{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# **<center>Décision dans l'incertain</center>**\n",
    "\n",
    "## <center> Le calcul du prix d'une option \"américaine\" <center>\n",
    "$\\newcommand\\indi[1]{{\\mathbf 1}_{\\displaystyle #1}}$\n",
    "$\\newcommand\\inde[1]{{\\mathbf 1}_{\\displaystyle\\left\\{ #1 \\right\\}}}$\n",
    "$\\newcommand{\\ind}{\\inde}$\n",
    "$\\newcommand\\E{{\\mathbf E}}$\n",
    "$\\newcommand\\Cov{{\\mathrm Cov}}$\n",
    "$\\newcommand\\Var{{\\mathrm Var}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Le cas européen (calcul d'espérance)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On considère le  modèle de Cox-Ross:\n",
    "$$\n",
    "X_0=x_0, X_{n+1}= X_{n} \\left(d\\inde{U_{n+1}=P}+u\\inde{U_{n+1}=F}\\right).\n",
    "$$\n",
    "On choisit les valeurs numériques de la façon suivante\n",
    "$$\n",
    "   x_0=100, r_0=0,1, \\sigma=0,3.\n",
    "$$\n",
    "et l'on définit $p$, $r$, $u$ et $d$ en fonction de $N$ de la façon suivante~:\n",
    "$$\n",
    "p=1/2,\\;r=r_0/N,\\;u=1+\\frac{\\sigma}{\\sqrt{N}}\\; \\mbox{ et }\\; d=1-\\frac{\\sigma}{\\sqrt{N}}.\n",
    "$$\n",
    "On cherche à évaluer $\\E(f(N,X_N))$ où \n",
    "$$\n",
    " f(N,x)=\\frac{1}{(1+r)^N}\\max(K-x,0),\n",
    "$$\n",
    "avec $K=x_0=100$ et $r=0,05$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Question 1.1.__ Calculer ces prix d'options européennes (call et put) se ramène\n",
    "  à des calculs d'espérance d'une fonction d'une chaîne de Markov. On\n",
    "  implémente ici la méthode de calcul d'espérance par ''programmation\n",
    "  dynamique''."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import math\n",
    "import random\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd\n",
    "\n",
    "\n",
    "def prix_eu(x_0,r,u,d,p,N,gain):\n",
    "# Calcul du prix européen à l'instant 0\n",
    "\n",
    "    U=np.zeros([N+1,N+1])\n",
    "    # U[n,k] = u(n,x^n_k) avec x^n_k = x_0 * u**k * d**(n-k)\n",
    "    \n",
    "    # la condition finale en N\n",
    "    for k in range(0,N+1): # range(0,N+1) = 0,1,2 ... , N\n",
    "        U[N,k] = gain(x_0 * u**k * d**(N-k))/(1+r)**N;\n",
    "    #le temps décroît de N-1 à 0\n",
    "    for n in range(N-1,-1,-1): # range(N-1,-1,-1) = {N-1,N-2,...,0} !\n",
    "        for k in range(n+1):\n",
    "        # écrire l'équation de programmation dynamique\n",
    "        # U[n,k] = u(n,x^n_k) avec x^n_k = x_0 * u**k * d**(n-k)\n",
    "        \n",
    "        ######  A vous de jouer  .....\n",
    "\n",
    "    return U[0,0]\n",
    "\n",
    "def prix_eu_n(n,x_0,r,u,d,p,N,gain):\n",
    "# Calcul du prix a l'instant n\n",
    "\n",
    "    # Pour calculer ce prix on ne change rien si ce n'est N en N-n\n",
    "    return prix_eu(x_0,r,u,d,p,N-n,gain)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " On peut vérifier que lorsque $p=1/2$ (ou $r=0$) et $K=x_0$ le prix\n",
    "  des puts et des calls coïncident.  On le vérifie.\n",
    "  \n",
    "  Pour le choix classique\n",
    "  $p=(1+r-d)/(u-d)$ (et $r\\not=0$), les prix sont différents, ce que\n",
    "  l'on vérifie aussi."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def main_1():\n",
    "    r_0=0.05;sigma=0.3;\n",
    "    N=50;\n",
    "    d=1-sigma/math.sqrt(N);u=1+sigma/math.sqrt(N);\n",
    "    r=r_0/N;\n",
    "\n",
    "    x_0=100;K=100;\n",
    "\n",
    "    # Lorsque p=1/2 et K=x_0 le prix du call = le prix du put (exercice!)\n",
    "    p=1/2;K=x_0;\n",
    "    \n",
    "    def gain_put(x): return max(K-x,0) # Payoff du put  \n",
    "    def gain_call(x): return max(x-K,0) # Payoff du call \n",
    "\n",
    "    p_put  = prix_eu_n(0,x_0,r,u,d,p,N,gain_put);\n",
    "    p_call = prix_eu_n(0,x_0,r,u,d,p,N,gain_call);\n",
    "    if (abs(p_put - p_call) >= 0.000001) :\n",
    "        print(\"WARNING: ces deux prix devrait coincider: \",p_put,\" <> \",p_call)\n",
    "    else:\n",
    "        print(\"Les deux prix coincident: \",p_put,\" <> \",p_call,end='')\n",
    "        print(\". Parfait!\");\n",
    "\n",
    "    p= (1+r-d)/(u-d)\n",
    "    print(\"Prix du call : \",prix_eu_n(0,x_0,r,u,d,p,N,gain_call))\n",
    "    print(\"Prix du put : \",prix_eu_n(0,x_0,r,u,d,p,N,gain_put))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Les deux prix coincident:  11.402142946815882  <>  11.40214294681589. Parfait!\n",
      "Prix du call :  14.285050131985198\n",
      "Prix du put :  9.410369101110671\n"
     ]
    }
   ],
   "source": [
    "main_1()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Le cas américain (arrêt optimal)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On s'intéresse au cas d'un option américaine qui promet (en valeur actualisée en $0$), si on l'exerce à l'instant $n$\n",
    "pour une valeur de $X_n$ valant $x$, une valeur $f(n,x)$\n",
    "$$\n",
    "   f(n,x) = \\frac{1}{(1+r)^n}\\max(K-x,0).\n",
    "$$\n",
    "On cherche à calculer son prix donné par \n",
    "$$\n",
    "u(0,x_0)=\\sup_{\\tau \\mbox{ t.a.} \\leq N} \\E(f(\\tau,X_\\tau)).\n",
    "$$\n",
    "On sait (voir le cours) que $u$ se calcule grace à l'équation de programmation dynamique suivante\n",
    " \\begin{equation}\\label{eq:rec} \n",
    "    \\left\\{\n",
    "      \\begin{array}{l}\n",
    "        u(n,x) = \\displaystyle \\max\\left[p u(n+1,xu)+(1-p) u(n+1,xd),\\frac{1}{(1+r)^n}(K-x)_+\\right], n<N, x\\in E\\\\\n",
    "        u(N,x) = \\frac{1}{(1+r)^N}(K-x)_+,  x\\in E.\n",
    "      \\end{array}\n",
    "    \\right.   \n",
    "  \\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Question 2.1__ On pose $v(n,x)=(1+r)^n u(n,x)$. Vérifier que $v$ est solution de \n",
    " \n",
    " $$\n",
    "    \\left\\{\n",
    "      \\begin{array}{l}\n",
    "        v(n,x) = \\displaystyle \\max\\left[\\frac{p v(n+1,xu)+(1-p) v(n+1,xd)}{1+r},(K-x)_+\\right], n<N, x\\in E\\\\\n",
    "        v(N,x) = (K-x)_+,  x\\in E.\n",
    "      \\end{array}\n",
