Create .gitignore and improved numerics

.gitignore

@@ -0,0 +1,159 @@
+########################################
+## PYTHON
+########################################
+# Byte-compiled / optimized / DLL files
+__pycache__/
+*.py[cod]
+*$py.class
+
+# C extensions
+*.so
+
+# Distribution / packaging
+.Python
+env/
+build/
+develop-eggs/
+dist/
+downloads/
+eggs/
+.eggs/
+lib/
+lib64/
+parts/
+sdist/
+var/
+wheels/
+*.egg-info/
+.installed.cfg
+*.egg
+
+# PyInstaller
+#  Usually these files are written by a python script from a template
+#  before PyInstaller builds the exe, so as to inject date/other infos into it.
+*.manifest
+*.spec
+
+# Installer logs
+pip-log.txt
+pip-delete-this-directory.txt
+
+# Unit test / coverage reports
+htmlcov/
+.tox/
+.coverage
+.coverage.*
+.cache
+nosetests.xml
+coverage.xml
+*,cover
+.hypothesis/
+
+# Translations
+*.mo
+*.pot
+
+# Django stuff:
+*.log
+local_settings.py
+
+# Flask stuff:
+instance/
+.webassets-cache
+
+# Scrapy stuff:
+.scrapy
+
+# Sphinx documentation
+docs/_build/
+
+# PyBuilder
+target/
+
+# Jupyter Notebook
+.ipynb_checkpoints
+
+# pyenv
+.python-version
+
+# celery beat schedule file
+celerybeat-schedule
+
+# SageMath parsed files
+*.sage.py
+
+# dotenv
+.env
+
+# virtualenv
+.venv
+venv/
+ENV/
+
+# Spyder project settings
+.spyderproject
+
+# Rope project settings
+.ropeproject
+
+# mkdocs documentation
+/site
+
+
+########################################
+## JETBRAINS / PYCHARM
+########################################
+
+# User-specific stuff
+.idea/**/workspace.xml
+.idea/**/tasks.xml
+.idea/**/usage.statistics.xml
+.idea/**/dictionaries
+.idea/**/shelf
+
+# Generated files
+.idea/**/contentModel.xml
+
+# Sensitive or high-churn files
+.idea/**/dataSources/
+.idea/**/dataSources.ids
+.idea/**/dataSources.local.xml
+.idea/**/sqlDataSources.xml
+.idea/**/dynamic.xml
+.idea/**/uiDesigner.xml
+
+# Gradle
+.idea/**/gradle.xml
+.idea/**/libraries
+
+# CMake
+cmake-build-debug/
+cmake-build-release/
+
+# Mongo Explorer plugin
+.idea/**/mongoSettings.xml
+
+# File-based project format
+*.iws
+
+# IntelliJ
+out/
+
+# mpeltonen/sbt-idea plugin
+.idea_modules/
+
+# JIRA plugin
+atlassian-ide-plugin.xml
+
+# Cursive Clojure plugin
+.idea/replstate.xml
+
+# Crashlytics plugin (for Android Studio and IntelliJ)
+com_crashlytics_export_strings.xml
+crashlytics.properties
+crashlytics-build.properties
+fabric.properties
+
+# Editor-based Rest Client
+.idea/httpRequests
+

SABR - Small Time Mass.ipynb

@@ -27,7 +27,7 @@
    "outputs": [],
    "source": [
     "import numpy as np\n",
-    "from scipy.special import gamma, gammainc, hyp1f1\n",
+    "from scipy.special import gamma, gammaincc, hyp1f1\n",
     "from scipy.integrate import quad\n",
     "from scipy.optimize import bisect\n",
     "import pandas as pd\n",
@@ -112,7 +112,13 @@
     "###Computes the function m by direct integration (quadrature)###\n",
     "################################################################\n",
     "def integrand_m(y, z, u):\n",
-    "    temp = np.exp(-z*np.cosh(2.*u))*hyp1f1(0.75, 1.5, 2.*z*np.sinh(u)*np.sinh(u))*np.sin(np.pi*u/y)\n",
+    "    # Instead of the original express,\n",
+    "    # temp = np.exp(-z*np.cosh(2.*u))*hyp1f1(0.75, 1.5, 2.*z*np.sinh(u)*np.sinh(u))*np.sin(np.pi*u/y)\n",
+    "    # we can simplify the expression using \n",
+    "    # Kummer's transformation, M(a,b,z) = e^z M(b-a,b,-z) and cosh(2x) = 1 + 2*sinh(x)^2\n",
+    "    # This new expression also gets around the problem of M(*,*,z) -> infty for large z\n",
+    "    \n",
+    "    temp = np.exp(-z)*hyp1f1(0.75, 1.5, z*(1.-np.cosh(2*u)))*np.sin(np.pi*u/y)\n",
     "    return np.exp(-u*u/y)*temp*np.sinh(2.*u)\n",
     "\n",
     "def m(y, z):\n",
@@ -132,7 +138,7 @@
     "def ppX(r, beta, x0):\n",
     "    ## mass at zero in the CEV model\n",
     "    b1 = 1.-beta\n",
-    "    return 1. - gammainc(0.5/b1, 0.5*x0**(2.*b1)/(r*b1*b1))\n",
+    "    return gammaincc(0.5/b1, 0.5*x0**(2.*b1)/(r*b1*b1))\n",
     "################################################################\n",
     "################################################################"
    ]
@@ -466,7 +472,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.7.6"
+   "version": "3.7.3"
   },
   "zanadu": {
    "authors": [