Lecture13.ipynb

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      "Populating the interactive namespace from numpy and matplotlib\n"
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    "%pylab inline"
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    "#Lecture 13: Numerical Methods for Partial Differential Equations\n",
    "\n",
    "## Topics\n",
    "\n",
    "* More on the stability of FDM for Black-Scholes\n",
    "* Multi spatial variables: ADI method\n",
    "* Numerical examples\n",
    "\n",
    "    \n",
    "    \n",
    "\n",
    "\n"
   ]
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    "\n",
    "# More on the stability of FDM for Black-Scholes:\n",
    "\n",
    "* When $r \\neq 0$:\n",
    "\n",
    "\\begin{aligned}\n",
    "e^{a\\triangle t} & = \\left\\{ \\frac{1}{2}\\sigma^2 x_j^2 \\frac{\\triangle t}{\\triangle x^2} - r x_j \\frac{\\triangle t}{2\\triangle x} \\right\\} e^{-il_m\\triangle x}\n",
    "\\\\\n",
    "& + \\left\\{ 1 - \\sigma^2 x_j^2 \\frac{\\triangle t}{\\triangle x^2}  -r \\triangle t \\right\\}\n",
    "\\\\\n",
    "& + \\left\\{ \\frac{1}{2}\\sigma^2 x_j^2 \\frac{\\triangle t}{\\triangle x^2} + r x_j \\frac{\\triangle t}{2\\triangle x} \\right\\} e^{il_m\\triangle x}\n",
    "\\end{aligned}\n",
    "\n",
    "\n",
    "\n"
   ]
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    "\n",
    "* which is\n",
    "\n",
    "$$\n",
    "e^{a\\triangle t}  = 1 - r\\triangle t + \\sigma^2 x_j^2 \\frac{\\triangle t}{\\triangle x^2} \\cdot 2\\sin^2(l_m\\triangle x/2) + i\\; r x_j \\frac{\\triangle t}{\\triangle x} \\sin(l_m\\triangle x)\n",
    "$$\n",
    "\n",
    "\n",
    "* To satyisfiy $ |e^{a\\triangle t}| < 1 $, we require\n",
    "\n",
    "\\begin{aligned}\n",
    "\\renewcommand{PDut}{\\frac{\\partial u}{\\partial t}}\n",
    "\\renewcommand{PDux}{\\frac{\\partial u}{\\partial x}}\n",
    "\\renewcommand{PDutt}{\\frac{\\partial ^2u}{\\partial t^2}}\n",
    "\\renewcommand{PDuxx}{\\frac{\\partial ^2u}{\\partial x^2}}\n",
    "\\renewcommand{FDut}{\\frac{u_{i,k+1}-u_{i,k}}{\\triangle t}}\n",
    "\\renewcommand{FDutb}{\\frac{u_{i,k}-u_{i,k-1}}{\\triangle t}}\n",
    "\\renewcommand{FDutc}{\\frac{u_{i,k+1}-u_{i,k-1}}{2\\triangle t}}\n",
    "\\renewcommand{FDutt}{\\frac{u_{i,k+1}-2u_{i,k}+u_{i,k-1}}{\\triangle t^2}}\n",
    "\\renewcommand{FDux}{\\frac{u_{i+1,k}-u_{i,k}}{\\triangle x}}\n",
    "\\renewcommand{FDuxb}{\\frac{u_{i,k}-u_{i-1,k}}{\\triangle x}}\n",
    "\\renewcommand{FDuxc}{\\frac{u_{i+1,k}-u_{i-1,k}}{2\\triangle x}}\n",
    "\\renewcommand{FDuxx}{\\frac{u_{i+1,k}-2u_{i,k}+u_{i-1,k}}{\\triangle x^2}}\n",
    "& r >0,\n",
    "\\\\\n",
    "& \\triangle t < \\frac{\\triangle x^2}{ \\sigma^2 x_{max}^2 },\n",
    "\\\\\n",
    "& \\triangle t < \\frac{ \\sigma^2 }{r^2 }.\n",
    "\\end{aligned}\n",
    "\n",
    "\n",
    "* The first condition is trivial (until the FED decrees all rates to be negative!)\n",
    "\n",
    "* The last condition does not have much impact in practice unless the volatility is very small.\n",
    "\n",
    "* So the important one is the second condition, which is what we went through last lecture.\n",
    "\n"
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    "# The Case of more than one spatial variables\n",
    "\n",
    "\n",
    "* Most numerical methods, FDM included, suffer the \"dimentionality\" curse, i.e. the size, complexity of the problem grow exponentially with the dimension.\n",
    "\n",
    "\n",
    "* Usually, Monte Carlo method is the only practical option in dimension $\\geq 3$.\n",
    "\n",
    "\n",
    "* But for two dimensional problems, it is worthwhile to explore the FDM further.\n",
    "\n",
    "\n",
    "* Finance examples that you will need many sptial variables: stochastic vol model, convertible bonds, credit risky bonds, variable annuities, etc.\n",
    "\n"
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    "## Heston Stochastic Volatility Model\n",
    "\n",
    "* ** Heston Stochastic Vol Model  **\n",
    "\n",
    "\n",
    "\\begin{aligned}\n",
    "& dS_t = rS_t dt + \\sqrt{\\nu_t}S_t dW_t^1\n",
    "\\\\\n",
    "& d\\nu_t = \\kappa(\\theta - \\nu_t) + \\xi\\sqrt{\\nu_t}  dW_t^2\n",
    "\\\\\n",
    "& \\hspace{0.2in} \\left< dW_t^1, dW_t^2 \\right> = \\rho dt\n",
    "\\end{aligned}\n",
    "\n",
    "\n",
    "\n"
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    "### PDE for Heston Stochastic Vol Model:\n",
    "\n",
    "\n",
    "\\begin{aligned}\n",
    "\\renewcommand{PDuS}{\\frac{\\partial u}{\\partial S}}\n",
    "\\renewcommand{PDuSS}{\\frac{\\partial ^2u}{\\partial S^2}}\n",
    "\\PDut &+ rS\\PDuS\n",
    "+ [\\kappa(\\theta - \\nu)-\\lambda \\nu]\\frac{\\partial u}{\\partial \\nu}\n",
    "\\\\\n",
    "&+ \\frac{1}{2}\\nu S^2\\PDuSS\n",
    " + \\rho\\xi\\nu S\\frac{\\partial^2 u}{\\partial S\\partial \\nu} \n",
    " + \\frac{1}{2}\\xi^2\\nu\\frac{\\partial^2 u}{\\partial \\nu^2} - ru = 0\n",
    "\\end{aligned}\n",
    "\n",
    "\n"
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    "## Alternating Direction Implicit Method\n",
    "\n",
    "* Write the Crank-Nicolson method as\n",
    "\n",
    "\\begin{aligned}\n",
    "\\small\n",
    "\\renewcommand{fD}{\\mathfrak{D}}\n",
    "\\FDut  = & \\frac{1}{4}  \\sigma^2 x_i^2 \\left\\{ \\FDuxx + \\frac{u_{i+1,k+1}-2u_{i,k+1}+u_{i-1,k+1}}{\\triangle x^2} \\right\\}\n",
    "\\\\ \n",
    "    & + \\frac{1}{2} r x_i \\left\\{ \\FDuxc + \\frac{u_{i+1,k+1}-u_{i-1,k+1}}{2\\triangle x} \\right\\} \n",
    "\\\\\n",
    "    & - \\frac{1}{2} r \\left\\{  u_{i,k} +  u_{i,k+1} \\right\\} \n",
    "\\\\\n",
    "    = & \\frac{1}{2} \\mathfrak{D}\\cdot (  u_{i,k} +  u_{i,k+1} )\n",
    "\\end{aligned}\n",
    "\n",
    "\n",
    "where $ \\mathfrak{D}\\cdot  u_{i,k} $ can be considered as the Crank-Nicolson finite difference operator on $u_{i,k}$.\n"
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    "### Crank-Nicolson in operator format\n",
    "\n",
    "\n",
    "* The Crank-Nicolson scheme can be denoted as\n",
    "\n",
    "$$\n",
    "(1 + \\frac{1}{2}\\triangle t\\fD)\\cdot u_{i,k+1} = (1-\\frac{1}{2}\\triangle t\\fD)\\cdot u_{i,k}\n",
    "$$\n",
    "\n",
    "* This is also applicable when the spatial variable $x_i$ is a vector.\n",
    "\n",
    "* For the one spatial variable case, the operator $\\fD$ involves three points in one time slice.\n",
    "\n",
