OptionPricing.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# オプション価格"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### オプション価格を求めるには\n",
    "\n",
    "* 原資産価格\n",
    "* 権利行使価格\n",
    "* 満期までの期間\n",
    "* 金利\n",
    "* 想定する変動率(ボラティリティ)\n",
    "* 価格を算出するためのモデル"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "from scipy import stats\n",
    "import matplotlib.pyplot as plt\n",
    "import ivolat3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "np.random.seed(1)\n",
    "s_df = pd.read_pickle(\"./data/s.pickle\")\n",
    "op_pn = pd.read_pickle(\"./data/op1.pickle\")\n",
    "t0 = pd.to_datetime(\"2018-02-26 15:15\")\n",
    "t1 = pd.to_datetime(\"2018-03-09 09:00\")\n",
    "s0 = s_df.loc[t0, \"1803\"]\n",
    "r = 0.001\n",
    "delta_t = t1 - t0\n",
    "t = (delta_t.days / 365) + (delta_t.seconds / 31536000)\n",
    "op_df = op_pn.loc[\"2018-02-26 15:15\"].dropna().copy()\n",
    "op_df[\"k\"] = op_df.index\n",
    "delta_call = op_df.apply(\n",
    "    lambda x: ivolat3.delta(s0, x[\"k\"], r, 0, t, x[\"iv\"], \"C\"), axis=1\n",
    ")\n",
    "\n",
    "delta_call = pd.Series(delta_call.values, index=op_df[\"k\"])\n",
    "k = delta_call.map(lambda x: abs(x - 0.25)).idxmin()\n",
    "price, sigma = op_df.loc[k, [\"price\", \"iv\"]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### 2018年02月26日のコールオプション"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "原資産価格: 22214.1\n",
      "権利行使価格: 22625.0\n",
      "満期までの期間: 0.03年(10.74日)\n",
      "プレミアム: 108.20\n",
      "ボラティリティ: 17.22%\n"
     ]
    }
   ],
   "source": [
    "print(\"原資産価格: {}\".format(s0))\n",
    "print(\"権利行使価格: {}\".format(k))\n",
    "print(\"満期までの期間: {:.2f}年({:.2f}日)\".format(t, t * 365))\n",
    "print(\"プレミアム: {:.2f}\".format(price))\n",
    "print(\"ボラティリティ: {:.2f}%\".format(sigma * 100))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### 幾何ブラウン運動\n",
    "\n",
    "> 「[幾何ブラウン運動](https://ja.wikipedia.org/wiki/幾何ブラウン運動)」（2017年3月2日 (木) 04:18　UTCの版）『ウィキペディア日本語版』。\n",
    "\n",
    "$dS_t = \\mu S_t\\,dt + \\sigma S_t\\,dB_t$\n",
    "\n",
    "$S_t = S_0\\exp\\left( \\left(\\mu - \\frac{\\sigma^2}{2} \\right)t + \\sigma B_t\\right),$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "$s_0 = 22214.1$  \n",
    "$\\mu = 0$  \n",
    "$sigma = 0.1722$  \n",
    "$t = 0.03$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "幾何ブラウン運動に従う乱数を用いて$S_t$を算出\n",
    "\n",
    "```python\n",
    "def get_st(s0, mu, sigma, t):\n",
    "    return s0 * np.exp(\n",
    "        np.random.normal(\n",
    "            (mu - 0.5 * sigma ** 2) * t, sigma * np.sqrt(t))\n",
    "    )\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "```python\n",
    ">>> get_st(s0, 0, sigma, t)\n",
    "23295.801603657848\n",
    ">>> get_st(s0, 0, sigma, t)\n",
    "21806.759778023705\n",
    ">>> get_st(s0, 0, sigma, t)\n",
    "21860.668118808022\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "def get_st(s0, mu, sigma, t):\n",
    "    return s0 * np.exp(\n",
    "        np.random.normal(\n",
    "            (mu - 0.5 * sigma ** 2) * t, sigma * np.sqrt(t))\n",
