Bayesian Machine Learning – Stochastic Volatility Model:

Metadata:

This repo includes the Python code for a Stochastic Volatility Model of Apple stock returns from 2007-2022.

I developed this project for the DS6040 (Bayesian Machine Learning) course in the Data Science masters program at the University of Virginia.

Synopsis:

Apple stock returns exhibit high time-varying volatility – quite stable sometimes, highly variable at others, and the distribution of market returns is highly non-gaussian. Thus, sampling Apple stock volatilities would be very difficult.

Stochastic Volatility (SV) models help capture such dynamic volatility, creating a more realistic representation of changing market risks. Leveraging Bayesian methods, the model provides uncertainty estimates, enabling better risk management.

To develop the model, I pull the data from the yfinance package into a Pandas dataframe and calculate a 'returns' column as the difference in the logarithmic returns, which I then use to model the volatility.

I model the logarithm of the daily returns with a Student-T distribution, parameterized by:

• the degrees of freedom ($𝜈$) following an exponential distribution
• volatility ($𝑠_𝑖$), where ($𝑖$) is the time index

The volatility follows a Gaussian random walk across all time steps, parameterized by a common variance given by an exponential distribution.

I model the logarithmic returns at each time point.

The model allows the volatility to change over time, such that the volatility at each time point is controlled by a parameter for that time point ($𝑠_𝑖$).

However, the scale parameters ($𝑠_𝑖$) at each time point cannot be completely independent; otherwise, the model would overfit the data.

One thing worth noting is that I have a single variance (𝜎) for the volatility process across all time, which may not represent the true nature of stock return behavior.

I use the PyMC package to develop the SV model by writing a basic function that takes the Pandas dataframe as its input and returns the PyMC model. The model is parametrized by the stochastic process previously described to capture the volatility dynamics.

def sv_model(data):
    with pm.Model(coords={"time": data.index.values}) as model:
        𝜎 = pm.Exponential("𝜎", 10)
        volatility = pm.GaussianRandomWalk(
            "volatility", sigma=𝜎, 
            dims="time", init_dist=pm.Normal.dist(0, 100)
        )
        𝜈 = pm.Exponential("𝜈", 0.1)
        r = pm.StudentT(
            "r", nu=𝜈, 
            lam=np.exp(-2 * volatility), 
            observed=data["return"], dims="time"
        )
    return model


svol_model = sv_model(df)

The notebook includes several visualizations with markdown cells above them providing descriptions.

Data:

I pulled the Apple stock data from the Yahoo Finance package for 2007 through 2022.

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Main Packages and Dependencies:

PyMC: 5.6.1

Numpy: 1.23.5

Pandas: 2.0.3

Arviz: 0.16.1

Sklearn: 1.3.0

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Manifest:

Files in repo:

• SVM.ipynb
• apple2.csv
• LICENSE
• README.md