1. Getting Started¶

1.1. SdePy¶

The SdePy package provides tools to state and numerically integrate Ito Stochastic Differential Equations (SDEs), including equations with time-dependent parameters, time-dependent correlations, and stochastic jumps, and to compute with, and extract statistics from, their realized paths.

Several preset processes are provided, including lognormal, Ornstein-Uhlenbeck, Hull-White n-factor, Heston, and jump-diffusion processes.

Computations are fully vectorized across paths, via NumPy and SciPy, making live sessions with 100000 paths reasonably fluent on single cpu hardware.

This package came out of practical need, so expect a flexible tool that gets real-life things done. On the other hand, not every part of it is clean and polished, so expect rough edges, and the occasional bug (please report!).

Developers are committed to the stability of the public API, here again out of practical need to safeguard dependencies.

1.1.1. Start here¶

1.1.2. License¶

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.

Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.

Neither the name of the copyright holder nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS “AS IS” AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

This package reuses the compatibly licensed files listed below.

File: sdepy/doc/_templates/autosummary/class.rst License: 3-clause BSD

For details, see sdepy/doc/_templates/autosummary/LICENSE.txt

1.2. Quick Guide¶

1.2.1. Install and import¶

Install using pip install sdepy, or copy the package source code in a directory in your Python path.

Import as

>>> import sdepy
>>> from sdepy import *  # safe and handy for interactive sessions
>>> import numpy as np
>>> import scipy
>>> import matplotlib.pyplot as plt  # optional, if plots are needed

1.2.2. How to state an SDE¶

Here follows a bare-bone definition of a Stochastic Differential Equation (SDE), in this case a Ornstein-Uhlenbeck process:

>>> @integrate
... def my_process(t, x, theta=1., k=1., sigma=1.):
...     return {'dt': k*(theta - x), 'dw': sigma}

This represents the SDE dX = k*(theta - X)*dt + sigma*dW(t), where theta, k and sigma are parameters and dW(t) are Wiener process increments. A further 'dn' or 'dj' entry in the returned dictionary would allow for Poisson or compound Poisson jumps.

A number of preset processes are provided, including lognormal processes, Hull-White n-factor processes, Heston processes, and jump-diffusion processes.

1.2.3. How to integrate an SDE¶

Now my_process is a class, a subclass of the cooperating SDE and integrator classes:

>>> issubclass(my_process, integrator), issubclass(my_process, SDE)
(True, True)

It is to be instantiated with a number of parameters, including the SDE parameters theta, k and sigma; its instances are callable, given a timeline they will integrate and return the process along it. Decorating my_process with kunfc allows for more concise handling of parameters:

>>> myp = kfunc(my_process)
>>> iskfunc(myp)
True

It is best explained by examples:

Scalar process in 100000 paths, with default parameters, computed at 5 time points, using 100 steps in between:

>>> coarse_timeline = (0., 0.25, 0.5, 0.75, 1.0)
>>> np.random.seed(1)  # make doctests predictable
>>> x = my_process(x0=1, paths=100*1000,
...                steps=100)(coarse_timeline)
>>> x.shape
(5, 100000)

Vector process with three components and correlated Wiener increments (same parameters, paths, timeline and steps as above):

>>> corr = ((1, .2, -.3), (.2, 1, .1), (-.3, .1, 1))
>>> x = my_process(x0=1, vshape=3, corr=corr,
...                paths=100*1000, steps=100)(coarse_timeline)
>>> x.shape
(5, 3, 100000)

Vector process with time-dependent parameters and correlations, computed on a fine-grained timeline and 10000 paths, using one integration step for each point in the timeline (no steps parameter):

>>> timeline = np.linspace(0., 1., 101)
>>> corr = lambda t: ((1, .2, -.1*t), (.2, 1, .1), (-.1*t, .1, 1))
>>> theta, k, sigma = (lambda t: 2-t, lambda t: 2/(t+1), lambda t: np.sin(t/2))
>>> x = my_process(x0=1, vshape=3, corr=corr,
...                theta=theta, k=k, sigma=sigma, paths=10*1000)(timeline)
>>> x.shape
(101, 3, 10000)
>>> gr = plt.plot(timeline, x[:, 0, :4])  # inspect a few paths
>>> plt.show()  