    "    \\right.   \n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Question 2.2.__ Ecrire un algorithme récursif (et inefficace ...) permettant de calculer $v$ en recopiant l'équation précédente."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "def prix_recursif_am(x,n,r,u,d,p,N,gain):\n",
    "    if n==N:\n",
    "        # en N c'est facile ...\n",
    "        return gain(x)\n",
    "    else:\n",
    "        # on écrit l'équation de programmation dynamique\n",
    "        # on n'oublie ni l'actualisation ni le max avec le gain\n",
    "        \n",
    "        ######  A vous de jouer  .....\n",
    "\n",
    "        return res\n",
    "\n",
    "def prix_slow_am(x,r,u,d,p,N,gain):\n",
    "    return prix_recursif_am(x,0,r,u,d,p,N,gain)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Question 2.3.__ Ecrire un algorithme efficace de calcul de $v(n,x)$ (le prix de\n",
    "  l'option américaine en $n$)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "def prix_am(x_0,r,u,d,p,N,gain):\n",
    "# Calcul du prix americain a l'instant 0\n",
    "    U=np.zeros([N+1,N+1])\n",
    "    # U[n,k] représente dans la suite\n",
    "    # la valeur de v à l'instant n\n",
    "    # au point x=x_0 * u**k * d**(n-k) k=0,...,n\n",
    "    \n",
    "    # on initialise U[N,k] à gain(x) avec x=x_0 * u**k * d**(N-k) \n",
    "    for k in range(N+1):\n",
    "        x=x_0 * u**k * d**(N-k) # le point de calcul\n",
    "        U[N,k]=gain(x);# Valeur de U en N\n",
    "\n",
    "    for n in range(N-1,-1,-1): # le temps decroit de N-1 a 0\n",
    "        for k in range(n+1): # k varie de 0 a n\n",
    "            x = x_0 * u**k * d**(n-k) # le point de calcul\n",
    "            # on écrit l'équation de programmation dynamique\n",
    "            # en version itérative cette fois\n",
    "            # U[n,k] = v(n,x^n_k) avec x^n_k = x_0 * u**k * d**(n-k)\n",
    "       \n",
    "            ######  A vous de jouer  .....\n",
    "\n",
    "    return U[0,0];\n",
    "\n",
    "def prix_am_n(n,x_0,r,u,d,p,N,gain): \n",
    "    # Calcul du prix a l'instant n\n",
    "\n",
    "    # Pour calculer ce prix on ne change rien si ce n'est N en N-n\n",
    "    return prix_am(x_0,r,u,d,p,N-n,gain)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " __Question 2.4.__ Pour $N=10$, comparer les $2$ méthodes pour vérifier que tout\n",
    "  fonctionne. Effectuer des calculs de prix avec des $N$ plus grands\n",
    "  (uniquement pour la deuxième méthode, bien sûr)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Ces deux prix devrait coincider (ou presque) :  8.461120344261216  <>  8.461120344261218. C'est parfait!\n"
     ]
    }
   ],
   "source": [
    "def main_2():\n",
    "    sigma=0.3; r_0=0.1;\n",
    "    K=100;x_0=100;\n",
    "\n",
    "    N=10;\n",
    "    r=r_0/N;\n",
    "    d=1-sigma/math.sqrt(N);\n",
    "    u=1+sigma/math.sqrt(N);\n",
    "    p= (1+r-d)/(u-d);#p=1/2;\n",
    "    \n",
    "    def gain_put(x): return max(K-x,0) # Payoff du put  \n",
    "    def gain_call(x): return max(x-K,0) # Payoff du call \n",
    "\n",
    "    prix_am(x_0,r,u,d,p,N,gain_put)\n",
    "    prix_slow_am(x_0,r,u,d,p,N,gain_put)\n",
    "\n",
    "    # Les deux algos font ils le même chose ?\n",
    "    # on verifie : prix_slow(x_0,N) \\approx prix(x_0,N)\n",
    "    p1=prix_slow_am(x_0,r,u,d,p,N,gain_put);\n",
    "    p2=prix_am(x_0,r,u,d,p,N,gain_put);\n",
    "    print(\"Ces deux prix devrait coincider (ou presque) : \",p1,\" <> \",p2,end='');\n",
    "    if (abs(p1 - p2) >= 0.00001):\n",
    "         print(\"WARNING: ces deux prix devrait coincider : \",p1,\" <> \",p2);\n",
    "    else:\n",
    "         print(\". C'est parfait!\")\n",
    "\n",
    "    N=1000;d=1-sigma/math.sqrt(N);u=1+sigma/math.sqrt(N);\n",
    "    prix_am(x_0,r,u,d,p,N,gain_put)\n",
    "\n",
    "main_2()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " __Question 2.5.__ Tracer les courbes de prix américaines et européennes  $x\\to v(0,x)$ pour $x\\in [80,120]$ et les\n",
    "  supperposer au \"payoff\".\n",
    "  \n",
    "  On constate que le prix est toujours plus grand que le prix européen et que le gain immédiat."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt;\n",
    "\n",
    "def main_3():\n",
    "    # tracé de courbes\n",
    "    N=50\n",
    "    sigma=0.3\n",
    "    p=1/2;d=1-sigma/math.sqrt(N);u=1+sigma/math.sqrt(N);\n",
    "    K=100;x_0=100;\n",
    "    r_0=0.1;\n",
    "    r=r_0/N;\n",
    "\n",
    "    def gain_put(x): return max(K-x,0) # Payoff du put  \n",
    "    def gain_call(x): return max(x-K,0) # Payoff du call \n",
    "  \n",
    "    vmin=50;\n",
    "    vmax=150;   \n",
    "    courbe_am=np.zeros(vmax-vmin+1)\n",
    "    courbe_eu=np.zeros(vmax-vmin+1)\n",
    "    obstacle=np.zeros(vmax-vmin+1)\n",
    "\n",
    "    n=0;\n",
    "    x=range(vmin,vmax+1,1)\n",
    "    for current_x in x:\n",
    "        #courbe_am[n]= le prix américain en current_x\n",
    "        #courbe_eu[n]= le prix européen en current_x\n",
    "        #obstacle[n] = l'obstacle en current_x\n",
    "        \n",
    "        ######  A vous de jouer  .....\n",
    "\n",
    "        n=n+1; \n",
    "        \n",
    "    # On compare les courbes \"Américaines\" et \"Européennes\"\n",
    "    plt.plot(x,courbe_eu,label='Prix \"européen\"')\n",
    "    plt.plot(x,obstacle,label='Obstacle')\n",
    "    plt.plot(x,courbe_am,color='red',label='Prix \"américain\"')\n",
    "    plt.legend(loc='upper right')\n",
    "\n",
    "main_3()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Question 2.6.__ Regardez comment le prix évolue au fils du temps~: tracez les\n",
    "  courbes $x\\to v(n,x)$, pour $n=0,2,5,20,50$. Ce prix est égal à $(K-x)_+$ en zéro, il croit avec $n$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def main_4():\n",
    "    sigma=0.3;\n",
    "    K=100;x_0=100;\n",
    "\n",
    "    N=50;\n",
    "    p=1/2;d=1-sigma/math.sqrt(N);u=1+sigma/math.sqrt(N);\n",
    "    r_0=0.1;r=r_0/N;\n",
    "\n",
    "    def gain_put(x): return max(K-x,0) # Payoff du put  \n",
    "    def gain_call(x): return max(x-K,0) # Payoff du call \n",
    "\n",
    "    vmin=50;vmax=150;\n",
    "    # évolution de la courbe en fonction de n\n",
    "    liste=[0,2,5,20,50]\n",
    "    courbe_am=np.zeros([np.size(liste),vmax-vmin+1])\n",
    "    index_n=0\n",
    "    x=range(vmin,vmax+1,1)\n",
    "    for N in liste:\n",
    "        i=0;\n",
    "        # construction de la courbe de prix si l'echéance est N\n",
    "        for current_x in x:\n",
    "            \n",
    "            # courbe_am[index_n,i] = prix du put en current_x si l'echéance est N\n",
    "        \n",
    "            ######  A vous de jouer  .....\n",
    "\n",
    "            i=i+1\n",
    "        index_n=index_n+1\n",
    "    \n",
    "    for n in range(index_n):\n",
    "        text='n = '+str(liste[n])\n",
    "        plt.plot(x,courbe_am[n,:],label=text)\n",
    "    plt.legend(loc='upper right')\n",
    "        \n",
    "main_4()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Question 2.7__ Que constatez vous lorsque $N$ augmente ($N=10,100,200,500$) et\n",
    "  que l'on choisit $r$, $u$ et $d$ en fonction de $N$ comme définis\n",
    "  plus haut ($r=r_0/N$, $d=1-\\sigma/\\sqrt{N}$, $u=1+\\sigma/\\sqrt{N}$).\n",
    "  \n",
    "__Commentaire:__ lorsque $N$ tend vers $+\\infty$ dans ces\n",
    "  conditions, les prix convergent vers le prix d'un modèle continu (le\n",
    "  célèbre modèle de Black et Scholes).  Dans le cas européen, le\n",
    "  résultat se prouve grâce au théorème de la limite centrale (cours \n",
    "  du premier semestre). Dans le cas américain qui nous intéresse, c'est \n",
    "  plus compliqué, mais ca reste vrai !"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def main_5():\n",
    "    # Avec cet algorithme on peut augmenter N\n",
    "    # mais il faut renormaliser convenablement u et d pour obtenir une convergence.\n",
    "    # Essayer avec N=10,100,200,...,1000\n",
    "\n",
    "    sigma=0.3\n",
    "    K=100\n",
    "    x_0=100\n",
    "    r_0=0.1\n",
    "    p=1/2\n",
    "    N=50\n",
    "\n",
    "    def gain_put(x): return max(K-x,0) # Payoff du put  \n",
    "\n",
    "    vmin=50;vmax=150;\n",
    "    courbe=np.zeros(vmax-vmin+1)\n",
    "    x=range(vmin,vmax+1,1)\n",
    "    for N in [2,5,20,100]:\n",
    "        # Une renormalisation convenable\n",
    "        # pour converger vers un modèle de Black et Scholes\n",
    "        r=r_0/N;\n",
    "        d=1-sigma/math.sqrt(N);\n",
    "        u=1+sigma/math.sqrt(N);\n",
    "        n=0;\n",
    "        for current_x in x:\n",
    "            text='N = '+str(N)\n",
    " \n",
    "            # courbe[n]= prix américain avec les paramètres r,d,u\n",
    "\n",
    "            ######  A vous de jouer  .....\n",
    "\n",
    "            n=n+1; \n",
    "        plt.plot(x,courbe,label=text);\n",
    "    plt.legend(loc='upper right')\n",
    "\n",
    "main_5()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simulation selon la loi du temps optimal"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On va chercher à se faire une idée du temps d'exercice optimal en simulant sa loi. On verra que cette loi a des formes variées en fonction de la valeur initiale."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Question 2.8__ Ecrire une fonction qui simule le vecteur\n",
    "  $(Bin(0),Bin(1),\\ldots,Bin(N))$ du nombre de piles cumulé successif dans des $N$\n",
    "  tirages à pile ou face $(p,1-p)$. Par convention $Bin(0)=0$.\n",
    "\n",
    "  Le vecteur $Bin$ permet d'exprimer $X(n)$, la valeur en $n$ du modèle de Cox-Ross, sous la forme:\n",
    "  $$\n",
    "   X(n)=x_0 u^{Bin(n)} d^{n-Bin(n)}.\n",
    "  $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1 1 2 3 4]\n"
     ]
    }
   ],
   "source": [
    "def simul_bin(N,p):\n",
    "# sommes partielles du nombre de tirages \"up\" (=u) \n",
    "# dans N tirages à pile ou face (p,1-p)\n",
    "\n",
    "    unif=np.random.rand(N+1) # N+1 tirages uniformes dans [0,1], on n'utilise pas le tirage \"0\"\n",
    "    Bin=np.zeros(N+1,dtype=int) \n",
    "    # Bin(0)=0, on commence donc a n=1 \n",
    "    for n in range(1,N+1):# range(1,N+1) = 1, ..., N\n",
    "        # tirages a pile ou face (p,1-p)\n",
    " \n",
    "        # On cumule des tirages de Bernouilli successifs\n",
    "\n",
    "        ######  A vous de jouer  .....\n",
    "\n",
    "    return Bin;\n",
    "\n",
    "# simul_bin(5,1/2) est un vecteur de dimension 6 : Bin(0),...,Bin(5). Bin(0) vaut toujours 0.\n",
    "print(simul_bin(5,1/2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Question 2.9__ Calculer la prix américain en conservant dans un vecteur $V(n,k)$ les valeurs en l'instant $n$ et au point $X[k]=x_0 u^{k} d^{n-k}$. $k$ varie de $0$ à $n$. Il suffit de modifier legérement l'algorithme de la __question 2.3__ en stockant les valeurs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [],
   "source": [
    "def prix_am_vect(x_0,r,u,d,p,N,gain):\n",
    "# On calcule les valeurs de \"v(n,x)\" (voir question précédente)\n",
    "# mais, ici, on les conserve dans un vecteur \"V\"\n",
    "# ce qui évitera d'avoir à les re-calculer un grand \n",
    "# nombre de fois lors des simulations.\n",
    "    V=np.zeros([N+1,N+1])\n",
    "    for k in range(N+1):\n",
    "        x=x_0 * u**k * d**(N-k) # le point de calcul\n",
    "        V[N,k]=gain(x) # Valeur de U en N\n",
    "\n",
    "    for n in range(N-1,-1,-1): # le temps decroit de N-1 a 0\n",
    "        for k in range(n+1): # k varie de 0 a n\n",
    "            x = x_0 * u**k * d**(n-k) # le point de calcul\n",
    " \n",
    "            # V[n,k]= ...\n",
    "\n",
    "            ######  A vous de jouer  .....\n",
    "\n",
    "    return V"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Question 2.10__ A partir d'une trajectoire du modèle de Cox-Ross donnée par le vecteur\n",
