0.69314718056 до 0.69314718056\n"
     ]
    }
   ],
   "source": [
    "M = 2**16\n",
    "\n",
    "# потому что dy нужно править много где, пока запомним, что dy = dx\n",
    "dx = 0.001\n",
    "\n",
    "x_space = np.linspace( -M*dx/2, M*dx/2, num = M, endpoint = False)\n",
    "u_space = np.linspace(-pi/dx, pi/dx, num = M, endpoint = False)\n",
    "du = u_space[1] - u_space[0]\n",
    "\n",
    "print(\"В вычислениях будет использовано всего\", M, \"точек\")\n",
    "print(\"du = \", du)\n",
    "print(\"Частота меняется от \", -pi/dx, \"до\", pi/dx)\n",
    "\n",
    "print(\"dx = \", dx)\n",
    "print(\"Пространственная переменная меняется от \", x_space[0], \"до\", x_space[M-1])\n",
    "\n",
    "print(\"Отношение длин диапазонов пространство/частота:\", (x_space[M-1] -  x_space[0])/(2*pi/dx))\n",
    "print(\"Отношение длин диапазонов частота/пространство:\", (2*pi/dx)/(x_space[M-1] -  x_space[0]))\n",
    "\n",
    "L = 1  # scaling coefficient\n",
    "y_min = L * log(0.5)\n",
    "y_max = L * log(2.0)\n",
    "\n",
    "K = log(K_original/H_original)\n",
    "H = log(H_original/H_original)\n",
    "y_space = x_space # prices array\n",
    "\n",
    "print('L =', L)\n",
    "print(\"Диапазон пространства в лог. шкале в терминах L был бы от \", y_min, 'до', y_max)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "\n",
    "# plt.plot(y_space, color = 'y')\n",
    "# plt.ylabel('price line in logscale')\n",
    "# plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Теперь дискретизация по времени:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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eipKkuW54eI1XAk2hCsnKjEREF8XUzxmZeftY83SPX7lyJT09Pdu09fX10dfX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\nRZIkVdr/B18cP5i8EkjYAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f796ab18ba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "original_prices_array = H_original * exp(y_space)\n",
    "# time discretization\n",
    "N = 20  # number_of_time_steps\n",
    "delta_t = T/N\n",
    "\n",
    "plt.plot(original_prices_array, color = 'y')\n",
    "plt.ylabel('prices in linear scale')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Величины, связанные с факторизацией. В этой модели всё просто - при постоянной вариации нет необходимости учитывать эффекты решётки вариации.\n",
    "\n",
    "Действие, которое мы хотим произвести на каждом шаге по времени, выглядит так:\n",
    "$$f_n(y) = ({\\color{green}{q\\Delta t}}) ^{-1} \\mathcal{E}_{\\color{green}{q}}^- 1_{y>ln(H)} \\mathcal{E}_{\\color{green}{q}}^+ f_{n+1}(y)$$\n",
    "\n",
    "Получить хотим $f_0(y)$, там для каждого y и ответ лежит\n",
    "\n",
    "Зелёные - константы. Определим их."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "q =  20.0953101798\n",
      "Множитель каждого шага = 0.995257093374\n"
     ]
    }
   ],
   "source": [
    "q = 1.0/delta_t + r\n",
    "print('q = ', q)\n",
    "factor = (q*delta_t)**(-1)\n",
    "print('Множитель каждого шага =', factor)\n",
    "\n",
    "# Зададим начальные уловия\n",
    "f_N = array([G(y_element, H, K) for y_element in y_space]) # = g(x) i.e. payoff func\n",
    "\n",
    "# plt.plot(original_prices_array, f_N,  'y')\n",
    "# plt.ylabel('initial conditions, payoff func')\n",
    "# plt.xlabel('stock price', color = 'y')\n",
    "# plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "В уравнении выше есть нигде пока не определённные функции (на самом деле, операторы)\n",
    "$$ \\mathcal{E}^\\pm_q f(x) = (2\\pi)^{-1} \\int_{-\\infty}^{+\\infty} e^{ix\\xi} \\phi^\\pm_q (\\xi) \\hat{f}(\\xi) d\\xi $$\n",
    "\n",
    "\n",
    "А в них - пока не определённые $\\phi_q^\\pm$\n",
    "\n",
    "Подробно процедуры для их вычисления описаны с сопутствующими соображениями в блокноте Mex-BS new mex symbol comparison. Здесь же составим краткие версии процедур. Сперва нужное нам потом везде преобразование Фурье, определённое \"как надо\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def make_rad_fft(f_x, d):\n",
    "    '''\n",
    "    Вычисляет преобразование Фурье без двух \"пи\" в показателе экспоненты\n",
    "    \n",
    "    Параметры:\n",
    "    \n",
    "    f_x - массив значений функции, задаваемый \"как есть\", без особенностей хранения\n",
    "    и определённый на диапазоне [-M*dx/2, M*dx/2] в M = 2**N равноудалённых точках, без последней.\n",
    "    \n",
    "    d = dx - желаемый шаг по переменной x\n",