    "* For two dimension case (the Heston model above), the operator will involve five points in one time slice.\n",
    "\n",
    "* So instead of solving a tridiagonal system, now the linear system has five nonzero diagonals, whcih is much more costly  to solve.\n",
    "\n",
    "\n"
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    "### Operator split \n",
    "\n",
    "\n",
    "* For the general $n$ spatial variables case, the way to ease the linear system solving (still can't get rid of the problem of the number of discretization points exploded!) is **splitting** the operator $\\fD$:\n",
    "\n",
    "\\begin{aligned}\n",
    "(1 + \\frac{1}{2}\\triangle t\\fD^1)\\cdot \\tilde{u}^1_{i,k+1} &= (1-\\frac{1}{2}\\triangle t\\fD^1)\\cdot u_{i,k}\n",
    "\\\\\n",
    "(1 + \\frac{1}{2}\\triangle t\\fD^2)\\cdot \\tilde{u}^2_{i,k+1} &= (1-\\frac{1}{2}\\triangle t\\fD^2)\\cdot \\tilde{u}^1_{i,k}\n",
    "\\\\\n",
    "\\vdots\n",
    "\\\\\n",
    "(1 + \\frac{1}{2}\\triangle t\\fD^n)\\cdot \\tilde{u}^n_{i,k+1} &= (1-\\frac{1}{2}\\triangle t\\fD^n)\\cdot \\tilde{u}^{n-1}_{i,k}\n",
    "\\end{aligned}\n",
    "\n",
    "and set\n",
    "\n",
    "$$\n",
    "u_{i+1,k} = \\tilde{u}^n_{i,k}\n",
    "$$\n",
    "\n"
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    "\n",
    "\n",
    "* Essentially, this says trying to solve the problem in a multistep aproach: each step is equivalent to the one dimensional Crank-Nicolson method.\n",
    "\n",
    "\n",
    "* This is  merely the basic form, the strategy can be customized to further improve the efficiency (not all steps are implicit) or accuracy (high order)\n"
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   "source": [
    "# Numerical Examples \n",
    "\n",
    "* Black Scholes formulae and FDM methods for vanilla European call and up-and-out Barrier Call.\n",
    "\n",
    "\n",
    "\n"
   ]
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EPVoiUqpYPcPubRpxwVHR/BoUEamIyUs28uyUFVxxXCcOa9MosjiUaIlIqR4K6xmOHpLa\n9QxFROLlFxRx/YszaNekLr89permzCqJTh2KSIlWbtrJQ+8vYnB61DMUEdnr0Q8Xs3Dddv516VHU\nqxVtqqMeLREp0d56hmemfj1DEZGYpet38PeJCxjYqzXf61a1c2aVRImWiHxLfD3Dtk3Stp6hiGQZ\nd+dPL8+kZvVqjBrcI+pwACVaIlJMrJ7hgU0zp56hiGSHcdNX8+GC9fzxjK60imDOrJIo0RKRb4jV\nM7xhUNrXMxSRLLJl5x5ufnU2fQ5szIX9O0Qdzl4aDC8ie2VYPUMRySJ3vDGXTTv3MOayfil1lbR6\ntERkr3StZygi2e3zZRt5ZvJyLvtuR3q0bRx1ON+gREtEgK/rGV58dIe0q2coItlrT2ER178wk7aN\n63BVCpYJ06lDEcHduVH1DEUkDT364WLmrd3GY5f0pX7t1Etr1KMlIrw2Yw2TlmzkmtO7qp6hiKSN\n5Rt28veJCzijR2tO6Z6a40qVaIlkOdUzFJF05O7c8PJMqpsxakj3qMMplRItkSyneoYiko5e/WoN\nH8zP5ZrTu9KmcepOrKxESySLqZ6hiKSjLbv2cNOrs+nVrjGXHNMx6nD2KfVGjYlIlblt/BzMUD1D\nEUkrd785lw3b8/jXpUelfE+8erREslSsnuEvBxyieoYikja+WL6JpyYt59JjO9GzXWrNmVUSJVoi\nWaigsIibXlE9QxFJL8GcWTNo3agOvz8tPaaiUaIlkoWenrycuTnbuGHQYapnKCJp47EPlzA3Zxs3\nDulBgxScM6sklUq0zOx3ZjbLzGaa2TNmVsfMmpnZBDNbEN43TVSwIlJ536xn2DrqcEREymTZhh38\n9e35nNGjNaelUdtV4UTLzNoBvwH6untPoDpwATACmOjuXYCJ4XMRSRH3TZiveoYiklbcnZEvzqRm\n9WqMHtIj6nDKpbKnDmsAdc2sBlAPWA0MBcaE68cAZ1fyPUQkQWav3spTk5apnqGIpJWXpq3io4Xr\nufaMrrRuXCfqcMqlwomWu68C7gGWA2uALe7+FtDK3deEm+UAJc6Jb2bDzWyqmU3Nzc2taBgiUkaq\nZygi6WjjjnxufnUORx7UhAv7d4g6nHKrzKnDpgS9V52AtkB9M7sofht3d8BLer27P+Lufd29b8uW\nLSsahoiU0fgZOapnKCJp59bX5rB11x5uP7c31VJ8zqySVObU4SnAEnfPdfc9wAvAscBaM2sDEN6v\nq3yYIlIZu/ILufW12apnKCJp5eOF63n+i5VceWJnurZOz+EOlUm0lgNHm1k9C0bUngzMAcYBw8Jt\nhgEvVy5EEaks1TMUkXSze08h1784g47N6/Grkw6JOpwKq/AkFO4+ycyeA74ACoAvgUeABsBYM7sc\nWAacn4hARaRiVM9QRNLRP95ZwLINO3n6iv5pPd9fpWb7cvdRwKhii/MIerdEJAWonmHpzOx3wBUE\nY0lnAD8huIL6v0BHYClwvrtviihEkaw0N2crD7+/mO9/50COPaRF1OFUimaGF8lgsXqGv1A9w2/R\nXIAiqamwyBnx/Awa1a3JyIGHRR1OpSnREslQ8fUMh6ueYWk0F6BIinlq0jKmrdjMn8/qTtP6taIO\np9KUaIlkqGdUz3CfNBegSOpZs2UXd70xj+O7tGDo4W2jDichlGiJZKBNO/K55635HNtZ9QxLo7kA\nRVLPqJdnUVBUxK1n98qYEmFKtEQykOoZlonmAhRJIW/MzOGt2Wu56pRDOah5vajDSRglWiIZZs6a\nr+sZpusEf1VEcwGKpIhtu/cwatxMDmvTiMuP6xR1OAlVqekdRCS1uDujx6meYVloLkCR1HH3m/NY\nty2PRy7uS83qmdUHpERLJIPE6hnecnZP1TMsA80FKBK9z5dt4j+fLWPYMR3p075J1OEkXGaljSJZ\nbFd+IbeNn8NhbRrxo36qZygiqW9PYRHXvzCD1o3qcM3pXaMOJynUoyWSIR56fxGrNu/iLz88XPUM\nRSQtPPLBYuat3cZjl/SlQe3MTEnUoyWSAWL1DM/q3Ub1DEUkLSxZv4O/TVzAwF6tOaV7idPVZQQl\nWiIZ4PbxczGD6zOgXIWIZD53Z+SLM6hdoxqjB/eIOpykUqIlkuY+WbSe12asUT1DEUkbz3+xik8W\nbWDEmd04oFGdqMNJKiVaImlM9QxFJN1s2J7HLa/Npm+HpvzoqMy/cEeJlkgaUz1DEUk3t742hx15\nBdx2bi+qZcGFO0q0RNLUph353DtB9QxFJH18MD+XF75cxc9P7MyhrbKjcoUSLZE0dd+E+WzbrXqG\nIpIeduQVcP2LMzi4ZX1+8b1Dog6nyijREklDsXqGF/U/SPUMRSQt3PvWfFZu2sUd5/bOqqEOSrRE\n0oy7c+MrYT3DU1XPUERS35fLN/GvT5Zw0dEHZd1cf0q0RNLM+Bk5fLZ4I1ef1pUm9WpFHY6IyD7l\nFxQx4vkZtGpYh2vP6BZ1OFUuM+e7F8lQqmcoIunmofcX7S2z07BO9hW7V4+WSBp5+IOgnuHowd1V\nz1BEUt7Cddu4/52FnNW7TUaX2dkXJVoiaWLlpp08+F5Qz7D/wc2jDkdEZJ+Kipxrn59BvdrVGT0k\ns8vs7IsSLZE0oXqGIpJOnpy0jM+XbeJPg7rTokHtqMOJjBItkTTw6aINqmcoImlj1eZd3Pn6XI7v\n0oJzj2wXdTiRUqIlkuIKCou48ZVZqmcoImnB3bnhxRkUOdx2Tq+sn1BZiZZIiovVMxw5UPUMRST1\njZu+mnfn5XLN6V1p36xe1OFETomWSAqL1TM85uDmnNFT9QxFJLVt3JHPja/Mpk/7Jlx6bMeow0kJ\nSrREUtjeeoZDumd997uIpL5bXp3N1l17uPO8XpqCJqRESyRFxdcz7Na6UdThiIjs03vz1vHCl6v4\nxYDOarPiKNESSUGqZygi6WRHXgEjX5xJ55b1+eVJh0QdTkqpVKJlZk3M7Dkzm2tmc8zsGDNrZmYT\nzGxBeN80UcGKZIvXZ6qeoYikj7vfnMfqLbu487ze1K6hi3biVbZH62/AG+7eDegDzAFGABPdvQsw\nMXwuImW0K7+QW19TPUMRSQ9fLN/EmE+XcvHRHejbsVnU4aScCidaZtYYOAF4HMDd8919MzAUGBNu\nNgY4u7JBimQT1TMUkXSRX1DEiOe/onWjOvzh9K5Rh5OSKtOj1QnIBf5lZl+a2WNmVh9o5e5rwm1y\ngBKrSJrZcDObamZTc3NzKxGGSOZQPUMRSScPvLeQ+Wu3c+s5PWlYp2bU4aSkyiRaNYAjgQfd/Qhg\nB8VOE7q7A17Si939EXfv6+59W7ZsWYkwRDKH6hmKSLqYv3Yb//fuQgb3actJ3UrsUxEql2itBFa6\n+6Tw+XMEiddaM2sDEN6vq1yIItkhVs/w5yeqnqGIpLaCwiL+8L/pNKxTk9GDu0cdTkqrcKLl7jnA\nCjOLnZQ9GZgNjAOGhcuGAS9XKkKRLBCrZ9iuSV1+dqLqGVYVXTktUjGPf7SE6Su3cOOQHjRvUDvq\ncFJaZa86/DXwlJl9BRwO3AbcAZxqZguAU8LnIrIPsXqGNwxSPcMqpiunRcppUe527p0wn9O6t+Ks\n3m2iDifl1ajMi919GtC3hFUnV2a/Itlk807VM4xC3JXTl0Jw5TSQb2ZDgQHhZmOA94Brqz5CkdRT\nWORc+9xX1K1ZnVvO7qnSYGWgmeFFInbfhPls3bVH9Qyrnq6cFimnf3+6lKnLNvHns7pzQKM6UYeT\nFpRoiURozpqtPPnZMi4+uoNqg1U9XTktUg7LN+zkrjfmMaBrS849sl3U4aQNJVoiEVE9w8jpymmR\nMnJ3rn3+K6pXM247p5d638tBiZZIRGL1DH+veoaR0JXTImX39OTlfLp4AyMHHabpZ8qpUoPhRaRi\n4usZ/lj1DKMUu3K6FrAY+AnBD9CxZnY5sAw4P8L4RCK3avMubh8/l+8e0pwLjmofdThpR4mWSARi\n9QzvO7+P6hlGSFdOi+ybu3P9CzMocueOc3vrlGEF6NShSBVbtXkXD72/iEGqZygiKe65z1fy/vxc\nrj2jG+2b1Ys6nLSkREukit02fg6geoYiktrWbt3Nza/Opl/HZlx8dIeow0lbSrREqtCnizbw2ldB\nPcN2GlAqIinK3Rn54kzyCoq48/u9qaYhDhWmREukiqieoYiki1e+WsPbc9ZyzWld6dSiftThpDUl\nWiJV5JkpK5ibs42RqmcoIils/fY8Rr08k8PbN+Gy4zpFHU7aU6IlUgU278zn3rfmcczBzTlT9QxF\nJEW5O39+eSY78gq5+/u9dVV0AijREqkCqmcoIungla/WMH5GDled2oUurRpGHU5GUKIlkmRzc4J6\nhhepnqGIpLB123bz5/CU4fDjNY40UZRoiSSRuzN63Cwa1a3J71XPUERSVGxi0l35hdx7fh9qVFd6\nkCj6JEWSKFbP8GrVMxSRFPbCF6t4e846/nB6Vzq3bBB1OBlFiZZIksTqGXZr3VD1DEUkZa3ZsovR\nr8ziqI5N+cl3dZVhoqnWoUiSxOoZPjv8aF25IyIpyd259vkZFBQ69/xAtVeTQT1aIkkQX8/waNUz\nFJEU9eyUFXwwP5frBnajQ3NNTJoMSrREkkD1DEUk1a3ctJNbXp3NsZ2bc1F/1TJMFiVaIgn22eKg\nnuGVJ3ZWPUMRSUlFRc4fn/sKgDvPUy3DZFKiJZJABYVFjB4X1DO88sTOUYcjIlKiJyct45NFG7jh\nrO60b1Yv6nAymhItkQRSPUMRSXVL1+/g9vFzOfHQllxwVPuow8l4SrREEiRWz/Dog5upnqGIpKTC\nIucPz02nRnXjjvN6qSRYFVCiJZIgsXqGo4f0UOMlIinpXx8vYcrSTYwa3IM2jTWGtCoo0RJJANUz\nFJFUt2DtNu56cx6nHHYA5x3ZLupwsoYSLZFKcnduHDdb9QxFJGXlFxRx1X+n0aB2DW4/t7d63auQ\nEi2RSnp9Zg6fLt6geoYikrL++vZ8Zq3eyh3n9qJlw9pRh5NVlGiJVMLuPapnKCKpbcrSjTz0/iJ+\n2Lc9p/XQhTpVTbUORSrh4fcXq56hiKSs7XkF/H7sNNo1rcufBnePOpyspERLpIJWbd7Fg+8vVD1D\nEUlZN78ym1WbdjH2Z8fQoLb+y49CpU8dmll1M/vSzF4NnzczswlmtiC8b1r5MEVSj+oZikgqe2tW\nDv+duoIrT+xM347Nog4nayVijNZvgTlxz0cAE929CzAxfC6SUVTPUERSWe62PK57YQY92jbiqlN0\nNXSUKpVomdmBwCDgsbjFQ4Ex4eMxwNmVeQ+RVBNfz/BnJ6ieoYikFnfnuhe+YlteAX/94eHUqqHr\n3qJU2U//r8AfgaK4Za3cfU34OAdoVdILzWy4mU01s6m5ubmVDEOk6sTXM6xbS/UMRSS1PDtlBW/P\nWceIM7rRpVXDqMPJehVOtMzsLGCdu39e2jbu7oCXsu4Rd+/r7n1btmxZ0TBEqpTqGWYWjTGVTLN0\n/Q5ufnU23z2kOZce2zHqcITK9Wh9FxhiZkuBZ4GTzOxJYK2ZtQEI79dVOkqRFPGXsJ7hqMGqZ5gh\nNMZUMkZBYRG/HzuNGtWMe37Qh2qaciYlVDjRcvfr3P1Ad+8IXAC84+4XAeOAYeFmw4CXKx2lSAqY\nm7OV/4T1DA9ro3qG6U5jTCXT/P2dhXyxfDO3nNNLBaNTSDJGyN0BnGpmC4BTwuciaU31DDNShceY\ngsaZSmqZvGQj97+zgPOOPJAhfdpGHY7ESUii5e7vuftZ4eMN7n6yu3dx91PcfWMi3kMkSm/E6hme\neqjqGWaAyo4xDddrnKmkhC0793DVs19yULN63Di0R9ThSDGaJlZkP3bvKeSWsJ7hj1TPMFPExpgO\nBOoAjeLHmLr7Go0xlXTg7lz34les25bH8z8/VrO/pyBNriGyH7F6hqOH9KBGdf3JZAKNMZVMMXbq\nCsbPyOHq07rSp32TqMOREuh/DZF92FvPsJfqGWYJjTGVtLEodzujx83m2M7N+dkJB0cdjpRCfYwi\n+3Db+Dm4w3UDu0UdiiSJu78HvBc+3gCcHGU8ImWRV1DIb575kjo1q3Hf+YdrKocUph4tkVLE6hn+\nfEBnDmwwHtwiAAAUBElEQVRaL+pwRET2uufNecxavZU7z+tN68Z1og5H9kGJlkgJVM9QRFLV+/Nz\nefTDJVx09EGc1kMVKlKdEi2REsTqGV4/UPUMRSR1rNu2m6vHTqfLAQ24YVD3qMORMtAYLZFi4usZ\nDuylX4sikhoKi5yrnp3G9rw9PHlFP+rU1I/AdKBES6QY1TMUkVT0j3cW8MmiDdx1Xm+6tVYZsHSh\nU4cicWL1DC/sr3qGIpI6Plm4nr9NXMC5R7TjB30PjDocKQclWiIh1TMUkVS0bttufvPsNA5uUZ+b\nz+6pnvY0o0RLJBRfz7BpfdUzFJHoFRY5v/tvMC7rgQu/Q32V2Ek7+sZEUD1DEUlN97+zkI8XBuOy\nurZuGHU4UgFKtET4up7hMz89WvUMRSQlfLJwPX+dOF/jstKc/keRrBdfz/CYzqpnKCLR07iszKEe\nLcl6t6ueoYikkPhxWU9d0V/jstKcerQkq322eAOvfrWGK09UPUMRSQ33vDWPjxdu4KYhPTUuKwMo\n0ZKsVVjk3PjKbNo1qcuVJ6qeoYhE742ZOTz43iJ+1K895x/VPupwJAGUaEnWembycuas2ap6hiKS\nEhau2841/5tOn/ZNGD2kR9ThSIIo0ZKstHlnPve8NY/+nVTPUESitz2vgCuf/JzaNarx4IVHUruG\nfvxlCo2wk6wUq2c4eojqGYpItNydPz43ncW523ny8v60bVI36pAkgdSjJVlnbs5Wnpy0XPUMRSQl\nPPrhYsbPyOHaM7px7CEtog5HEkyJlmSVWD3DBrVrqJ6hiETuk0XrueP1uQzs1ZrhJxwcdTiSBEq0\nJKvE6hlec5rqGYpItFZs3Mmvnv6Sg1s24K7v99EwhgylREuyhuoZikiq2J5XwBVjplJQWMSjl/Sl\ngSYlzVj6ZiVrPPKB6hmKSPSKipyrnp3GwtztPPGTo+jUon7UIUkS6X8byQqrNu/igfcWMrBXa9Uz\nFJFI3fPWPN6es5Y/DTqM47u0jDocSTIlWpIVYvUMrx94WNShiEgWe3naKh54bxE/6ncQw47tGHU4\nUgWUaEnGm6R6hiKSAqat2MwfnvuKfp2acaPm8MsaSrQkoxUWOaNfmU3bxnVUz1BEIpOzZTfD/z2V\nAxrW5qGLvkOtGvrvN1vom5aMFqtnOHJQd9UzFJFI7Mgr4Ip/T2FHXgGPDetLM00tk1UqnGiZWXsz\ne9fMZpvZLDP7bbi8mZlNMLMF4X3TxIUrUnabd+Zzr+oZikiECgqL+NXTXzBnzTbu//GRdGutahTZ\npjI9WgXA1e7eHTga+KWZdQdGABPdvQswMXwuUuX+MmE+W1TPUEQi4u6MGjeLd+flctPQHnyv2wFR\nhyQRqHCi5e5r3P2L8PE2YA7QDhgKjAk3GwOcXdkgRcprXs421TOUfVKvvCTbwx8s5qlJy7nyxM5c\n2L9D1OFIRBIyRsvMOgJHAJOAVu6+JlyVA7Qq5TXDzWyqmU3Nzc1NRBgiQFjP8JVZqmco+6NeeUma\ncdNXc8frcxncpy1/PL1r1OFIhCqdaJlZA+B54Cp33xq/zt0d8JJe5+6PuHtfd+/bsqUmbJPEeWNm\nDp8s2sDVqmco+6BeeUmWzxZv4Jqx0zmqY1Pu/n5vqlXT0IVsVqlEy8xqEiRZT7n7C+HitWbWJlzf\nBlhXuRBFyi6+nuGPVc9Qyki98pIoM1dt4YoxUzmoeT0eubgvdWrqaudsV5mrDg14HJjj7vfFrRoH\nDAsfDwNernh4IuUTq2c4anAP1TOUMlGvvCTKkvU7GPbPyTSuW5P/XN5PPeoCVK5H67vAxcBJZjYt\nvA0E7gBONbMFwCnhc5GkUz1DKS/1ykui5GzZzUWPTcKBf1/ejzaN60YdkqSIGhV9obt/BJR24vnk\niu5XpKJUz1DKowy98negXnkpg80787nkn5PYvDOfZ4cfQ+eWDaIOSVJIhRMtkVQSq2f425O7qJ6h\nlFWsV36GmU0Ll11PkGCNNbPLgWXA+RHFJ2lge14BP3liCkvX7+SJy46i14GNow5JUowSLUl7qmco\nFaFeeamsHXkF/ORfk/lq5RYeuPBIju3cIuqQJAVptLCkvVg9w+sHHaZ6hiJSJXbmF3DZE1P4Yvlm\n/n7BEZzeQ2W+pGRKtCStbdm5Z289w0G92kQdjohkgV35hVwxZipTlm7kvvP7MKi32h4pnU4dSlr7\ny9uqZygiVWf3nkKG/2cqny7ewH3n92Ho4e2iDklSnHq0JG3Ny9nGfz5bxo/7H6R6hiKSdDvzC/jp\nv6fy4YL13Hleb8454sCoQ5I0oB4tSUvx9QyvPlV1xEQkubbs2sNlT0zhy+WbuPv7vflB3/ZRhyRp\nQomWpKU3ZwX1DG8a2kOzL4tIUm3YnsfFj09mwbpt3P/jIxmo8aBSDkq0JO3s3lPIza+qnqGIJN+a\nLbu46LFJrNy0i0cu6cv3uh4QdUiSZpRoSdqJ1TN8+qf9Vc9QRJJmce52LvnnZDbv3MO/L+tH/4NV\n2kvKT4mWpJXVcfUMNTmgiCTL5CUbGf6fqVQz46kr+tOnfZOoQ5I0pURL0sptqmcoIkk2bvpqrhk7\nnQOb1uVfPzmKDs3rRx2SpDElWpI2YvUMf6N6hiKSBO7OA+8t4u4359GvYzMeueQ7NKmni22kcpRo\nSVqIr2f4c9UzFJEEyyso5M8vzeK/U1cw9PC23PX93tSuoZJeUnlKtCQtxOoZ3v/jI1TPUEQSKmfL\nbn7+1Od8uXwzvz7pEH5/6qGqNCEJo0RLUp7qGYpIskxespFfPPUFO/MLePDCIzlTbYwkmBItSXmx\neoajBqueoYgkRlGR8/hHS7jzjbm0b1aPp3/an0NbNYw6LMlASrQkpcXXM+zeVvUMRaTy1m/P4+qx\n03l/fi6ndW/F3T/oQ+O6NaMOSzKUEi1JWapnKCKJ9tGC9fxu7DS27NrDzUN7cNHRHdRTLkmlREtS\nluoZikiibM8r4I7X5/DkZ8s55IAG/PuyfhzWRr3kknxKtCQl7d5TyC2vqZ6hiFTeJwvX84fnvmL1\nll1ccVwnrj6tq65eliqjREtS0iMfLGblJtUzFJGKW789jzten8tzn6/k4Bb1ee7KY/hOh2ZRhyVZ\nRomWpJxYPcMze6qeoYiUX0FhEU9+tox7J8xn955CrjyxM1ed0oU6NdWLJVVPiZaknNtfn6t6hiJS\nbu7Ou/PWcdcb85ibs43jDmnB6CE9OOSABlGHJllMiZaklEmLN/DK9NX85uQutG+meoYiUjafL9vI\nna/PY/LSjRzUrB4PXHgkZ/ZsrSsKJXJKtCRlqJ6hiJSHu/Ppog08+P4iPlywnpYNa3Pz2T35Yd/2\n1KqhsZ2SGpRoScp4dorqGYrI/uUXFPHW7Bwe/WAx01duoWXD2ow4sxuXHNOBerX035qkFv2LlJSw\nZece7nlT9QxFpHQrNu7kmcnLGTt1Jeu359GxeT1uO6cX5x7ZTgPdJWUp0ZKUoHqGIlKSDdvzGD8z\nh1emr2bK0o0YcFK3VlzY/yBOOLQl1aupvZDUpkRLIqd6hiIS4+4sWLed9+fl8u68dUxaspHCIueQ\nAxrwu1MO5fvfOZC2TepGHaZ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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f0b67fb3ef0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "from scipy.stats import norm\n",
    "import time \n",
    "\n",
    "#Black and Scholes\n",
    "def BlackScholesFormula(type, S0, K, r, sigma, T):\n",
    "    dtmp1 = np.log(S0 / K)\n",
    "    dtmp2 = 1.0/(sigma * np.sqrt(T))\n",
    "    sigsq = 0.5 * sigma * sigma\n",
    "    d1 =  dtmp2 * (dtmp1 + T * (r + sigsq))\n",
    "    d2 =  dtmp2 * (dtmp1 + T * (r - sigsq))\n",
    "    if type==\"C\":\n",
    "        return S0 * norm.cdf(d1) - K * np.exp(-r * T) * norm.cdf(d2)\n",
    "    else:\n",
    "        return K * np.exp(-r * T) * stats.cdf(-d2) - S0 * norm.cdf(-d1)\n",
    "\n",
    "K = 100.0\n",
    "r = 0.05\n",
    "sigma = 0.35\n",
    "T = 1\n",
    "putCall ='C'\n",
    "\n",
    "Smin = 0.0\n",
    "Smax = 200.0\n",
    "ns = 201\n",
    "Ss = np.linspace(Smin, Smax, ns, endpoint=True)\n",
    "Ss = Ss[1:-1]\n",