    "    )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "23295.80160365785"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "get_st(s0, 0, sigma, t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "21806.759778023705"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "get_st(s0, 0, sigma, t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "21860.668118808022"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "get_st(s0, 0, sigma, t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "21511.68039021118"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "get_st(s0, 0, sigma, t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "上記を10000回繰り返した結果のヒストグラム"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "st = np.array([get_st(s0, 0, sigma, t) for _ in range(10000)])\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.hist(st, bins=100);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "$k = 22625$を超えた価格がITMとなる"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "itm = st[st - k > 0]\n",
    "\n",
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(1, 1, 1)\n",
    "ax.hist(st, range=(st.min(), st.max()), bins=100, label=\"OTM\");\n",
    "ax.hist(itm, range=(st.min(), st.max()), bins=100, label=\"ITM\");\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "権利行使価格$k$で得られる損益は$s_t - k$または0の最大値をとる\n",
    "\n",
    "$max(s_t - k, 0)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "payoff = [max(s - k, 0) for s in st]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(1, 1, 1)\n",
    "_ = ax.hist(payoff)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "この差分の平均を算出するとコールオプションの価格となる"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "109.49959424558575"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean([max(s - k, 0) for s in st])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "十分なサンプルを得るには大数の法則を得るまで(数万回)試行する必要がある\n",
    "* 大変\n",
    "* コンピュータへの負荷\n",
    "* 時間がかかる"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### ブラックショールズ方程式\n",
    "\n",
    "> 「[ブラック–ショールズ方程式](https://ja.wikipedia.org/wiki/ブラック–ショールズ方程式)」（2016年10月10日 (月) 15:39 　UTCの版）『ウィキペディア日本語版』。\n",
    "\n",
    "下記の式を用いるとコールオプションの価格が一発で求められる\n",
    "\n",
    "$C(S_t, t) = S_t N(d_1) - Ke^{-r(T-t)} N(d_2)$\n",
    "\n",
    "プットオプションの価格はプットコールパリティから求められる"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "$N(d)$は確率密度関数と呼ばれ、先に試行した分布(ヒストグラム)の色が変わった部分を求められる"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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8re3z7iRjnCU3vwXa+qL2+CdJZo7a5w2t3t9I8ryh8jNa2WyS84fKT0lyYyv/UDtJPhELtPftSb7e3r+PJTl+aJ8e2/s3ra03J/l0kie38jV7LLe6zNneofWvS1JJNrTHXbY3yQVJDrb39+YkZw7tc2yP56qayh9gM/DMtvx44JvAqcDbgPNb+fnAW9vymcC/M5jSfDpwYys/Ebiz/T6hLZ/Q1t3Utk3b9/lT1tZfB34V+BwwM7T9qcBXgUcDpwDfYnDS+7i2/BTgUW2bU9s+VwNnt+X3AH86he/t7wGPaOVvHXpve23vE4a2eTXwnrV+LC/U3vZ4K4NJGncBG3puL3AB8JdzbH/Mj+ep7dFX1aGq+kpb/hFwO4MrbncCl7fNLgde0JZ3AlfUwA3A8Uk2A88Drquq/66q7wPXAWe0dU+oqhtq8Ne7Yui5jqn52lpVt1fVXBeS7QSuqqr7q+rbwCyD20789NYTVfV/wFXAztbbeTbw4bb/8N/tmFugvZ+uqgfaZjcwuP4C+m3vD4c2eyzw0AmzNXssw4L/dgEuBv6Kh9sKfbd3Lsf8eJ7aoB+WZBvwDOBGYFNVHWqrvgNsastz3XZhyyLlB+Yon6ij2jqfpbb1ScAPhkJ0KtoKC7b3lQx6atBxe5NcmORu4KXAX7fNujiW4cj2JtkJHKyqrx61WZftbUXnteGoy9KGmZnA8Tz1QZ/kccBHgNce1QOifZp3M21oobb2aL72JnkT8ABw5aTqNg5ztbeq3lRVWxm09bxJ1m+1DbeXwfv5Rh7+MOvOHO/vJcBTgR3AIeAdk6rbVAd9kkcy+MNdWVUfbcX3tP+60X7f28rnu+3CQuUnzVE+EfO0dT5Lbev3GPx3+BFHlU/MfO1N8sfA7wMvbR/k0HF7h1wJ/EFbXtPHMszZ3qcyGI/+apL9DOr4lSS/SJ/tparuqaoHq+onwHsZDM3AJI7n1T4xsVo/DE6yXAH8/VHlb+fIk7Fva8tnceQJnZvq4RM632ZwMueEtnxizX1C58xpauvQ+s9x5MnYp3HkyZw7GZzIeURbPoWHT+Y8re3zrxx5MufPpvC9PYPBraw3HlXea3u3Dy3/OfDhtX4sj3I8t2328/DJ2C7bC2weWv4LBuPyEzmeJ/KHGfGP9zsMhmX2ATe3nzMZjFddD9wBfGbojQ+DLzv5FvA1jgzGVzI44TELvGKofAa4pe3zD7QLyKaorS9kMB53P3AP8Kmhfd7U6v0NhmYctP2+2da9aaj8Ke0fx2w7aB49he/tLIMxyofK3tN5ez/Sjr99wL8xOEG7po/lhdp71Db7eTjou2wv8P7Wnn0M7vU1HPzH9Hj2ylhJ6txUj9FLklbOoJekzhn0ktQ5g16SOmfQS1LnDHpJ6pxBL0mdM+glqXP/D2fvm2+yaNVUAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "_ = plt.hist(st, range=(st.min(), st.max()), bins=100, label=\"OTM\")\n",
