Heston,.wh factorization, explicit formulae.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from math import log, sqrt\n",
    "import numpy as np\n",
    "from numpy import exp, fft, array, pi, zeros\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Процедуры, которые возвращают дерево для процесса CIR"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def compute_f(r, omega):\n",
    "    return 2*sqrt(r)/omega\n",
    "\n",
    "\n",
    "def compute_v(R, omega):\n",
    "    if R > 0:\n",
    "        return (R**2) * (omega**2)/4.0\n",
    "    else:\n",
    "        return 0.0\n",
    "\n",
    "\n",
    "def build_volatility_tree(T, v0, kappa, theta, omega, N):\n",
    "    div_by_zero_counter = 0\n",
    "    f = zeros((N+1, N+1))\n",
    "    f[0, 0] = compute_f(v0, omega)\n",
    "    dt = float(T)/float(N)\n",
    "    sqrt_dt = sqrt(dt)\n",
    "    V = zeros((N+1, N+1))\n",
    "    V[0, 0] = compute_v(f[0, 0], omega)\n",
    "    f[1, 0] = f[0, 0]-sqrt_dt\n",
    "    f[1, 1] = f[0, 0]+sqrt_dt\n",
    "    V[1, 0] = compute_v(f[1, 0], omega)\n",
    "    V[1, 1] = compute_v(f[1, 1], omega)\n",
    "\n",
    "    for i in range(1, N):\n",
    "        for j in range(i+1):\n",
    "            f[i+1, j] = f[i, j] - sqrt_dt\n",
    "            f[i+1, j+1] = f[i, j] + sqrt_dt\n",
    "            V[i+1, j] = compute_v(f[i+1, j], omega)\n",
    "            V[i+1, j+1] = compute_v(f[i+1, j+1], omega)\n",
    "\n",
    "    f_down = zeros((N+1, N+1))\n",
    "    f_up = zeros((N+1, N+1))\n",
    "    pu_f = zeros((N+1, N+1))\n",
    "    pd_f = zeros((N+1, N+1))\n",
    "    for i in range(0, N):\n",
    "        for j in range(i+1):\n",
    "            # /*Compute mu_f*/\n",
    "            v_curr = V[i][j]\n",
    "            mu_r = kappa*(theta-v_curr)\n",
    "            z = 0\n",
    "            while V[i, j] + mu_r*dt < V[i+1, j-z] and j-z >= 0:\n",
    "                z += 1\n",
    "            f_down[i, j] = -z\n",
    "            Rd = V[i+1, j-z]  # the next low vertice we can reach\n",
    "            z = 0\n",
    "            while V[i, j] + mu_r*dt > V[i+1, j+z] and j+z <= i:\n",
    "                z += 1\n",
    "            Ru = V[i+1, j+z]  # the next high vertice we can reach\n",
    "            f_up[i, j] = z\n",
    "            if Ru == Rd:\n",
    "                div_by_zero_counter += 1\n",
    "            pu_f[i, j] = (V[i, j]+mu_r*dt-Rd)/(Ru-Rd)\n",
    "\n",
    "            if Ru-1.e-9 > V[i+1, i+1] or j+f_up[i][j] > i+1:\n",
    "                pu_f[i][j] = 1.0\n",
    "                f_up[i][j] = i+1-j\n",
    "                f_down[i][j] = i-j\n",
    "\n",
    "            if Rd+1.e-9 < V[i+1, 0] or j+f_down[i, j] < 0:\n",
    "                pu_f[i, j] = 0.0\n",
    "                f_up[i, j] = 1 - j\n",
    "                f_down[i, j] = 0 - j\n",
    "            pd_f[i, j] = 1.0 - pu_f[i][j]\n",
    "    return [V, pu_f, pd_f, f_up, f_down]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Процедуры для fft в радианах"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "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": [
    "Код для вычисления цены"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "rho_hat = 0.8660254037844386\n",
      "q =  5.095310179804325\n",
      "Множитель каждого шага = 0.9812945284112251\n"
     ]
    }
   ],
   "source": [
    "# option parameters\n",
    "T = 1\n",
    "H_original = 90  # limit\n",
    "K_original = 100.0  # strike\n",
    "r_premia = 10  # annual interest rate\n",
    "r = log(r_premia/100 + 1)\n",
    "\n",
    "# Heston model parameters\n",
    "V0 = 0.1  # initial volatility\n",
    "kappa = 2.0  # heston parameter, mean reversion\n",
    "theta = 0.1  # heston parameter, long-run variance\n",