    "#print Ss\n",
    "\n",
    "t=time.time()\n",
    "px = BlackScholesFormula(putCall, Ss, K, r, sigma, T)\n",
    "elapsed=time.time()-t\n",
    "print(px)\n",
    "print(\"Elapsed Time:\", elapsed)\n",
    "#idx = 200-1\n",
    "#print Ss[idx]\n",
    "#print px[idx]\n",
    "\n",
    "payoff = clip(Ss-K, 0.0, 1e600)\n",
    "#print \"payoff = \", payoff\n",
    "\n",
    "figure(figsize=[10, 4])\n",
    "subplot(1, 2, 1)\n",
    "plot(Ss, payoff)\n",
    "xlabel('Stock', fontsize=14);\n",
    "title('Payoff' , fontsize=16);\n",
    "\n",
    "subplot(1, 2, 2)\n",
    "plot(Ss, px)\n",
    "xlabel('Stock', fontsize=14);\n",
    "title('Analytical Solution' , fontsize=16);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ns = 201\n",
      "dx = 1.0\n",
      "sigma = 0.35\n",
      "by CFL, dt < 0.00020408163265306126\n",
      "mt_min ~=  4900\n"
     ]
    }
   ],
   "source": [
    "from scipy import sparse\n",
    "import scipy.sparse.linalg.dsolve as linsolve\n",
    "\n",
    "class BS_FDM_explicit:\n",
    "  def __init__(self, \n",
    "               r, \n",
    "               sigma, \n",
    "               maturity, \n",
    "               Smin, \n",
    "               Smax, \n",
    "               Fl, \n",
    "               Fu, \n",
    "               payoff, \n",
    "               nt, \n",
    "               ns):\n",
    "    self.r  = r \n",
    "    self.sigma = sigma \n",
    "    self.maturity  = maturity\n",
    "\n",
    "    self.Smin = Smin     \n",
    "    self.Smax = Smax\n",
    "    self.Fl = Fl        \n",
    "    self.Fu = Fu\n",
    "\n",
    "    self.nt  = nt\n",
    "    self.ns  = ns\n",
    "    \n",
    "    self.dt = float(maturity)/nt\n",
    "    self.dx = float(Smax-Smin)/(ns+1)\n",
    "    self.xs = Smin/self.dx\n",
    "\n",
    "    self.u = empty((nt + 1, ns))\n",
    "    self.u[0,:] = payoff\n",
    "\n",
    "    ## Building Coefficient matrix:        \n",
    "    A = sparse.lil_matrix((self.ns, self.ns))\n",
    "\n",
    "    for j in range(0, self.ns):\n",
    "      xd = j + 1 + self.xs\n",
    "      sx = self.sigma * xd\n",
    "      sxsq = sx * sx\n",
    "      \n",
    "      dtmp1 = self.dt * sxsq\n",
    "      dtmp2 = self.dt * self.r\n",
    "      A[j,j] = 1.0 - dtmp1 - dtmp2\n",
    "      dtmp1 = 0.5 * dtmp1\n",
    "      dtmp2 = 0.5 * dtmp2 * xd\n",
    "      if j > 0:\n",
    "        A[j,j-1] = dtmp1 - dtmp2\n",
    "      if j < self.ns - 1:\n",
    "        A[j,j+1] = dtmp1 + dtmp2\n",
    "\n",
    "    self.A = A.tocsr()\n",
    "\n",
    "    ### Building bc_coef:\n",
    "    nxl = 1 + self.xs\n",
    "    sxl = self.sigma * nxl\n",
    "    nxu = self.ns + self.xs\n",
    "    sxu = self.sigma * nxu\n",
    "    \n",
    "    self.blcoef = 0.5 * self.dt * (sxl * sxl - self.r * nxl)\n",
    "    self.bucoef = 0.5 * self.dt * (sxu * sxu + self.r * nxu)\n",
    "    \n",
    "  def solve(self):\n",
    "    for i in range(0, m):\n",
    "        self.u[i+1,:]          = self.A * self.u[i,:]\n",
    "        self.u[i+1,0]         += self.blcoef * self.Fl[i]\n",
    "        self.u[i+1,self.ns-1] += self.bucoef * self.Fu[i]\n",
    "\n",
    "    return self.u\n",
    "\n",
    "dx = (Smax - Smin)/(ns-1)\n",
    "print(\"ns =\", ns)\n",
    "print(\"dx =\", dx)\n",
    "print(\"sigma =\", sigma)\n",
    "dt_max = dx*dx/(sigma*sigma*Smax*Smax)\n",
    "print(\"by CFL, dt <\", dt_max)\n",
    "mt_min = int(T/dt_max)+1\n",
    "print(\"mt_min ~= \", mt_min)\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Elapsed Time1: 0.04476022720336914\n",
      "[  4.07772413e-23   1.58474399e-20   1.73965059e-18   8.74535265e-17\n",
      "   2.49568562e-15   4.58093482e-14   5.87935695e-13   5.60665034e-12\n",
      "   4.16258352e-11   2.49732862e-10   1.24803027e-09   5.32745593e-09\n",
      "   1.98368066e-08   6.55753768e-08   1.95342914e-07   5.31044478e-07\n",
      "   1.33172299e-06   3.10914675e-06   6.81146334e-06   1.40982161e-05\n",
      "   2.77310391e-05   5.21031331e-05   9.39269921e-05   1.63093911e-04\n",
      "   2.73710967e-04   4.45312032e-04   7.04229658e-04   1.08510517e-03\n",
      "   1.63250565e-03   2.40260928e-03   3.46491532e-03   4.90393153e-03\n",
      "   6.82079120e-03   9.33475279e-03   1.25845383e-02   1.67294712e-02\n",
      "   2.19503804e-02   2.84502447e-02   3.64545569e-02   4.62113982e-02\n",
      "   5.79912176e-02   7.20863197e-02   8.88100703e-02   1.08495836e-01\n",
      "   1.31495676e-01   1.58178815e-01   1.88929915e-01   2.24147190e-01\n",
      "   2.64240376e-01   3.09628608e-01   3.60738215e-01   4.18000478e-01\n",
      "   4.81849371e-01   5.52719318e-01   6.31042986e-01   7.17249139e-01\n",
      "   8.11760571e-01   9.14992136e-01   1.02734890e+00   1.14922439e+00\n",
      "   1.28099903e+00   1.42303865e+00   1.57569319e+00   1.73929554e+00\n",
      "   1.91416052e+00   2.10058402e+00   2.29884230e+00   2.50919136e+00\n",
      "   2.73186658e+00   2.96708236e+00   3.21503195e+00   3.47588741e+00\n",
      "   3.74979962e+00   4.03689847e+00   4.33729310e+00   4.65107220e+00\n",
      "   4.97830447e+00   5.31903906e+00   5.67330611e+00   6.04111737e+00\n",
      "   6.42246685e+00   6.81733144e+00   7.22567170e+00   7.64743254e+00\n",
      "   8.08254400e+00   8.53092204e+00   8.99246929e+00   9.46707585e+00\n",
      "   9.95462009e+00   1.04549694e+01   1.09679810e+01   1.14935027e+01\n",
      "   1.20313735e+01   1.25814247e+01   1.31434803e+01   1.37173575e+01\n",
      "   1.43028680e+01   1.48998182e+01   1.55080097e+01   1.61272404e+01\n",
      "   1.67573046e+01   1.73979938e+01   1.80490973e+01   1.87104022e+01\n",
      "   1.93816947e+01   2.00627597e+01   2.07533816e+01   2.14533451e+01\n",
      "   2.21624346e+01   2.28804354e+01   2.36071337e+01   2.43423168e+01\n",
      "   2.50857736e+01   2.58372947e+01   2.65966726e+01   2.73637021e+01\n",
      "   2.81381802e+01   2.89199068e+01   2.97086840e+01   3.05043172e+01\n",
      "   3.13066146e+01   3.21153874e+01   3.29304500e+01   3.37516204e+01\n",
      "   3.45787194e+01   3.54115716e+01   3.62500049e+01   3.70938507e+01\n",
      "   3.79429437e+01   3.87971226e+01   3.96562292e+01   4.05201090e+01\n",
      "   4.13886111e+01   4.22615879e+01   4.31388956e+01   4.40203938e+01\n",
      "   4.49059452e+01   4.57954165e+01   4.66886774e+01   4.75856011e+01\n",
      "   4.84860639e+01   4.93899457e+01   5.02971292e+01   5.12075005e+01\n",
