9879HW08_Chen_Weiyi.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Student: Chen Weiyi, weiyi.alan.chen@gmail.com"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# MTH9879 Homework 8\n",
    "\n",
    "Assigned: April 14, 2015.\n",
    "Due: April 28, 2015 by 6pm. \n",
    "\n",
    "Late homework **will not be accepted**.\n",
    "\n",
    "$$\n",
    "\\newcommand{\\supp}{\\mathrm{supp}}\n",
    "\\newcommand{\\E}{\\mathbb{E}}\n",
    "\\newcommand{\\Eof}[1]{\\mathbb{E}\\left[ #1 \\right]}\n",
    "\\def\\Cov{{ \\mbox{Cov} }}\n",
    "\\def\\Var{{ \\mbox{Var} }}\n",
    "\\newcommand{\\1}{\\mathbf{1} }\n",
    "\\newcommand{\\PP}{\\mathbb{P} }\n",
    "%\\newcommand{\\Pr}{\\mathrm{Pr} }\n",
    "\\newcommand{\\QQ}{\\mathbb{Q} }\n",
    "\\newcommand{\\RR}{\\mathbb{R} }\n",
    "\\newcommand{\\DD}{\\mathbb{D} }\n",
    "\\newcommand{\\HH}{\\mathbb{H} }\n",
    "\\newcommand{\\spn}{\\mathrm{span} }\n",
    "\\newcommand{\\cov}{\\mathrm{cov} }\n",
    "\\newcommand{\\sgn}{\\mathrm{sgn} }\n",
    "\\newcommand{\\HS}{\\mathcal{L}_{\\mathrm{HS}} }\n",
    "%\\newcommand{\\HS}{\\mathrm{HS} }\n",
    "\\newcommand{\\trace}{\\mathrm{trace} }\n",
    "\\newcommand{\\LL}{\\mathcal{L} }\n",
    "%\\newcommand{\\LL}{\\mathrm{L} }\n",
    "\\newcommand{\\s}{\\mathcal{S} }\n",
    "\\newcommand{\\ee}{\\mathcal{E} }\n",
    "\\newcommand{\\ff}{\\mathcal{F} }\n",
    "\\newcommand{\\hh}{\\mathcal{H} }\n",
    "\\newcommand{\\bb}{\\mathcal{B} }\n",
    "\\newcommand{\\dd}{\\mathcal{D} }\n",
    "\\newcommand{\\g}{\\mathcal{G} }\n",
    "\\newcommand{\\p}{\\partial}\n",
    "\\newcommand{\\half}{\\frac{1}{2} }\n",
    "\\newcommand{\\T}{\\mathcal{T} }\n",
    "\\newcommand{\\bi}{\\begin{itemize}}\n",
    "\\newcommand{\\ei}{\\end{itemize}}\n",
    "\\newcommand{\\beq}{\\begin{equation}}\n",
    "\\newcommand{\\eeq}{\\end{equation}}\n",
    "\\newcommand{\\beas}{\\begin{eqnarray*}}\n",
    "\\newcommand{\\eeas}{\\end{eqnarray*}}\n",
    "\\newcommand{\\cO}{\\mathcal{O}}\n",
    "\\newcommand{\\cF}{\\mathcal{F}}\n",
    "\\newcommand{\\cL}{\\mathcal{L}}\n",
    "\\newcommand{\\BS}{\\text{BS}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<font color = \"red\">Homework is to be done by each student individually.  To receive full credit, you must email a completed copy of this iPython notebook to Yu Gan (yugan323@gmail.com), Fubo Shi (fubo.shi.baruch@gmail.com), and Tai-Ho Wang (tai-ho.wang@baruch.cuny.edu) by the due date and time.  All R-code must run correctly and solutions must be written up neatly in Markdown/LaTeX format.\n",
    "\n",
    "<font color=\"blue\">If you encounter problems with Markdown/LaTeX or iPython notebook, please contact your TAs Yu Gan and/or Fubo Shi.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 0. (0 points) \n",
    "Read Section 6.7 of Bouchaud, Farmer and Lillo."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Predictability of order flow"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1. (6 points)\n",
    "\n",
    "Suppose that the distribution of meta order sizes $Q$ is exponential so that\n",
    "\n",
    "$$\n",
    "\\Pr(Q>V)= \\frac{1}{\\lambda}\\,e^{-\\lambda\\,V}\n",
    "$$\n",
    "\n",
    "Assuming slices of  constant size $\\bar v$, compute the conditional probability that a hidden order has more slices given that $n$ slices have already been observed.  Compare your result with the case where distribution of metaorder size is power-law."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<font color=\"darkblue\">\n",
    "\n",
    "### Solution\n",
    "\n",
    "The conditional probability that a hidden order has more slices given that n slices have already been observed, \n",