    "sigma = omega = 0.2  # heston parameter, volatility of variance.\n",
    "# Omega is used in variance tree, sigma - everywhere else\n",
    "rho = 0.5  # heston parameter #correlation\n",
    "\n",
    "# method parameters\n",
    "N = 5  # number_of_time_steps\n",
    "delta_t = T/N\n",
    "\n",
    "rho_hat = sqrt(1 - rho**2)\n",
    "print('rho_hat =' , rho_hat)\n",
    "\n",
    "q = 1.0/delta_t + r\n",
    "print('q = ', q)\n",
    "factor = (q*delta_t)**(-1)\n",
    "print('Множитель каждого шага =', factor)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "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  -0.6931471805599453 до 0.6931471805599453\n"
     ]
    }
   ],
   "source": [
    "def G(S, K):\n",
    "    \"\"\"the payoff function of a put option. Nothing to do with barrier\"\"\"\n",
    "    return max(K-S, 0)\n",
    "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": "markdown",
   "metadata": {},
   "source": [
    "### Замечание о естественных ценах\n",
    "\n",
    "Мы подбираем равномерную сетку в логарифмической шкале, но она не будет таковой в линейной.\n",
    "\n",
    "Существует грубая оценка на корректность работы алгоритма при различных параметрах, которую я принёс на бумаге. Её нужно либо перенести сюда, либо отвергнуть.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Строим дерево с параметрами. Отметим, что при малых N = (1,2) процедура не работает из-за делений на ноль"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "if 2 * kappa * theta < sigma**2:\n",
    "    print(\"Warning, Novikov condition is not satisfied, the volatility values could be negative\")\n",
    "r = log(r_premia/100 + 1)\n",
    "omega = sigma\n",
    "\n",
    "first_step_of_return = [elem + V0*rho/sigma for elem in x_space]\n",
    "original_prices_array = H_original * exp(first_step_of_return)\n",
    "# making volatilily tree\n",
    "markov_chain = build_volatility_tree(T, V0, kappa, theta, omega, N)\n",
    "V = markov_chain[0]\n",
    "pu_f = markov_chain[1]\n",
    "pd_f = markov_chain[2]\n",
    "f_up = markov_chain[3]\n",
    "f_down = markov_chain[4]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Теперь, когда дерево у нас есть, можно создавать пространственную переменную.\n",
    "Проблема состоит в том, что пространство меняется в каждой вершине цепи так, чтобы барьер оказывался в центре.\n",
    "Поэтому нужно хранить много пространств, что не щадит память совершенно.\n",
    "\n",
    "$ y_j (n,k) = x_j - {\\rho \\over \\sigma} V (n,k) $\n",
    "\n",
    "$y = \\ln(S/H) - {\\rho \\over \\sigma} V$\n",
    "\n",
    "$y[j][n][k]$ начальное сосотяние процессов цены после замены\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y_j_n_k consctuction completed\n"
     ]
    }
   ],
   "source": [
    "y = zeros((len(x_space), len(V), len(V[N])))\n",
    "for j in range(len(x_space)):\n",
    "    for n in range(len(V)):\n",
    "        for k in range(len(V[N])):\n",
    "            y[j][n][k] = x_space[j] - rho/sigma * V[n, k]\n",
    "\n",
    "print(\"y_j_n_k consctuction completed\")\n",
    "\n",
    "xi_space = u_space\n",
    "\n",
    "F = zeros((len(y), len(V), len(V[N])), dtype=complex)\n",
    "for j in range(len(y)):\n",
    "    for k in range(len(V[N])):\n",
    "        F[j, N, k] = G(H_original * exp(y[j][N][k] + rho / sigma * V[N, k]), K_original)\n",
    "\n",
    "# when the variance is less than that, is is reasonable to assume it to be zero, which leads to simpler calculations\n",
    "treshold = 1e-6\n",
    "discount_factor = exp(r*delta_t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Можно попробовать визуализировать начальные условия:\n",
    "# for k in range(len(V[N])):\n",