A scalar process with path-dependent initial conditions and parameters, integrated backwards (i0=-1):

>>> x0 = np.random.random(10*1000)
>>> sigma = 1 + np.random.random(10*1000)
>>> x = my_process(x0=x0, sigma=sigma, paths=10*1000,
...                i0=-1)(timeline)
>>> x.shape
(101, 10000)
>>> (x[-1, :] == x0).all()
True

A scalar process computed on a 10 x 15 grid of parameters sigma and k (note that the shape of the initial conditions and of each parameter should be broadcastable to the values of the process across paths, i.e. to shape vshape + (paths,)):

>>> sigma = np.linspace(0., 1., 10).reshape(10, 1, 1)
>>> k = np.linspace(1., 2., 15).reshape(1, 15, 1)
>>> x = my_process(x0=1, theta=2, k=k, sigma=sigma, vshape=(10, 15),
...                paths=10*1000)(coarse_timeline)
>>> x.shape
(5, 10, 15, 10000)
>>> gr = plt.plot(coarse_timeline, x[:, 5, ::2, :].mean(axis=-1))
>>> plt.show() 

In the example above, set steps=100 to go from inaccurate and fast, to meaningful and slow (the plot illustrates the k-dependence of average process values).

Processes generated using integration results as stochasticity sources (mind using consistent vshape and paths, and synchronizing timelines):

>>> my_dw = integrate(lambda t, x: {'dw': 1})(vshape=1, paths=10000)(timeline)
>>> p = myp(dw=my_dw, vshape=3, paths=10000,
...         x0=1, sigma=((1,), (2,), (3,)))  # using myp = kfunc(my_process)
>>> x = p(timeline)
>>> x.shape
(101, 3, 10000)

Now, x1, x2, x3 = = x[:, 0], x[:, 1], x[:, 2] have different sigma, but share the same dw increments, as can be seen plotting a path:

>>> k = 0  # path to be plotted
>>> gr = plt.plot(timeline, x[:, :, k])
>>> plt.show()  # doctest: +SKIP

If more integrations steps are needed between points in the output timeline, use steps to keep the integration timeline consistent with the one of my_dw:

>>> x = p(coarse_timeline, steps=timeline)
>>> x.shape
(5, 3, 10000)

Using stochasticity sources with memory (mind using consistent vshape and paths):

>>> my_dw = true_wiener_source(paths=10000)
>>> p = myp(x0=1, k=1, sigma=1, dw=my_dw, paths=10000)

>>> t1 = np.linspace(0., 1.,  30)
>>> t2 = np.linspace(0., 1., 100)
>>> t3 = t = np.linspace(0., 1., 300)
>>> x1, x2, x3 = p(t1), p(t2), p(t3)
>>> y1, y2, y3 = p(t, theta=1.5), p(t, theta=1.75), p(t, theta=2)

These processes share the same underlying Wiener increments: x1, x2, x3 illustrate SDE integration convergence as steps become smaller, and y1, y2, y3 illustrate how k affects paths, all else being equal:

>>> i = 0 # path to be plotted
>>> gr = plt.plot(t, x1(t)[:, i], t, x2(t)[:, i], t, x3(t)[:, i])
>>> gr = plt.plot(t, y1[:, i], t3, y2[:, i], t3, y3[:, i])
>>> plt.show() 

1.2.4. How to handle the integration output¶

SDE integrators return process instances, a subclass of np.ndarray with a timeline stored in the t attribute (note the shape of x, repeatedly used in the examples below):

>>> coarse_timeline = (0., 0.25, 0.5, 0.75, 1.0)
>>> timeline = np.linspace(0., 1., 101)
>>> x = my_process(x0=1, vshape=3, paths=1000)(timeline)
>>> x.shape
(101, 3, 1000)

x is a process instance, based on the given timeline:

>>> type(x)
<class 'sdepy.infrastructure.process'>
>>> np.isclose(timeline, x.t).all()
True

Whenever possible, a process will store references, not copies, of timeline and values. In fact,

