AMM-Analysis.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_nfts(num_nfts):\n",
    "  return np.linspace(1, num_nfts, num_nfts)\n",
    "\n",
    "# Given number of nfts and price, returns the buy/sell price at each instantaneous point\n",
    "# Returns instantaneous price, buy price, sell price (per row), slippage\n",
    "def constant_product(num_nfts, nft_price):\n",
    "  x = num_nfts\n",
    "  y = num_nfts*nft_price\n",
    "  k = x*y\n",
    "  xs = get_nfts(num_nfts*2)\n",
    "  ys = np.zeros(len(xs))\n",
    "  prices = np.zeros((len(xs), 4))\n",
    "  for index, x_value in enumerate(xs):\n",
    "    y_value = k/x_value\n",
    "    ys[index] = y_value\n",
    "  for index, y_value in enumerate(ys):\n",
    "    buy_price = 0\n",
    "    sell_price = 0\n",
    "    if index > 0:\n",
    "      buy_price = ys[index-1]-ys[index]\n",
    "    if index < len(ys)-1:\n",
    "      sell_price = ys[index]-ys[index+1]\n",
    "    instant_price = y_value/(index+1)\n",
    "    prices[index] = [instant_price, buy_price, sell_price, (buy_price-sell_price)/instant_price]\n",
    "  return prices\n",
    "\n",
    "# Given a list of prices (like the kind `constant_product` above returns)\n",
    "# a starting price, and a multiplier, will return the prices that fall into this range\n",
    "def liquidity_depth(prices, spot_price, multiplier):\n",
    "  min_price = spot_price/multiplier\n",
    "  max_price = spot_price*multiplier\n",
    "  start_index = -1\n",
    "  end_index = -1\n",
    "  for index, p in enumerate(prices):\n",
    "    buy_price = p[1]\n",
    "    sell_price = p[2]\n",
    "    if start_index == -1:\n",
    "      if (buy_price != 0 and buy_price < max_price) or sell_price < max_price:\n",
    "        start_index = index\n",
    "    if end_index == -1:\n",
    "      if (buy_price != 0 and buy_price < min_price) or sell_price < min_price:\n",
    "        end_index = index\n",
    "  if end_index == -1:\n",
    "    end_index = len(prices)\n",
    "  if start_index == -1:\n",
    "    start_index = 0\n",
    "  return min_price, max_price, prices[start_index:end_index+1,:]\n",
    "\n",
    "# Given a list of prices, a starting price, and the number of nfts to buy/sell,\n",
    "# will return the average price to buy/sell that number of nfts\n",
    "def avg_price(prices, spot_price, num_nfts_to_transact, is_buy):\n",
    "  middle_index = int(len(prices)/2)-1\n",
    "#   assert(prices[middle_index][0] == spot_price)\n",
    "  if is_buy:\n",
    "    total = np.sum(prices[middle_index-num_nfts_to_transact+1:middle_index+1, 1])\n",
    "  else:\n",
    "    total = np.sum(prices[middle_index:middle_index+num_nfts_to_transact, 2])\n",
    "  return total/num_nfts_to_transact"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.3888888888888893"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "num_nfts = 10\n",
    "spot_price = 1\n",
    "depth_multiplier = 1.5\n",
    "\n",
    "prices = constant_product(num_nfts, spot_price)\n",