    "  $Bin$ et des valeurs de $V$ précédemment calculées, évaluer le\n",
    "  temps d'arrêt optimal associé à cette trajectoire. \n",
    "  \n",
    "  Pour des raisons techniques, on est obligé de rajouter \"& (gain_immediat > 0)\" à la condition \n",
    "  d'exercice classique (le gain immediat = la valeur courrante de $v$). Cette condition est naturelle, \n",
    "  puisque si le gain immédiat est nul, on n'a aucun intérêt à exercer en cet instant, bien sûr."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [],
   "source": [
    "def temps_optimal_option(x_0,u,d,Bin,V,gain):\n",
    "    # Bin est la somme cumulée de tirages de Bernouilli indépendants\n",
    "    # Calcule le temps d'arrêt optimal associé à la trajectoire \n",
    "    # (X(0),X(1),...,X(N)) où  X(n) = x_0 * u**Bin(n) * d**(n-Bin(n))\n",
    "\n",
    "    for n in range(np.size(Bin)):\n",
    "        x_n=x_0 * u**Bin[n] * d**(n-Bin[n]) # Valeur de X(n) pour ce tirage\n",
    "        Valeur = V[n,Bin[n]] # Calcul de V(n,X(n)) que l'on a pre-calculé\n",
    "        gain_immediat = gain(x_n) # le gain si on exerce en n\n",
    "         \n",
    "        # condition d'exercice\n",
    "        # if (...) & (gain_immediat > 0) : return n\n",
    "        \n",
    "        ######  A vous de jouer  .....\n",
    "\n",
    "    return N"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "  Voici un programme qui permet de tracer des histogrammes. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [],
   "source": [
    "def histo_discret(samples, maxsize,titre):\n",
    "# histogramme de tirages selon une loi discrète à valeurs entières\n",
    "# supposé prendre des valeurs entre 0 et maxsize.\n",
    "    size, scale = 20000, 100\n",
    "    my_histo = pd.Series(samples)\n",
    "    my_histo.plot.hist(grid=True, bins=maxsize+1, rwidth=2, color='#607c8e',range=[0,maxsize])\n",
    "    plt.title(' ')\n",
    "    plt.xlabel(titre)\n",
    "    plt.ylabel(' ')\n",
    "    plt.grid(axis='y', alpha=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Simuler un grand nombre de trajectoires du modèle de Cox-Ross et\n",
    "  les valeurs des temps d'arrêt associés à ces trajectoires.  Tracer\n",
    "  un histogramme de la loi du temps d'arrêt optimal.  Faire varier les\n",
    "  valeurs de $x_0$ ($x_0=60,70,80,100,120)$) et voir l'influence de\n",
    "  ce choix sur le loi du temps d'arrêt optimal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [],
   "source": [
    "def un_test(x_0,N):\n",
    "    # paramètres du modèle et de l'option\n",
    "    sigma=0.3\n",
    "    p=1/2;d=1-sigma/math.sqrt(N);u=1+sigma/math.sqrt(N)\n",
    "    K=100\n",
    "    r_0=0.1\n",
    "    r=r_0/N\n",
    "    # payoff (ou gain) de l'option put\n",
    "    def gain_put(x): return max(K-x,0) \n",
    "\n",
    "    # calcul de la fonction v\n",
    "    V=prix_am_vect(x_0,r,u,d,p,N,gain_put)\n",
    "\n",
    "    # \"Nbre\" tirages selon la loi du temps d'arrêt optimal\n",
    "    Nbre=1000;\n",
    "    tau=np.zeros(Nbre,dtype=int)\n",
    "    for j in range(Nbre):\n",
    "         \n",
    "        # commencez par simuler des Bernouillis cumulées (voir plus haut),\n",
    "        # puis en déduire un tirage selon la loi du temps d'arrêt optimal\n",
    "        \n",
    "        ######  A vous de jouer  .....\n",
    "\n",
    "    histo_discret(tau,N,'Temps optimal')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Expérimenter pour diverses valeurs de $x_0$. On vous suggère $x_0=K=100$, puis $x_0=60, 70, 80, 100, 120$. Vous constaterez que la loi de $\\tau$ varie considérablement."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# un test simple \"at the money\" |$(K=x_0)$|\n",
    "N=50\n",
    "x_0=100;tau=un_test(x_0,N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "    N=50\n",
    "    x_0=60;tau=un_test(x_0,N) # lorsque x_0 << K, on exerce toujours en 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "    N=50\n",
    "    x_0=70;tau=un_test(x_0,N) # on augmente x_0, on n'exerce plus (jamais) en 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYgAAAEWCAYAAAB8LwAVAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDIuMi4zLCBodHRwOi8vbWF0cGxvdGxpYi5vcmcvIxREBQAAFflJREFUeJzt3X+w5XV93/HnSyBauCDqyi2CdtUhDkp1kVtiNdW7rLVonRgzasTUrKhdmcGOJrb1R6USrR1blc10kphZI5UkymJDUEJtlRCvSCNJds0qS5AE6MYuy+xGQOEqGoF3/zjfq2cvn/11d885957zfMzcuef7Od8f78+es+d1P9/zPZ+TqkKSpMUeNeoCJEnLkwEhSWoyICRJTQaEJKnJgJAkNRkQkqQmA0KS1GRASJKaDAhJUpMBIUlqMiAkSU0GhCSpyYCQJDUZEJKkJgNCktRkQEiSmgwISVKTASFJajIgJElNBoQkqcmAkCQ1HT3qAg7HqlWravXq1Uve/nvf+x7HHXfckStomZu0/oJ9nhT2+dBs3br121X1xAOtt6IDYvXq1WzZsmXJ28/NzTE7O3vkClrmJq2/YJ8nhX0+NEn+9mDW8xSTJKnJgJAkNRkQkqQmA0KS1GRASJKaDAhJUpMBIUlqMiAkSU0GhCSpaUV/klqSxt35F21stq9fd+bAj+0IQpLUZEBIkpoMCElSkwEhSWoyICRJTQaEJKlpYAGR5NIke5Js72u7Ism27mdHkm1d++okD/Td99uDqkuSdHAG+TmITwK/AfzuQkNV/eLC7SQfBb7bt/7tVbVmgPVIkg7BwAKiqq5Psrp1X5IArwHOGdTxJUmHJ1U1uJ33AuKaqjpjUfsLgUuqaqZvvZuBvwbuA95bVV/Zxz43ABsApqenz9q8efOS65ufn2dqamrJ2680k9ZfsM+TYpz7vGPXnmb7qhOOXXKf165du3Xh9Xd/RjXVxnnA5X3LdwFPqaq7k5wFfDbJs6rqvsUbVtUmYBPAzMxMHc4XlU/aF51PWn/BPk+Kce7z/qbaGHSfh34VU5KjgV8Arlhoq6ofVtXd3e2twO3ATw+7NknST4ziMtcXA9+sqp0LDUmemOSo7vbTgNOAO0ZQmySpM8jLXC8Hvgo8I8nOJG/q7note59eAngh8I0kXwf+ALigqu4ZVG2SpAMb5FVM5+2j/Q2NtiuBKwdViyTp0PlJaklSkwEhSWoyICRJTQaEJKnJgJAkNRkQkqQmA0KS1GRASJKaDAhJUpMBIUlqMiAkSU0GhCSpyYCQJDUZEJKkJgNCktRkQEiSmgwISVKTASFJajIgJElNAwuIJJcm2ZNke1/bxUnuTLKt+3lZ333vTnJbkluT/ItB1SVJOjiDHEF8Eji30b6xqtZ0P58HSPJM4LXAs7ptfivJUQOsTZJ0AAMLiKq6HrjnIFd/BbC5qn5YVf8XuA04e1C1SZIO7OgRHPOtSX4Z2AK8o6ruBU4BbuxbZ2fX9ghJNgAbAKanp5mbm1tyIfPz84e1/Uozaf0F+zwpxrnPLzq9+VI4lD4POyA+BnwAqO73R4E3AmmsW60dVNUmYBPAzMxMzc7OLrmYubk5Dmf7lWbS+gv2eVKMc5/Pv2hjs339ujMH3uehXsVUVbur6qGqehj4OD85jbQTeHLfqqcCu4ZZmyRpb0MNiCQn9y2+Eli4wulq4LVJHp3kqcBpwJ8PszZJ0t4GdoopyeXALLAqyU7gfcBskjX0Th/tAN4CUFU3J/kM8FfAg8CFVfXQoGqTJB3YwAKiqs5rNH9iP+t/EPjgoOqRJB0aP0ktSWoyICRJTQaEJKnJgJAkNRkQkqQmA0KS1GRASJKaDAhJUpMBIUlqMiAkSU0GhCSpyYCQJDUZEJKkJgNCktRkQEiSmgwISVKTASFJajIgJElNBoQkqWlgAZHk0iR7kmzva/twkm8m+UaSq5Kc2LWvTvJAkm3dz28Pqi5J0sEZ5Ajik8C5i9quBc6oqmcDfw28u+++26tqTfdzwQDrkiQdhIEFRFVdD9yzqO2LVfVgt3gjcOqgji9JOjypqsHtPFkNXFNVZzTu+yPgiqr6/W69m+mNKu4D3ltVX9nHPjcAGwCmp6fP2rx585Lrm5+fZ2pqasnbrzST1l+wz5NinPu8Y9eeZvuqE45dcp/Xrl27tapmDrTe0Uva+2FK8h+AB4FPdU13AU+pqruTnAV8Nsmzquq+xdtW1SZgE8DMzEzNzs4uuY65uTkOZ/uVZtL6C/Z5Uoxzn8+/aGOzff26Mwfe56FfxZRkPfBy4JeqG75U1Q+r6u7u9lbgduCnh12bJOknhhoQSc4F3gn8XFV9v6/9iUmO6m4/DTgNuGOYtUmS9jawU0xJLgdmgVVJdgLvo3fV0qOBa5MA3NhdsfRC4P1JHgQeAi6oqnuaO5YkDcXAAqKqzms0f2If614JXDmoWiRJh85PUkuSmgwISVKTASFJajIgJElNBoQkqcmAkCQ1GRCSpCYDQpLUZEBIkpoMCElSkwEhSWoyICRJTQaEJKnJgJAkNRkQkqQmA0KS1GRASJKaDAhJUpMBIUlqGmhAJLk0yZ4k2/vaHp/k2iR/0/1+XNeeJP8tyW1JvpHkuYOsTZK0f4MeQXwSOHdR27uA66rqNOC6bhngpcBp3c8G4GMDrk2StB8DDYiquh64Z1HzK4DLutuXAT/f1/671XMjcGKSkwdZnyRp31JVgz1Ashq4pqrO6Ja/U1Un9t1/b1U9Lsk1wIeq6oau/TrgnVW1ZdH+NtAbYTA9PX3W5s2bl1zb/Pw8U1NTS95+pZm0/oJ9nhTj3Ocdu/Y021edcOyS+7x27dqtVTVzoPWOXtLeByONtkekV1VtAjYBzMzM1Ozs7JIPODc3x+Fsv9JMWn/BPk+Kce7z+RdtbLavX3fmwPs8iquYdi+cOup+L8TjTuDJfeudCuwacm2SpM4oAuJqYH13ez3wub72X+6uZnoe8N2qumsE9UmSGPAppiSXA7PAqiQ7gfcBHwI+k+RNwLeAV3erfx54GXAb8H3g/EHWJknav4EGRFWdt4+71jXWLeDCQdYjSTp4fpJaktRkQEiSmgwISVKTASFJajIgJElNBoQkqcmAkCQ1GRCSpCYDQpLUZEBIkpoMCElSkwEhSWoyICRJTQaEJKnJgJAkNRkQkqQmA0KS1GRASJKaDAhJUtNAv5O6JckzgCv6mp4G/EfgROBfA3/Xtb+nqj4/5PIkSZ2hB0RV3QqsAUhyFHAncBVwPrCxqj4y7JokSY806lNM64Dbq+pvR1yHJGmRVNXoDp5cCnytqn4jycXAG4D7gC3AO6rq3sY2G4ANANPT02dt3rx5ycefn59nampqyduvNJPWX7DPk2Kc+7xj155m+6oTjl1yn9euXbu1qmYOtN7IAiLJTwG7gGdV1e4k08C3gQI+AJxcVW/c3z5mZmZqy5YtS65hbm6O2dnZJW+/0kxaf8E+T4px7vP5F21stq9fd+aS+5zkoAJilKeYXkpv9LAboKp2V9VDVfUw8HHg7BHWJkkTb5QBcR5w+cJCkpP77nslsH3oFUmSfmzoVzEBJDkW+OfAW/qa/2uSNfROMe1YdJ8kachGEhBV9X3gCYvaXj+KWiRJbaO+zFWStEwZEJKkJgNCktRkQEiSmgwISVKTASFJajIgJElNBoQkqWkkH5TT8J1/0UZedPopj5j4679/4FdGVJGk5c4RhCSpyYCQJDUZEJKkJgNCktRkQEiSmgwISVKTASFJajIgJElNBoQkqcmAkCQ1jWyqjSQ7gPuBh4AHq2omyeOBK4DVwA7gNVV176hqlKRJNuoRxNqqWlNVM93yu4Drquo04LpuWZI0Asttsr5XALPd7cuAOeCdoypG42fxZIULnLRQeqRRjiAK+GKSrUk2dG3TVXUXQPf7pJFVJ0kTLlU1mgMnT6qqXUlOAq4F/g1wdVWd2LfOvVX1uEXbbQA2AExPT5+1efPmJdcwPz/P1NTUkrdfSXbs2sPxjzmG+3/wo73aVz9pvDN48WO8Y9ee5nrj9O8wSc/rBePc5309Z1edcOyS+7x27dqtfaf292lkp5iqalf3e0+Sq4Czgd1JTq6qu5KcDDziX6aqNgGbAGZmZmp2dnbJNczNzXE4268kC98H8eVb7tyr/Q2ve82IKhqOxY/xvk4xjdO/wyQ9rxeMc5/39Zxdv+7Mgfd5JKeYkhyX5PiF28BLgO3A1cD6brX1wOdGUZ8kaXQjiGngqiQLNXy6qv53kr8APpPkTcC3gFePqD5JmngjCYiqugN4TqP9bmDd8CuSJC026s9BSJKWKQNCktRkQEiSmgwISVLTcptqQ8uEU1JIMiDGzL5e2CUN1zj8keUpJklSkyMIHRHj8NeSpL05gpAkNTmCkMZA/wjuRaef8uNlR3A6HI4gJElNBoQkqcmAkCQ1GRCSpCbfpJb2w8t3x5eP7YEZEFoR/M8sDZ8BoZHwBV/a23KcJseAWOZ8IZXGy3IMgn3xTWpJUpMjiCFzRCBppRj6CCLJk5N8KcktSW5O8rau/eIkdybZ1v28bNi1SZJ+YhQjiAeBd1TV15IcD2xNcm1338aq+sgIapIkLTL0gKiqu4C7utv3J7kFOGXYdUiS9i9VNbqDJ6uB64EzgF8F3gDcB2yhN8q4t7HNBmADwPT09FmbN29e8vHn5+eZmppa8vZLsWPXnmb76iedNND1AY5/zDHc/4MfDfW4o1p/weLHeFjHHbb+Ovsf5+VW56As5f/yoB/b/f1fPBJWnXDskl+/1q5du7WqZg603sjepE4yBVwJvL2q7kvyMeADQHW/Pwq8cfF2VbUJ2AQwMzNTs7OzS65hbm6Ow9l+Kfb1JvUbXveaga4PvWmgv3zLnUM97qjWX7D4MR7WcYdt8XTfC4/zcqtzUJbyf3nQj+2gL2ddv+7Mgb9+jSQgkhxDLxw+VVV/CFBVu/vu/zhwzShqkw7HoV6ltpKvapukvk6qoQdEkgCfAG6pqkv62k/u3p8AeCWwfdi1aXwsvBj1f3mOxkPr8XzR6b6NOQijGEG8AHg9cFOSbV3be4Dzkqyhd4ppB/CWEdQmaZlYKcG+UupcilFcxXQDkMZdnx92LZKkfXOqDUlSkwEhSWpyLiZpArXOm3s1kRZzBCFJajIgJElNnmKSVpBRXFK50i/jXOn1j5IjCElSkyMISToIkzgScQQhSWoyICRJTQaEJKnJgJAkNRkQkqQmA0KS1GRASJKaDAhJUpMBIUlqMiAkSU0GhCSpadkFRJJzk9ya5LYk7xp1PZI0qZZVQCQ5CvhN4KXAM4HzkjxztFVJ0mRaVgEBnA3cVlV3VNXfA5uBV4y4JkmaSKmqUdfwY0leBZxbVW/ull8P/ExVvbVvnQ3Ahm7xGcCth3HIVcC3D2P7lWbS+gv2eVLY50Pzj6rqiQdaabl9H0QabXslWFVtAjYdkYMlW6pq5kjsayWYtP6CfZ4U9nkwltsppp3Ak/uWTwV2jagWSZpoyy0g/gI4LclTk/wU8Frg6hHXJEkTaVmdYqqqB5O8FfgCcBRwaVXdPMBDHpFTVSvIpPUX7POksM8DsKzepJYkLR/L7RSTJGmZMCAkSU0TGRCTMJ1HkkuT7Emyva/t8UmuTfI33e/HjbLGIy3Jk5N8KcktSW5O8raufWz7neQxSf48yde7Pv9a1/7UJH/W9fmK7qKPsZHkqCR/meSabnnc+7sjyU1JtiXZ0rUN/Hk9cQExQdN5fBI4d1Hbu4Drquo04LpueZw8CLyjqk4Hngdc2D2249zvHwLnVNVzgDXAuUmeB/wXYGPX53uBN42wxkF4G3BL3/K49xdgbVWt6fvsw8Cf1xMXEEzIdB5VdT1wz6LmVwCXdbcvA35+qEUNWFXdVVVf627fT+8F5BTGuN/VM98tHtP9FHAO8Add+1j1OcmpwL8EfqdbDmPc3/0Y+PN6EgPiFOD/9S3v7NomwXRV3QW9F1PgpBHXMzBJVgNnAn/GmPe7O92yDdgDXAvcDnynqh7sVhm35/ivA/8eeLhbfgLj3V/ohf4Xk2ztphuCITyvl9XnIIbkgNN5aGVLMgVcCby9qu7r/YE5vqrqIWBNkhOBq4DTW6sNt6rBSPJyYE9VbU0yu9DcWHUs+tvnBVW1K8lJwLVJvjmMg07iCGKSp/PYneRkgO73nhHXc8QlOYZeOHyqqv6wax77fgNU1XeAOXrvv5yYZOEPwHF6jr8A+LkkO+idHj6H3ohiXPsLQFXt6n7vofdHwNkM4Xk9iQExydN5XA2s726vBz43wlqOuO5c9CeAW6rqkr67xrbfSZ7YjRxI8g+AF9N77+VLwKu61camz1X17qo6tapW0/u/+ydV9UuMaX8BkhyX5PiF28BLgO0M4Xk9kZ+kTvIyen91LEzn8cERl3TEJbkcmKU3JfBu4H3AZ4HPAE8BvgW8uqoWv5G9YiX5WeArwE385Pz0e+i9DzGW/U7ybHpvUB5F7w++z1TV+5M8jd5f2I8H/hL4V1X1w9FVeuR1p5j+bVW9fJz72/Xtqm7xaODTVfXBJE9gwM/riQwISdKBTeIpJknSQTAgJElNBoQkqcmAkCQ1GRCSpKZJ/CS1xlB3yd913eI/BB4C/q5bPrubd2tZS3IO8P2qurFbvpDeFBKfOgL73gmc0X2YTjooBoTGQlXdTW82U5JcDMxX1UdGWtShOwf4NnAjQFX95mjL0aTzFJPGXpL13XcmbEvyW0keleToJN9J8uEkX0vyhSQ/k+TLSe7oPkxJkjcnuaq7/9Yk7+3aj0/yv7rvYdie5FWN4z63+46CbyS5Mslju/Ybkvx6kq92c/zPJHk68Gbg33V1Pj/Jf0ry9r5tLknylSR/1W1zVfddABf3HfOPugndbk7y5iH882qMGRAaa0nOAF4JPL+q1tAbNb+2u/uxwBer6rnA3wMXA+uAVwPv79vN2d02zwVel2QN8DJgR1U9p6rOoDeL6mK/T+/7KZ4N3Apc1Hffo6vqn9L7XoPfqarb6U1f/eFuzv8/bezvgar6Z/SmE/kscAHwj4ENC9NtAOur6izgnwC/OogvkdHkMCA07l5M78VySzcl9ouAp3f3PVBVCy/sNwFz3ZTRNwGr+/bxhaq6t6q+R++F+WeBb9D7cp4PJXlBVX23/6DdeyKPqaobuqbLgBf2rXI5QFX9CXBSNwPtgSzMGXYTcFNV7a6qHwA76E1QB/ArSb4OfLVre/oj9iIdJN+D0LgLvfm2LtqrsTfzZ/8b1w/T+3a2hdv9/zcWz0dTVXVLkhl6I4kPJ7mmqv7zouPuzyP2eYD1WVRf/zxDDwNHJ3kxvRB6XlU9kOQG4DEHsV+pyRGExt0fA69Jsgp6f9knecoh7uMlSU5Mciy9b/H6P0lOofdG+O8Bl9A7/fRjVfVt4IEkz++aXg98uW+VX+zqmQV2d6OT+4HjD7G2fo8F7unC4Vn0Rk7SkjmC0FirqpuS/Brwx0keBfyI3rn7Q/m+gBuAT9M7XfN7VbWtexP7Q0kepjcSuaCx3euBj3XTcN8GnN93331J/pReICy0fw74H0l+AbjwEOpb8D/pvR/xdeCb9GaxlZbM2Vyl/eiuBDqjqt5+BPd5A/DWqtp2pPYpDYKnmCRJTY4gJElNjiAkSU0GhCSpyYCQJDUZEJKkJgNCktT0/wEBMVf2z8O+AwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "N=50\n",
    "x_0=80;tau=un_test(x_0,N) # On exerce de plus en plus tard ..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "N=50\n",