>>> timeline is x.t
True

The first axis is reserved for the timeline, the last for paths, and axes in the middle match the shape of process values:

>>> x.shape == x.t.shape + x.vshape + (x.paths,)
True

Calling processes interpolates in time (the result is an array, not a process):

>>> y = x(coarse_timeline)

>>> y.shape
(5, 3, 1000)

>>> type(y)
<class 'numpy.ndarray'>

All array methods, including indexing, work as usual (no overriding), and return NumPy arrays:

>>> type(x[0])
<class 'numpy.ndarray'>
>>> type(x.mean(axis=0))
<class 'numpy.ndarray'>

You can slice processes along time, values and paths with special indexing:

>>> y = x['t', ::2]  # time indexing
>>> y.shape
(51, 3, 1000)
>>> y = x['v', 0]  # values indexing
>>> y.shape
(101, 1000)
>>> y = x['p', :10]  # paths indexing
>>> y.shape
(101, 3, 10)

The output of a special indexing operation is a process:

>>> isinstance(y, process)
True

Smart indexing is allowed. To select paths that cross x=0 at some point and for some component, use:

>>> i_negative = x.min(axis=(0, 1)) < 0
>>> y = x['p', i_negative]
>>> y.shape == (101, 3, i_negative.sum())
True

You can do algebra with processes that either share the same timeline, or are constant (a process with a one-point timeline is assumed to be constant), and either have the same number of paths, or are deterministic (with one path):

>>> x_const = x['t', 0]  # a constant process
>>> x_one_path = x['p', 0]  # a process with one path

>>> y = np.exp(x) - x_const
>>> z = np.maximum(x, x_one_path)

>>> isinstance(y, process), isinstance(z, process)
(True, True)

When integrating SDEs, the SDE parameters and/or stochasticity sources accept processes as valid values (mind using deterministic processes, or synchronizing the number of paths, and make sure that the shape of values do broadcast together). To use a realization of my_process as the volatility of a 3-component lognormal process, do as follows:

>>> stochastic_vol = my_process(x0=1, paths=10*1000)(timeline)
>>> stochastic_vol_x = lognorm_process(x0=1, vshape=3, paths=10*1000,
...     mu=0, sigma=stochastic_vol)(timeline)

Processes have specialized methods, and may be analyzed, and their statistics cumulated across multiple runs, using the montecarlo class. Some examples follow:

Cumulative probability distribution function at t=0.5 of the process values of x across paths:
```
>>> cdf = x.cdf(0.5, x=np.linspace(-2, 2, 100))  # an array
```

Characteristic function at t=0.5 of the same distribution:

>>> chf = x.chf(0.5, u=np.linspace(-2, 2, 100))  # an array

Standard deviation across paths:

>>> std = x.pstd()  # a one-path process
>>> std.shape
(101, 3, 1)

Maximum value reached along the timeline:

>>> xmax = x.tmax()  # a constant process
>>> xmax.shape
(1, 3, 1000)

A linearly interpolated, or Gaussian kernel estimate (default) of the probability distribution function (pdf) and its cumulated values (cdf) across paths, at a given time point, may be obtained using the montecarlo class:

>>> y = x(1)[0]  # 0-th component of x at time t=1
>>> a = montecarlo(y, bins=30)
>>> ygrid = np.linspace(y.min(), y.max(), 200)
>>> gr = plt.plot(ygrid, a.pdf(ygrid), ygrid, a.cdf(ygrid))
>>> gr = plt.plot(ygrid, a.pdf(ygrid, method='interp', kind='nearest'))
>>> plt.show()  # doctest: +SKIP

A montecarlo instance can be used to cumulate the results of multiple simulations, across multiple components of process values:

>>> p = my_process(x0=1, vshape=3, paths=10*1000)
>>> a = montecarlo(bins=100)  # empty montecarlo instance
>>> for _ in range(10):
...     x = p(timeline)  # run simulation
...     a.update(x(1))  # cumulate x values at t=1
>>> a.paths
100000
>>> gr = plt.plot(ygrid, a[0].pdf(ygrid), ygrid, a[0].cdf(ygrid))
>>> gr = plt.plot(ygrid, a[0].pdf(ygrid, method='interp', kind='nearest'))
>>> plt.show()  