    "avg_price(prices, spot_price, 1, True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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q28LiUqb9pG5lCJckSXXb0dvc7MbN9pO6lSFckiTVrdl5v50vXHomQ7gkSarb8O6d7Bvoa6jP/oE+n2gprdHSEB4R3xsRH4+Iv4uIpyrLP4qI11VpeyAi7ouIuYh4IiLORcTbI+I5raxJkiS11uGRIXqivrY9AYdGhtpbkJRDLQvhEfFu4DRwPfAp4D8Dvw/sAm5c0/YNq9p+AvhV4LnA+4ATrapJkiS1XmGwn7sO7tkwiPcEHD2416EoUhUtuUsiIt4M/EfgT4CDKaX5Ndv/yar/fRnwIeBp4MaU0ucr698D3A/cEhG3pZQM45Ikdahbr7uSK3ZtZ2x8iokq84bvH+jj0MiQAVyqYdMhPCJ6gF8GngD+zdoADpBS+saqt7cALwR+eyWAV9osVq6mjwNvwyvikiR1tMJgP4XBfiZn5ylOl1hYXGJH7zYKg/2OAZc20Ior4QeAAeAe4B8i4geAlwOLwJmU0ufWtL+psvxUlX2dphzmD0TEJSmlp1pQnyRJaqPh3TsN3VKDWhHCr6ssZ4EvAHtWb4yI08AtKaW/r6x6aWU5uXZHKaWliJgBXgZcBTy83gdHxIM1Nl1dX+mSJElS9lpxY+a3VZY/BVwK/AtgJ+Wr4X9I+ebL31vV/vLK8rEa+1tZ//wW1CZJkiR1nFZcCV+ZUjAoX/H+75X3/29EvJHyFe8bIuLVVYamVLNyr3XaqGFK6dqqOyhfIX9FHZ8lSZIkZa4VV8L/obJ8ZFUAByCl9CTlq+EA+yrLlSvdl1PdZWvaSZIkSV2lFSH8i5Xl12psXwnpl65pP7y2YURso3yT5xLwSAtqkyRJkjpOK0L4acqheSginltl+8sry0cry/sry9dWaXs9sB34rDOjSJIkqVttOoSnlErAScrDS35x9baI+JfA91MeWrIyJeE9QAm4LSJeuaptL/DeytsPbrYuSZIkqVO15ImZwDuA/cAvRMT1wBngO4A3Un4y5u0ppa8BpJQej4jbKYfxByLiBDAH3Ex5+sJ7KId6SZIkqSu1YjgKKaWvUA7h7wO+HThE+aE8fwB8b0rp99a0vxe4gfJQljcBPwN8g3KYvy2ltOHMKJIkSVJetepKOCmlOcoh+h11ti8Cr2vV50uSdLHysfFS/rQshEuSpGwVp0scG5/izMzcs7btG+jj8MgQhcH+LahM0kZaMhxFkiRl6+TZ84wen6gawAHOzMwxenyCu89eyLgySfUwhEuSlDPF6RJ3nHqI5Q3uoFpOcOTUOYrTpWwKk1Q3Q7gkSTlzbHxqwwC+YjnB2PhUewuS1DBDuCRJOTI5O19zCEotEzNzTM7Ot6kiSc0whEuSlCPNDi1xSIrUWQzhkiTlyMLiUqb9JLWHIVySpBzZ0dvc7MLN9pPUHoZwSZJypNl5v50vXOoshnBJknJkePdO9g30NdRn/0CfT9CUOowhXJKknDk8MkRP1Ne2J+DQyFB7C5LUMEO4JEk5Uxjs566DezYM4j0BRw/udSiK1IG8S0OSpBy69boruWLXdsbGp5ioMm/4/oE+Do0MGcClDmUIlyQppwqD/RQG+5mcnac4XWJhcYkdvdsoDPY7BlzqcIZwSZJybnj3TkO3lDOOCZckSZIyZgiXJEmSMmYIlyRJkjJmCJckSZIy5o2ZkiRtkrOTSGqUIVySpCYVp0scG5/iTJV5uvcN9HHYebol1eBwFEmSmnDy7HlGj09UDeAAZ2bmGD0+wd1nL2RcmaQ8MIRLktSg4nSJO049xHJav91ygiOnzlGcLmVTmKTcMIRLktSgY+NTGwbwFcsJxsan2luQpNwxhEuS1IDJ2fmaQ1BqmZiZY3J2vk0VScojQ7gkSQ1odmiJQ1IkrWYIlySpAQuLS5n2k9SdDOGSJDVgR29zs/s2209SdzKES5LUgGbn/Xa+cEmrGcIlSWrA8O6d7Bvoa6jP/oE+n6Ap6RkM4ZIkNejwyBA9UV/bnoBDI0PtLUhS7hjCJUlqUGGwn7sO7tkwiPcEHD2416Eokp7Fu0QkSWrCrdddyRW7tjM2PsVElXnD9w/0cWhkyAAuqSpDuCRJTSoM9lMY7Gdydp7idImFxSV29G6jMNjvGHBJ6zKES5K0ScO7dxq6JTXEMeGSJElSxloSwiPi0YhINV5frtHnQETcFxFzEfFERJyLiLdHxHNaUZMkSZLUqVo5HOUx4P1V1i+sXRERbwA+DiwCJ4E54PXA+4AC8OYW1iVJkiR1lFaG8K+llO7cqFFEXAZ8CHgauDGl9PnK+vcA9wO3RMRtKaUTLaxNkiRJ6hhbMSb8FuCFwImVAA6QUloE3l15+7YtqEuSJEnKRCuvhF8SEW8BrgT+ETgHnE4pPb2m3U2V5aeq7OM08ARwICIuSSk91cL6JEldyikCJeVNK0P4i4CPrlk3ExE/mlL601XrXlpZTq7dQUppKSJmgJcBVwEPr/eBEfFgjU1X11eyJCnPitMljo1PcabKw3L2DfRx2IflSOpQrRqO8mFghHIQfx6wB/h14CXAJyPiu1a1vbyyfKzGvlbWP79FtUmSutDJs+cZPT5RNYADnJmZY/T4BHefvZBxZZK0sZZcCU8p/dKaVX8J/FRELADvBO4E3ljn7mJlt3V87rVVd1C+Qv6KOj9PkpQzxekSd5x6iOUN/qRYTnDk1DlevOtSr4hL6ijtvjHz1yrL61etW7nSfTnVXbamnSRJz3BsfGrDAL5iOcHY+FR7C5KkBrU7hH+lsnzeqnVfrCyH1zaOiG3AALAEPNLe0iRJeTQ5O19zCEotEzNzTM7Ot6kiSWpcu0P4qyvL1YH6/srytVXaXw9sBz7rzCiSpGqK06VM+0lSO2w6hEfEyyKir8r67wB+pfL2Y6s23QOUgNsi4pWr2vcC7628/eBm65IkdaeFxaVM+0lSO7Tixsw3A0ci4tPADDAPfCfwA0AvcB/wn1Yap5Qej4jbKYfxByLiBOXH1t9MefrCeyg/yl6SpGfZ0dvcH13N9pOkdmjFL9KnKYfn76Y8/OR5wNeAP6M8b/hHU0rPuH0mpXRvRNwA/ALwJsphfRp4BzC2tr0kSSuaneXE2VEkdZJNh/DKg3j+dMOGz+5XBF632c+XJF1chnfvZN9AX0M3Z+4f6PMJmpI6SrtvzJQkqeUOjwzRExu3A+gJODQy1N6CJKlBhnBJUu4UBvu56+CeDYN4T8DRg3sdiiKp43iXiiQpl2697kqu2LWdsfEpJqoMTdk/0MehkSEDuKSOZAiXJOVWYbCfwmA/k7PzFKdLLCwusaN3G4XBfseAS+pohnBJUu4N795p6JaUK44