issue #102: adding more maths for HW calibration

ccjeremylo · Nov 22, 2023 · abe9a3c · abe9a3c
1 parent acd28a2
commit abe9a3c
Showing 1 changed file with 7 additions and 7 deletions.
diff --git a/src/python/Lecture6/hull_white_model.ipynb b/src/python/Lecture6/hull_white_model.ipynb
@@ -189,18 +189,18 @@
     "& \\;\\;\\;\\;\\;\\;\\; + \\underbrace{\\frac{1}{2} \\int_t^T \\sigma^2(u) \\, G^2(u,T) \\, du + G(t,T) \\int_0^t \\sigma^2(u) \\, G(u,t) \\, \\partial_t G(u,t) \\, du + \\frac{1}{2} G^2(t,T) \\int_0^t \\sigma^2(u) \\, \\partial_t G^2(u,t) \\, du}_{\\text{from ($\\color{blue}{\\text{B}}$)}} \\\\\n",
     "&= -f(0,t) G(t,T) \n",
     "+ r_0 \\big(G(0,T) - G(0,t)\\big)\n",
-    "-\\underbrace{\\int_t^T \\tilde{\\theta}(u) G(u,T) \\, du}_{\\text{what we want}}\n",
-    "+ \\frac{1}{2} \\int_t^T \\sigma^2(u) \\, G^2(u,T) \\, du\n",
+    "-\\underbrace{\\int_t^T \\tilde{\\theta}(u) G(u,T) \\, du}_{\\text{what we want}} \\\\\n",
+    "& \\;\\;\\;\\;\\;\\; + \\frac{1}{2} \\int_t^T \\sigma^2(u) \\, G^2(u,T) \\, du\n",
     "+ \\frac{1}{2} G^2(t,T) \\int_0^t \\sigma^2(u) \\, \\partial_t G^2(u,t) \\, du\n",
     "\\end{align}\n",
     "Rearranging, we now have an analytical expression of the integral $\\int_t^T \\tilde{\\theta}(u) G(u,T) \\, du$ in terms of day-0 market quantities:\n",
     "\\begin{align}\n",
     "\\implies\n",
     "\\underbrace{\\int_t^T \\tilde{\\theta}(u) G(u,T) \\, du}_{\\text{what we want}}\n",
-    "= -\\ln \\frac{B(0,T)}{B(0,t)}\n",
+    "&= -\\ln \\frac{B(0,T)}{B(0,t)}\n",
     "-f(0,t) G(t,T) \n",
-    "+ r_0 \\big(G(0,T) - G(0,t)\\big)\n",
-    "+ \\frac{1}{2} \\int_t^T \\sigma^2(u) \\, G^2(u,T) \\, du\n",
+    "+ r_0 \\big(G(0,T) - G(0,t)\\big) \\\\\n",
+    "& \\;\\;\\;\\;\\;\\;\\; + \\frac{1}{2} \\int_t^T \\sigma^2(u) \\, G^2(u,T) \\, du\n",
     "+ \\frac{1}{2} G^2(t,T) \\int_0^t \\sigma^2(u) \\, \\partial_t G^2(u,t) \\, du \n",
     "\\end{align}"
    ]
@@ -210,9 +210,9 @@
    "id": "3aac081d-5c31-456b-8749-b5a03f22efa7",
    "metadata": {},
    "source": [
-    "Putting everything back into our HW ZCB equation:\n",
+    "Substituting the integral back into our HW ZCB equation:\n",
     "\\begin{align}\n",
-    "B(t,T) = \\exp \\bigg( -r_t \\, G(t,T) - \\int_t^T \\tilde{\\theta}(u) \\, G(u,T) \\, du + \\frac{1}{2} \\int_t^T \\sigma^2(u) \\, G^2(u,T) \\, du  \\bigg)\n",
+    "B(t,T) = \\exp \\bigg( -r_t \\, G(t,T) - \\underbrace{\\int_t^T \\tilde{\\theta}(u) \\, G(u,T) \\, du}_{\\text{substitude here}} + \\frac{1}{2} \\int_t^T \\sigma^2(u) \\, G^2(u,T) \\, du  \\bigg)\n",
     "\\end{align}\n",
     "We now have\n",
     "\\begin{align}\n",