    "#     answers_local = array([F[j][N][k] for j in range(len(F))])\n",
    "#     original_prices_space_local = array([H_original * exp(y[j][N][k] + rho / sigma * V[N, k]) for j in range(len(F))])\n",
    "#     plt.plot(original_prices_space_local[(original_prices_space_local > 0) & (original_prices_space_local<200)], \n",
    "#              answers_local[(original_prices_space_local > 0) & (original_prices_space_local<200)].real)\n",
    "#     plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4of4\n"
     ]
    },
    {
     "data": {
      "image/png": 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Uh95JkqRmJnqNiCRJ2rMZRCRJUjMGEUmS1IxBRJIkNWMQkSRJzRhEJElSMwYR\nSZLUjEFEkiQ1YxCRJEnNGEQkSVIzBhFJktTM/wMYG3zCCtmLZAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3b67b09278>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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Uh95JkqRmJnqNiCRJ2rMZRCRJUjMGEUmS1IxBRJIkNWMQkSRJzRhEJElSMwYR\nSZLUjEFEkiQ1YxCRJEnNGEQkSVIzBhFJktTM/wMYG3zCCtmLZAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3b68107b38>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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Uh95JkqRmJnqNiCRJ2rMZRCRJUjMGEUmS1IxBRJIkNWMQkSRJzRhEJElSMwYR\nSZLUjEFEkiQ1YxCRJEnNGEQkSVIzBhFJktTM/wMYG3zCCtmLZAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3b67db1fd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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Uh95JkqRmJnqNiCRJ2rMZRCRJUjMGEUmS1IxBRJIkNWMQkSRJzRhEJElSMwYR\nSZLUjEFEkiQ1YxCRJEnNGEQkSVIzBhFJktTM/wMYG3zCCtmLZAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3b67a92f28>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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Uh95JkqRmJnqNiCRJ2rMZRCRJUjMGEUmS1IxBRJIkNWMQkSRJzRhEJElSMwYR\nSZLUjEFEkiQ1YxCRJEnNGEQkSVIzBhFJktTM/wMYG3zCCtmLZAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3b67c73ef0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3of4\n"
     ]
    },
    {
     "data": {
      "image/png": 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Uh95JkqRmJnqNiCRJ2rMZRCRJUjMGEUmS1IxBRJIkNWMQkSRJzRhEJElSMwYR\nSZLUjEFEkiQ1YxCRJEnNGEQkSVIzBhFJktTM/wMYG3zCCtmLZAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3b67d1bf28>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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Uh95JkqRmJnqNiCRJ2rMZRCRJUjMGEUmS1IxBRJIkNWMQkSRJzRhEJElSMwYR\nSZLUjEFEkiQ1YxCRJEnNGEQkSVIzBhFJktTM/wMYG3zCCtmLZAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3b68059e80>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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Uh95JkqRmJnqNiCRJ2rMZRCRJUjMGEUmS1IxBRJIkNWMQkSRJzRhEJElSMwYR\nSZLUjEFEkiQ1YxCRJEnNGEQkSVIzBhFJktTM/wMYG3zCCtmLZAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3b6806ee48>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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Uh95JkqRmJnqNiCRJ2rMZRCRJUjMGEUmS1IxBRJIkNWMQkSRJzRhEJElSMwYR\nSZLUjEFEkiQ1YxCRJEnNGEQkSVIzBhFJktTM/wMYG3zCCtmLZAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3b6b99c2b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2of4\n"
     ]
    },
    {
     "data": {
      "image/png": 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Uh95JkqRmJnqNiCRJ2rMZRCRJUjMGEUmS1IxBRJIkNWMQkSRJzRhEJElSMwYR\nSZLUjEFEkiQ1YxCRJEnNGEQkSVIzBhFJktTM/wMYG3zCCtmLZAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3b6803af98>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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Uh95JkqRmJnqNiCRJ2rMZRCRJUjMGEUmS1IxBRJIkNWMQkSRJzRhEJElSMwYR\nSZLUjEFEkiQ1YxCRJEnNGEQkSVIzBhFJktTM/wMYG3zCCtmLZAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3b67d96e80>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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Uh95JkqRmJnqNiCRJ2rMZRCRJUjMGEUmS1IxBRJIkNWMQkSRJzRhEJElSMwYR\nSZLUjEFEkiQ1YxCRJEnNGEQkSVIzBhFJktTM/wMYG3zCCtmLZAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3b6b9dfe48>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1of4\n"