    "_ = plt.hist(itm, range=(st.min(), st.max()), bins=100, label=\"ITM\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "```python\n",
    "def d1(S0, k, r, sigma, t):\n",
    "    return (np.log(S0 / k) + (r + sigma ** 2 / 2) * t) / (sigma * np.sqrt(t))\n",
    "\n",
    "\n",
    "def d2(S0, K, r, sigma, t):\n",
    "    return (np.log(S0 / k) + (r - sigma ** 2 / 2) * t) / (sigma * np.sqrt(t))\n",
    "\n",
    "\n",
    "def bs_call(S0, k, r, sigma, t):\n",
    "    return S0 * stats.norm.cdf(d1(S0, k, r, sigma, t)) - k * np.exp(\n",
    "        -r * t\n",
    "    ) * stats.norm.cdf(d2(S0, k, r, sigma, t))\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "```python\n",
    ">> bs_call(s0, k, r, sigma, t)\n",
    "108.19777523292942\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "def d1(S0, k, r, sigma, t):\n",
    "    return (np.log(S0 / k) + (r + sigma ** 2 / 2) * t) / (sigma * np.sqrt(t))\n",
    "\n",
    "\n",
    "def d2(S0, K, r, sigma, t):\n",
    "    return (np.log(S0 / k) + (r - sigma ** 2 / 2) * t) / (sigma * np.sqrt(t))\n",
    "\n",
    "\n",
    "def bs_call(S0, k, r, sigma, t):\n",
    "    return S0 * stats.norm.cdf(d1(S0, k, r, sigma, t)) - k * np.exp(\n",
    "        -r * t\n",
    "    ) * stats.norm.cdf(d2(S0, k, r, sigma, t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "108.19777523292942"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bs_call(s0, k, r, sigma, t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### インプライド・ボラティリティ\n",
    "\n",
    "* オプションのプレミアムからボラティリティを逆算できる\n",
    "* プレミアムからボラティリティを求める式は存在しない  \n",
    "  ボラティリティの値を変化させてプレミアムと等価になるまで繰り返し演算する\n",
    "* 結局はコンピュータが必要になる"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### グリークス\n",
    "\n",
    "ブラックショールズの式から偏微分して求められる"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### デルタ\n",
    "$$ \\Delta := \\frac{\\partial C}{\\partial S_t} = N(d_1) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### ガンマ\n",
    "$$ \\Gamma := \\frac{\\partial^2 C}{\\partial S_t^2} = \\dfrac{1}{\\sigma S_t\\sqrt{T-t}}\\frac{1}{\\sqrt{2\\pi}}\\mathrm{exp}\\left\\{-\\frac{1}{2}d_1^2\\right\\} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### ベガ\n",
    "$$ \\kappa := \\frac{\\partial C}{\\partial \\sigma} = S_t\\sqrt{T-t}\\frac{1}{\\sqrt{2\\pi}}\\mathrm{exp}\\left\\{-\\frac{1}{2}d_1^2\\right\\} $$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### セータ\n",
    "$$ \\Theta := \\frac{\\partial C}{\\partial t} = - rKe^{-r(T-t)}N(d_2) - \\dfrac{\\sigma S_t}{2\\sqrt{T-t}}\\frac{1}{\\sqrt{2\\pi}}\\mathrm{exp}\\left\\{-\\frac{1}{2}d_1^2\\right\\} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### ブラックショールズモデルの課題\n",
    "\n",
    "* 原資産価格の価格が幾何ブラウン運動(対数正規分布)に従う前提となっている  \n",
    "  実際の分布はどうなのか？ -> いくつかのモデルが考案されているが真の分布は誰もわからない\n",
    "* 権利行使価格がATMから離れるほどボラティリティが変化する  \n",
    "  この曲線をボラティリティスマイルまたはボラティリティサーフェイスと呼ばれる"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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