      "   5.21209489e+01   5.30373665e+01   5.39566484e+01   5.48786928e+01\n",
      "   5.58034006e+01   5.67306756e+01   5.76604244e+01   5.85925560e+01\n",
      "   5.95269824e+01   6.04636179e+01   6.14023794e+01   6.23431862e+01\n",
      "   6.32859600e+01   6.42306247e+01   6.51771066e+01   6.61253340e+01\n",
      "   6.70752380e+01   6.80267498e+01   6.89798084e+01   6.99343360e+01\n",
      "   7.08903187e+01   7.18475437e+01   7.28064348e+01   7.37654549e+01\n",
      "   7.47290004e+01   7.56839518e+01   7.66677184e+01   7.75754125e+01\n",
      "   7.86933146e+01   7.92584878e+01   8.12530925e+01   7.96565621e+01\n",
      "   8.68588532e+01   7.30811291e+01   1.07990554e+02   3.30302708e+01\n",
      "   1.99034047e+02  -1.48175809e+02   5.65252293e+02  -8.46307420e+02\n",
      "   1.86517555e+03  -3.16154189e+03   5.84112652e+03  -9.68847278e+03\n",
      "   1.60815607e+04  -2.49405793e+04   3.75380247e+04  -5.31645025e+04\n",
      "   7.17829508e+04  -9.05316708e+04   1.06627979e+05  -1.14453853e+05\n",
      "   1.09689360e+05  -8.78344405e+04   4.96751069e+04]\n",
      "Elapsed Time2: 0.046859025955200195\n",
      "Elapsed Time3: 0.045465946197509766\n"
     ]
    },
    {
     "data": {
      "image/png": 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lZcc6NU8np2AnDeum0LR+HZqmK2EUkejI21HMg+8tpH+HJlx6TIf9X1CDlDCKiHd7zDD6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\nkqSOHDBKkiRJkjr6O1xHLL/PH1uTAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f0b94415860>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = ns-2\n",
    "X = linspace(0.0, Smax, n+2)\n",
    "X = X[1:-1]\n",
    "\n",
    "payoff = clip(X-K, 0.0, 1e600)\n",
    "#print \"payoff = \", payoff\n",
    "  \n",
    "m = 4555 \n",
    "Fl = zeros((m+1,))\n",
    "Fu = Smax - K*exp(-r * linspace(0.0, T, m+1))\n",
    "    \n",
    "t = time.time()\n",
    "bs1 = BS_FDM_explicit(r, sigma, T, Smin, Smax, Fl, Fu, payoff, m, n)\n",
    "px_fd_mat = bs1.solve()\n",
    "elapsed=time.time()-t\n",
    "print(\"Elapsed Time1:\", elapsed)\n",
    "\n",
    "##print  px_fd_mat.shape\n",
    "nrow = len(px_fd_mat[:,1])\n",
    "print(px_fd_mat[nrow-1,:])\n",
    "\n",
    "figure(figsize=[15, 4]);\n",
    "subplot(1, 3, 1)\n",
    "plot(X, px_fd_mat[nrow-1,:])\n",
    "xlabel('Stock', fontsize=14);\n",
    "title('nx = %.f'%m, fontsize=16);\n",
    "\n",
    "m = 4560\n",
    "Fl = zeros((m+1,))\n",
    "Fu = Smax - K*exp(-r * linspace(0.0, T, m+1))\n",
    "\n",
    "t = time.time()\n",
    "bs2 = BS_FDM_explicit(r, sigma, T, Smin, Smax, Fl, Fu, payoff, m, n)\n",
    "px_fd_mat = bs2.solve()\n",
    "elapsed=time.time()-t\n",
    "print(\"Elapsed Time2:\", elapsed)\n",
    "\n",
    "subplot(1, 3, 2)\n",
    "nrow = len(px_fd_mat[:,1])\n",
    "plot(X, px_fd_mat[nrow-1,:])\n",
    "xlabel('Stock', fontsize=14);\n",
    "title('nx = %.f'%m, fontsize=16);\n",
    "\n",
    "m = 4565\n",
    "Fl = zeros((m+1,))\n",
    "Fu = Smax - K*exp(-r * linspace(0.0, T, m+1))\n",
    "    \n",
    "t = time.time()\n",
    "bs3 = BS_FDM_explicit(r, sigma, T, Smin, Smax, Fl, Fu, payoff, m, n)\n",
    "px_fd_mat = bs3.solve()\n",
    "elapsed=time.time()-t\n",
    "print(\"Elapsed Time3:\", elapsed)\n",
    "\n",
    "subplot(1, 3, 3)\n",
    "nrow = len(px_fd_mat[:,1])\n",
    "plot(X, px_fd_mat[nrow-1,:])\n",
    "xlabel('Stock', fontsize=14);\n",
    "title('nx = %.f' % m, fontsize=16);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Elapsed Time1: 0.04440593719482422\n",
      "[  1.01704397e-28   5.07283379e-26   7.18984489e-24   4.68830891e-22\n",
      "   1.74072660e-20   4.16234095e-19   6.95467412e-18   8.61238308e-17\n",
      "   8.26824165e-16   6.37765319e-15   4.06932208e-14   2.20044937e-13\n",
      "   1.02914359e-12   4.23570647e-12   1.55695816e-11   5.17651596e-11\n",
      "   1.57382418e-10   4.41714179e-10   1.15387760e-09   2.82564962e-09\n",
      "   6.52727942e-09   1.43015144e-08   2.98649516e-08   5.96924592e-08\n",
      "   1.14628045e-07   2.12191890e-07   3.79775698e-07   6.58934313e-07\n",
      "   1.11098825e-06   1.82414722e-06   2.92234764e-06   4.57596663e-06\n",
      "   7.01453177e-06   1.05414905e-05   1.55510379e-05   2.25469294e-05\n",
      "   3.21631260e-05   4.51860447e-05   6.25781067e-05   8.55022075e-05\n",
      "   1.15346670e-04   1.53750189e-04   2.02626238e-04   2.64186378e-04\n",
      "   3.40961911e-04   4.35823312e-04   5.51996896e-04   6.93078223e-04\n",
      "   8.63041768e-04   1.06624646e-03   1.30743677e-03   1.59173903e-03\n",
      "   1.92465290e-03   2.31203775e-03   2.76009409e-03   3.27534000e-03\n",
      "   3.86458272e-03   4.53488566e-03   5.29353117e-03   6.14797920e-03\n",
      "   7.10582258e-03   8.17473908e-03   9.36244092e-03   1.06766222e-02\n",
      "   1.21249047e-02   1.37147828e-02   1.54535680e-02   1.73483331e-02\n",
      "   1.94058580e-02   2.16325751e-02   2.40345184e-02   2.66172723e-02\n",
      "   2.93859255e-02   3.23450254e-02   3.54985376e-02   3.88498080e-02\n",
      "   4.24015289e-02   4.61557096e-02   5.01136502e-02   5.42759210e-02\n",
      "   5.86423455e-02   6.32119876e-02   6.79831445e-02   7.29533428e-02\n",
      "   7.81193390e-02   8.34771256e-02   8.90219392e-02   9.47482747e-02\n",
      "   1.00649901e-01   1.06719884e-01   1.12950607e-01   1.19333799e-01\n",
      "   1.25860566e-01   1.32521423e-01   1.39306324e-01   1.46204708e-01\n",
      "   1.53205527e-01   1.60297295e-01   1.67468123e-01   1.74705766e-01\n",
      "   1.81997660e-01   1.89330975e-01   1.96692647e-01   2.04069434e-01\n",
      "   2.11447949e-01   2.18814709e-01   2.26156179e-01   2.33458809e-01\n",
      "   2.40709078e-01   2.47893532e-01   2.54998828e-01   2.62011762e-01\n",
      "   2.68919314e-01   2.75708677e-01   2.82367293e-01   2.88882878e-01\n",
      "   2.95243460e-01   3.01437400e-01   3.07453417e-01   3.13280617e-01\n",
      "   3.18908511e-01   3.24327033e-01   3.29526565e-01   3.34497943e-01\n",
      "   3.39232480e-01   3.43721975e-01   3.47958722e-01   3.51935520e-01\n",
      "   3.55645682e-01   3.59083036e-01   3.62241932e-01   3.65117244e-01\n",
      "   3.67704370e-01   3.69999230e-01   3.71998265e-01   3.73698434e-01\n",
      "   3.75097207e-01   3.76192563e-01   3.76982977e-01   3.77467420e-01\n",
      "   3.77645341e-01   3.77516663e-01   3.77081772e-01   3.76341503e-01\n",
      "   3.75297128e-01   3.73950345e-01   3.72303264e-01   3.70358392e-01\n",