    "    \n",
    "    Области определения:\n",
    "    \n",
    "    xi_space = np.linspace( -pi/dx, pi/dx, num = M, endpoint = False)\n",
    "    x_space = np.linspace( -M*dx/2, M*dx/2, num = M, endpoint = False)\n",
    "    '''\n",
    "    dx = d\n",
    "    M = len(f_x)\n",
    "    \n",
    "    sign_change_k = np.array([(-1)**k for k in range(0, M)])\n",
    "    sign_change_l = np.array([(-1)**l for l in range(0, M)])\n",
    "    #учитываем порядок хранения\n",
    "    sign_change_l = fft.fftshift(sign_change_l)\n",
    "\n",
    "    f = sign_change_k * f_x\n",
    "    f_hat = dx * sign_change_l * fft.fft(f)\n",
    "    \n",
    "    # избегаем особенностей хранения результатов fft, нам они не нужны.\n",
    "    return f_hat\n",
    "\n",
    "def make_rad_ifft(f_hat_xi, d):\n",
    "    '''\n",
    "    Вычисляет обратное преобразование Фурье без двух \"пи\" в показателе экспоненты\n",
    "    \n",
    "    Параметры:\n",
    "    \n",
    "    f_xi - массив значений функции, задаваемый \"как есть\", без особенностей хранения\n",
    "    и определённый на диапазоне [-pi/d, pi/d] в M = 2**N равноудалённых точках, без последней.\n",
    "    \n",
    "    d = dx - желаемый шаг по переменной x, после того, как ifft отработает\n",
    "    \n",
    "    Области определения:\n",
    "    \n",
    "    xi_space = np.linspace( -pi/dx, pi/dx, num = M, endpoint = False)\n",
    "    x_space = np.linspace( -M*dx/2, M*dx/2, num = M, endpoint = False)\n",
    "\n",
    "    '''\n",
    "    dx = d\n",
    "    M = len(f_hat_xi)\n",
    "\n",
    "    sign_change_k = np.array([(-1)**k for k in range(0, M)])\n",
    "    sign_change_l = np.array([(-1)**l for l in range(0, M)])\n",
    "\n",
    "    f =  (1/(dx)) * sign_change_k * fft.ifft(sign_change_l * f_hat_xi)\n",
    "    return f"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Фи-минус"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\phi^-_q(\\xi)$ будут иметь вид:\n",
    "\n",
    "$$ \\phi^-_q(\\xi) = exp[-i\\xi F^-(0) - \\xi^2 \\hat{F}^-(\\xi)] $$\n",
    "\n",
    "$$ F^-(x) = 1_{[0,+\\infty)}(x)(2\\pi)^{-1} \\int_{-\\infty + i\\omega_+}^{+\\infty + i\\omega_+} e^{ix\\eta} \\frac{ln(q+\\psi(\\eta))}{\\eta^2} d\\eta =$$\n",
    "$$= 1_{[0,+\\infty)}(x)(2\\pi)^{-1} \\int_{-\\infty}^{+\\infty} e^{ix(\\upsilon+i\\omega_+)} \\frac{ln(q+\\psi(\\upsilon+i\\omega_+))}{(\\upsilon+i\\omega_+)^2} d(\\upsilon) =  $$\n",
    "$$= 1_{[0,+\\infty)}(x)e^{-x\\omega_+}\\frac{1}{2\\pi} \\int_{-\\infty}^{+\\infty} e^{ix(\\upsilon)} \\frac{ln(q+\\psi(\\upsilon+i\\omega_+))}{(\\upsilon+i\\omega_+)^2} d(\\upsilon) $$\n",
    "$$ \\hat{F}^-(\\xi) = \\int_{-\\infty}^{+\\infty} e^{-ix\\xi} F^-(x) dx $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "omega_plus = 1\n",
    "\n",
    "def integrand_minus(upsilon_array):\n",
    "    \"\"\"\n",
    "    принимает и возвращает массив длиной в степень двойки, исходя из логики дальнейшего использования\n",
    "    \"\"\"\n",
    "    return np.array([log(q + psi(upsilon + 1j*omega_plus)) / (upsilon + 1j*omega_plus)**2 for upsilon in upsilon_array])\n",
    "\n",
    "def F_minus_capital():\n",
    "    indicator = np.where(x_space >= 0, 1, 0)\n",
    "    trimmed_x_space = indicator*x_space # чтобы при \"сильно отрицательных\" x не росла экспонента\n",
    "    integral = make_rad_ifft(integrand_minus(u_space), dx)\n",
    "    exponent = exp(-trimmed_x_space * omega_plus)\n",
    "    return indicator * exponent * integral \n",
    "\n",
    "fm = F_minus_capital()\n",
    "\n",
    "F_m_hat = make_rad_fft(fm, dx)\n",
    "\n",
    "def make_phi_minus_array(xi_array):\n",
    "    first_term = - 1j * xi_array * (fm[M//2])\n",
    "    second_term = - xi_array * xi_array * F_m_hat\n",
    "    return exp(first_term + second_term)\n",
    "\n",
    "mex_symbol_minus = make_phi_minus_array(u_space)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Фи-плюс"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\phi^+_q(\\xi)$ будут иметь вид:\n",
    "\n",
    "$$ \\phi^+_q(\\xi) = exp[i\\xi F^+(0) - \\xi^2 \\hat{F}^+(\\xi)], $$\n",
    "$$ F^+(x) = 1_{(-\\infty, 0]}(x)(2\\pi)^{-1} \\int_{-\\infty + i\\omega_-}^{+\\infty + i\\omega_-} e^{ix\\eta} \\frac{ln(q+\\psi(\\eta))}{\\eta^2} d\\eta = $$\n",
    "$$ = 1_{(-\\infty, 0]}(x)(2\\pi)^{-1} \\int_{-\\infty}^{+\\infty} e^{ix(\\upsilon+i\\omega_-)} \\frac{ln(q+\\psi(\\upsilon+i\\omega_-))}{(\\upsilon+i\\omega_-)^2} d(\\upsilon) = $$ \n",