1.2.5. Example - Stochastic Runge-Kutta¶

Minimal implementation of a basic stochastic Runge-Kutta integration, scheme, as a subclass of integrator (the A and dZ methods below are the standardized way in which equations are exposed to integrators):

>>> from numpy import sqrt
>>> class my_integrator(integrator):
...     def next(self):
...         t, new_t = self.itervars['sw']
...         x, new_x = self.itervars['xw']
...         dt = new_t - t
...         A, dZ = self.A(t, x), self.dZ(t, dt)
...         a, b, dw = A['dt'], A['dw'], dZ['dw']
...         b1 = self.A(t, x + a*dt + b*sqrt(dt))['dw']
...         new_x[...] = x + a*dt + b*dw + (b1 - b)/2 * (dw**2 - dt)/sqrt(dt)

SDE of a lognormal process, as a subclass of SDE, and classes that integrate it with the default integration method (p1) and via my_integrator (p2):

>>> class my_SDE(SDE):
...     def sde(self, t, x): return {'dt': 0, 'dw': x}
>>> class p1(my_SDE, integrator): pass
>>> class p2(my_SDE, my_integrator): pass

Comparison of integration errors, as the integration from t=0 to t=1 is carried out with an increasing number of steps:

>>> np.random.seed(1)
>>> args = dict(dw=true_wiener_source(paths=100), paths=100, x0=10)
>>> timeline = (0, 1)
>>> steps = np.array((2, 3, 5, 10, 20, 30, 50, 100,
...                   200, 300, 500, 1000, 2000, 3000))
>>> exact = lognorm_process(mu=0, sigma=1, **args)(timeline)[-1].mean()
>>> errors = np.abs(np.array([
...     [p1(**args, steps=s)(timeline)[-1].mean()/exact - 1,
...      p2(**args, steps=s)(timeline)[-1].mean()/exact - 1]
...     for s in steps]))
>>> ax = plt.axes(label=0); ax.set_xscale('log'); ax.set_yscale('log')
>>> gr = ax.plot(steps, errors)
>>> plt.show()  
>>> print('euler error: {:.2e}\n   rk error: {:.2e}'.format(errors[-1,0], errors[-1,1]))
euler error: 1.70e-03
   rk error: 8.80e-06

1.2.6. Example - Fokker-Planck Equation¶

Monte Carlo integration of partial differential equations, illustrated in the simplest example of the heat equation diff(u, t) - k*diff(u, x, 2) == 0, for the function u(x, t), i.e. the Fokker-Planck equation for the SDE dX(t) = sqrt(2*k)*dW(t). Initial conditions at t=t0, two examples:

u(x, t0) = 1 for lb < x < hb and 0 otherwise,
u(x, t0) = sin(x).

Setup:

>>> from numpy import exp, sin
>>> from scipy.special import erf
>>> from scipy.integrate import quad
>>> np.random.seed(1)
>>> k = .5
>>> x0, x1 = 0, 10;
>>> t0, t1 = 0, 1
>>> lb, hb = 4, 6

Exact green function and solutions, to be checked against results:

>>> def green_exact(y, s, x, t):
...     return exp(-(x - y)**2/(4*k*(t - s)))/sqrt(4*np.pi*k*(t - s))
>>> def u1_exact(x, t):
...     return (erf((x - lb)/2/sqrt(k*(t - t0))) - erf((x - hb)/2/sqrt(k*(t - t0))))/2
>>> def u2_exact(x, t):
...     return exp(-k*(t - t0))*sin(x)

Realization of the needed stochastic process, by backward integration from a grid of final values of x at t=t1, using the preset wiener_process class (the steps keyword is added as a reminder of the setup needed for less-than-trivial equations, it does not actually make a difference here):

>>> xgrid = np.linspace(x0, x1, 51)
>>> tgrid = np.linspace(t0, t1, 5)
>>> xp = wiener_process(paths=10000,
...             sigma=sqrt(2*k), steps=100,
...             vshape=xgrid.shape, x0=xgrid[..., np.newaxis],
...             i0=-1)(timeline=tgrid)