      "   3.68118620e-01   3.65587208e-01   3.62767768e-01   3.59664251e-01\n",
      "   3.56280934e-01   3.52622398e-01   3.48693515e-01   3.44499436e-01\n",
      "   3.40045570e-01   3.35337570e-01   3.30381318e-01   3.25182908e-01\n",
      "   3.19748632e-01   3.14084962e-01   3.08198542e-01   3.02096154e-01\n",
      "   2.95784760e-01   2.89271303e-01   2.82563285e-01   2.75666908e-01\n",
      "   2.68592676e-01   2.61338692e-01   2.53939079e-01   2.46326719e-01\n",
      "   2.38712943e-01   2.30562309e-01   2.23293232e-01   2.13339863e-01\n",
      "   2.09528451e-01   1.90671166e-01   2.06361812e-01   1.43753470e-01\n",
      "   2.52562922e-01  -3.99580310e-03   4.94288821e-01  -5.19976227e-01\n",
      "   1.40031289e+00  -2.18719234e+00   4.21080703e+00  -6.85231546e+00\n",
      "   1.14799447e+01  -1.77278041e+01   2.67320910e+01  -3.78361940e+01\n",
      "   5.10864915e+01  -6.44524111e+01   7.58680118e+01  -8.14990126e+01\n",
      "   7.80272499e+01  -6.25720798e+01   3.52918730e+01]\n",
      "Elapsed Time2: 0.044289588928222656\n",
      "Elapsed Time3: 0.0435185432434082\n"
     ]
    },
    {
     "data": {
      "image/png": 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LTvO9myaSn5MR1vd2JQm/vm06qclJfP3pD/B47ZkjFJbMGWPaFbiNqr+jpSVzxkTGtqNn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\nVMyJiIiIiIiUof8PD28KX8x5E+wAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f0b977b9e10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Smax = 120\n",
    "n = ns-2\n",
    "X = linspace(0.0, Smax, n+2)\n",
    "X = X[1:-1]\n",
    "\n",
    "payoff = clip(X-K, 0.0, 1e600)\n",
    "#print \"payoff = \", payoff\n",
    "  \n",
    "m = 4570\n",
    "Fl = zeros((m+1,))\n",
    "Fu = zeros((m+1,))\n",
    "    \n",
    "t = time.time()\n",
    "bs1 = BS_FDM_explicit(r, sigma, T, Smin, Smax, Fl, Fu, payoff, m, n)\n",
    "px_fd_mat = bs1.solve()\n",
    "elapsed=time.time()-t\n",
    "print(\"Elapsed Time1:\", elapsed)\n",
    "\n",
    "##print  px_fd_mat.shape\n",
    "nrow = len(px_fd_mat[:,1])\n",
    "print(px_fd_mat[nrow-1,:])\n",
    "\n",
    "figure(figsize=[15, 4]);\n",
    "subplot(1, 3, 1)\n",
    "plot(X, px_fd_mat[nrow-1,:])\n",
    "xlabel('Stock', fontsize=14);\n",
    "title('nx = %.f'%m, fontsize=16);\n",
    "\n",
    "m = 4573\n",
    "Fl = zeros((m+1,))\n",
    "Fu = zeros((m+1,))\n",
    "\n",
    "t = time.time()\n",
    "bs2 = BS_FDM_explicit(r, sigma, T, Smin, Smax, Fl, Fu, payoff, m, n)\n",
    "px_fd_mat = bs2.solve()\n",
    "elapsed=time.time()-t\n",
    "print(\"Elapsed Time2:\", elapsed)\n",
    "\n",
    "subplot(1, 3, 2)\n",
    "nrow = len(px_fd_mat[:,1])\n",
    "plot(X, px_fd_mat[nrow-1,:])\n",
    "xlabel('Stock', fontsize=14);\n",
    "title('nx = %.f'%m, fontsize=16);\n",
    "\n",
    "m = 4580\n",
    "Fl = zeros((m+1,))\n",
    "Fu = zeros((m+1,))\n",
    "    \n",
    "t = time.time()\n",
    "bs3 = BS_FDM_explicit(r, sigma, T, Smin, Smax, Fl, Fu, payoff, m, n)\n",
    "px_fd_mat = bs3.solve()\n",
    "elapsed=time.time()-t\n",
    "print(\"Elapsed Time3:\", elapsed)\n",
    "\n",
    "subplot(1, 3, 3)\n",
    "nrow = len(px_fd_mat[:,1])\n",
    "plot(X, px_fd_mat[nrow-1,:])\n",
    "xlabel('Stock', fontsize=14);\n",
    "title('nx = %.f' % m, fontsize=16);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "from scipy import sparse\n",
    "import scipy.sparse.linalg.dsolve as linsolve\n",
    "\n",
    "class BS_FDM_implicit:\n",
    "  def __init__(self, \n",
    "               r, \n",
    "               sigma, \n",
    "               maturity, \n",
    "               Smin, \n",
    "               Smax, \n",
    "               Fl, \n",
    "               Fu, \n",
    "               payoff, \n",
    "               nt, \n",
    "               ns):\n",
    "    self.r  = r \n",
    "    self.sigma = sigma \n",
    "    self.maturity  = maturity\n",
    "\n",
    "    self.Smin = Smin     \n",
    "    self.Smax = Smax\n",
    "    self.Fl = Fl        \n",
    "    self.Fu = Fu\n",
    "\n",
    "    self.nt  = nt\n",
    "    self.ns  = ns\n",
    "    \n",
    "    self.dt = float(maturity)/nt\n",
    "    self.dx = float(Smax-Smin)/(ns+1)\n",
    "    self.xs = Smin/self.dx\n",
    "\n",
    "    self.u = empty((nt + 1, ns))\n",
    "    self.u[0,:] = payoff\n",
    "\n",
    "    ## Building Coefficient matrix:        \n",
    "    A = sparse.lil_matrix((self.ns, self.ns))\n",
    "\n",
    "    for j in range(0, self.ns):\n",
    "      xd = j + 1 + self.xs\n",
    "      sx = self.sigma * xd\n",
    "      sxsq = sx * sx\n",
    "      \n",
    "      dtmp1 = self.dt * sxsq\n",
    "      dtmp2 = self.dt * self.r\n",
    "      A[j,j] = 1.0 + dtmp1 + dtmp2\n",
    "        \n",
    "      dtmp1 = -0.5 * dtmp1\n",
    "      dtmp2 = -0.5 * dtmp2 * xd\n",
    "      if j > 0:\n",
    "        A[j,j-1] = dtmp1 - dtmp2\n",
    "      if j < self.ns - 1:\n",
    "        A[j,j+1] = dtmp1 + dtmp2\n",
    "\n",
    "    self.A = linsolve.splu(A)\n",
    "    self.rhs = empty((self.ns, ))\n",
    "    \n",
    "    ### Building bc_coef:\n",
    "    nxl = 1 + self.xs\n",
    "    sxl = self.sigma * nxl\n",
    "    nxu = self.ns + self.xs\n",
    "    sxu = self.sigma * nxu\n",
    "    \n",
    "    self.blcoef = 0.5 * self.dt * (- sxl * sxl + self.r * nxl)\n",
    "    self.bucoef = 0.5 * self.dt * (- sxu * sxu - self.r * nxu)    \n",
    "    \n",
    "  def solve(self):\n",
    "    for i in range(0, m):\n",
    "        self.rhs[:] = self.u[i,:]\n",
    "        self.rhs[0]         -= self.blcoef * self.Fl[i]\n",
    "        self.rhs[self.ns-1] -= self.bucoef * self.Fu[i]\n",
    "        self.u[i+1,:] = self.A.solve(self.rhs)\n",
    "\n",
    "    return self.u\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/opt/anaconda3/lib/python3.6/site-packages/scipy/sparse/linalg/dsolve/linsolve.py:253: SparseEfficiencyWarning: splu requires CSC matrix format\n",
      "  warn('splu requires CSC matrix format', SparseEfficiencyWarning)\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Elapsed Time1: 0.07251667976379395\n",
      "Elapsed Time2: 0.017630577087402344\n",
      "Elapsed Time3: 0.004076242446899414\n",
      "[  2.98641926e-21   7.95294389e-19   6.07132757e-17   2.16141936e-15\n",
      "   4.46125332e-14   6.06271497e-13   5.90512355e-12   4.38258315e-11\n",
      "   2.59592502e-10   1.27238057e-09   5.31142971e-09   1.93277995e-08\n",