    "x_0=100;tau=un_test(x_0,N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "N=50\n",
    "x_0=120;tau=un_test(x_0,N) # Le plus souvent en N"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " __Question 2.11__ Le cas du call (l'option dont le gain instantané vaut $(x-K)_+$) est très particulier. On peut montrer que dans ce cas, l'option ne s'exerce qu'à échéance. On le vérifie, ici, par simulation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Teste le cas du call (x-K)_+\n",
    "# qui ne s'exerce jamais avant l'écheance N.\n",
    "# Le fait que le prix eu du call soit toujours plus grand \n",
    "# que l'obstacle permet de le prouver rigoureusement (pourquoi?).\n",
    "  \n",
    "def main_8():\n",
    "    sigma=0.3; r_0=0.1\n",
    "    K=100;x_0=100\n",
    "    N=50\n",
    "    r=r_0/N\n",
    "    u=(1+r)*math.exp(sigma/math.sqrt(N));d=(1+r)*math.exp(-sigma/math.sqrt(N))\n",
    "    p=1/2;  #p= (1+r-d)/(u-d); # en principe pour Cox-Ross\n",
    "    \n",
    "    def gain_call(x): return max(x-K,0) # Payoff du call \n",
    " \n",
    "    V_call=prix_am_vect(x_0,r,u,d,p,N,gain_call)\n",
    "    Nbre=1000\n",
    "    tau=np.zeros(Nbre)\n",
    "    for j in range(Nbre):\n",
    "        Bin=simul_bin(N,p)\n",
    "        tau[j]=temps_optimal_option(x_0,u,d,Bin,V_call,gain_call)\n",
    "    histo_discret(tau,N,'Temps optimal')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "main_8() # un call sans dividende s'exerce toujours à l'instant N"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Question 2.12__ Probabilité d'exercice anticipé."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " Évaluer la probabilité d'exercice anticipé (i.e. $\\P(\\tau < N)$,\n",
    "  si $\\tau$ est le temps d'arrêt optimal. Vérifier par simulation que\n",
    "  cette probabilité est strictement positive (pour le put). Elle décroit en fonction de $x_0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [],
   "source": [
    "def compute_proba(x_0,N):\n",
    "    # Proba que tau<N\n",
    "    sigma=0.3;\n",
    "    p=1/2;d=1-sigma/math.sqrt(N);u=1+sigma/math.sqrt(N)\n",
    "    K=100\n",
    "    r_0=0.1\n",
    "    r=r_0/N\n",
    " \n",
    "    def gain_put(x): return max(K-x,0) # Payoff du put  \n",
    "\n",
    "    V=prix_am_vect(x_0,r,u,d,p,N,gain_put);\n",
    "\n",
    "    Nbre=1000;\n",
    "    tau=np.zeros(Nbre,dtype=int)\n",
    "    for j in range(Nbre):\n",
    "        Bin=simul_bin(N,p)\n",
    "        tau[j]=temps_optimal_option(x_0,u,d,Bin,V,gain_put)\n",
    "    \n",
    "    res=0\n",
    "    for j in range(Nbre):\n",
    "         \n",
    "        # exprimer le condition d'exercice anticipé (tau est strictement inferieur à N)\n",
    "        \n",
    "        ######  A vous de jouer  .....\n",
    "\n",
    "    return res / Nbre\n",
    "\n",
    "def main_9():\n",
    "    N = 50\n",
    "    for x_0 in [60,70,80,100,120]:\n",
    "         print(\"x_0 =\",x_0,\": proba d'exercice = \",compute_proba(x_0,N))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x_0 = 60 : proba d'exercice =  1.0\n",
      "x_0 = 70 : proba d'exercice =  0.937\n",
      "x_0 = 80 : proba d'exercice =  0.805\n",
      "x_0 = 100 : proba d'exercice =  0.533\n",
      "x_0 = 120 : proba d'exercice =  0.271\n"
     ]
    }
   ],
   "source": [
    "main_9()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Question 2.12__ Tracer la frontière d'exercice en fonction du temps. \n",
    "\n",
    "C'est la courbe $n\\to s(n)$ telle que $V(n,x) > (K-x)_+$ si et seulement si $x>s(n)$. Autrement \n",
    "dit le point où $V(n,x)$ se détache de $(K-x)_+$.\n",
    "\n",
    "On doit exercer l'option lorsque $X_n < s(n)$, attendre sinon.\n",
    "\n",
    "Pour mieux comprendre, consulter le dessin correspondant à la __question 2.5__."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [],
   "source": [
    "def frontiere(N,x_0,u,d,V,gain):\n",
    "    # Calcule la frontière d'exercice t -> s(t) t\\in[0,N]\n",
    "    # à partir du tableau V[n,j]\n",
    "    s=np.zeros(N+1);\n",
    "    for n in range(N):\n",
    "        for k in range(n+1):\n",
    "            # Dans le cas du put, on cherche le premier point ou le prix est strictement \n",
    "            # supérieur au gain immédiat. \n",
    "            x=x_0*u**k *d**(n-k);\n",
    "        \n",
    "            ######  A vous de jouer  .....\n",
    "\n",
    "    # Le cas n=N est particulier (puisque V[N,j] est exactement égal à gain(x_j)),\n",
    "    # on prolonge avec la valeur précédente pour avoir un beau dessin.\n",
    "    s[N]=s[N-1] \n",
    "    return s\n",
    "\n",
    "def main_10(): \n",
    "    N=1000;\n",
    "    r_0=0.05;r=r_0/N;\n",
    "    sigma=0.3;\n",
    "    K=100;\n",
    "    x_0=60;# pour avoir un joli dessin sans artéfacts, \n",
    "           # il est préférable de partir d'une valeur x_0 proche de la valeur de la fontière en 0.\n",
    "           # Vous pouvez expérimenter avec d'autres valeurs plus petites et plus grande \n",
    "           # chercher à interpréter ce qui se passe. Ce n'est pas si facile ...\n",
    "    p=1/2;d=1-sigma/math.sqrt(N);u=1+sigma/math.sqrt(N);\n",
    "    def gain_put(x): return max(K-x,0) # Payoff du put  \n",
    "\n",
    "    V=prix_am_vect(x_0,r,u,d,p,N,gain_put);\n",
    "    front=frontiere(N,x_0,u,d,V,gain_put);\n",
    "    plt.plot(front,label='frontière');\n",
    "    plt.legend(loc='upper right')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "main_10()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3. Un problème de recrutement"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On reçoit, consécutivement, $N$ candidats à un poste. Les\n",
    "circonstances m'imposent de décider tout de suite du recrutement (soit\n",
    "on recrute la personne que l'on vient de recevoir, soit on la refuse\n",
    "définitivement). On cherche à maximiser la proabilité de recruter le\n",