     ]
    },
    {
     "data": {
      "image/png": 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Uh95JkqRmJnqNiCRJ2rMZRCRJUjMGEUmS1IxBRJIkNWMQkSRJzRhEJElSMwYR\nSZLUjEFEkiQ1YxCRJEnNGEQkSVIzBhFJktTM/wMYG3zCCtmLZAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3b67bb3828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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Uh95JkqRmJnqNiCRJ2rMZRCRJUjMGEUmS1IxBRJIkNWMQkSRJzRhEJElSMwYR\nSZLUjEFEkiQ1YxCRJEnNGEQkSVIzBhFJktTM/wMYG3zCCtmLZAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3b67d4e710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0of4\n"
     ]
    },
    {
     "data": {
      "image/png": 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Uh95JkqRmJnqNiCRJ2rMZRCRJUjMGEUmS1IxBRJIkNWMQkSRJzRhEJElSMwYR\nSZLUjEFEkiQ1YxCRJEnNGEQkSVIzBhFJktTM/wMYG3zCCtmLZAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3b6810e6d8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for n in range(len(V[N]) - 2, -1, -1):\n",
    "    print(str(n) + \"of\" + str(len(V[N]) - 2))\n",
    "    for k in range(n+1):\n",
    "        answers_local = array([F[j][n][k] for j in range(len(F))])\n",
    "        original_prices_space = array([H_original * exp(y[j][N][k] + rho / sigma * V[N, k]) for j in range(len(F))])\n",
    "        plt.plot(original_prices_space[(original_prices_space > 0) & (original_prices_space < 400)], \n",
    "             answers_local[(original_prices_space > 0) & (original_prices_space < 400)].real)\n",
    "        plt.show()\n",
    "        # to calculate the binomial expectation one should use Antonino's matrices f_up and f_down\n",
    "        # the meaning of the containing integers are as follows - after (n,k) you will be in\n",
    "        # either (n+1, k + f_up) or (n+1, k - f_down). We use k_u and k_d shorthands, respectively\n",
    "        k_u = k + int(f_up[n][k])\n",
    "        k_d = k + int(f_down[n][k])\n",
    "\n",
    "        # initial condition of a step\n",
    "        f_n_plus_1_k_u = array([F[j][n + 1][k_u] for j in range(len(F))])\n",
    "        f_n_plus_1_k_d = array([F[j][n + 1][k_d] for j in range(len(F))])\n",
    "\n",
    "        H_N_k = - (rho / sigma) * V[n, k]  # modified barrier\n",
    "        local_domain = array([y[j][n][k] for j in range(len(y))])\n",
    "        indicator = np.where((local_domain>=0), 1, 0)\n",
    "\n",
    "        if V[n, k] >= treshold:\n",
    "            # set up variance-dependent parameters for a given step\n",
    "            sigma_local = rho_hat * sqrt(V[n, k])\n",
    "            gamma = r - 0.5 * V[n, k] - rho/sigma * kappa * (theta - V[n, k]) # also local\n",
    "\n",
    "            # beta_plus and beta_minus\n",
    "            beta_minus = - (gamma + sqrt(gamma**2 + 2*sigma_local**2 * q))/sigma_local**2\n",
    "            beta_plus = - (gamma - sqrt(gamma**2 + 2*sigma_local**2 * q))/sigma_local**2\n",
    "\n",
    "            # factor functions\n",
    "            phi_plus_array = array([beta_plus/(beta_plus - 1j*2*pi*xi) for xi in xi_space])\n",
    "            phi_minus_array = array([-beta_minus/(-beta_minus + 1j*2*pi*xi) for xi in xi_space])\n",
    "\n",
    "            # factorization calculation\n",
    "            f_n_k_u = factor * \\\n",
    "                      fft.ifft(phi_minus_array *\n",
    "                               fft.fft(indicator * \n",
    "                                       fft.ifft(phi_plus_array * \n",