    "$$ = 1_{(-\\infty, 0]}(x)e^{-x\\omega_-}\\frac{1}{2\\pi} \\int_{-\\infty}^{+\\infty} e^{ix(\\upsilon)} \\frac{ln(q+\\psi(\\upsilon+i\\omega_-))}{(\\upsilon+i\\omega_-)^2} d(\\upsilon);$$\n",
    "$$ \\hat{F}^+(\\xi) = \\int_{-\\infty}^{+\\infty} e^{-ix\\xi} F^+(x) dx. $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\phi^+_q(\\xi) = exp[i\\xi F^+(0) - \\xi^2 \\hat{F}^+(\\xi)] $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "omega_minus = -1\n",
    "\n",
    "def integrand_plus(upsilon_array):\n",
    "    \"\"\"\n",
    "    принимает и возвращает массив длиной M,\n",
    "    работает (должна) на upsilon_space по умолчанию\n",
    "    тоже должна возвращать\n",
    "    \"\"\"\n",
    "    return np.array([log(q + psi(upsilon + 1j*omega_minus)) / (upsilon + 1j*omega_minus)**2 for upsilon in upsilon_array])\n",
    "\n",
    "def F_plus_capital():\n",
    "    indicator = np.where(x_space <= 0, 1, 0)\n",
    "    trimmed_x_space = indicator*x_space # чтобы при \"сильно положительных\" x не росла экспонента\n",
    "    integral = make_rad_ifft(integrand_plus(u_space), dx)\n",
    "    exponent = exp(-trimmed_x_space * omega_minus)\n",
    "    return indicator * exponent * integral \n",
    "\n",
    "fp = F_plus_capital()\n",
    "F_p_hat = make_rad_fft(fp, dx)\n",
    "\n",
    "def make_phi_plus_array(xi_array):\n",
    "    first_term = 1j *  xi_array * fp[M//2]\n",
    "    second_term = - xi_array * xi_array * F_p_hat\n",
    "    return exp(first_term + second_term)\n",
    "\n",
    "mex_symbol_plus = make_phi_plus_array(u_space)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Теперь появляется возможность перейти к итеративной процедуре. В контексте модели Б-Ш она довольно проста.\n",
    "\n",
    "На входе каждого из шагов есть $ \\color{cyan}{f_n(y)} $\n",
    "\n",
    "\n",
    "$$ f_{n+1} =  \\color{green}{(q\\cdot\\Delta_t)^{-1}} \\color{blue}{\\mathcal{E}^-_q} \\mathcal{1}_{0,+\\infty}(y) \\color{orange}{\\mathcal{E}^+_q}\\color{cyan}{f_n(y)}.$$\n",
    "\n",
    "\n",
    "Принимая во внимание, что \n",
    "\n",
    "$$ \\mathcal{E}^\\pm_q (\\color{cyan}{f(y)}) = (2\\pi)^{-  1} \\int_{-\\infty}^{+\\infty} e^{iy\\xi} \\phi^\\pm_q (\\xi) \\color{magenta}{\\hat{f}(\\xi)} d\\xi,$$\n",
    "\n",
    "можно раскрыть выражение выше в виде (черту стоит понимать как преобразование Фурье):\n",
    "\n",
    "$$ f_{n+1} =  \\color{green}{(q\\cdot\\Delta_t)^{-1}} \n",
    "\\color{blue}{(2\\pi)^{-1} \\int_{-\\infty}^{+\\infty} e^{ix\\xi} \\phi^-_q (\\xi) (\n",
    "\\overline{\n",
    "\\color{black}{\\mathcal{1}_{0,+\\infty}(y)}\n",
    "\\color{orange}{(2\\pi)^{-1} \\int_{-\\infty}^{+\\infty} e^{ix\\xi} \\phi^+_q (\\xi) \\color{red}{\\hat{f}(\\xi)} d\\xi})}d\\xi}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Перед тем, как продолжить, стоит сказать ещё несколько слов о том, как эта конструкция может выглядеть в виде алгоритма. Внешняя простота интегралов выше обманчива.\n",
    "\n",
    "Прямое преобразование Фурье мы определяем как:\n",
    "\n",
    "$$ \\color{magenta}{\\hat{f}(\\xi)} = \\int_{-\\infty}^{+\\infty} e^{-ix\\xi} \\color{cyan}{f(x)} dx $$\n",
    "\n",
    "Обратное преобразование Фурье - в виде\n",
    "\n",
    "$$ \\color{cyan}{f(x)} = (2\\pi)^{-1} \\int_{-\\infty}^{+\\infty} e^{ix\\xi} \\color{magenta}{\\hat{f}(\\xi)} dx $$\n",
    "\n",
    "Мы раньше записали операторы следующим образом: \n",
    "\n",
    "$$ \\mathcal{E}^\\pm_q (\\color{cyan}{f(x)}) = (2\\pi)^{-  1} \\int_{-\\infty}^{+\\infty} e^{ix\\xi} \\phi^\\pm_q (\\xi) \\color{magenta}{\\hat{f}(\\xi)} d\\xi$$\n",
    "\n",
    "Полностью же это можно записать так:\n",
    "\n",
    "$$ \\mathcal{E}^\\pm_q (\\color{cyan}{f(x)}) = (2\\pi)^{-  1} \\int_{-\\infty}^{+\\infty} e^{ix\\xi} \\phi^\\pm_q (\\xi) \\color{magenta}{(\\int_{-\\infty}^{+\\infty} e^{-ix\\xi} \\color{cyan}{f(x)} dx)} d\\xi,$$\n",
    "\n",
    "Далее, рассмотрим ближе конструкцию из итеративной процедуры:\n",
    "\n",
    "$$ f_{n+1} =  \\color{green}{(q\\cdot\\Delta_t)^{-1}} \\color{blue}{\\mathcal{E}^-_q} \\mathcal{1}_{0,+\\infty}(y) \\color{orange}{\\mathcal{E}^+_q}\\color{cyan}{f_n(y)}.$$\n",
    "\n",
    "$$ f_{n+1} =  \\color{green}{(q\\cdot\\Delta_t)^{-1}} \n",
    "\\color{blue}{(2\\pi)^{-1} \\int_{-\\infty}^{+\\infty} e^{ix\\xi} \\phi^-_q (\\xi) (\n",
    "\\overline\n",
    "{\n",
    "\\color{black}{\\mathcal{1}_{0,+\\infty}(y)}\n",
    "\\color{orange}{(2\\pi)^{-1}\n",