JlyRJkjJmCJckSZIyZgiXJEmSMmYIlyRJkjLmjZmSpE1zdhJJaowhXJLUtOJ0iWPjU1UfIb9voI/DztMtSVU5HEWS1JSTZ88zenyiagAHODMzx+jxCe4+eyHjyiSp8xnCJUkNK06XuOPUQyyn9dstJzhy6hzF6VI2hUlSThjCJUkNOzY+tWEAX7GcYGx8qr0FSVLOGMIlSQ2ZnJ2vOQSllomZOSZn59tUkSTljyFcktSQZoeWOCRFkr7FEC5JasjC4lKm/SSpGxnCJUkN2dHb3Oy2zfaTpG5kCJckNaTZeb+dL1ySvsUQLklqyPDunewb6Guoz/6BPp+gKUmrGMIlSQ07PDJET9TXtifg0MhQewuSpJwxhEuSGlYY7Oeug3s2DOI9AUcP7nUoiiSt4V0ykqSm3HrdlVyxaztj41NMVJk3fP9AH4dGhgzgklSFIVyS1LTCYD+FwX4mZ+cpTpdYWFxiR+82CoP9jgGXpHUYwiVJmza8e6ehW5Ia4JhwSZIkKWOGcEmSJCljhnBJkiQpY4ZwSZIkKWOGcEmSJCljhnBJkiQpY4ZwSZIkKWOGcEmSJCljhnBJkiQpYz4xU5K6gI+Nl6R8MYRLUo4Vp0scG5/izMzcs7btG+jj8MgQhcH+LahMkrSetgxHiYjRiEiV14/XaHMgIu6LiLmIeCIizkXE2yPiOe2oSZK6zcmz5xk9PlE1gAOcmZlj9PgEd5+9kHFlkqSNtDyER8S3Ax8AFtZp8wbgNHA98AngV4HnAu8DTrS6JknqNsXpEneceojltH675QRHTp2jOF3KpjBJUl1aGsIjIoAPA18Ffq1Gm8uADwFPAzemlH4spfS/AP8c+BxwS0Tc1sq6JKnbHBuf2jCAr1hOMDY+1d6CJEkNafWV8EPATcCPAv9Yo80twAuBEymlz6+sTCktAu+uvH1bi+uSpK4xOTtfcwhKLRMzc0zOzrepIklSo1p2Y2ZEXAMcBY6llE5HxE01mq6s/1SVbaeBJ4ADEXFJSumpDT7zwRqbrq6nZknKo2aHlhSnS86YIkkdoiVXwiNiG/BR4Dzw8xs0f2llObl2Q0ppCZih/JeDq1pRmyR1m4XFpUz7SZJar1VXwn8R+G7ge1JKT27Q9vLK8rEa21fWP3+jD00pXVttfeUK+Ss26i9JebSjt7mf7mb7SZJab9NXwiNiH+Wr3/85pfS5zZdEVJZ13nIkSReXZuf9dr5wSeocmwrhq4ahTALvqbPbypXuy2tsv2xNO0nSKsO7d7JvoK+hPvsH+hwPLkkdZLNXwncAw8A1wOKqB/Qk4N9X2nyosu79lfdfrCyH1+6sEuoHgCXgkU3WJkld6/DIED2xcTuAnoBDI0PtLUiS1JDNDhB8CjheY9srKI8T/zPKwXtlqMr9wL8FXgv87po+1wPbgdMbzYwiSRezwmA/dx3cs+EDe3oCjh7c61AUSeowmwrhlZswaz2W/k7KIfy/ppR+c9Wme4BfBm6LiA+szBUeEb3AeyttPriZuiTpYnDrdVdyxa7tjI1PMVFl3vD9A30cGhkygEtSB8r8VvmU0uMRcTvlMP5ARJwA5oCbKU9feA9wMuu6JCmPCoP9FAb7mZydpzhdYmFxiR292ygM9jsGXJI62JbMV5VSujcibgB+AXgT0AtMA+8AxlJKzowiSQ0Y3r3T0C1JOdK2EJ5SuhO4c53tReB17fp8ScqSV6IlSY3wyQ2StAnF6RLHxqc4U2VM9r6BPg47JluSVEVLHlsvSRejk2fPM3p8omoABzgzM8fo8QnuPnsh48okSZ3OEC5JTShOlzacHhBgOcGRU+coTpeyKUySlAuGcElqwrHxqQ0D+IrlBGPjU+0tSJKUK4ZwSWrQ5Ox8zSEotUzMzDE5O9+miiRJeWMIl6QGNTu0xCEpkqQVhnBJatDC4lKm/SRJ3ccQLkkN2tHb3OyuzfaTJHUfQ7gkNajZeb+dL1yStMIQLkkNGt69k30DfQ312T/Q5xM0JUnfZAiXpCYcHhmiJ+pr2xNwaGSovQVJknLFEC5JTSgM9nPXwT0bBvGegKMH9zoURZL0DN4lJElNuvW6K7li13bGxqeYqDJv+P6BPg6NDBnAJUnPYgiXpE0oDPZTGOxncnae4nSJhcUldvRuozDY7xhwSVJNhnBJaoHh3TsN3ZKkujkmXJIkScqYIVySJEnKmCFckiRJypghXJIkScqYN2ZK6grOTiJJyhNDuKRcK06XODY+xZkq83TvG+jjsPN0S5I6kMNRJOXWybPnGT0+UTWAA5yZmWP0+AR3n72QcWWSJK3PEC4pl4rTJe449RDLaf12ywmOnDpHcbqUTWGSJNXBEC4pl46NT20YwFcsJxgbn2pvQZIkNcAQLil3Jmfnaw5BqWViZo7J2fk2VSRJUmMM4ZJyp9mhJQ5JkSR1CkO4pNxZWFzKtJ8kSa1mCJeUOzt6m5tdtdl+kiS1miFcUu40O++384VLkjqFIVxS7gzv3sm+gb6G+uwf6PMJmpKkjmEIl5RLh0eG6In62vYEHBoZam9BkiQ1wBAuKZcKg/3cdXDPhkG8J+Dowb0ORZEkdRTvUpKUW7dedyVX7NrO2PgUE1XmDd8/0MehkSEDuCSp4xjCJeVaYbCfwmA/k7PzFKdLLCwusaN3G4XBfseAS5I6liFcUlcY3r3T0C1Jyo2WjAmPiF+OiPGIuBART0bEXET8eUT8+4h4QY0+ByLivkrbJyLiXES8PSKe04qaJEmSpE7VqhszfxZ4HvDHwDHgd4Al4E7gXER8++rGEfEG4DRwPfAJ4FeB5wLvA060qCZJkiSpI7VqOMplKaXFtSsj4n8Ffh64A/ifK+suAz4EPA3cmFL6fGX9e4D7gVsi4raUkmFckiRJXaklV8KrBfCKuyvL1RP03gK8EDixEsBX7ePdlbdva0VdkiRJUidq942Zr68sz61ad1Nl+akq7U8DTwAHIuKSlNJT7SxOUus4O4kkSfVraQiPiHcBO4DLgVcC30M5gB9d1eylleXk2v4ppaWImAFeBlwFPLzB5z1YY9PVjVUuqVnF6RLHxqc4U2We7n0DfRx2nm5Jkp6l1VfC3wXsXvX+U8BbU0p/v2rd5ZXlYzX2sbL++a0tTVKrnTx7njtOPcRyqr79zMwco8cnOHpwLz903bdXbyRJ0kWopSE8pfQigIjYDRygfAX8zyPiX6eUvlDnblYeQl3jj/VnfN61VXdQvkL+ijo/T1ITitOldQP4iuUER06d48W7LvWKuCRJFa2aovAZUkqzKaVPAN8HvAD47VWbV650X/6sjmWXrWknqQMdG5/aMICvWE4wNj7V3oIkScqRtoTwFSmlvwH+CnhZRKxcAvtiZTm8tn1EbAMGKM8x/kg7a5PUvMnZ+apjwNczMTPH5Ox8myqSJClf2hrCK/5pZfl0ZXl/ZfnaKm2vB7YDn3VmFKlzFadLmfaTJKnbbDqER8TVEfGiKut7Kg/r+TbKofofKpvuAUrAbRHxylXte4H3Vt5+cLN1SWqfhcWlTPtJktRtWnFj5muB/z0iTgP/H/BVyjOk3EB5msEvA7evNE4pPR4Rt1MO4w9ExAlgDriZ8vSF9wAnW1CXpDbZ0dvcT0ez/SRJ6jat+BPxT4DfAArAd1GeWvAfKc8D/lFgLKX0jMGjKaV7I+IG4BeANwG9wDTwjkr7Om/3krQVmp3lxNlRJEkq23QITyn9JfDTTfQrAq/b7OdLyt7w7p3sG+hr6ObM/QN9PkFTkqSKLG7MlNSFDo8M0RMbtwPoCTg0MtTegiRJyhFDuKSmFAb7uevgng2DeE/A0YN7HYoiSdIq3iUlqWm3XnclV+zaztj4FBNVhqbsH+jj0MiQAVySpDUM4ZI2pTDYT2Gwn8nZeYrTJRYWl9jRu43CYL9jwCVJqsEQLqklhnfvNHRLklQnx4RLkiRJGTOES5IkSRkzhEuSJEkZM4RLkiRJGTOES5IkSRkzhEuSJEkZM4RLkiRJGTOES5IkSRkzhEuSJEkZ84mZUpfwsfGSJOWHIVzKueJ0iWPjU5yZmXvWtn0DfRweGaIw2L8FlUmSpFocjiLl2Mmz5xk9PlE1gAOcmZlj9PgEd5+9kHFlkiRpPYZwKaeK0yXuOPUQy2n9dssJjpw6R3G6lE1hkiRpQ4ZwKaeOjU9tGMBXLCcYG59qb0GSJKluhnAphyZn52sOQallYmaOydn5NlUkSZIaYQiXcqjZoSUOSZEkqTMYwqUcWlhcyrSfJElqLUO4lEM7epubXbTZfpIkqbUM4VIONTvvt/OFS5LUGQzhUg4N797JvoG+hvrsH+jzCZqSJHUIQ7iUU4dHhuiJ+tr2BBwaGWpvQZIkqW6GcCmnCoP93HVwz4ZBvCfg6MG9DkWRJKmDeJeWlGO3XnclV+zaztj4FBNV5g3fP9DHoZEhA7gkSR3GEC7lXGGwn8JgP5Oz8xSnSywsLrGjdxuFwX7HgEuS1KEM4VKXGN6909AtSVJOOCZckiRJypghXJIkScqYIVySJEnKmCFckiRJypghXJIkScqYs6NILeD0gJIkqRGbDuER8QLgjcAPAHuAFwNfBx4CPgx8OKW0XKXfAeDdwKuAXmAa+C3gAymlpzdbl5SF4nSJY+NTnKnyoJx9A30c9kE5kiSpilYMR3kz8CFgPzABvB/4OPBy4DeBuyPiGQ/Wjog3AKeB64FPAL8KPBd4H3CiBTVJbXfy7HlGj09UDeAAZ2bmGD0+wd1nL2RcmSRJ6nStGI4yCdwM/MHqK94R8fPAGeBNwEHKwZyIuIxyaH8auDGl9PnK+vcA9wO3RMRtKSXDuDpWcbrEHaceYjmt3245wZFT53jxrku9Ii5Jkr5p01fCU0r3p5R+f+2Qk5TSl4Ffq7y9cdWmW4AXAidWAnil/SLl4SkAb9tsXVI7HRuf2jCAr1hOMDY+1d6CJElSrrR7dpRvVJZLq9bdVFl+qkr708ATwIGIuKSdhUnNmpydrzkEpZaJmTkmZ+fbVJEkScqbts2OEhHbgB+uvF0duF9aWU6u7ZNSWoqIGeBlwFXAwxt8xoM1Nl3dWLVS/YrTpab7OWOKJEmC9l4JP0r55sz7Ukp/uGr95ZXlYzX6rax/fpvqkjZlYXFp40Yt7CdJkrpPW66ER8Qh4J3AXwOjjXavLDcccZtSurbG5z8IvKLBz5XqsqO3uf9smu0nSZK6T8uvhEfETwPHgL8CXpNSWjt4duVK9+VUd9madlJHaXaWE2dHkSRJK1oawiPi7cCvAH9JOYB/uUqzL1aWw1X6bwMGKN/I+Ugra5NaZXj3TvYN9DXUZ/9An+PBJUnSN7UshEfEz1F+2M5fUA7gX6nR9P7K8rVVtl0PbAc+m1J6qlW1Sa12eGSInti4HUBPwKGRofYWJEmScqUlIbzyoJ2jwIPASEppvekj7gFKwG0R8cpV++gF3lt5+8FW1CW1S2Gwn7sO7tkwiPcEHD2416EokiTpGTZ9p1hE/AjwHyg/AfMzwKE1T6kHeDSl9BGAlNLjEXE75TD+QEScAOYoP3XzpZX1Jzdbl9Rut153JVfs2s7Y+BQTVeYN3z/Qx6GRIQO4JEl6llZM1zBQWT4HeHuNNn8KfGTlTUrp3oi4AfgFyo+17wWmgXcAYymlOp9FKG2twmA/hcF+JmfnKU6XWFhcYkfvNgqD/Y4BlyRJNW06hKeU7gTubKJfEXjdZj9f6gTDu3cauiVJUt3a/dh6SZIkSWsYwiVJkqSMGcIlSZKkjBnCJUmSpIy1YnYUacs5O4kkScoTQ7hyrThd4tj4FGeqzNO9b6CPw87TLUmSOpDDUZRbJ8+eZ/T4RNUADnBmZo7R4xPcffZCxpVJkiStzxCuXCpOl7jj1EMsb/BYp+UER06dozhdyqYwSZKkOhjClUvHxqc2DOArlhOMjU+1tyBJkqQGGMKVO5Oz8zWHoNQyMTPH5Ox8myqSJElqjCFcudPs0BKHpEiSpE5hCFfuLCwuZdpPkiSp1Qzhyp0dvc3NrNlsP0mSpFYzhCt3mp332/nCJUlSpzCEK3eGd+9k30BfQ332D/T5BE1JktQxDOHKpcMjQ/REfW17Ag6NDLW3IEmSpAYYwpVLhcF+7jq4Z8Mg3hNw9OBeh6JIkqSO4p1qyq1br7uSK3ZtZ2x8iokq84bvH+jj0MiQAVySJHUcQ7hyrTDYT2Gwn8nZeYrTJRYWl9jRu43CYL9jwCVJUscyhKsrDO/eaeiWJEm54ZhwSZIkKWOGcEmSJCljhnBJkiQpY4ZwSZIkKWPemKmWcHYSSZKk+hnCtSnF6RLHxqc4U2We7n0DfRx2nm5JkqRncTiKmnby7HlGj09UDeAAZ2bmGD0+wd1nL2RcmSRJUmczhKspxekSd5x6iOW0frvlBEdOnaM4XcqmMEmSpBwwhKspx8anNgzgK5YTjI1PtbcgSZKkHDGEq2GTs/M1h6DUMjEzx+TsfJsqkiRJyhdDuBrW7NASh6RIkiSVGcLVsIXFpUz7SZIkdRtDuBq2o7e5mS2b7SdJktRtDOFqWLPzfjtfuCRJUpkhXA0b3r2TfQN9DfXZP9DnEzQlSZIqDOFqyuGRIXqivrY9AYdGhtpbkCRJUo60JIRHxC0R8YGI+ExEPB4RKSI+tkGfAxFxX0TMRcQTEXEuIt4eEc9pRU1qr8JgP3cd3LNhEO8JOHpwr0NRJEmSVmnVnXLvBr4LWAC+BFy9XuOIeAPwcWAROAnMAa8H3gcUgDe3qC610a3XXckVu7YzNj7FRJV5w/cP9HFoZMgALkmStEarQvjPUg7f08ANwKdrNYyIy4APAU8DN6aUPl9Z/x7gfuCWiLgtpXSiRbWpjQqD/RQG+5mcnac4XWJhcYkdvdsoDPY7BlySJKmGloTwlNI3Q3fEhgOFbwFeCPz2SgCv7GMxIt4NjANvAwzhDdjqEDy8e6ehW5IkqU5bMXHzTZXlp6psOw08ARyIiEtSSk+tt6OIeLDGpnWHw3ST4nSJY+NTVR8jv2+gj8MOB5EkSeo4WzE7yksry8m1G1JKS8AM5b8cXJVlUXl08ux5Ro9PVA3gAGdm5hg9PsHdZy9kXJkkSZLWsxVXwi+vLB+rsX1l/fM32lFK6dpq6ytXyF/RcGU5Upwucceph1hO67dbTnDk1DlevOtSr4hLkiR1iE6cJ3xlUPkG8fLidmx8asMAvmI5wdj4VHsLkiRJUt22IoSvXOm+vMb2y9a00xqTs/M1h6DUMjEzx+TsfJsqkiRJUiO2IoR/sbIcXrshIrYBA8AS8EiWReVJcbqUaT9JkiS11laE8Psry9dW2XY9sB347EYzo1zMFhaXMu0nSZKk1tqKEH4PUAJui4hXrqyMiF7gvZW3H9yCunJjR29z99M220+SJEmt1ZJUFhE/CPxg5e2LKstXR8RHKv+7lFJ6F0BK6fGIuJ1yGH8gIk5Qfmz9zZSnL7yH8qPsVUOzs5w4O4okSVJnaNWl0X8O/MiadVfxrbm+/wZ418qGlNK9EXED8AvAm4Beyo+8fwcwllJyZpR1DO/eyb6BvoZuztw/0OcTLSVJkjpES4ajpJTuTCnFOq+XVOlTTCm9LqW0K6V0aUppT0rpfSmlp1tRU7c7PDJET2zcDqAn4NDIUHsLkiRJUt06cZ5w1aEw2M9dB/dsGMR7Ao4e3OtQFEmSpA7inXo5dut1V3LFru2MjU8xUWVoyv6BPg6NDBnAJUmSOowhPOcKg/0UBvuZnJ2nOF1iYXGJHb3bKAz2OwZckiSpQxnCu8Tw7p2GbkmSpJxwTLgkSZKUMUO4JEmSlDFDuCRJkpQxQ7gkSZKUMUO4JEmSlDFDuCRJkpQxQ7gkSZKUMUO4JEmSlDFDuCRJkpQxQ7gkSZKUMR9b3yKTs/MUp0ssLC6xo3cbhcF+HyMvSZKkqgzhm1ScLnFsfIozM3PP2rZvoI/DI0MUBvu3oDJJkiR1KoejbMLJs+cZPT5RNYADnJmZY/T4BHefvZBxZZIkSepkhvAmFadL3HHqIZbT+u2WExw5dY7idCmbwiRJktTxDOFNOjY+tWEAX7GcYGx8qr0FSZIkKTcM4U2YnJ2vOQSllomZOSZn59tUkSRJkvLEEN6EZoeWOCRFkiRJYAhvysLiUqb9JEmS1F0M4U3Y0dvczI7N9pMkSVJ3MYQ3odl5v50vXJIkSWAIb8rw7p3sG+hrqM/+gT6foClJkiTAEN60wyND9ER9bXsCDo0MtbcgSZIk5YYhvEmFwX7uOrhnwyDeE3D04F6HokiSJOmbvFNwE2697kqu2LWdsfEpJqrMG75/oI9DI0MGcEmSJD2DIXyTCoP9FAb7mZydpzhdYmFxiR292ygM9jsGXJIkSVUZwltkePdOQ7ckSZLq4phwSZIkKWOGcEmSJCljhnBJkiQpY4ZwSZIkKWOGcEmSJCljhnBJkiQpY4ZwSZIkKWOGcEmSJCljhnBJkiQpY4ZwSZIkKWORUtrqGlouIr566aWX9l1zzTVbXYokSZK62MMPP8yTTz45l1J6QSP9ujWEzwCXAY9ucSl5c3Vl+ddbWkV+efw2z2O4OR6/zfH4bY7Hb3M8fpuzlcfvJcDjKaWBRjp1ZQhXcyLiQYCU0rVbXUseefw2z2O4OR6/zfH4bY7Hb3M8fpuTx+PnmHBJkiQpY4ZwSZIkKWOGcEmSJCljhnBJkiQpY4ZwSZIkKWPOjiJJkiRlzCvhkiRJUsYM4ZIkSVLGDOGSJElSxgzhkiRJUsYM4ZIkSVLGDOGSJElSxgzhkiRJUsYM4V0qIl4QET8eEZ+IiOmIeDIiHouIP4uIH4uIuv+/j4hHIyLVeH25nd9jK7Xye0fEFRHxWxHxPyLiqcq+3x8Ru9pV/1aKiLeuc+xWXk/Xua+uPv8i4paI+EBEfCYiHq98r49t0OdARNwXEXMR8UREnIuIt0fEc5r4/Jbtays0cvwiYigifi4i7o+ICxHx9YiYjYj/FhGvafBzX7LB+X2iNd+wvRo8fi3/zhfZ+feROn4Xx+v83Nyff9FkTumm379tWXyItsSbgQ8Cfwd8GjgP7AYOAr8J/KuIeHOq/2lNjwHvr7J+YfOldrRNf++I+E7gs8C3Af8N+GtgH3AYeG1EFFJKX918qR3lL4BfqrHte4GbgE82sL9uPv/eDXwX5e/yJeDq9RpHxBuAjwOLwElgDng98D6gQPm//bq0cl9bqJHj9x+BW4G/Au6j/H1fCtwM3BwRh1NKYw1+/n8H7q2y/i8b3M9Waej8q2jJd74Iz797gUdrbBsFrqKx30XI9/nXcE7put+/lJKvLnxRDjmvB3rWrH8R5RM9AW+qc1+PAo9u9XfagmPYku8N/GHleP/MmvX/pbL+17b6u2Z8XD9X+d43Z/n/Q6e+gNcAQ0AAN1aOzcdqtL0M+ArwFPDKVet7Kf9FLwG31fm5LdtXjo7fW4HvrrL+BuDrlWPxP9X5uS+pfNZHtvoYZHj8WvadL8bzb519PB94onIs+uvsk/vzr9Gc0o2/fw5H6VIppftTSr+fUlpes/7LwK9V3t6YeWEXmYi4Cvg+ykHyV9ds/vfAPwKjEfG8jEvbEhHxcuBVwN8Cf7DF5XSElNKnU0pTqfInwAZuAV4InEgpfX7VPhYpX5EDeFudH93KfW2ZRo5fSukjKaU/r7L+T4EHgOcCB1pfZedq8PxrpYvu/FvHKHApcCqlVGpRaR2viZzSdb9/Dke5OH2jslxqoM8lEfEW4ErKwfEccDqlVNe43hzb7Pe+qbL8oyo/NPMRUaQc0l8F1DUWMOd+srI83uC5c7Gef2utnE+fqrLtNOWraQci4pKU0lMZ7qsbNPO7CPBPI+IngRcAXwU+l1I619LKOk8rvrPn37fcXln+RhN9u/X8q/bfY9f9/hnCLzIRsQ344crbaidfLS8CPrpm3UxE/GjlKlK32uz3fmllOVlj+xTlED5Ml4fwiLgUeAuwTHm8XyMu1vNvrZrnU0ppKSJmgJdRHlv6cIb7yrWI+A5ghPIfvKcb7P4vK6/V+3sA+JGU0vmWFNh5WvGdPf+AiHg1sAeYTCl9uolddN35t05O6brfP4ejXHyOAi8H7ksp/WGdfT5M+Q+oFwHPo/yD8euUx6R9MiK+qw11doJWfO/LK8vHamxfWf/8pqvMjx+i/D0/mVK60EC/i/X8q6aV55PnJhARlwC/A1wC3JlS+oc6uz5B+UbPa4FdldcNlG8wuxEY78JhZq38zp5/ZT9RWX6owX7dfP7Vyild9/tnCL+IRMQh4J2UZ+cYrbdfSumXKmO3ZlNKT6SU/jKl9FOUbyy8FLizLQVvsYy+d6x8XAv21elW/rD59UY6XaznX5NaeT51/blZmYbso5RnQjgJ/Kd6+6aUvpJS+sWU0hdSSl+rvE5T/petCWAQ+PF21L1VMv7OF8P5dznlixNfBz7SSN9uPf+azSkr3SvL3Pz+GcIvEhHx08AxylNzvSalNNeC3a7cOHF9C/aVJ41875W/TV9eY/tla9p1pYj4Z5RvePsS5anhWuFiPP9aeT5d1OdmJYB/jPI0ZHcDb2nFzYkppSW+Ndzqojg3m/zOF/X5V/EWYDstvCEzz+dfHTml637/DOEXgYh4O/ArlOcNfU3lzuNW+Eplmdd/8mpWI9/7i5XlcI3tQ5VlrTHj3aLZGzLXczGefzXPp8o4ygHKNzI9kvG+cqXy/X4XuA34P4F/UwkvrfL3leXFdG42+p0v2vNvlZUbMhv618E65O78qzOndN3vnyG8y0XEz1GeeP4vKJ/YX1m/R0NeXVl2849kNY1875Ubbb5v7dO/ImIn5X8GfxL4f1pXXmeJiF7K/6y4DBxv4a4vxvPv/srytVW2XU/5qtpn67ybv5X7yo2IeC5wD+Ur4L8NjLZhlp1XVZYX07nZ6He+KM+/FRGxn/JDfiZTSg+0ePe5Ov8ayCnd9/uXOmDCdl/teQHvoTye6fNA3wZt/wnlJ31955r1L6vWF/gOyjN7JODnt/q7tuHYNfS9ax2/yraL+mE9lAN4An5/nTaef+XvdSMbP6zn72ngAROU/7n1atY8hKaZfXX6q47jdwnl+ekT5X+y76ljn7WO337guVXa30T5CXwJOLDVx6TFx6/h7+z5V7Pt8Urbd17M5x+N5ZSu+/2Lyoeqy0TEj1C+0eNp4ANUH9f0aErpI5X2LwFmgL9JKb1k1X7uBI5QvqI7A8wD3wn8AOWT9T7gjSmlr7fli2yRRr93reNX2bb2sfUPU/4BfQ3lYSgHUvc9tv6bIuIzwPdQfkLm79do8xIu0vMvIn4Q+MHK2xcB30/5CtZnKutKKaV3rWl/D+U/aE9QftTyzZSn3LoH+KG06oc9It5KeYaZ/5pSemuVz657X52okeMXER+m/NTMEvB/UP2mqwfSqiuTtY5fZRq4l1F+yM+XKqv38q35h9+TUnpv018sIw0evwdo8Dt7/j3zv99Kn8uA/0H54sOL0zrjwbv5/Gs0p1T6/CDd9Pu31X8L8tWeF+UZI9IGrwdWtX9JZd2ja/ZzA+Wxk38NfI3yBPp/D/wx5Xk8Y6u/a5uOX0Pfu9bxW7X92yn/EPwd5Tvh/4byDSjr/s0/7y/gmspxuQA8Z512F+35V8d/q886pygPY7oP+AfKw5keAn622jGmHDoTNR5v3ci+OvHVyPGjHFg2+l28s57jB/wY8H9TfhruAuUraucpz7LyvVt9XNp0/Br+zp5/Vf/7fVtl2+/Wsf+uPf/qOHbPyCnNnDOdfv55JVySJEnKmDdmSpIkSRkzhEuSJEkZM4RLkiRJGTOES5IkSRkzhEuSJEkZM4RLkiRJGTOES5IkSRkzhEuSJEkZM4RLkiRJGTOES5IkSRkzhEuSJEkZM4RLkiRJGTOES5IkSRkzhEuSJEkZM4RLkiRJGTOES5IkSRkzhEuSJEkZ+/8BTduHOkvxtK4AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 248,
       "width": 368
      },
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Liquidity around a price (+/-) X% scales linearly with the amount of liquidity \n",
    "# placed into the pool for constant product\n",
    "range_lengths = []\n",
    "for i in range(1, 21):\n",
    "  num_nfts = i*10\n",
    "  prices = constant_product(num_nfts, spot_price)\n",
    "  min_price, max_price, price_range = liquidity_depth(prices, spot_price, depth_multiplier)\n",
    "  range_lengths.append(len(price_range))\n",
    "plt.scatter(get_nfts(20), range_lengths)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Given number of nfts, price, delta\n",
    "# Returns instantaneous price, buy price, sell price (per row), slippage\n",
    "def linear_curve(num_nfts, nft_price, delta):\n",
    "    prices = np.zeros((num_nfts*2, 4))\n",
    "    for i in range(-num_nfts, num_nfts):\n",
    "      buy_price = nft_price + (i+1)*delta\n",
    "      sell_price = nft_price + (i)*delta\n",
