rebate_model_notes.ipynb

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   "id": "searching-integration",
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   "source": [
    "## Rebate Model"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "focused-upgrade",
   "metadata": {},
   "source": [
    "Every trade pays spread and rebate:\n",
    "\n",
    "- If a MO hits the ask-quote of the MM: $q \\to q - 1 $ and $x \\to x + (S + \\delta^+) + \\epsilon$\n",
    "\n",
    "- If a MO hits the bid-quote of the MM: $q \\to q + 1 $ and $x \\to x - (S - \\delta^-) + \\epsilon$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "greater-stationery",
   "metadata": {},
   "source": [
    "## Problem Setup\n",
    "\n",
    "- The MM seeks the strategy $(\\delta^{+}, \\delta^{-})$ that maximizes cash at the terminal time $T$. \n",
    "- At time $T$ the MM liquidates her terminal inventory $Q_T$ using an MO at a price which is worse than the mid-price to account for liquidity taking fees and market impact. \n",
    "\n",
    "- The MM caps her inventory so that it is bounded above by $q_{max} > 0$ and below by $q_{min} < 0$.\n",
    "\n",
    "- The MM penalizes non-zero inventory with a running inventory penalty.\n",
    "\n",
    "**The performance criterion**:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "H^{\\delta^{-}, \\delta^{+}}(t, x, s, q)=\\mathbb{E}_t \\left[ X_T + (S_t - \\gamma Q_T)Q_T - \\phi \\int_{t}^{T}Q_s^2 ds \\ | \\ X_t\\right]\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "where the superscript on $H$ highlights the fact that the performance criterion is controlled by the MM by changing the quoted spreads $\\delta^{-}, \\delta^{+}$.\n",
    "\n",
    "**The MMs value function**:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "H(t, x, s, q) = \\underset{\\delta^{-}, \\delta^{+} \\in \\mathcal{A}}{max}\\ H^{\\delta^{-}, \\delta^{+}}(t, x, s, q)\n",
    "\\end{align}\n",
    "$$\n"
   ]
  },
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   "cell_type": "markdown",
   "id": "explicit-pursuit",
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   "source": [
    "## HJB equation\n",
    "\n",
    "\n",
    "The value function $H(t, x, s, q)$ satisfies HJB equation of the form:\n",
    "\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "0 = \\underset{\\delta^{-}, \\delta^{+} \\in \\mathcal{A}}{max} \\left( \\mathcal{L}H^{\\delta^{-}, \\delta^{+}}(t, x, s, q) - \\phi q_t^2 \\right)\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "**Derivation of $\\mathcal{L}H^{\\delta^{-}, \\delta^{+}}(t, x, s, q)$**:\n",
    "\n",
    "Let's recall the definition of the infinitesimal generator:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\mathcal{L}H^{\\delta^{-}, \\delta^{+}}(t, x, s, q) = \\lim_{dt \\to 0}\\frac{\\mathbb{E}(\\partial H^{\\delta^{-}, \\delta^{+}}(t, x, s, q)}{dt}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "We start by applying Ito's lemma to $H^{\\delta^{-}, \\delta^{+}}(t, x, s, q)$:\n",
    "\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\partial H^{\\delta^{-}, \\delta^{+}}(t, x, s, q) &= \\partial_t H^{\\delta^{-}, \\delta^{+}} dt + \\partial_s H^{\\delta^{-}, \\delta^{+}} dS_t + \\frac{1}{2}\\partial_{ss}H^{\\delta^{-}, \\delta^{+}} dS_t^2 \\\\\n",
    "& + \\left[H^{\\delta^{-}, \\delta^{+}}(t, x-(s-\\delta^-) + \\epsilon, s, q+1) - H^{\\delta^{-}, \\delta^{+}}(t, x, s, q) \\right]dN_t^{-} \\\\\n",
    "& + \\left[H^{\\delta^{-}, \\delta^{+}}(t, x+(s+\\delta^+) + \\epsilon, s, q-1) - H^{\\delta^{-}, \\delta^{+}}(t, x, s, q) \\right]dN_t^{+}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Next, we transform the fill counting processes into martingales:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "N_t^{-} = M_t^{-} + \\int_{0}^{t}\\lambda^{-}_s ds \\to dN_t^{-} = dM_t^{-} + \\lambda^{-}_t dt\\\\\n",