Computation of the green function and of the solution u(x, t1) (note the liberal use of scipy.integrate.quad below, enabled by the smoothness of the Gaussian kernel estimate a[i, j].pdf):

>>> a = montecarlo(xp, bins=100)
>>> def green(y, i, j):
...     """green function from (y=y, s=tgrid[i]) to (x=xgrid[j], t=t1)"""
...     return a[i, j].pdf(y)
>>> u1, u2 = np.empty(51), np.empty(51)
>>> for j in range(51):
...     u1[j] = quad(lambda y: green(y, 0, j), lb, hb)[0]
...     u2[j] = quad(lambda y: sin(y)*green(y, 0, j), -np.inf, np.inf)[0]

Comparison against exact values:

>>> y = np.linspace(x0, x1, 500)
>>> for i, j in ((1, 20), (2, 30), (3, 40)):
...     gr = plt.plot(y, green(y, i, j),
...                   y, green_exact(y, tgrid[i], xgrid[j], t1), ':')
>>> plt.show()  
>>> gr = plt.plot(xgrid, u1, y, u1_exact(y, t1), ':')
>>> gr = plt.plot(xgrid, u2, y, u2_exact(y, t1), ':')
>>> plt.show()  
>>> print('u1 error: {:.2e}\nu2 error: {:.2e}'.format(
...     np.abs(u1 - u1_exact(xgrid, t1)).mean(),
...     np.abs(u2 - u2_exact(xgrid, t1)).mean()))
u1 error: 2.49e-03
u2 error: 5.51e-03

1.2.7. Example - Basket Lookback Option¶

Take a basket of 4 financial securities, with risk-neutral probabilities following lognormal processes in the Black-Sholes framework. Correlations, dividend yields and term structure of volatility (will be linearly interpolated) are given below:

>>> corr = [
...     [1,    0.50, 0.37, 0.35],
...     [0.50,    1, 0.47, 0.46],
...     [0.37, 0.47,    1, 0.19],
...     [0.35, 0.46,  0.19,   1]]

>>> dividend_yield = process(c=(0.20, 4.40, 0., 4.80))/100
>>> riskfree = 0  # to keep it simple

>>> vol_timepoints = (0.1, 0.2, 0.5, 1, 2, 3)
>>> vol = np.array([
...     [0.40, 0.38, 0.30, 0.28, 0.27, 0.27],
...     [0.31, 0.29, 0.22, 0.16, 0.18, 0.21],
...     [0.24, 0.22, 0.19, 0.19, 0.21, 0.22],
...     [0.35, 0.31, 0.21, 0.18, 0.19, 0.19]])
>>> sigma = process(t=vol_timepoints, v=vol.T)
>>> sigma.shape
(6, 4, 1)

The prices of the securities at the end of each quarter for the next 2 years, simulated across 50000 independent paths and their antithetics (odd_wiener_source is used), are:

>>> maturity = 2
>>> timeline = np.linspace(0, maturity, 4*maturity + 1)
>>> p = lognorm_process(x0=100, corr=corr, dw=odd_wiener_source,
...                     mu=(riskfree - dividend_yield),
...                     sigma=sigma,
...                     vshape=4, paths=100*1000, steps=maturity*250)
>>> np.random.seed(1)
>>> x = p(timeline)
>>> x.shape
(9, 4, 100000)

A call option knocks in if any of the securities reaches a price below 80 at any quarter (starting from 100), and pays the lookback maximum attained by the basket (equally weighted), minus 105, if positive. Its price is:

>>> x_worst = x.min(axis=1)
>>> x_mean = x.mean(axis=1)
>>> down_and_in_paths = (x_worst.min(axis=0) < 80)
>>> lookback_x_mean = x_mean.max(axis=0)
>>> payoff = np.maximum(0, lookback_x_mean - 105)
>>> payoff[np.logical_not(down_and_in_paths)] = 0
>>> a = montecarlo(payoff, use='even')
>>> print(a)  
    4.997 +/- 0.027