    "      # insert in sequential order \n",
    "      prices[-(i+num_nfts)-1] = [num_nfts-i,\n",
    "                                 buy_price, sell_price, (buy_price-sell_price)/(buy_price+sell_price)]\n",
    "    return prices\n",
    "\n",
    "# Given number of nfts, price, delta\n",
    "# Returns instantaneous price, buy price, sell price (per row), slippage\n",
    "def exponential_curve(num_nfts, nft_price, delta):\n",
    "    assert(delta > 1)\n",
    "    prices = np.zeros((num_nfts*2, 4))\n",
    "    for i in range(-num_nfts, num_nfts):\n",
    "      buy_price = nft_price * np.power([delta], i)\n",
    "      sell_price = nft_price * np.power([delta], i-1)\n",
    "      # insert in sequential order \n",
    "      prices[-(i+num_nfts)-1] = [num_nfts-i,\n",
    "                                 buy_price, sell_price, (buy_price-sell_price)/(buy_price+sell_price)]\n",
    "    return prices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [],
   "source": [
    "num_nfts = 10\n",
    "prices = [constant_product(num_nfts, spot_price),\n",
    "         linear_curve(num_nfts, spot_price, spot_price/num_nfts),\n",
    "         exponential_curve(num_nfts, spot_price, 1+1/(num_nfts))]\n",
    "\n",
    "range_multipliers = [1.25, 1.5, 1.75, 2.0, 3.0, 4.0]\n",
    "\n",
    "coverage = np.zeros((len(prices), len(range_multipliers)))\n",
    "\n",
    "for x, m in enumerate(range_multipliers):\n",
    "  for y, p in enumerate(prices):\n",
    "    p1, p2, ranges = liquidity_depth(p, spot_price, m)\n",
    "    coverage[y, x] = len(ranges)/len(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.15, 0.25, 0.3 , 0.4 , 0.6 , 0.8 ],\n",
       "       [0.3 , 0.45, 0.65, 0.8 , 0.85, 0.9 ],\n",
       "       [0.3 , 0.5 , 0.6 , 0.8 , 1.  , 1.  ]])"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coverage"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.        ,  2.35794769,  2.14358881,  0.04761905],\n",
       "       [ 2.        ,  2.14358881,  1.9487171 ,  0.04761905],\n",
       "       [ 3.        ,  1.9487171 ,  1.771561  ,  0.04761905],\n",
       "       [ 4.        ,  1.771561  ,  1.61051   ,  0.04761905],\n",
       "       [ 5.        ,  1.61051   ,  1.4641    ,  0.04761905],\n",
       "       [ 6.        ,  1.4641    ,  1.331     ,  0.04761905],\n",
       "       [ 7.        ,  1.331     ,  1.21      ,  0.04761905],\n",
       "       [ 8.        ,  1.21      ,  1.1       ,  0.04761905],\n",
       "       [ 9.        ,  1.1       ,  1.        ,  0.04761905],\n",
       "       [10.        ,  1.        ,  0.90909091,  0.04761905],\n",
       "       [11.        ,  0.90909091,  0.82644628,  0.04761905],\n",
       "       [12.        ,  0.82644628,  0.7513148 ,  0.04761905],\n",
       "       [13.        ,  0.7513148 ,  0.68301346,  0.04761905],\n",
       "       [14.        ,  0.68301346,  0.62092132,  0.04761905],\n",
       "       [15.        ,  0.62092132,  0.56447393,  0.04761905],\n",
       "       [16.        ,  0.56447393,  0.51315812,  0.04761905],\n",
       "       [17.        ,  0.51315812,  0.46650738,  0.04761905],\n",
       "       [18.        ,  0.46650738,  0.42409762,  0.04761905],\n",