    "N_t^{+} = M_t^{+} + \\int_{0}^{t}\\lambda^{}_s ds \\to dN_t^{+} = dM_t^{+} + \\lambda^{+}_t dt\\\\\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Substituting these into the expression for Ito's lemma yields:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\partial H^{\\delta^{-}, \\delta^{+}}(t, x, s, q) &= \\partial_t H^{\\delta^{-}, \\delta^{+}} dt + \\partial_s H^{\\delta^{-}, \\delta^{+}} dS_t + \\frac{1}{2}\\partial_{ss}H^{\\delta^{-}, \\delta^{+}} dS_t^2 \\\\\n",
    "& + \\left[H^{\\delta^{-}, \\delta^{+}}(t, x-(s-\\delta^-)+ \\epsilon, s, q+1) - H^{\\delta^{-}, \\delta^{+}}(t, x, s, q) \\right](dM_t^{-} + \\lambda^{-}_t dt) \\\\\n",
    "& + \\left[H^{\\delta^{-}, \\delta^{+}}(t, x+(s+\\delta^+)+ \\epsilon, s, q-1) - H^{\\delta^{-}, \\delta^{+}}(t, x, s, q) \\right](dM_t^{+} + \\lambda^{+}_t dt)\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Since we assume Bachelier type model for the spot price, $dS_t = \\sigma dW_t$ and $dS_t^2 = \\sigma^2 dt$. Substituting these into the above gives:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\partial H^{\\delta^{-}, \\delta^{+}}(t, x, s, q) &= \\partial_t H^{\\delta^{-}, \\delta^{+}} dt + \\partial_s H^{\\delta^{-}, \\delta^{+}} \\sigma dW_t + \\frac{1}{2}\\partial_{ss}H^{\\delta^{-}, \\delta^{+}} \\sigma^2 dt \\\\\n",
    "& + \\left[H^{\\delta^{-}, \\delta^{+}}(t, x-(s-\\delta^-)+ \\epsilon, s, q+1) - H^{\\delta^{-}, \\delta^{+}}(t, x, s, q) \\right](dM_t^{-} + \\lambda^{-}_t dt) \\\\\n",
    "& + \\left[H^{\\delta^{-}, \\delta^{+}}(t, x+(s+\\delta^+)+ \\epsilon, s, q-1) - H^{\\delta^{-}, \\delta^{+}}(t, x, s, q) \\right](dM_t^{+} + \\lambda^{+}_t dt)\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "\n",
    "\n",
    "Now, since we made the explicit assumption that $\\lambda_t = \\lambda(\\delta) = \\lambda_0 e^{-\\kappa \\delta}$ we get that:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\partial H^{\\delta^{-}, \\delta^{+}}(t, x, s, q) &= \\partial_t H^{\\delta^{-}, \\delta^{+}} dt + \\partial_s H^{\\delta^{-}, \\delta^{+}} \\sigma dW_t + \\frac{1}{2}\\partial_{ss}H^{\\delta^{-}, \\delta^{+}} \\sigma^2 dt \\\\\n",
    "& + \\left[H^{\\delta^{-}, \\delta^{+}}(t, x-(s-\\delta^-)+ \\epsilon, s, q+1) - H^{\\delta^{-}, \\delta^{+}}(t, x, s, q) \\right](dM_t^{-} + \\lambda_0^{-} e^{-\\kappa^{-} \\delta^{-}} dt) \\\\\n",
    "& + \\left[H^{\\delta^{-}, \\delta^{+}}(t, x+(s+\\delta^+)+ \\epsilon, s, q-1) - H^{\\delta^{-}, \\delta^{+}}(t, x, s, q) \\right](dM_t^{+} + \\lambda_0^{+} e^{-\\kappa^{+} \\delta^{+}} dt)\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "\n",
    "Finally, taking expectation and dividing by $dt$ yields:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\mathcal{L}H^{\\delta^{-}, \\delta^{+}}(t, x, s, q) &= \\partial_t H^{\\delta^{-}, \\delta^{+}}  +  \\frac{1}{2}\\partial_{ss}H^{\\delta^{-}, \\delta^{+}} \\sigma^2  \\\\\n",
    "& + \\lambda_0^{-} e^{-\\kappa^{-} \\delta^{-}} \\left[H^{\\delta^{-}, \\delta^{+}}(t, x-(s-\\delta^-)+ \\epsilon, s, q+1) - H^{\\delta^{-}, \\delta^{+}}(t, x, s, q) \\right] \\\\\n",
    "& + \\lambda_0^{+} e^{-\\kappa^{+} \\delta^{+}}\\left[H^{\\delta^{-}, \\delta^{+}}(t, x+(s+\\delta^+)+ \\epsilon, s, q-1) - H^{\\delta^{-}, \\delta^{+}}(t, x, s, q) \\right]\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "**HJB equation**:\n",
    "\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "0 &= \\partial_t H^{\\delta^{-}, \\delta^{+}}  +  \\frac{1}{2}\\partial_{ss}H^{\\delta^{-}, \\delta^{+}} \\sigma^2  \\\\\n",
    "& + \\lambda_0^{-} \\underset{\\delta^{-}}{max} \\left( e^{-\\kappa^{-} \\delta^{-}} \\left[H^{\\delta^{-}, \\delta^{+}}(t, x-(s-\\delta^-)+ \\epsilon, s, q_t+1) - H^{\\delta^{-}, \\delta^{+}}(t, x, s, q_t) \\right] \\right)\\\\\n",