    "                                                fft.fft(f_n_plus_1_k_u))))\n",
    "\n",
    "            f_n_k_d = factor * \\\n",
    "                      fft.ifft(phi_minus_array *\n",
    "                               fft.fft(indicator * \n",
    "                                       fft.ifft(phi_plus_array * \n",
    "                                                fft.fft(f_n_plus_1_k_d))))\n",
    "\n",
    "        elif V[n, k] < treshold:\n",
    "            f_n_plus_1_k_u = [F[j][n+1][k_u] for j in range(len(F))]\n",
    "            f_n_k_u = discount_factor * f_n_plus_1_k_u\n",
    "\n",
    "            f_n_plus_1_k_d = [F[j][n + 1][k_d] for j in range(len(F))]\n",
    "            f_n_k_d = discount_factor * f_n_plus_1_k_d\n",
    "\n",
    "        f_n_k = f_n_k_u * pu_f[n, k] + f_n_k_d * pd_f[n, k]\n",
    "\n",
    "#         for j in range(len(f_n_k)):\n",
    "#             # here we try some cutdown magic. The procedure without it returns great bubbles to the right\n",
    "#             # from the strike. And the more L the greater this bubble grows.\n",
    "#             # what we are going to do there is to try to cut off all the values on prices greater than, say,\n",
    "#             # 4 times bigger then the strike\n",
    "#             # we use S>4K and, therefore, y > ln(4K/H) + (pho/sigma)*V inequality to do this\n",
    "#             if y[j][n][k] < log(3.5*K_original/H_original + (rho/sigma) * V[n][k]):\n",
    "#                 F[j][n][k] = f_n_k[j]\n",
    "#             else:\n",
    "#                 F[j][n][k] = complex(0)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<class 'numpy.ndarray'>\n"
     ]
    },
    {
     "data": {
      "image/png": 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BXDTA+CVJ0gLZZ4idJjkMOBaYqKqv9W2vBz6f5M1VtX5M1zcCZ1TVpX2fE4ENwHOAT1bV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\nSc0ZSCRJUnP/H3Ddqg9fEnrOAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fea07e20978>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  8.09111204e-70   8.17242907e-70   8.25456334e-70 ...,   9.72772306e-02\n",
      "   9.82548830e-02   9.92423610e-02]\n"
     ]
    }
   ],
   "source": [
    "answers_list = array([F[j][0][0] for j in range(len(F))])\n",
    "print(type(original_prices_array))\n",
    "# for j in range(len(F)):\n",
    "#    tree_to_csv_file(F[j], \"../output/routine/answers/F\" + str(original_prices_array[j]) + \".csv\")\n",
    "plt.plot(original_prices_array[(original_prices_array > 0) & (original_prices_array < 0.1)], \n",
    "         answers_list[(original_prices_array > 0) & (original_prices_array < 0.1)].real)\n",
    "plt.show()\n",
    "\n",
    "print(original_prices_array[(original_prices_array > 0) & (original_prices_array < 0.1)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/basil/anaconda3/lib/python3.5/site-packages/numpy/core/numeric.py:482: ComplexWarning: Casting complex values to real discards the imaginary part\n",
      "  return array(a, dtype, copy=False, order=order)\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fea0a04a518>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.+0.j  0.+0.j  0.+0.j  0.+0.j  0.+0.j  0.+0.j  0.+0.j  0.+0.j  0.+0.j\n",
      "  0.+0.j  0.+0.j  0.+0.j  0.+0.j  0.+0.j  0.+0.j  0.+0.j  0.+0.j  0.+0.j\n",
      "  0.+0.j  0.+0.j  0.+0.j  0.+0.j  0.+0.j  0.+0.j  0.+0.j  0.+0.j  0.+0.j\n",
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     ]
    }
   ],
   "source": [
    "plt.plot(original_prices_array, answers_list)\n",
    "plt.show()\n",
    "print(answers_list * np.where(answers_list!=complex(0,0), 1, np.nan))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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