       "       [19.        ,  0.42409762,  0.38554329,  0.04761905],\n",
       "       [20.        ,  0.38554329,  0.3504939 ,  0.04761905]])"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prices[2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "prices[2][:, 3] = prices[2][:, 3]*100 \n",
    "table = pd.DataFrame(prices[2][:, 1:],\n",
    "             index=[str(i) + \" NFTs in Pool\" for i in range(1, num_nfts*2+1)],\n",
    "             columns=[\"ETH Price to Buy 1 NFT\", \"ETH Price to Sell 1 NFT\", \"Slippage % between Buy/Sell\"]\n",
    "            )\n",
    "table = table.round(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "buy_prices = []\n",
    "sell_prices = []\n",
    "\n",
    "for i in range(1, 6):\n",
    "  buy_prices.append(avg_price(prices[2], spot_price, i, True))\n",
    "  sell_prices.append(avg_price(prices[2], spot_price, i, False))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [],
   "source": [
    "table = pd.DataFrame(np.array([buy_prices, sell_prices]),\n",
    "             columns=[\"For 1 NFT\", \"For 2 NFTs\", \"For 3 NFTs\", \"For 4 NFTs\", \"For 5 NFTs\"],\n",
    "             index=[\"Avg Buying Price\", \"Avg Selling Price\"])\n",
    "table = table.round(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "multipliers = [1.25, 1.5, 1.75, 2, 3, 4]\n",
    "ranges = []\n",
    "for m in multipliers:\n",
    "  p1, p2, price = liquidity_depth(prices[2], spot_price, m)\n",
    "  ranges.append(len(price)/len(prices[2]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.3333333333333333,\n",
       " 3,\n",
       " array([[ 1.        ,  2.35794769,  2.14358881,  0.04761905],\n",
       "        [ 2.        ,  2.14358881,  1.9487171 ,  0.04761905],\n",
       "        [ 3.        ,  1.9487171 ,  1.771561  ,  0.04761905],\n",
       "        [ 4.        ,  1.771561  ,  1.61051   ,  0.04761905],\n",
       "        [ 5.        ,  1.61051   ,  1.4641    ,  0.04761905],\n",
       "        [ 6.        ,  1.4641    ,  1.331     ,  0.04761905],\n",
       "        [ 7.        ,  1.331     ,  1.21      ,  0.04761905],\n",
       "        [ 8.        ,  1.21      ,  1.1       ,  0.04761905],\n",
       "        [ 9.        ,  1.1       ,  1.        ,  0.04761905],\n",
       "        [10.        ,  1.        ,  0.90909091,  0.04761905],\n",
       "        [11.        ,  0.90909091,  0.82644628,  0.04761905],\n",
       "        [12.        ,  0.82644628,  0.7513148 ,  0.04761905],\n",
       "        [13.        ,  0.7513148 ,  0.68301346,  0.04761905],\n",
       "        [14.        ,  0.68301346,  0.62092132,  0.04761905],\n",
       "        [15.        ,  0.62092132,  0.56447393,  0.04761905],\n",
       "        [16.        ,  0.56447393,  0.51315812,  0.04761905],\n",
       "        [17.        ,  0.51315812,  0.46650738,  0.04761905],\n",
       "        [18.        ,  0.46650738,  0.42409762,  0.04761905],\n",
       "        [19.        ,  0.42409762,  0.38554329,  0.04761905],\n",
       "        [20.        ,  0.38554329,  0.3504939 ,  0.04761905]]))"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "liquidity_depth(prices[2], spot_price, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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