    "\n",
    "\\begin{equation}\n",
    "    q_t = \\left.\\Pr(T \\geq t+1 \\right|T \\ge t) = \\frac{\\Pr(T \\ge t+1)}{\\Pr(T \\ge t)} = e^{-\\lambda}\n",
    "\\end{equation}\n",
    "\n",
    "If distribution of metaorder size is power-law, then similary (from lecture note)\n",
    "\n",
    "\\begin{equation}\n",
    "    q_t = \\left.\\Pr(T \\geq t+1 \\right|T \\ge t) = \\frac{\\Pr(T \\ge t+1)}{\\Pr(T \\ge t)} \\sim \\left(\\frac{t}{t+1}\\right)^\\lambda\n",
    "\\end{equation}\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Price manipulation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2. (10 points)\n",
    "\n",
    "Consider the price process\n",
    "\n",
    "$$\n",
    "S_t=S_0+\\int_0^t\\,f(\\dot{x}_s)\\,G(t-s)\\,ds+\\int_0^t\\,\\sigma\\,dZ_s\n",
    "$$\n",
    "\n",
    "where ${\\dot x}_s$ is the rate of trading in dollars at time $s < t$, $f(\\dot{x}_s)$ represents the impact of trading at time $s$ and $G(t-s)$ is a decay factor.\n",
    "\n",
    "(a) With the choices\n",
    "\n",
    "$$\n",
    "f(v)=\\sqrt{v},\\quad G(\\tau)=\\frac{1}{1+\\tau},\n",
    "$$\n",
    "    \n",
    "repeat the computations in Lecture 8  to derive an expression for the expected cost of accumulating shares at the rate $v_1$ and liquidating them at the rate $v_2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<font color=\"darkblue\">\n",
    "\n",
    "### Solution\n",
    "\n",
    "From lecture note equation (7), and given the conditions \n",
    "\n",
    "$$\n",
    "f(v)=\\sqrt{v},\\quad G(\\tau)=\\frac{1}{1+\\tau},\n",
    "$$\n",
    "\n",
    "we have\n",
    "\n",
    "\\begin{eqnarray*}\n",
    "C_{11}&=&v_1\\,f\\left(v_1\\right)\\,\\int_0^{\\theta\\,T}\\,dt\\,\\int_0^t\\,G(t-s)\\,ds  =v_1^{3/2}\\int_0^{\\theta\\,T}\\,dt\\,\\int_0^t\\,\\frac{1}{1+t-s}\\,ds &=& v_1^{3/2}(1+\\theta\\,T)\\ln(1+\\theta\\,T) - v_1^{3/2}\\theta\\,T \n",
    "\\end{eqnarray*}\n",
    "\n",
    "Similarly,\n",
    "\n",
    "\\begin{eqnarray*}\n",
    "C_{22} &=& v_2^{3/2}(1+T-\\theta\\,T)\\ln(1+T-\\theta\\,T) + v_2^{3/2}(\\theta\\,T-T)\\nonumber\\\\\n",
    "C_{12} &=& v_2\\,\\sqrt v_1\\, \\left[(1+T)\\ln(1+T) - (1+\\theta\\,T) \\ln(1+\\theta\\,T) - (1+T-\\theta\\,T)\\ln(1+T-\\theta\\,T)\\right]\n",
    "\\end{eqnarray*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) Verify that with $T=1$, $v_1=1/4$, $v_2=1$, the expected cost of this round-trip strategy is negative, thus demonstrating that price manipulation is possible. (Recall from the lecture slides that price manipulation is always possible if $f(\\cdot)$ is nonlinear and the decay kernel $G(\\cdot)$  is finite at the origin.)\n",
    "\n",
    "Hint: You might consider using Mathematica or Wolfram Alpha to do the integrations.\n",
    "\n",
    "<font color=\"darkblue\">\n",
    "\n",
    "### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "C_11:\t0.032251999603\n",
      "C_22:\t0.0187858681527\n",
      "C_12:\t0.0547462480717\n",
      "C:\t-0.00370838031594\n",
      "C pos?: False\n"
     ]
    }
   ],
   "source": [
    "import math\n",
    "\n",
    "v_1   = 1. / 4.\n",
    "v_2   = 1.\n",
    "T     = 1.\n",
    "theta = v_2 / (v_1 + v_2)\n",
    "\n",
    "C_11 = v_1**(1.5) * (1 + theta*T) * math.log(1+theta*T) - v_1**(1.5) * theta * T\n",
    "C_22 = v_2**(1.5) * (1 + T - theta*T) * math.log(1+T-theta*T) + v_2**(1.5) * (theta*T - T)\n",
    "C_12 = v_2 * v_1**0.5 * ((1+T)*math.log(1+T) - (1+theta*T)*math.log(1+theta*T) - (1+T-theta*T)*math.log(1+T-theta*T))\n",
    "C    = C_11 + C_22 - C_12 \n",
    "\n",
    "print 'C_11:\\t', C_11\n",
    "print 'C_22:\\t', C_22\n",
    "print 'C_12:\\t', C_12\n",
    "print 'C:\\t'   , C\n",