      "   6.24868999e-08   1.82323286e-07   4.86411656e-07   1.19954343e-06\n",
      "   2.75978915e-06   5.97002567e-06   1.22239975e-05   2.38272713e-05\n",
      "   4.44331425e-05   7.96114517e-05   1.37564347e-04   2.29997480e-04\n",
      "   3.73148280e-04   5.88965351e-04   9.06425082e-04   1.36296400e-03\n",
      "   2.00599845e-03   2.89449764e-03   4.10057163e-03   5.71103357e-03\n",
      "   7.82889451e-03   1.05747499e-02   1.40880193e-02   1.85280050e-02\n",
      "   2.40747384e-02   3.09295924e-02   3.93156382e-02   4.94777381e-02\n",
      "   6.16823668e-02   7.62171638e-02   9.33902216e-02   1.13529124e-01\n",
      "   1.36979747e-01   1.64104854e-01   1.95282487e-01   2.30904205e-01\n",
      "   2.71373176e-01   3.17102172e-01   3.68511464e-01   4.26026688e-01\n",
      "   4.90076661e-01   5.61091218e-01   6.39499062e-01   7.25725659e-01\n",
      "   8.20191211e-01   9.23308698e-01   1.03548203e+00   1.15710433e+00\n",
      "   1.28855625e+00   1.43020460e+00   1.58240089e+00   1.74548019e+00\n",
      "   1.91976005e+00   2.10553959e+00   2.30309872e+00   2.51269751e+00\n",
      "   2.73457569e+00   2.96895229e+00   3.21602540e+00   3.47597205e+00\n",
      "   3.74894819e+00   4.03508884e+00   4.33450820e+00   4.64730004e+00\n",
      "   4.97353796e+00   5.31327594e+00   5.66654874e+00   6.03337257e+00\n",
      "   6.41374561e+00   6.80764873e+00   7.21504618e+00   7.63588627e+00\n",
      "   8.07010218e+00   8.51761268e+00   8.97832294e+00   9.45212526e+00\n",
      "   9.93889993e+00   1.04385160e+01   1.09508320e+01   1.14756967e+01\n",
      "   1.20129500e+01   1.25624237e+01   1.31239419e+01   1.36973221e+01\n",
      "   1.42823756e+01   1.48789083e+01   1.54867214e+01   1.61056120e+01\n",
      "   1.67353733e+01   1.73757958e+01   1.80266675e+01   1.86877743e+01\n",
      "   1.93589007e+01   2.00398302e+01   2.07303457e+01   2.14302298e+01\n",
      "   2.21392656e+01   2.28572363e+01   2.35839265e+01   2.43191215e+01\n",
      "   2.50626084e+01   2.58141760e+01   2.65736149e+01   2.73407182e+01\n",
      "   2.81152810e+01   2.88971013e+01   2.96859798e+01   3.04817200e+01\n",
      "   3.12841285e+01   3.20930149e+01   3.29081921e+01   3.37294765e+01\n",
      "   3.45566877e+01   3.53896488e+01   3.62281863e+01   3.70721305e+01\n",
      "   3.79213151e+01   3.87755775e+01   3.96347584e+01   4.04987026e+01\n",
      "   4.13672580e+01   4.22402765e+01   4.31176133e+01   4.39991273e+01\n",
      "   4.48846808e+01   4.57741397e+01   4.66673732e+01   4.75642541e+01\n",
      "   4.84646585e+01   4.93684656e+01   5.02755581e+01   5.11858218e+01\n",
      "   5.20991457e+01   5.30154219e+01   5.39345454e+01   5.48564144e+01\n",
      "   5.57809297e+01   5.67079952e+01   5.76375175e+01   5.85694058e+01\n",
      "   5.95035723e+01   6.04399313e+01   6.13784001e+01   6.23188981e+01\n",
      "   6.32613474e+01   6.42056721e+01   6.51517989e+01   6.60996564e+01\n",
      "   6.70491758e+01   6.80002898e+01   6.89529337e+01   6.99070444e+01\n",
      "   7.08625609e+01   7.18194239e+01   7.27775760e+01   7.37369617e+01\n",
      "   7.46975269e+01   7.56592194e+01   7.66219884e+01   7.75857847e+01\n",
      "   7.85505608e+01   7.95162703e+01   8.04828685e+01   8.14503119e+01\n",
      "   8.24185583e+01   8.33875669e+01   8.43572980e+01   8.53277132e+01\n",
      "   8.62987752e+01   8.72704478e+01   8.82426958e+01   8.92154852e+01\n",
      "   9.01887830e+01   9.11625570e+01   9.21367761e+01   9.31114100e+01\n",
      "   9.40864295e+01   9.50618059e+01   9.60375116e+01   9.70135196e+01\n",
      "   9.79898040e+01   9.89663392e+01   9.99431006e+01   1.00920064e+02\n",
      "   1.01897207e+02   1.02874505e+02   1.03851938e+02]\n"
     ]
    },
    {
     "data": {
      "image/png": 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514ptjO8fypm92zodR02cSH30+vdx7M4o4LObR9PITzfkRcQdEtLzeO37OC4Y1J5Tw9s4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\nSZJ6xEGcJEmSJPXI/wO1NJ6zfMmMPgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f0b944156d8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Smax = 200\n",
    "n = ns-2\n",
    "X = linspace(0.0, Smax, n+2)\n",
    "X = X[1:-1]\n",
    "\n",
    "payoff = clip(X-K, 0.0, 1e600)\n",
    "  \n",
    "m = 4555 \n",
    "Fl = zeros((m+1,))\n",
    "Fu = Smax - K*exp(-r * linspace(0.0, T, m+1))\n",
    "    \n",
    "t = time.time()\n",
    "bs1 = BS_FDM_implicit(r, sigma, T, Smin, Smax, Fl, Fu, payoff, m, n)\n",
    "px_fd_mat = bs1.solve()\n",
    "elapsed=time.time()-t\n",
    "print(\"Elapsed Time1:\", elapsed)\n",
    "\n",
    "#print  px_fd_mat.shape\n",
    "\n",
    "figure(figsize=[15, 4]);\n",
    "subplot(1, 3, 1)\n",
    "nrow = len(px_fd_mat[:,1])\n",
    "plot(X, px_fd_mat[nrow-1,:])\n",
    "xlabel('Stock', fontsize=14);\n",
    "title('nx = %.f'%m,  fontsize=16);\n",
    "\n",
    "m = 1000\n",
    "Fl = zeros((m+1,))\n",
    "Fu = Smax - K*exp(-r * linspace(0.0, T, m+1))\n",
    "\n",
    "t = time.time()\n",
    "bs2 = BS_FDM_implicit(r, sigma, T, Smin, Smax, Fl, Fu, payoff, m, n)\n",
    "px_fd_mat = bs2.solve()\n",
    "elapsed=time.time()-t\n",
    "print(\"Elapsed Time2:\", elapsed)\n",
    "\n",
    "subplot(1, 3, 2)\n",
    "nrow = len(px_fd_mat[:,1])\n",
    "plot(X, px_fd_mat[nrow-1,:])\n",
    "xlabel('Stock', fontsize=14);\n",
    "title('nx = %.f'%m, fontsize=16);\n",
    "\n",
    "m = 100\n",
    "Fl = zeros((m+1,))\n",
    "Fu = Smax - K*exp(-r * linspace(0.0, T, m+1))\n",
    "    \n",
    "t = time.time()\n",
    "bs3 = BS_FDM_implicit(r, sigma, T, Smin, Smax, Fl, Fu, payoff, m, n)\n",
    "px_fd_mat = bs3.solve()\n",
    "elapsed=time.time()-t\n",
    "print(\"Elapsed Time3:\", elapsed)\n",
    "\n",
    "subplot(1, 3, 3)\n",
    "nrow = len(px_fd_mat[:,1])\n",
    "plot(X, px_fd_mat[nrow-1,:])\n",
    "xlabel('Stock', fontsize=14);\n",
    "title('nx = %.f'%m, fontsize=16);\n",
    "\n",
    "\n",
    "print(px_fd_mat[nrow-1,:])\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/opt/anaconda3/lib/python3.6/site-packages/scipy/sparse/linalg/dsolve/linsolve.py:253: SparseEfficiencyWarning: splu requires CSC matrix format\n",
      "  warn('splu requires CSC matrix format', SparseEfficiencyWarning)\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Elapsed Time1: 0.07329130172729492\n",
      "Elapsed Time2: 0.018253803253173828\n",
      "Elapsed Time3: 0.004925966262817383\n",