    "& + \\lambda_0^{+} \\underset{\\delta^{+}}{max} \\left( e^{-\\kappa^{+} \\delta^{+}}\\left[H^{\\delta^{-}, \\delta^{+}}(t, x+(s+\\delta^+)+ \\epsilon, s, q_t-1) - H^{\\delta^{-}, \\delta^{+}}(t, x, s, q_t) \\right]\\right) \\\\\n",
    "&- \\phi q_t^2\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Since the terminal condition for $H$ is $H(T, X_T, S_T, Q_T) = X_T + (S_T - \\gamma Q_T)Q_T = X_T + S_T Q_T - \\gamma Q_T^2$ we use an ansatz:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "H(t,x,s,q) &= x + sq + h(t, q_t) \\\\\n",
    " h(T,q_T) &= - \\gamma q_T^2\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Substituting this into the HJB gives:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "0 &= \\partial_t h(t, q_t) \\\\\n",
    "& + \\lambda_0^{-} \\underset{\\delta^{-}}{max} \\left( e^{-\\kappa^{-} \\delta^{-}} \\left[ \\delta^{-}+ \\epsilon + h(t, q_t+1) - h(t, q_t) \\right] \\right)\\\\\n",
    "& + \\lambda_0^{+} \\underset{\\delta^{+}}{max} \\left( e^{-\\kappa^{+} \\delta^{+}}\\left[ \\delta^{+}+ \\epsilon + h(t, q_t-1) - h(t, q_t) \\right]\\right) \\\\\n",
    "&- \\phi q_t^2\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "\n"
   ]
  },
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   "cell_type": "markdown",
   "id": "suspended-guess",
   "metadata": {},
   "source": [
    "## Finite Difference scheme for the HJB equation\n",
    "\n",
    "\n",
    "We may solve the HJB equation using a finite difference scheme by discretizing the time derivative $\\partial_t h(t, q_t)$:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\partial_t h(t, q) \\approx \\frac{h(t, q) - h(t - dt, q)}{dt}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Substituting this into the HJB equation gives:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "0 &= \\frac{h(t, q) - h(t - dt, q)}{dt} \\\\\n",
    "& + \\lambda_0^{-} \\underset{\\delta^{-}}{max} \\left( e^{-\\kappa^{-} \\delta^{-}} \\left[ \\delta^{-}+ \\epsilon + h(t, q_t+1) - h(t, q_t) \\right] \\right)\\\\\n",
    "& + \\lambda_0^{+} \\underset{\\delta^{+}}{max} \\left( e^{-\\kappa^{+} \\delta^{+}}\\left[ \\delta^{+}+ \\epsilon + h(t, q_t-1) - h(t, q_t) \\right]\\right) \\\\\n",
    "&- \\phi q_t^2\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Solving for $ h(t - dt, q) $ gives the following backward scheme for $h$:\n",
    "\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "h(t - dt, q) &= h(t, q) \\\\\n",
    "& + \\lambda_0^{-} \\underset{\\delta^{-}}{max} \\left( e^{-\\kappa^{-} \\delta^{-}} \\left[ \\delta^{-}+ \\epsilon + h(t, q_t+1) - h(t, q_t) \\right] \\right)dt\\\\\n",
    "& + \\lambda_0^{+} \\underset{\\delta^{+}}{max} \\left( e^{-\\kappa^{+} \\delta^{+}}\\left[ \\delta^{+}+ \\epsilon + h(t, q_t-1) - h(t, q_t) \\right]\\right)dt \\\\\n",
    "&- \\phi q_t^2 dt\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Now, we define q-grid such that $q_j \\in \\{q_{min}, q_{min}+1, ..., q_{max}-1, q_{max} \\}$ and a t-grid where $t_i \\in \\{0, \\Delta t, ..., T-2\\Delta t, T-\\Delta t, T \\}$. \n",
    "\n",
    "\n",
    "The following solves $h(t_i, q_j)$ over the grid:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "h(t_{i-1}, q_j) &= h(t_{i}, q_j) \\\\\n",
    "& + \\mathbb{1}_{q_j < q_{max}}\\lambda_0^{-} \\underset{\\delta^{-}}{max} \\left( e^{-\\kappa^{-} \\delta^{-}} \\left[ \\delta^{-}+ \\epsilon + h(t_i, q_j+1) - h(t_i, q_j) \\right] \\right)\\Delta t \\\\\n",
    "& + \\mathbb{1}_{q_j < q_{max}}\\lambda_0^{+} \\underset{\\delta^{+}}{max} \\left( e^{-\\kappa^{+} \\delta^{+}}\\left[ \\delta^{+}+ \\epsilon + h(t_i, q_j-1) - h(t_i, q_j) \\right]\\right)\\Delta t \\\\\n",
    "&- \\phi q_j^2 \\Delta t\n",
    "\\end{align}\n",
    "$$"
   ]
  }
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