    "meilleur candidat. Quelle est la meilleure stratégie ?\n",
    "\n",
    "On a vu en cours que l'on peut se ramener à $(S_1,\\ldots,S_N)$ est une\n",
    "suite de variables aléatoires indépendantes de Bernouilli $1/n$, qui\n",
    "est une chaîne de Markov sur l'espace $E=\\{0,1\\}$, non homogène, de\n",
    "matrice de transition, dépendant du temps, $P_n$\n",
    "$$\n",
    "\\begin{array}{lcl}\n",
    "  P_n(0,0)&=&P_n(1,0)=1-\\frac{1}{n+1}=\\frac{n}{n+1},\\\\\n",
    "  P_n(0,1)&=&P_n(1,1)=\\frac{1}{n+1}.\n",
    "\\end{array}\n",
    "$$\n",
    "On chercher à maximiser $\\P(\\tau\\;\\mbox{ est le\n",
    "  meilleur})=\\E\\left(\\frac{\\tau}{N} S_\\tau\\right)$ (voir les transparents pour une preuve) parmi tous les\n",
    "temps d'arrêt."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Question 3.1__ Ecrire un programme _Python_ qui calcule la solution\n",
    "  $(u(n,0\\mbox{ ou } 1),0\\leq n \\leq N)$ de l'équation de\n",
    "  programmation dynamique donnée par (voir cours)\n",
    "  $$\n",
    "  \\left\\{\n",
    "    \\begin{array}{l}\n",
    "      u(n,x) = \\max\\Big\\{ \\frac{n}{n+1} u(n+1,0) + \\frac{1}{n+1} u(n+1,1),\\frac{n}{N} x\\Big\\}, n<N,\\\\\n",
    "      u(N,x) = x,\n",
    "    \\end{array}\n",
    "  \\right.   \n",
    "  $$\n",
    "  On calcule $(u(n,\\{0,1\\}),0\\leq n \\leq N)$ à l'aide de l'équation de\n",
    "  programmation dynamique. On désigne l'\"obstacle\" par $(f(n,\\{0,1\\}),0\\leq n \\leq N)$.\n",
    "  Notez que $f(n,0)=0$ et $f(n,1)=n/N$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [],
   "source": [
    "  def compute_u(N):\n",
    "    u=np.zeros([N+1,2]);\n",
    "    u[N,0]=0;u[N,1]=1; # initialisation des valeurs terminales en N\n",
    "    for n in range(N-1,-1,-1): # = N-1,N-2, ..., 0\n",
    "         \n",
    "        # écrire l'équation précédente.\n",
    "        \n",
    "        ######  A vous de jouer  .....\n",
    "\n",
    "        \n",
    "    return u"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " __Question 3.2__ Tracer les courbes $u(n,0)$, $u(n,1)$ ainsi que \"l'obstacle\"\n",
    "  $f(n,1)=\\frac{n}{N}$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [],
   "source": [
    "def main_11():\n",
    "    N=1000;\n",
    "    obstacle=np.zeros(N+1)\n",
    "    for k in range(N+1):\n",
    "        obstacle[k]=k/N;\n",
    "\n",
    "    u=compute_u(N);\n",
    "    plt.plot(range(N+1),u[:,0],label='valeur pour x=0');# u(n,0)\n",
    "    plt.plot(range(N+1),obstacle,label='obstacle');\n",
    "    plt.plot(range(N+1),u[:,1],label='valeur pour x=1');# u(n,1)\n",
    "    plt.legend(bbox_to_anchor=(1.05, 1), loc='upper left', borderaxespad=0.)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 179,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "main_11()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Question 3.3__ Interprétez $u(n=0,s=1)$ comme la probabilité de choisir\n",
    "le meilleur candidat avec la stratégie optimale. Tracer la courbe\n",
    "$N$ donne\n",
    "$$\n",
    "  \\P(\\mbox{Le candidat choisi par une stratégie optimale est le\n",
    "    meilleur}),\n",
    "$$\n",
    "pour $N=10,25,50,100,250,500,1000$. Vérifier numériquement qu'elle converge vers $1/e\\approx 37\\%$.\n",
    "  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 180,
   "metadata": {},
   "outputs": [],
   "source": [
    "def main_12():\n",
    "    # u(0,1) = P(succés pour la stratégie optimale)\n",
    "    # On vérifie que cette proba tends vers 1/e ~ 37\\%\n",
    "    valeurs=[10,15,20,25,30,40,50,60,70,80,100,150,200,250,500,1000];\n",
    "    courbe=np.zeros(np.size(valeurs));\n",
    "\n",
    "    i=0;\n",
    "    for N in valeurs:\n",
    "         \n",
    "        # calculer la probabilité d'obtenir le meilleur, avec la stratégie optimale, pour N candidats\n",
    "        \n",
    "        ######  A vous de jouer  .....\n",
    "\n",
    "        i=i+1;\n",
    "\n",
    "    plt.plot(valeurs,courbe)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 181,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "main_12()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Calcul et simulation du temps d'arrêt optimal"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vérifier que l'on a $u(n,0) > 0$ pour tout $n<N$ et donc que\n",
    "$0=f(n,0)\\not=u(n,0)$. Par ailleurs, on a bien sûr, $u(N,0)=0=f(N,0)$.\n",
    "\n",
    "Nous savons (voir le cours) qu'un temps d'arret optimal est donné par\n",
    "  $$\n",
    "  \\tau = \\inf\\{n\\geq 0, u(n,S_n)=f(n,S_n)\\}.\n",
    "  $$\n",
    "Se convaincre, à partir de cette formule, que le temps optimal $\\tau$ sera toujours \n",
    "postérieur à $\\tau_{min}$, le premier temps où $u(n,1)=f(n,1)=n/N$, (puisque, pour $n<N$, \n",
    "$u(n,0) > 0 = f(n,0)$, et le temps optimal ne peut donc être atteint avec $S_n=0$ lorsque $n<N$). \n",
    "\n",
    "Le temps d'arrêt optimal $\\tau$ sera donc la premier instant après $\\tau_{min}$ ($\\tau_{min}$ compris) \n",
    "pour lequel $S_n=1$ et $u(n,1)=n/N$ si $n<N$.\n",
    "\n",
    "Dans le cas où $S_n=0$ pour tous les $n$ entre $\\tau_{min}$ et $N$, on est obligé de choisir\n",
    "$\\tau=N$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Question 3.4__ \n",
    "Ecrire un algorithme de calcul du temps (déterministe) $\\tau_{min}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 182,
   "metadata": {},
   "outputs": [],
   "source": [
    "  def temps_min(N):\n",
    "    # Calcule le\n",
    "    # premier temps où |$u(n,1)=f(n,1)=n/N$|\n",
    "    # à |$\\epsilon$| près.\n",
    "    epsilon= 0.00001;\n",
    "    u=compute_u(N);\n",
    "    for n in range(N+1):\n",
    "        if (abs(u[n,1] - (n/N)) < epsilon): break;\n",
    "    return n;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Question 3.5__ Vérifier par simulation que le temps (déterministe) $\\tau_{min}$ \"vaut environ\"\n",
    "  $N/e$ pour $N$ grand.  On peut le démontrer \"sans trop de difficulté\"\n",