      "[  7.86391997e-26   2.43630771e-23   2.17528767e-21   9.09456815e-20\n",
      "   2.21047505e-18   3.54181257e-17   4.06680009e-16   3.55313181e-15\n",
      "   2.47154878e-14   1.41799741e-13   6.90208717e-13   2.91621919e-12\n",
      "   1.08981438e-11   3.65882728e-11   1.11800634e-10   3.14357643e-10\n",
      "   8.20959540e-10   2.00717719e-09   4.62568425e-09   1.01075920e-08\n",
      "   2.10483475e-08   4.19585339e-08   8.03800929e-08   1.48488656e-07\n",
      "   2.65321208e-07   4.59783507e-07   7.74601788e-07   1.27138672e-06\n",
      "   2.03697314e-06   3.19118575e-06   4.89615859e-06   7.36730436e-06\n",
      "   1.08859900e-05   1.58139276e-05   2.26092360e-05   3.18440739e-05\n",
      "   4.42236828e-05   6.06066236e-05   8.20259326e-05   1.09710872e-04\n",
      "   1.45108906e-04   1.89907492e-04   2.46055265e-04   3.15782142e-04\n",
      "   4.01617915e-04   5.06408855e-04   6.33331909e-04   7.85906063e-04\n",
      "   9.68000495e-04   1.18383919e-03   1.43800172e-03   1.73541999e-03\n",
      "   2.08137073e-03   2.48146367e-03   2.94162538e-03   3.46807874e-03\n",
      "   4.06731814e-03   4.74608068e-03   5.51131338e-03   6.37013683e-03\n",
      "   7.32980557e-03   8.39766541e-03   9.58110837e-03   1.08875253e-02\n",
      "   1.23242571e-02   1.38985441e-02   1.56174758e-02   1.74879390e-02\n",
      "   1.95165671e-02   2.17096898e-02   2.40732837e-02   2.66129244e-02\n",
      "   2.93337411e-02   3.22403726e-02   3.53369270e-02   3.86269431e-02\n",
      "   4.21133563e-02   4.57984671e-02   4.96839136e-02   5.37706478e-02\n",
      "   5.80589157e-02   6.25482418e-02   6.72374173e-02   7.21244924e-02\n",
      "   7.72067733e-02   8.24808223e-02   8.79424628e-02   9.35867870e-02\n",
      "   9.94081690e-02   1.05400280e-01   1.11556106e-01   1.17867972e-01\n",
      "   1.24327569e-01   1.30925975e-01   1.37653695e-01   1.44500684e-01\n",
      "   1.51456391e-01   1.58509791e-01   1.65649421e-01   1.72863426e-01\n",
      "   1.80139595e-01   1.87465404e-01   1.94828057e-01   2.02214531e-01\n",
      "   2.09611616e-01   2.17005958e-01   2.24384105e-01   2.31732543e-01\n",
      "   2.39037742e-01   2.46286193e-01   2.53464452e-01   2.60559173e-01\n",
      "   2.67557150e-01   2.74445350e-01   2.81210949e-01   2.87841363e-01\n",
      "   2.94324281e-01   3.00647696e-01   3.06799929e-01   3.12769658e-01\n",
      "   3.18545941e-01   3.24118239e-01   3.29476437e-01   3.34610862e-01\n",
      "   3.39512300e-01   3.44172009e-01   3.48581737e-01   3.52733727e-01\n",
      "   3.56620734e-01   3.60236024e-01   3.63573389e-01   3.66627143e-01\n",
      "   3.69392131e-01   3.71863730e-01   3.74037843e-01   3.75910904e-01\n",
      "   3.77479872e-01   3.78742226e-01   3.79695962e-01   3.80339584e-01\n",
      "   3.80672096e-01   3.80692997e-01   3.80402265e-01   3.79800354e-01\n",
      "   3.78888175e-01   3.77667089e-01   3.76138892e-01   3.74305803e-01\n",
      "   3.72170448e-01   3.69735846e-01   3.67005398e-01   3.63982863e-01\n",
      "   3.60672354e-01   3.57078312e-01   3.53205495e-01   3.49058961e-01\n",
      "   3.44644053e-01   3.39966381e-01   3.35031804e-01   3.29846420e-01\n",
      "   3.24416543e-01   3.18748691e-01   3.12849567e-01   3.06726049e-01\n",
      "   3.00385168e-01   2.93834095e-01   2.87080130e-01   2.80130679e-01\n",
      "   2.72993250e-01   2.65675428e-01   2.58184871e-01   2.50529291e-01\n",
      "   2.42716442e-01   2.34754109e-01   2.26650094e-01   2.18412206e-01\n",
      "   2.10048248e-01   2.01566006e-01   1.92973243e-01   1.84277682e-01\n",
      "   1.75487001e-01   1.66608823e-01   1.57650706e-01   1.48620136e-01\n",
      "   1.39524521e-01   1.30371177e-01   1.21167330e-01   1.11920100e-01\n",
      "   1.02636504e-01   9.33234429e-02   8.39876998e-02   7.46359341e-02\n",
      "   6.52746771e-02   5.59103273e-02   4.65491469e-02   3.71972583e-02\n",
      "   2.78606405e-02   1.85451268e-02   9.25640199e-03]\n"
     ]
    },
    {
     "data": {
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ePdqMGxanIS3eWO12J2OxCO5cnoX3ajvQOxS+G2VOg3MTGcJrx71da+81wQUT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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f0b636f7ef0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Smax = 120\n",
    "\n",
    "n = ns-2\n",
    "X = linspace(0.0, Smax, n+2)\n",
    "X = X[1:-1]\n",
    "\n",
    "payoff = clip(X-K, 0.0, 1e600)\n",
    "  \n",
    "m = 4555 \n",
    "Fl = zeros((m+1,))\n",
    "Fu = zeros((m+1,))\n",
    "    \n",
    "t = time.time()\n",
    "bs1 = BS_FDM_implicit(r, sigma, T, Smin, Smax, Fl, Fu, payoff, m, n)\n",
    "px_fd_mat = bs1.solve()\n",
    "elapsed=time.time()-t\n",
    "print(\"Elapsed Time1:\", elapsed)\n",
    "\n",
    "#print  px_fd_mat.shape\n",
    "\n",
    "figure(figsize=[15, 4]);\n",
    "subplot(1, 3, 1)\n",
    "nrow = len(px_fd_mat[:,1])\n",
    "plot(X, px_fd_mat[nrow-1,:])\n",
    "xlabel('Stock', fontsize=14);\n",
    "title('nx = %.f'%m,  fontsize=16);\n",
    "\n",
    "m = 1000\n",
    "Fl = zeros((m+1,))\n",
    "Fu = zeros((m+1,))\n",
    "\n",
    "t = time.time()\n",
    "bs2 = BS_FDM_implicit(r, sigma, T, Smin, Smax, Fl, Fu, payoff, m, n)\n",
    "px_fd_mat = bs2.solve()\n",
    "elapsed=time.time()-t\n",
    "print(\"Elapsed Time2:\", elapsed)\n",
    "\n",
    "subplot(1, 3, 2)\n",
    "nrow = len(px_fd_mat[:,1])\n",
    "plot(X, px_fd_mat[nrow-1,:])\n",
    "xlabel('Stock', fontsize=14);\n",
    "title('nx = %.f'%m, fontsize=16);\n",
    "\n",
    "m = 100\n",
    "Fl = zeros((m+1,))\n",
    "Fu = zeros((m+1,))\n",
    "    \n",
    "t = time.time()\n",
    "bs3 = BS_FDM_implicit(r, sigma, T, Smin, Smax, Fl, Fu, payoff, m, n)\n",
    "px_fd_mat = bs3.solve()\n",
    "elapsed=time.time()-t\n",
    "print(\"Elapsed Time3:\", elapsed)\n",
    "\n",
    "subplot(1, 3, 3)\n",
    "nrow = len(px_fd_mat[:,1])\n",
    "plot(X, px_fd_mat[nrow-1,:])\n",
    "xlabel('Stock', fontsize=14);\n",
    "title('nx = %.f'%m, fontsize=16);\n",
    "\n",
    "\n",
    "print(px_fd_mat[nrow-1,:])\n"
   ]
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   "source": [
    "# Homework \n",
    "\n",
    "* Implement the FDM for Black-Scholes PDE using the Crank-Nicolson Scheme and test your code using the same example here for both the vanilla European Call and the Up-and-out Barrier Call.\n",
    "\n",
    "\n",
    "\n"
   ]
  }
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