    "\\int_{-\\infty}^{+\\infty} e^{ix\\xi} \n",
    "\\phi^+_q (\\xi) \n",
    "\\color{magenta}{\\hat{f}(\\xi)} \n",
    "d\\xi})\n",
    "}\n",
    "d\\xi}\n",
    "$$\n",
    "\n",
    "Запишем внутренности $\\color{orange}{\\mathcal{E}^+_q}(\\color{cyan}{f(y)})$\n",
    "\n",
    "$$ f_{n+1} =  \\color{green}{(q\\cdot\\Delta_t)^{-1}} \n",
    "\\color{blue}{(2\\pi)^{-1} \\int_{-\\infty}^{+\\infty} e^{iy\\xi} \\phi^-_q (\\xi) (\n",
    "\\overline\n",
    "{\n",
    "\\color{black}{\\mathcal{1}_{0,+\\infty}(y)}\n",
    "\\color{orange}{\n",
    "(2\\pi)^{-1}\n",
    "\\int_{-\\infty}^{+\\infty} e^{iy\\xi} \n",
    "\\phi^+_q (\\xi) \n",
    "(\\color{magenta}{\\int_{-\\infty}^{+\\infty} e^{-iy\\xi} \\color{cyan}{f_n(y)} dy})\n",
    "d\\xi})\n",
    "}\n",
    "d\\xi}\n",
    "$$\n",
    "\n",
    "Нужно теперь отразить тот факт, что от выражения\n",
    "\n",
    "$$\n",
    "\\overline\n",
    "{\n",
    "\\color{black}{\\mathcal{1}_{0,+\\infty}(y)}\n",
    "\\color{orange}{\n",
    "(2\\pi)^{-1}\n",
    "\\int_{-\\infty}^{+\\infty} e^{iy\\xi} \n",
    "\\phi^+_q (\\xi) \n",
    "(\\color{magenta}{\\int_{-\\infty}^{+\\infty} e^{-iy\\xi} \\color{cyan}{f_n(y)} dy})\n",
    "d\\xi})\n",
    "}\n",
    "$$\n",
    "\n",
    "(которое и является функцией-аргументом для оператора $\\color{blue}{\\mathcal{E}^-_q}$) придётся взять преобразование Фурье.\n",
    "\n",
    "Получится вот что\n",
    "\n",
    "$$ f_{n+1} =  \\color{green}{(q\\cdot\\Delta_t)^{-1}} \n",
    "\\color{blue}{(2\\pi)^{-1} \\int_{-\\infty}^{+\\infty} e^{iy\\xi} \\phi^-_q (\\xi) (\n",
    "\\color{red}\n",
    "{\n",
    "\\int_{-\\infty}^{+\\infty} e^{-iy\\xi}(\n",
    "\\color{black}{\\mathcal{1}_{0,+\\infty}(y)}\n",
    "\\color{orange}{\n",
    "(2\\pi)^{-1}\n",
    "\\int_{-\\infty}^{+\\infty} e^{iy\\xi} \n",
    "\\phi^+_q (\\xi) \n",
    "(\\color{magenta}{\\int_{-\\infty}^{+\\infty} e^{-iy\\xi} \\color{cyan}{f_n(y)} dy})\n",
    "d\\xi})\n",
    "dy\n",
    "}\n",
    ")d\\xi}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Вычисление придётся проводить в средующем порядке.\n",
    "\n",
    "0. Подготовить массив значений функции $\\color{cyan}{f_n(y)}$\n",
    "1. <span style=\"color:magenta\">Вычислить от него преобразование Фурье</span>\n",
    "2. Результат будет определён в точках $\\xi$. В тех же точках нужно определить $\\color{orange}{\\phi^+_q (\\xi)}$, после чего выполнить поэлементное умножение массивов.\n",
    "3. <span style=\"color:orange\">От того, что получилось - вычислить обратное преобразование Фурье</span>\n",
    "4. Теперь результат будет определён в точках $y$ снова. В этих же точках нужно определить индикатор, в виде массива, после чего выполнить поэлементное умножение результата шага 3 на массив-индикатор\n",
    "5. <span style=\"color:red\">От того, что получилось - вычислить прямое преобразование Фурье</span>\n",
    "6. Результат будет определён в точках $\\xi$. В тех же точках нужно определить $\\color{blue}{\\phi^-_q (\\xi)}$, после чего выполнить поэлементное умножение массивов.\n",
    "7. <span style=\"color:blue\">От того, что получилось - вычислить обратное преобразование Фурье</span>\n",
    "8. <span style=\"color:green\">Полученный результат умножить на константу.</span>\n",
    "9. Результат, который мы получили, нужно положить в массив, который называется $\\color{cyan}{f_{n+1}(y)}$\n",
    "\n",
    "Все шаги повторить до тех пор, пока $n\\ne N$. Найдя $\\color{cyan}{f_{N}(y)}$ нужно провести интерполяцию и получить необходимые ответы."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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c+4bXcUQiWiDFzRVAP+fcl8EOIxKNsvKyaFSzkdcxJMQSExJ5ss+T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\nHQENbmfurqWbC/AocH3uswGvAJc1u24DYSEmfct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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7968b2c208>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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4WbwYZs7UUOg4aN8ehgyBu++GBRlff1akYKhwEYmT6dPBXS0ucXHuubBsWShe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\nAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f796ab410b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "phi_minus_array = make_phi_minus_array(u_space)\n",
    "phi_plus_array = make_phi_plus_array(u_space)\n",
    "\n",
    "#to cut the strange effects on high prices out\n",