    "  à l'aide de la série harmonique. On se contente de le vérifier numériquement ici."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 183,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def main_13():\n",
    "    # Vérification que temps_min vaut environ |$N/e$|.\n",
    "    Taille=1000;\n",
    "    x=np.zeros(Taille+1)\n",
    "    for N in range(1,Taille+1):\n",
    "        x[N]=temps_min(N)/N;\n",
    "    e=math.exp(1)\n",
    "    x=x-1/e;\n",
    "    plt.plot(range(Taille+1),x);\n",
    "    \n",
    "main_13()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ecrire une fonction qui calcule la suite des rangs d'insertion ($R$ dans le cours)\n",
    "  d'une permutation et une fonction qui calcule la permutation\n",
    "  définie par ses rangs d'insertion successifs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 184,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "rangs d'insertion:  [1 2 2 1 1 6 3 7 9 1]\n",
      "[ 5  7  6  3  2  9  4  8 10  1] ?=? [ 5  7  6  3  2  9  4  8 10  1]\n"
     ]
    }
   ],
   "source": [
    "def Omega2R(omega):\n",
    "# Calcule les rangs d'insertion d'une permutation omega donnée\n",
    "    R=np.zeros(np.size(omega),dtype=int) # crée u tableau d'entier\n",
    "    for n in range(np.size(omega)):\n",
    "        # classe le vecteur omega[1,...,n] en croissant\n",
    "        y=np.sort(omega[0:n+1]);\n",
    "        # R(n) = le classement de omega(n) parmi les n premiers\n",
    "        R[n]=np.where(y==omega[n])[0][0]+1\n",
    "    return R\n",
    "\n",
    "def R2Omega(R):\n",
    "    # Calcule omega connaissant les rangs d'insertion\n",
    "    iomega=np.zeros(0,dtype=int);# crée u tableau d'entier\n",
    "    for n in range(np.size(R)):\n",
    "        # J'insére n à l'indice R[n]\n",
    "        iomega=np.concatenate([iomega[0:int(R[n]-1)],[n+1],iomega[int(R[n]-1):n+1]])\n",
    "    # On inverse la permutation\n",
    "    omega=np.zeros(np.size(R),dtype=int)# crée u tableau d'entier\n",
    "    for n in range(np.size(omega)):\n",
    "        omega[int(iomega[n]-1)]=int(n+1)\n",
    "    return omega\n",
    "\n",
    "def main_14():\n",
    "    # test sur une permutation\n",
    "    N=10\n",
    "    omega=np.random.permutation(N)+1# tirage d'une permutation aléatoire.\n",
    "    R=Omega2R(omega)\n",
    "    print(\"rangs d'insertion: \",R)\n",
    "    \n",
    "    # Retrouve t'on omega ?\n",
    "    omega2=R2Omega(R)\n",
    "    print(omega,end='')\n",
    "    print(' ?=? ',end='')\n",
    "    print(omega2)\n",
    "\n",
    "main_14()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " __Question 3.6__ Vérifier par simulation que la probabilité d'obtenir le meilleur\n",
    "  candidat, lorsque l'on utilise la stratégie optimale, est de l'ordre\n",
    "  de $1/e\\approx 37\\%$. Ce qui n'est pas génial, mais c'est le mieux\n",
    "  que l'on puisse faire (sans connaitre l'avenir!)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 185,
   "metadata": {},
   "outputs": [],
   "source": [
    "def temps_optimal(omega,N):\n",
    "# Calcule le temps d'arret optimum\n",
    "# pour la permutation |$\\omega$| \n",
    "\n",
    "    # Calcule la suite R à partir de omega (omega->R)\n",
    "    R=Omega2R(omega);\n",
    "    # calcul de tau_min\n",
    "    u=compute_u(N);\n",
    "    tau_min=temps_min(N);\n",
    "    # Le temps optimal se situe après tau_min et c'est le premier instant\n",
    "    # où R(n)=S(n)=1 apres ce temps, sauf si n=N, auquel cas on est oblige\n",
    "    # de prendre le dernier candidat.\n",
    "    for n in range(tau_min,N+1): # = tau_min, tau_min+1, ... , N\n",
    "        if R[n-1]==1: # n-1 parce que les indices de R varie de 0 a n-1\n",
    "            if u[n,1] == n/N: # à vrai dire cette condition n'est pas indispensable\n",
    "                              # car après tau_min, cette égalité est forcément vérifiée\n",
    "                break\n",
    "    # Si on sort de cette boucle avec n=N et N est bien optimal.\n",
    "    return n\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On teste dans un cas particulier."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 186,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "probabilité d'obtenir le meilleur:  0.363 ~ 1/e =  0.36787944117144233\n",
      "erreur : 0.004879441171442345 erreur Monte-Carlo maximum probable : 0.02988882127355379\n"
     ]
    }
   ],
   "source": [
    "def main_15():\n",
    "    # Verification que la probabilité d'obtenir\n",
    "    # le meilleur est de l'ordre de 1/e\n",
    "    N=100;\n",
    "    Taille=1000;\n",
    "  \n",
    "    ss=0;\n",
    "    for i in range(Taille):\n",
    "        omega=np.random.permutation(N)+1 # on rajoute 1 pour avoir la notation classique d'une permutation\n",
    "        tirages=temps_optimal(omega,N)    \n",
    "        # tirages(i) est il le meilleur ?\n",
    "        if (omega[int(tirages-1)]==1):  ss=ss+1;\n",
    "    proba=ss/Taille;\n",
    "    \n",
    "    e=math.exp(1)\n",
    "    p=1/e;\n",
    "    print(\"probabilité d'obtenir le meilleur: \",proba,\"~ 1/e = \",p);\n",
    "    print('erreur :',abs(proba-p),'erreur Monte-Carlo maximum probable :', 1.96*math.sqrt(p*(1-p)/Taille))\n",
    "                \n",
    "main_15()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Thème de réflexion 1.__ Pourquoi le problème posé avec la\n",
    "    chaîne de Markov $(R_n,0\\leq k\\leq N)$ en lieu et place de\n",
    "    $(S_n,0\\leq k\\leq N)$ donnerait il le même résultat ?\n",
    " "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Thème de réflexion 2.__ Pourquoi le fait de donner une note\n",
    "   suivant une loi uniforme entre l'intervalle réel $[0,20]$ change le\n",
    "   résultat et augmente la probabilité d'obtenir le meilleur candidat\n",
    "   (bien que la loi induite sur les permutations reste la loi\n",
    "   uniforme) ?"
   ]
  }
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