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.buildinfo

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+# Sphinx build info version 1
+# This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done.
+config: 66b5179abbc7b26fba469d22fdf36f30
+tags: 645f666f9bcd5a90fca523b33c5a78b7

_sources/api/options.rst

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+==================================
+Options Calibration and Pricing
+==================================
+
+
+VolSurface
+==================
+
+.. module:: quantflow.options.surface
+
+.. autoclass:: VolSurface
+   :members:

_sources/api/overview.rst

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+
+API Reference
+==============
+
+.. toctree::
+   :maxdepth: 2
+
+   sp
+   options
+   utils

_sources/api/sp.rst

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+==================================
+Stochastic Process API Reference
+==================================
+
+
+.. module:: quantflow.sp.base
+
+
+StochasticProcess
+==================
+
+.. autoclass:: StochasticProcess
+   :members:
+
+
+StochasticProcess1d
+=====================
+
+.. autoclass:: StochasticProcess1d
+   :members:
+
+
+IntensityProcess
+=====================
+
+.. autoclass:: IntensityProcess
+   :members:
+
+
+WeinerProcess
+=====================
+
+.. module:: quantflow.sp.weiner
+
+.. autoclass:: WeinerProcess
+   :members:
+
+
+PoissonProcess
+=====================
+
+.. module:: quantflow.sp.poisson
+
+.. autoclass:: PoissonProcess
+   :members:
+
+
+CompoundPoissonProcess
+=======================
+
+.. module:: quantflow.sp.poisson
+
+.. autoclass:: CompoundPoissonProcess
+   :members:
+
+
+Heston
+=======================
+
+.. module:: quantflow.sp.heston
+
+.. autoclass:: Heston
+   :members:
+
+
+JumpDiffision
+=======================
+
+.. module:: quantflow.sp.jump_diffusion
+
+.. autoclass:: JumpDiffision
+   :members:
+
+.. autoclass:: Merton
+   :members:

_sources/api/utils.rst

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+===========
+Utilities
+===========
+
+
+Paths
+==================
+
+.. module:: quantflow.utils.paths
+
+.. autoclass:: Paths
+   :members:
+
+
+Marginal1D
+==================
+
+.. module:: quantflow.utils.marginal
+
+.. autoclass:: Marginal1D
+   :members:

_sources/applications/calibration.md

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+---
+jupytext:
+  formats: ipynb,md:myst
+  text_representation:
+    extension: .md
+    format_name: myst
+    format_version: 0.13
+    jupytext_version: 1.14.7
+kernelspec:
+  display_name: Python 3 (ipykernel)
+  language: python
+  name: python3
+---
+
+# Calibration
+
+Early pointers
+
+* https://github.com/rlabbe/filterpy
+* [filterpy book](https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python)
+
++++
+
+## Calibrating ABC
+
+For calibration we use {cite:p}`ukf`.
+Lets consider the Heston model as a test case
+
+```{code-cell} ipython3
+from quantflow.sp.heston import Heston
+
+pr = Heston.create(vol=0.6, kappa=1.3, sigma=0.8, rho=-0.6)
+pr.variance_process.is_positive
+```
+
+The Heston model is a classical example where the calibration of parameters requires to deal with the estimation of an unobserved random variable, the stochastic variance. The model can be discretized as follow:
+
+\begin{align}
+ d \nu_t &= \kappa\left(\theta -\nu_t\right) dt + \sigma \sqrt{\nu_t} d z_t \\
+ d s_t &= -\frac{\nu_t}{2}dt + \sqrt{\nu_t} d w_t \\
+ {\mathbb E}\left[d w_t d z_t\right] &= \rho dt
+\end{align}
+
+noting that
+
+\begin{equation}
+d z_t = \rho d w_t + \sqrt{1-\rho^2} d b_t
+\end{equation}
+
+which leads to
+
+\begin{align}
+d \nu_t &= \kappa\left(\theta -\nu_t\right) dt + \sigma \sqrt{\nu_t} \rho d w_t + \sigma \sqrt{\nu_t} \sqrt{1-\rho^2} d b_t \\
+d s_t &= -\frac{\nu_t}{2}dt + \sqrt{\nu_t} d w_t \\
+\end{align}
+
+and finally
+
+\begin{align}
+d \nu_t &= \kappa\left(\theta -\nu_t\right) dt + \sigma \rho \frac{\nu_t}{2} dt + \sigma \sqrt{\nu_t} \sqrt{1-\rho^2} d b_t + \sigma \rho d s_t\\
+d s_t &= -\frac{\nu_t}{2}dt + \sqrt{\nu_t} d w_t \\
+\end{align}
+
+Our problem is to find the *best* estimate of $\nu_t$ given by ths equation based on the observations $s_t$.
+
+The Heston model is a dynamic model which can be represented by a state-space form: $X_t$ is the state while $Z_t$ is the observable
+
+\begin{align}
+X_{t+1} &= f\left(X_t, \Theta\right) + B^x_t\\
+Z_t &= h\left(X_t, \Theta\right) + B^z_t \\
+B^x_t &= {\cal N}\left(0, Q_t\right) \\
+B^z_t &= {\cal N}\left(0, R_t\right) \\
+\end{align}
+
+$f$ is the *state transition equation* while $h$ is the *measurement equation*.
+
++++
+
+the state equation is given by
+
+\begin{align}
+X_{t+1} &= \left[\begin{matrix}\kappa\left(\theta\right) dt \\ 0\end{matrix}\right] + 
+\end{align}
+
+```{code-cell} ipython3
+[p for p in pr.variance_process.parameters]
+```
+
+```{code-cell} ipython3
+
+```
+
+## Calibration against historical timeseries
+
+We calibrate the Heston model agais historical time series, in this case the measurement is the log change for a given frequency.
+
+\begin{align}
+F_t &= \left[\begin{matrix}1 - \kappa\theta dt \\ 0\end{matrix}\right] \\
+Q_t &= \left[\begin{matrix}1 - \kappa\theta dt \\ 0\end{matrix}\right]  \\
+z_t &= d s_t
+\end{align}
+
+The observation vector is given by
+\begin{align}
+x_t &= \left[\begin{matrix}\nu_t && w_t && z_t\end{matrix}\right]^T \\
+\bar{x}_t = {\mathbb E}\left[x_t\right] &= \left[\begin{matrix}\nu_t && 0 && 0\end{matrix}\right]^T
+\end{align}
+
+```{code-cell} ipython3
+from quantflow.data.fmp import FMP
+frequency = "1min"
+async with FMP() as cli:
+    df = await cli.prices("ETHUSD", frequency)
+df = df.sort_values("date").reset_index(drop=True)
+df
+```
+
+```{code-cell} ipython3
+import plotly.express as px
+fig = px.line(df, x="date", y="close", markers=True)
+fig.show()
+```
+
+```{code-cell} ipython3
+import numpy as np
+from quantflow.utils.volatility import parkinson_estimator, GarchEstimator
+df["returns"] = np.log(df["close"]) - np.log(df["open"])
+df["pk"] = parkinson_estimator(df["high"], df["low"])
+ds = df.dropna()
+dt = cli.historical_frequencies_annulaized()[frequency]
+fig = px.line(ds["returns"], markers=True)
+fig.show()
+```
+
+```{code-cell} ipython3
+import plotly.express as px
+from quantflow.utils.bins import pdf
+df = pdf(ds["returns"], num=20)
+fig = px.bar(df, x="x", y="f")
+fig.show()
+```
+
+```{code-cell} ipython3
+g1 = GarchEstimator.returns(ds["returns"], dt)
+g2 = GarchEstimator.pk(ds["returns"], ds["pk"], dt)
+```
+
+```{code-cell} ipython3
+import pandas as pd
+yf = pd.DataFrame(dict(returns=g2.y2, pk=g2.p))
+fig = px.line(yf, markers=True)
+fig.show()
+```
+
+```{code-cell} ipython3
+r1 = g1.fit()
+r1
+```
+
+```{code-cell} ipython3
+r2 = g2.fit()
+r2
+```
+
+```{code-cell} ipython3
+sig2 = pd.DataFrame(dict(returns=np.sqrt(g2.filter(r1["params"])), pk=np.sqrt(g2.filter(r2["params"]))))
+fig = px.line(sig2, markers=False, title="Stochastic volatility")
+fig.show()
+```
+
+```{code-cell} ipython3
+class HestonCalibration:
+    
+    def __init__(self, dt: float, initial_std = 0.5):
+        self.dt = dt
+        self.kappa = 1
+        self.theta = initial_std*initial_std
+        self.sigma = 0.2
+        self.x0 = np.array((self.theta, 0))
+    
+    def prediction(self, x):
+        return np.array((x[0] + self.kappa*(self.theta - x[0])*self.dt, -0.5*x[0]*self.dt))
+    
+    def state_jacobian(self):
+        """THe Jacobian of the state equation"""
+        return np.array(((1-self.kappa*self.dt, 0),(-0.5*self.dt, 0)))
+```
+
+```{code-cell} ipython3
+
+```
+
+```{code-cell} ipython3
+c = HestonCalibration(dt)
+c.x0
+```
+
+```{code-cell} ipython3
+c.prediction(c.x0)
+```
+
+```{code-cell} ipython3
+c.state_jacobian()
+```

_sources/applications/hurst.md

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+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+    format_version: 0.13
+    jupytext_version: 1.14.7
+kernelspec:
+  display_name: Python 3 (ipykernel)
+  language: python
+  name: python3
+---
+
+# Hurst Exponent
+
+The [Hurst exponent](https://en.wikipedia.org/wiki/Hurst_exponent) is used as a measure of long-term memory of time series. It relates to the autocorrelations of the time series, and the rate at which these decrease as the lag between pairs of values increases.
+
+It is a statistics which can be used to test if a time-series is mean reverting or it is trending.
+
+```{code-cell} ipython3
+from quantflow.sp.cir import CIR
+
+p = CIR(kappa=1, sigma=1)
+```
+
+# Links
+
+* [Wikipedia](https://en.wikipedia.org/wiki/Hurst_exponent)
+* [Hurst Exponent for Algorithmic Trading
+](https://robotwealth.com/demystifying-the-hurst-exponent-part-1/)
+
+```{code-cell} ipython3
+
+```

_sources/applications/overview.md

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+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+    format_version: 0.13
+    jupytext_version: 1.14.7
+kernelspec:
+  display_name: Python 3 (ipykernel)
+  language: python
+  name: python3
+---
+
+# Applications
+
+Real-world applications of the library
+
+```{tableofcontents}
+```
+
+```{code-cell} ipython3
+
+```