Weiner Process

In this document, we use the term Weiner process \(w_t\) to indicate a Brownian motion with standard deviation given by the parameter \(\sigma\); that is to say, the one-dimensional Weiner process is defined as:

1. \(w_t\) is Lévy process

2. \(d w_t = w_{t+dt}-w_t \sim N\left(0, \sigma dt\right)\) where \(N\) is the normal distribution

The characteristic exponent of \(w\) is

(30)\[\begin{equation} \phi_{w, u} = \frac{\sigma^2 u^2}{2} \end{equation}\]

from quantflow.sp.weiner import WeinerProcess

pr = WeinerProcess(sigma=0.5)
pr

WeinerProcess(sigma=0.5)

from quantflow.utils import plot
# create the marginal at time in the future
m = pr.marginal(1)
plot.plot_characteristic(m, n=32)

from quantflow.utils import plot
import numpy as np
plot.plot_marginal_pdf(m, 128)

Test Option Pricing

from quantflow.options.pricer import OptionPricer
from quantflow.sp.weiner import WeinerProcess
pricer = OptionPricer(WeinerProcess(sigma=0.2))
pricer

OptionPricer(model=WeinerProcess(sigma=0.2), n=128, max_moneyness_ttm=1.5)

import plotly.express as px
import plotly.graph_objects as go
from quantflow.options.bs import black_call
pricer.reset()
r = pricer.maturity(0.005)
b = r.black()
fig = px.line(x=r.moneyness_ttm, y=r.time_value, markers=True, title="Time value")
fig.add_trace(go.Scatter(x=b.moneyness_ttm, y=b.time_value, name=b.name, line=dict()))
fig.show()

pricer.maturity(0.1).plot()

/home/runner/work/quantflow/quantflow/quantflow/options/bs.py:65: RuntimeWarning:

some derivatives were zero