    "trimmer = np.where(original_prices_array < 2.5*K_original, 1.0, 0.0)\n",
    "\n",
    "#indicator-function, for the y-domain\n",
    "indicator = np.where(y_space > 0, 1.0, 0.0)\n",
    "\n",
    "# main cycle (iterating over time)\n",
    "f_n = array(f_N)\n",
    "\n",
    "for i in range(N):\n",
    "   f_n_plus_1 = factor * make_rad_ifft(phi_minus_array *\n",
    "       make_rad_fft(indicator * \n",
    "               make_rad_ifft(phi_plus_array * make_rad_fft(f_n, dx), dx), dx), dx)\n",
    "   f_n = f_n_plus_1 #* trimmer\n",
    "\n",
    "plt.plot(original_prices_array[(original_prices_array > 0) & (original_prices_array < 300)], \n",
    "         f_n_plus_1.real[(original_prices_array > 0) & (original_prices_array < 300)], 'g')\n",
    "plt.xlabel('stock price', color = 'y')\n",
    "plt.ylabel('option price, real part')\n",
    "filename_real_pic = \"out_bsmex_l_re N %2d L %.1f M %2d.png\"  % (N, L, M)\n",
    "plt.savefig(\"out_pictures/\" + filename_real_pic)\n",
    "plt.show()\n",
    "\n",
    "plt.plot(original_prices_array[(original_prices_array > 0) & (original_prices_array < 300)], \n",
    "         f_n_plus_1.imag[(original_prices_array > 0) & (original_prices_array < 300)], 'r')\n",
    "plt.xlabel('stock price', color = 'y')\n",
    "plt.ylabel('option price, imag part')\n",
    "filename_imag_pic = \"out_bsmex_l_im N %2d L %.1f M %2d.png\"  % (N, L, M)\n",
    "plt.savefig(\"out_pictures/\" + filename_imag_pic)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "32517\n",
      "34127\n"
     ]
    },
    {
     "data": {
      "image/png": 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Z9bO5td+t3DztZq7odQVdmnUJOpKIyH6iKW7OAoa4+/uxDiOJrWhz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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7968b3e898>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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at1NOgQ03hBEjok4iIqLCRSJUtbhijx7R5pD6tWwJ554LDzwQVvIWEYmQCheJ\nTmUldO4MG2wQdRJpyLBhobOuFl8UkYipcJHozJihy0RxsfHGcPrpcPvt8OOPUacRkQKmwkWio6HQ\n8XLBBWHF6HvvjTqJiBQwFS4SjZ9+gtmzNaIoTjp3DnO63Hhj+O8nIhKBjBYuZrbKzF4xs+JMvq/k\noQ8/hFWr1OISN3/5C8yfHzrqiohEINMtLqcA/wHuyPD7Sr6pGlGkFpd46d4djj02TBj4889RpxGR\nApTRwsXdR7n7Fe6+eybfV/JQZSV07Aht20adRFJ1ySXwxRcwalTUSUSkAKVduJhZdzMbaGbrJR5r\npWVJXmWlWlviqkcPGDwYrrkGfvkl6jQiUmBSLlzMbGMzGwfMAp4HNk089YCZaWpNSY6GQsfbX/8K\nc+fCww9HnURECkw6LS4jgRXAFsDSavsfAw7IRCjJc8uXw6xZKlzirFcvOOIIuPrq8N9TRCRH0ilc\n9geGu/vnNfZ/CHRZ+0iS92bPDr/sdKko3v76V5gzBx59NOokIlJA0ilcWrJmS0uVtoCGGUjDZswI\nt2pxibc+fWDQoNDqsnJl1GlEpECkU7i8DpxY7bGbWRPgIuDVjKSS/FZZGUYTtW8fdRJZW5deGubk\nKSuLOomIFIh0CpeLgDPM7AVgHeB64D2gPzA8g9kkX1WNKNJAtPjbeWc45BC44gr1dRGRnEi5cHH3\n94BtgDeApwmXjv4H2MndZ2c2nuQlrVGUX/72t9BvSfO6iEgONEvnRe6+GLg6w1mkEKxcCTNnwkkn\nRZ1EMqV3bzj6aLjqqrCW0brrRp1IRPJYOvO4DDGzo2rZf5SZ6beR1O/TT8MCfWpxyS9XXgnz5sE9\n90SdRETyXDp9XP4MzK9l/wLgL2sXR/Ke1ijKT9tuG1rRrrkGliyJOo2I5LF0CpctgLm17P808ZxI\n3WbMgFatoFOnqJNIpl12GSxaBLfeGnUSEclj6RQuC4Ada9nfG/h27eJI3qvqmKsRRflnyy3hjDPg\n+uvhu++iTiMieSqdwqUMuNXMBphZ08S2D3AL8K/MxqudmQ0zszlmtszM3jGzXRo4fh0zu9rMPjGz\nn8zsYzM7ORdZpQYtrpjf/vKX0IdphJYtE5HsSKdwuRSYAIwHliW2l4BXyEEfFzM7GhgBXA7sBEwF\nxppZu3pe9gQwABhCGMpdAnyQ5ahSk7sWV8x3m20G55wDI0fC/Nq6womIrJ105nH5xd2PBnoAxwGH\nA1u5+ylRqaz3AAAgAElEQVTunos17kuBe9z9IXefCQwlLEFwSm0Hm9kBQD/gIHd/1d3nuvsEd387\nB1mlui++gB9+UOGS7y6+GJo3D5PSiYhkWDotLgC4+yx3f8Ldn3X3TzMZqi5m1hwoJrT2VOVwYBzQ\nt46XHQK8Cww3s8/N7AMzu8HMNNlErmlEUWFo2zYswHjffWHOHhGRDEpqAjozuwm41N2XJO7Xyd0v\nyEiy2rUDmvLr4djzgW3reE03QovLT8Chife4i7Ao5KnZiSm1qqwMk5NtuWXUSSTbzj4bbr89tL48\n9VTUaUQkjyQ7c+5OQPPE/SLA6ziurv1RagKsAo519x8BzOwC4AkzO8vdtaJ1rsyYAT16QNOmUSeR\nbGvRIqwafdxx8Prr0K9f1IlEJE8kVbi4+4Bq9/fOWpqGfQOsBDrU2N8B+KqO13wJfFFVtCTMAAzo\nBNS5vlJpaSlt2rRZY19JSQklJSUpxhZAaxQVmmOOgZtuggsvhLff1hB4kTxUVlZGWY3V4RcvXpzV\nz7TQRSTJg0Mfk2VAn8RiizlnZu8AE9z9vMRjI0yId6u731DL8acDI4H27r40se/3wJNAq9paXMys\nCCgvLy+nqKgoez9MIXGHdu2gtDT0f5DC8OqrsM8+8PjjcNSvVgoRkTxUUVFBcXExQLG7V2T6/VPq\nnOvuywlFQpRt/TcBp5vZiWbWA7gbWB8YBWBm15rZ6GrHP0qYGO9BM+tpZv2B64EHdJkoh77+GhYu\nVItLoRkwAA4+OPR1+SUXgw5FJN+lM6roauAaM2ub6TDJcPfHgT8BVwGTCbP4DnT3rxOHdAQ6Vzt+\nCbAfsCEwCXgYeBo4L4expWpEkQqXwnP99WFxTS0FICIZkGzn3OrOBroD88zsU2CNFdXcPevXVtz9\nTuDOOp4bUsu+WcDAbOeSelRWQrNmsNVWUSeRXNtuOzjzTLjqKjj+eOjYMepEIhJj6RQuGtsoqaus\nhG22CROTSeG58kooKwtLAvzjH1GnEZEYS7lwcfcrsxFE8pym+i9sbdvC3/4GZ50VWl92qXd5MRGR\nOqU9c66Z7WxmJyS24kyGkjykxRXljDNgxx3h3HNh1aqo04hITKVcuJhZJzN7HZhIWBH6FmCSmb1h\nZp0yHVDywKJF8NVXanEpdE2bhg6677wD//xn1GlEJKbSaXG5nzCLbk93b+vubYGeife6P5PhJE/M\nmBFuVbjIXnuF+VyGDw8LboqIpCidwmUv4Ex3/6BqR+L+OUD/TAWTPFJZCU2ahM65IjfcAIsXhz4v\nIiIpSqdw+YzV6xZV1xSYt3ZxJC9VVkK3bmGBRZEuXcLoopEjYfr0qNOISMykU7hcCNxmZjtX7Ujc\nv4UwMZzImrRGkdR04YXQvTsMHaqOuiKSknQKl1FAH2CCmf1sZj8DEwirRv/DzBZWbRnMKXGmodBS\nU4sWcNdd8NZbmtdFRFKSzgR052c8heSvH36AuXM1FFp+be+94aST4KKLYNAgaN8+6kQiEgPpTEA3\nuuGjRBJmzgy3anGR2txwAzzzTLh0NFr/tIhIw9KegE4kKVVDoXv0iDaHNE6bbBIWYXzoIXj11ajT\niEgMqHCR7KqshC22gFatok4ijdWQIbDnnvCHP8DSpVGnEZFGToWLZJdGFElDmjSB++4LfaEuvzzq\nNCLSyKlwkexS4SLJ6NEjrCB9000wYULUaUSkEVPhItmzbBnMmaPCRZLzxz9CcXG4dPTTT1GnEZFG\nKp1FFlua2d/M7C0z+8jMPq6+ZSOkxNSsWWFyMQ2FlmQ0axbmdPnoI7jqqqjTiEgjlc48LvcT1it6\nGPgS8IwmkvxRWRluVbhIsnr1gssugyuugCOOCC0wIiLVpFO4HAgc7O5vZjqM5JkZM2DTTWGjjaJO\nInEyfDj8+99w8skwaZLWuBKRNaTTx2URoOn8pWGVlWptkdQ1bx4mo5s1C/7616jTiEgjk07hcilw\nlZmtn+kwkmc0okjSteOOcM01MGIEjB8fdRoRaUTSKVz+CAwE5pvZdDOrqL5lOJ/E1fLl8OGHKlwk\nfaWlsM8+YT2jRYuiTiMijUQ6fVyeyngKyT8ffQQrVuhSkaSvSZNwyWiHHWDoUPjXv8As6lQiErF0\nFlm8MhtBJM9UjShSi4usjU6d4J574Oij4Xe/gxNOiDqRiEQs7QnozKzYzI5PbDtlMlQSnz3MzOaY\n2TIze8fMdknydb8xs+W6pJUD778P7dpB+/ZRJ5G4Gzw4FCzDhsHHmipKpNClMwFdezN7BZgE3JrY\nys1svJltkumAtXz+0cAI4HJgJ2AqMNbM2jXwujbAaGBctjMK6pgrmXX77WEl6cGD4eefo04jIhFK\np8XlNmADYHt3b+vubYFeQGtCEZNtpcA97v6Qu88EhgJLgVMaeN3dwCPAO1nOJxBaXLbfPuoUki9a\nt4YnnoDp0+FPf4o6jYhEKJ3C5QDgLHefUbXD3SuBYYTJ6bLGzJoDxcB/x0e6uxNaUfrW87ohQFdA\n/XNyYcUK+OADtbhIZhUVwciRofXlySejTiMiEUmncGkCLK9l//I03y8V7YCmwPwa++cDHWt7gZlt\nDVwDHOfuq7IbT4Awomj5crW4SOadeWa4XHTqqTB7dtRpRCQC6QyHfgW4xcxK3H0egJltDoykWktI\nY2BmTQiXhy5396p/5ZIeT1laWkqbNm3W2FdSUkJJSUnmQuYjjSiSbDGD++4LaxgddRS89ZaWBBCJ\nUFlZGWVlZWvsW7x4cVY/08KVlhReYNYZGANsD3yW2N0ZeA8Y5O6fZzThmp/dnNCf5Qh3H1Nt/yig\njbsfVuP4NoQlClawumBpkri/Atjf3f+vls8pAsrLy8spKirKwk+S5/72N7j5ZvjmG827IdkxZQr0\n7QvHHBNWlNb3TKTRqKiooDgskFrs7hkfxZvOPC6fJX6x/xbokdg9w92zPlrH3ZebWTmwL6F4wsws\n8bi2jsHfEzoOVzcMGAAcAXyStbCFrLIyXCbSLxPJlj594N574cQTQ9+Xc86JOpGI5Eg6l4qqOsS+\nnNhy7SZgVKKAmUgYZbQ+MArAzK4FNnP3kxI5K6u/2MwWAD9V71wsGVZZGf4aFsmmE06AyZPD0gA7\n7AB77x11IhHJgaQKFzM7F7jX3X9K3K+Tu2d1SLS7P56Ys+UqoAMwBRjo7l8nDulIuHQlUVixAmbO\nhNNOizqJFILrr4epU0N/l3ffhS5dok4kIlmWVB8XM5sD7Ozu3ybu18XdvVvG0kVEfVzWwqxZsO22\nMG4c7Ltv1GmkEHzzDeyyC2y0EbzxBqyvhetFotQo+ri4e9fa7ov8yvvvh1sNhZZcadcOnnoK9tgj\nXD564omwQKOI5KV0pvy/zMx+9SeNma1nZpdlJpbEVmVl+Mu3Q4eok0gh6d0bysrgf/8XLroo6jQi\nkkXp/FlyOdCqlv3rJ56TQvb++2H+Fo0oklwbNCgMwx8xAu66K+o0IpIl6YwqMqC2jjG9gYVrF0di\nr7ISdtst6hRSqM49N8yoe/bZoaPuQQdFnUhEMizpFhczW2RmCwlFyywzW1htW0wYGv14toJKDKxc\nGUYUacZcidJNN8HvfheWBpg8Oeo0IpJhqbS4nE9obfkH4ZJQ9Tl9fwE+cfe3M5hN4ubjj+Hnn9Ux\nV6LVtCk8+igMGAADB4aRRttsE3UqEcmQpAsXdx8N/x0a/Za717bQohQyrVEkjUXLlvD889C/P+y3\nH7z5JnTqFHUqEcmAlDvnuvtrwCozO9LMLk1sR5hZWrPwSh55/33YcEPYdNOok4iEYdIvvRTu779/\nmO9FRGIvneHQ2wOzgNHAYYltNPChmdVcF0gKSWWlRhRJ49KpE7z8cihaDjwQfvgh6kQispbSGQ59\nP/A+0Mndi9y9iDDF/jTg3kyGk5ipGgot0phssw2MHRtmdf7d7+DHH6NOJCJrIZ3CpQ/wZ3dfVLUj\ncf8SYKdMBZOYWbECZsyAXmp0k0Zop53ghRegogIOPhiWLIk6kYikKZ3CZRZhccOa2gMfrV0cia0P\nPwwjinbcMeokIrXbY4/Q8lJREeZ3UfEiEkvpFC5/Bm5NdM7tlNiOBG4GhptZ66ots1GlUZs+Pdzu\nsEO0OUTqo+JFJPbSGQn0bOL2cVbPoFvVG/OZao8daJp+NImVadPCaKJ27aJOIlK/quJl4MDQYfeZ\nZ6BNm6hTiUiS0ilcBmQ8hcTf9OlqbZH42GOPMFT6oIPCRHUvvgjt20edSkSSkHLhkpjHRWRN06bB\nEUdEnUIkeX37wmuvhZaXfv1CIdOlS9SpRKQB6fRxwcw2NLM/mtn9ia3UzNTWWqh++AE++UQdcyV+\ndtwxLAmwfDnsuWcYGScijVo6E9DtDMwGSoG2ie0CYLaZFWU2nsTCe++FW10qkjjaaqtQvGy4YWh5\nefPNqBOJSD3SaXEZCYwBtnT3w939cKArodPuzZkMJzExbVpY2K5nz6iTiKRns83gP/8JC4Tusw+U\nlUWdSETqkE7hsjPwd3dfUbUjcf/6xHNSaKZPD7OTrrtu1ElE0rfRRqGfy9FHw7HHwt/+Bu4Nv05E\nciqdUUXfA1sAM2vs7wxoIZBCpBFFki9atIDRo2HrreGyy+Cjj+Dee8N+EWkU0mlxeQx4wMyONrPO\nie0YwhpGal8tNO7hUpE65kq+MINLL4VHH4V//SsMl543L+pUIpKQTuHyJ+B/gIeATxLbKOBJYHiG\ncklcfPEFfPedWlwk/5SUhH4vc+dCURG8/nrUiUSENAoXd//F3c8DNiIsuNgHaOvupe7+c6YD1sbM\nhpnZHDNbZmbvmNku9Rx7mJm9ZGYLzGyxmb1lZvvnImdBmDYt3KpwkXy0225QXg49eoROu7feqn4v\nIhFLax4XAHdf6u7TE9vSTIaqj5kdDYwALiesRj0VGGtmdc013x94CTgQKAJeBZ4xs945iJv/pk+H\nDTbQxF2Svzp0gHHj4Nxz4bzz4Pjjw9xFIhKJtAuXCJUC97j7Q+4+ExgKLAVOqe3gREvQje5e7u6z\n3f0S4EPgkNxFzmPTpkGvXtAkjl8lkSQ1awYjRoRh0s88AzvtBJMmRZ1KpCDF6reNmTUHioHxVfvc\n3YFxQN8k38OADYCF2chYcCoqwj/iIoXgmGNg8mRo2zasd3TDDbBqVdSpRApKrAoXoB1hxen5NfbP\nBzom+R4XAi0Jq1vL2liyBD74IHRcFCkUVTPt/vGPcNFFcMABoZO6iORE3AqXtWJmxwKXAke5+zdR\n54m9adNCR0W1uEihWWcduO66MGHde++FGXdHj1bHXZEcSGcCuih9A6wEOtTY3wH4qr4XJuaauRc4\n0t1fTebDSktLadNmzbUjS0pKKCkpSTpwXps8OVz73377qJOIRGO//ULhcv75cPLJ8MQTcM89sPnm\nUScTyYmysjLKaiyRsXjx4qx+pnnM/kIws3eACYkh2VV9VuYCt7r7DXW8poQwQd7R7v5sEp9RBJSX\nl5dTpMsgdTvttDBUdPLkqJOIRO+ZZ+CMM2DZMhg5MhQyZlGnEsm5iooKiouLAYrdvSLT7x/HS0U3\nAaeb2Ylm1gO4G1