    "print 'C pos?:', (C > 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Set up R environment"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%load_ext rpy2.ipython"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Obizhaeva-Wang model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3. (8 points)\n",
    "\n",
    "In the Obizhaeva-Wang (OW) model, the price follows\n",
    "\n",
    "$$\n",
    "S_t=S_0+\\eta\\,\\int_0^t\\,{\\dot x}_s\\,e^{-\\rho\\,(t-s)}\\,ds+\\int_0^t\\,\\sigma\\,dZ_s\n",
    "$$\n",
    "\n",
    "(a) Derive an expression for the expected price path corresponding to a VWAP execution.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<font color=\"darkblue\">\n",
    "\n",
    "### Solution\n",
    "\n",
    "For a VWAP exection, we have given condition \n",
    "\n",
    "$${\\dot x} = \\frac{x}{T}$$\n",
    "\n",
    "Then\n",
    "$$\n",
    "S_t=S_0+\\eta\\,\\int_0^t\\,{\\frac{x_s}{T_s}}\\,e^{-\\rho\\,(t-s)}\\,ds+\\int_0^t\\,\\sigma\\,dZ_s\n",
    "$$\n",
    "\n",
    "When $0 \\leq t \\leq T$,\n",
    "\n",
    "$$\n",
    "\\mathbb{E}[S_t] = S_0 + \\eta \\frac{x}{T} \\int_0^t e^{-\\rho(t-s)}ds = S_0 + \\frac{\\eta x}{\\rho T}(1-e^{-\\rho t})\n",
    "$$\n",
    "\n",
    "\n",
    "But if $t > T$, \n",
    "$$\n",
    "\\mathbb{E}[S_t] = \\mathbb{E}[S_T] = S_0 + \\frac{\\eta x}{\\rho T}(1 - e^{-\\rho T})\n",
    "$$\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) With the duration $T$ of the meta order set to 1, plot the expected price path from time $t=0$ to time $t=2$ for $\\rho=0.5,1,$ and $2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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nipDtnj177GXnsWPHipatiNnSSCDKBOyc3yqVYIQjvDgNfZf/CtB9WIlRRJMm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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R\n",
    "\n",
    "# Initialize Paramters\n",
    "ls_timeStamps = seq(0,2,0.01)\n",
    "f_rho1 = 0.5\n",
    "f_rho2 = 1\n",
    "f_rho3 = 2\n",
    "\n",
    "# function to generate expected price path\n",
    "expectedPrice = function(t,f_rho){\n",
    "    ls_price = (1-exp(-f_rho*t)) / f_rho;\n",
    "    if (t > 1){\n",
    "        ls_price = exp(f_rho*(1-t))/f_rho - exp(f_rho*(-t))/f_rho;\n",
    "    }\n",
    "    return (ls_price)\n",
    "}\n",
    "\n",
    "# apply defined function for different rho\n",
    "S_1 = sapply(ls_timeStamps, function(t)expectedPrice(t, f_rho1))\n",
    "S_2 = sapply(ls_timeStamps, function(t)expectedPrice(t, f_rho2))\n",
    "S_3 = sapply(ls_timeStamps, function(t)expectedPrice(t, f_rho3))\n",
    "\n",
    "# plot\n",
    "plot(ls_timeStamps, S_1, type='l', xlab='Time Stamps', ylab='Expected Price', col='red')\n",
    "points(ls_timeStamps, S_2, type='l', col='blue')\n",
    "points(ls_timeStamps, S_3, type='l', col='green')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(c) What is the expected average cost per share $\\bar S_T$ of a VWAP order in the OW model?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<font color=\"darkblue\">\n",
    "\n",
    "### Solution\n",
    "\n",
    "According to lecture note, the expected average cost is\n",
    "\n",
    "$$\\bar S_T= \\frac{C[\\Pi]}{x} = \\frac{1}{x} \\int_0^T\\,{\\dot x}_t\\,dt\\,\\int_0^t\\,\\dot x e^{-\\rho\\,(t-s)}ds$$\n",
    "\n",
    "If indicated in part (a) with $\\dot x = \\frac{x}{T}$, we have \n",
    "\n",
    "$$C[\\Pi] = \\frac{\\eta\\,x^2}{\\rho\\,T} + \\frac{\\eta\\,x^2}{\\rho^2\\,T^2}\\,(e^{-\\rho\\,T}-1)$$\n",
    "\n",
    "Therefore,\n",
    "\n",
    "$$\\bar S_T = \\frac{\\eta\\,x}{\\rho\\,T} + \\frac{\\eta\\,x}{\\rho^2\\,T^2}\\,(e^{-\\rho\\,T}-1)$$"
   ]
  }
 ],
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