ifMAkeZnatmY2uOjhxeWg08Edgkpl1SGytcx89z0yerMtEIlUOOQTefx8GDYJT\nToG99lq9crqIZEzsChd3f5wwe+9VwGRgR2Cgu3+dOKQjYd2kKqcTOvTeAcyrtmkl67Xxyy/hH2UV\nLiKrtW0LDz0E48fDggXQpw9ceCH8+GPUyUTyRuwKFwB3v9Pdt3T39dy9r7u/W+25Ie6+T7XHA9y9\naS1brfO+SJIqK0PxoktpIr+2zz4wdSpcdRXcfjv07BnmgNHQaZG1FsvCRRqByZPD9fvemoBYpFYt\nWsBf/hKK/OJiOPZY2H13rXkkspZUuEh6Jk+GrbeGVq2iTiLSuHXtCk89Ba++Glpc+veHww+HDz+M\nOplILKlwkfRUVOgykUgq9t4bJk6Ehx+Gd9+F7baDoUPD6tMikjQVLpK6lSthyhQVLiKpatIkLNL4\nwQdw9dXw5JPQvTucdRZ8/nnU6URiQYWLpK6yMkz3v+uuUScRiaf11gvLBcyZA1deCY89FpYSOPts\n+OyzqNOJNGoqXCR1EyaEvxzDBEMikq4NNoA//zkUMJddBo8+Ct26hVaZqVOjTifSKKlwkdRNnBim\n+VfHXJHMaN0aLrkEPv0UbrwxLOLYpw/sv39YDylmM5yLZJMKF0ndxIm6TCSSDRtsAOedBx99FOZ9\n+fZbGDgQevWC226DLK8BIxIHKlwkNUuWhBlzVbiIZE+zZnDMMWH00SuvhBFIF1wAm20Gp58eRvWJ\nFCgVLpKaioowqmi33aJOIpL/zGDAgLDq9KefwsUXw4svhv5lu+4Kd94ZWmVECogKF0nNxIlhRMT2\n20edRKSwbLYZXHpp6Mj71FOwySZw7rmw6aZhQrv//V/4+eeoU4pknQoXSc3EieGvvWbNok4iUpia\nNYPf/x6eew6++AJuuCG0xhx+eChuhg6FceNgxYqok4pkhQoXSc3bb+sykUhj0aFD6MxbXh76np1+\nOowdC/vtBx07wqmnwgsvhAVRRfKEChdJ3ty5YXKsPfeMOomI1LT99nDddfDxx6FT7xlnhAUdDzoI\n2reHkpKw3MDXX0edVGStqHCR5FWtaqvCRaTxMguXc6+5JiwtMG1aaJX58EM48cTQSrP77nDVVTBp\nUlj4USRGVLhI8l5/HXr2hHbtok4iIskwgx12CMsKvPsufPkl/OMfsMUWcNNNYWRS+/ZwxBFw++3w\n/vua7E4aPfWwlOS98YZaW0TirGNHOPnksC1fHvqsjRsHr74a5olZvjyMVtp77zAMe889wxwyTZtG\nHFxkNRUukpxvvw1/jQ0fHnUSEcmE5s2hf/+wASxdCm+9FYqYV18NQ61XrAiz+e66a7i8tPvuoXP+\nJptEm10KmgoXSc6bb4bbfv2izSEi2bH++vDb34YNwizZkybBO++E7b774Oqrw3NbbRWKmT59YKed\nwq2KGckRFS6SnNdfh803hy5dok4iIrnQsmW4ZLT33uGxO3zySShi3n47DMEeMyYUOBD+fagqZHr3\nDqOcuncPLTsiGaTCRZIzbhzss0/o7CcihccMunYNW0lJ2LdqVVgQcsoUmDw53N5/P3z1VXi+WbNQ\nvPTsuXrbbjvYdttQGImkQYWLNGzBgvAP0gUXRJ1ERBqTJk1gm23CNnjw6v0LFsCMGVBZGW5nzIDR\no8NMv1U23xy6dQuXnbbaavX9bt3CyEX9kSR1UOEiDRs/PtxWXfsWEalP+/Zh22uvNfcvXgwzZ4ZC\nZvbssM2YAc8+C998s/q4DTYIBUyXLtCpE3TuvOZtp07QokVufyZpNFS4SMNefhl69QqLuYmIpKtN\nmzAqqbZlQ77/PiwgOXt2mP139uwwU/cbb8Dnn8PChWse3759KGA23zwM8+7QIWzV73foAK1bq/Um\nz8SycDGzYcCfgI7AVOAcd59Uz/F7AyOA7YG5wNXuPjoHUePPPRQuRx0VdRIRyWetW4dOvb171/78\nkiXhUtNnn4VCpur2889D/5r588NWc12mddddXcRssgm0bQsbbxxu67qvYqdRi13hYmZHE4qQM4CJ\nQCkw1sy2cfdvajl+S+BZ4E7gWOC3wP1mNs/dX85V7tiaOTP8w6DLRCISpZYtV/enqYs7fPfd6iJm\n/vzQUbjq/tdfh1ad8vIwN9XChWHSvZqaNg0FzEYbhVai1q3r32oe06pVyLvuuqEfkGRU7AoXQqFy\nj7s/BGBmQ4GDgVOA62s5/kzgY3e/KPH4AzPbM/E+Klwa8vTTYX6HAQOiTiIiUj+zUGxstBH06NHw\n8e6hJaeqiFm4cPX9qtsffgiXsb7/PhQ/VfertobWelp//bC1bJn87XrrhT48LVqE4qfm/fr2rbNO\n3rcWxapwMbPmQDFwTdU+d3czGwf0reNluwPjauwbC4zMSsh88/TTMHBg+B9JRCSfmIXWkVat0puj\nyj3MOPz996HjcdXtkiVhW7o0bFX3a97++GMohmo+t2wZ/Pxz+gtgVhU1VVvz5mFr1mz1/dq2dJ5v\n1iy0UFW//fzz9HInKVaFC9AOaArMr7F/PrBtHa/pWMfxrc2shbv/nNmIa2ncOBg1anXFbLZ6y+bj\nvn1DP5bqlfqXX4bJpkarO5CIyK+YhRaSli2zM3hhxQr46adQxNS8rW1fbc/9/HO4HLZiRbituVXf\nv2xZKL6SPb7q8cqVq29zIG6FS06VlpbSpk2bNfaVlJRQUjX5Ujb8+OPqatV9za3mvmSOSeZ1v/wC\nN98crgWfe+7qLGPGhAr64IOz9/OKiEjtmjVb3SLUSJWVlVFWVrZ6hzuLv/sujAbLEvMYLWGeuFS0\nFDjC3cdU2z8KaOPuh9XymteAcne/oNq+k4GR7r5RHZ9TBJSXl5dTVFSU2R+isRo6FJ54AubNWz0/\nwr77htuqeVxEREQaUFFRQXFxMUCxu1dk+v1j1d3Z3ZcD5cC+VfvMzBKP36rjZW9XPz5h/8R+qXLO\nOaEj2tix4fHcuWGF2OOPjzaXiIhINbEqXBJuAk43sxPNrAdwN7A+MArAzK41s+qdMu4GupnZ381s\nWzM7Czgy8T5SZfvtw/bYY+HxI4+EXupHHBFtLhERkWpi18fF3R83s3bAVUAHYAow0N2/ThzSEehc\n7fhPzOxgwiiic4HPgVPdveZIIznqKBgxIvRsf+ABOOywMCeBiIhIIxG7wgXA3e8kTChX23NDatn3\nH8IwaqnPkUfCFVfAmWeG6bYfeSTqRCIiImuI46UiyZbttoPiYnjoITjggNrXExEREYlQLFtcJEvM\n4MknoawMTj896jQiIiK/osJF1rTllvDnP0edQkREpFa6VCQiIiKxocJFREREYkOFi4iIiMSGChcR\nERGJDRUuIiIiEhsqXERERCQ2VLiIiIhIbKhwERERkdhQ4SIiIiKxocJFREREYkOFi4iIiMSGChcR\nERGJDRUuIiIiEhsqXERERCQ2VLiIiIhIbKhwERERkdhQ4SIiIiKxocJFREREYkOFi4iIiMSGChcR\nERGJjVgVLma2kZk9YmaLzWyRmd1vZi3rOb6Zmf3dzKaZ2Y9m9oWZjTazTXOZu5CUlZVFHSF2dM7S\no/OWOp2z9Oi8NS6xKlyAR4GewL7AwUB/4J56jl8f6ANcCewEHAZsCzyd3ZiFS/+Dp07nLD06b6nT\nOUuPzlvj0izqAMkysx7AQKDY3Scn9p0DPGdmf3L3r2q+xt2/T7ym+vucDUwws07u/nkOoouIiEiG\nxKnFpS+wqKpoSRgHOLBbCu+zYeI132Uwm4iIiORAnAqXjsCC6jvcfSWwMPFcg8ysBXAd8Ki7/5jx\nhCIiIpJVkV8qMrNrgeH1HOKEfi1r+znNgCcS73dWA4evCzBjxoy1/diCs3jxYioqKqKOESs6Z+nR\neUudzll6dN5SU+1357rZeH9z92y8b/IBzDYGNm7gsI+BE4Ab3f2/x5pZU+An4Eh3r7PDbbWiZUtg\nH3df1ECmY4FHkvoBREREpDbHufujmX7TyFtc3P1b4NuGjjOzt4ENzWynav1c9gUMmFDP66qKlm7A\ngIaKloSxwHHAJ4TCSERERJKzLqGhYGw23jzyFpdUmNnzQHvgTGAd4B/ARHc/odoxM4Hh7v50omj5\nN2FI9O9Ys4/MQndfnrPwIiIistYib3FJ0bHA7YTRRKuAJ4HzahyzNdAmcX9zQsECMCVxa4R+LgOA\n/2QzrIiIiGRWrFpcREREpLDFaTi0iIiIFDgVLiIiIhIbBVu4mNnlZraqxlZZ45irzGyemS01s5fN\nrHtUeaNiZv3MbExigcpVZjaolmPqPU9m1sLM7jCzb8zsBzN70sza5+6nyL2GzpuZPVjL9+/5GscU\nzHkzsz+b2UQz+97M5pvZ/5rZNrUcp+9aNcmcN33Xfs3MhprZ1MSCvYvN7C0zO6DGMfquVdPQOcvl\n96xgC5eE94AOhJl3OwJ7Vj1hZsOBs4EzgF2BJcBYM1sngpxRakno2HwWoVPzGpI8TzcTFsU8grAw\n5maE0V75rN7zlvACa37/Smo8X0jnrR9wG2H5jt8CzYGXzGy9qgP0XatVg+ctQd+1NX1GmPi0CCgG\nXgGeNrOeoO9aHeo9Zwm5+Z65e0FuwOVART3PzwNKqz1uDSwDBkedPcJztgoYlMp5Sjz+GTis2jHb\nJt5r16h/pgjP24PA/9TzmoI+b0C7xM+6Z7V9+q6ld970XUvu3H0LDEnc13ct9XOWs+9Zobe4bJ1o\nyp9tZv80s84AZtaVUC2OrzrQw0rTEwiLPQpJn6edCcPuqx/zATAXncu9E837M83sTjNrW+25Ygr7\nvFUthroQ9F1LwRrnrRp91+pgZk3M7BhgfeAtfdcaVvOcVXsqJ9+zuM3jkknvACcDHwCbAlcA/zGz\nXoQvrQPza7xmPkku6FggkjlPHYBfEv/j13VMIXqB0EQ6B9gKuBZ43sz6evhTpCMFet7MzAhNym+4\ne1W/M33XGlDHeQN912qV+Lf+bcIsrz8QWgI+MLO+6LtWq7rOWeLpnH3PCrZwcffqUxG/Z2YTgU+B\nwcDMaFJJoXD3x6s9fN/MpgOzgb2BVyMJ1XjcCWwH/CbqIDFT63nTd61OM4HehAlLjwQeMrP+0UZq\n9Go9Z+4+M5ffs0K/VPRf7r4YmAV0B74izLDbocZhHRLPSZDMefoKWMfMWtdzTMFz9znAN4TvHxTo\neTOz24GDgL3d/ctqT+m7Vo96ztuv6LsWuPsKd//Y3Se7+yXAVMJM7Pqu1aGec1bbsVn7nqlwSTCz\nVoQTPC9xwr8iLOJY9XxrQs/9t2p/h8KT5HkqB1bUOGZbYAtCk6MAZtaJsEp61S+dgjtviV++vycs\nhjq3+nP6rtWtvvNWx/EF/12rQxOghb5rKWkCtKjtiax+z6LulRzVBtxAGI7VBdgDeJlwrW3jxPMX\nER9XBnMAAAXUSURBVHpMHwLsADwFfAisE3X2HJ+nloSmwT6E3t/nJx53TvY8EZqw5xCaDIuBN4HX\no/7ZojpvieeuJ/xD2CXxP/K7wAygeSGet8TPuogwvLdDtW3dasfou5biedN3rc7zdk3inHUBehH6\nY6wA9tF3LfVzluvvWeQnI8L/CGXA54QhbnOBR4GuNY65gjAsbilhee7uUeeO4DztRfjFu7LG9o9k\nzxOhIr+N0Gz4A/AE0D7qny2q80bo2PYi4a+6n4CPgbvg/7d37yGaV3Ucx98HNSmkjdTSiiLc1lWz\nJY1wdclveIUiTRRB1C7oH0utiGIpRq39EZiCZUaFC9KiQpfRSjMcUb4YtaAhmpuXvKyoQbRtlKir\nq+7pj3MGHoaZeWZ31h1/+H7BMDO/c57fOXN4ZvhwLvNj/7fruM0yVm8A506r53ttB8bN99qs47au\nj8XWPjaT9NDie23Hx2x3v898yKIkSRoM97hIkqTBMLhIkqTBMLhIkqTBMLhIkqTBMLhIkqTBMLhI\nkqTBMLhIkqTBMLhIkqTBMLhIkqTBMLhIekvKLNszyxd2U1ubMssFu6MtSQuz52J3QNKwZZYbgCUR\n9bTF7ssCfAp4abE7IWk8g4ukt63MsldEfS2iblnsvkiaH4OLpLEyy+nAt4GltKflPgCcAnwD+BJQ\nM8t2oAKfjaj3ZpbDgR8AK/trJoCLIupLI/f9KnBRv+8WYCKizrhkk1muAM4DToqoG2co/w5wKu2p\ntN8C9gVuB86PqC/0OjcA7wHuB75Ge5LtQZllE3BNRL2211sCfL//jEuAJ4BLI+odvXwV8D3aTM1m\n4DfAZRH15R0YVkk7wT0ukuaUWQ4AbqY91n45cCxwC1CAq4Ff0h5p/37gQODPmeVd/doW4EjgdOB4\n2iPtp+67GrgO+ClwGPA54O+z9OFHwNnAqplCy4ilwBn9XicBnwR+PK3OccCy3p/Pz9BW6X1fCZwF\nHAJcArzRyw8C/gD8Cvg4cCZwzOjPJunN44yLpHEOBPYAbo2oz/Vrf5sqzCxbgXdE1M0j174C7A2c\nG1FfAR7NLF8Hbsss3+x1LweuiqjXjbT14LS298osNwErgGMi6j/H9HVv4JypepllDXB7Zrk4ov6r\n13kROC+ivj7LPU6gzaQsj6hP9WvPjJRfCtwYUaeCytOZ5UIgM8vqiLptTB8lLYDBRdI4DwF3Axsz\ny53AJPDriPrfOV6zHHioh5Ypf6LN8h6cWQA+ANwzpu1raMs5R0XU/8yjr89OCzcbaKHrYGAquDw8\nR2iBFpKeHwktM5UfnlnOHrlW+uePAo/Po5+SdpJLRZLmFFG3R9QTgZNpMy1rgMczy0cWcNut86w3\nCXywt72rjDs9NK5v+wA/Az5BCzEr+tfLgNnCjqRdxOAiaV4i6oaIegVt38g24Iu9aBttVmPUo8CK\nzPLOkWuraPtEHouoL9KWX44b0+zvaPtM1mWWM+fRzQ/3PTlTVvY2d2QW5K/AhzLL0lnKHwAOjaib\nIurT0z7mmsmRtAu4VCRpTpnl07SAMUlbbjkK2A94pFd5Bjgxsyyjbcb9H3ATsBb4eT8N9D7gWmB9\nRP13f91a4CeZZTNts+u7gaOn7Xkhov42s5wDrM8sr0fUiTm6+2pv8xLaaaAfAr8Y2d8yVj8R9Udg\nIrNcDDxJW/qqEfVO4EpgQ98wvI42g3MYcHxEXTPfdiTtHGdcJI3zAvAZ4Pe0mYvv0o41T/by6/v1\nv9CCzdERdSvtVM97gftoJ4/uoi0zARBR1wMXAquBjbTZldFZjjpSdwL4Mi28nDpHX5+gnXi6g3Yy\n6EHasedx6rTvT6Mdmb6Ztjx2Jf3vZUR9mHay6mPAvbQZmLXAP+bRjqQFKrVO/32VpOHp/8fllIh6\nxGL3RdKbxxkXSZI0GAYXSZI0GC4VSZKkwXDGRZIkDYbBRZIkDYbBRZIkDYbBRZIkDYbBRZIkDYbB\nRZIkDYbBRZIkDYbBRZIkDcb/AQr+470z8RtNAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7968c45f98>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "index_bot = 0\n",
    "while(original_prices_array[index_bot] < H_original-20):\n",
    "    index_bot+=1\n",
    "print(index_bot)\n",
    "\n",
    "index_top = 0\n",
    "while(original_prices_array[index_top] < 350):\n",
    "    index_top+=1\n",
    "print(index_top)\n",
    "\n",
    "plt.plot(original_prices_array[index_bot:index_top], f_n_plus_1.real[index_bot:index_top], 'g')\n",
    "plt.xlabel('stock price', color = 'y')\n",
    "plt.ylabel('option price, real part')\n",
    "filename_real_pic = \"out_bsmex_s_re N %2d L %.1f M %2d.png\"  % (N, L, M)\n",
    "plt.savefig(\"out_pictures/\" + filename_real_pic)\n",
    "plt.show()\n",
    "\n",
    "plt.plot(original_prices_array[index_bot:index_top], f_n_plus_1.imag[index_bot:index_top], 'r')\n",
    "plt.xlabel('stock price', color = 'y')\n",
    "plt.ylabel('option price, imag part')\n",
    "filename_imag_pic = \"out_bsmex_s_im N %2d L %.1f M %2d.png\"  % (N, L, M)\n",
    "plt.savefig(\"out_pictures/\" + filename_imag_pic)\n",
    "plt.savefig(\"out.png\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [conda root]",
   "language": "python",
   "name": "conda-root-py"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}