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In this notebook we will explore a machine learning approach to find anomalies in stock options pricing.
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# Unsupervised Learning for anomaly detection in options pricing
**Note**: This notebook is part of a broader work for predicting stock prices. The outcome (identified anomaly) is a feature (input) in a **LSTM** model (within a **GAN** architecture)- <a href='https://github.com/borisbanushev/stockpredictionai'>link to the repo</a>.
#### 1. Why using options pricing data and anomalies as inputs in the GAN?
Options valuation is a very difficult task. It entails using a lot of data points (some are listed below) and some of them are quite subjective (such as the implied volatility - see below) and difficult to calculate precisely. As an example let us check the calculation for the call's _Theta_ - $\theta$:
<center><img src='imgs/Options/calltheta.png'></img></center>
where (among others) $N(d_1)$ and $N(d_2)$ are cumulative distribution functions for a standard normal distribution, respectively:
<center><img src='imgs/Options/optionsinfo.png'></img></center>
and the **Black-Scholes** formula which is used for calculating the options price itself. The formula for calculating the price of a European call option with maturity in $t$ and current price of $S_0$ is:
<center><img src='imgs/Options/bsmcall.png'></img></center>
Second, the Black-Scholes model, famously known as _binomial tree_, requires a lot of assumptions to be met in order for the model to be accurate. These rules, however, most often cannot be met in real life. Some of them include:
- the model assumes that the underlying volatility ($\sigma$) is constant over the life of the option and stays unaffected by changes in the underlying stock price levels. Many times the volatility changes with changes in the strike price - the higher the difference between strike and the underlying price, the higher the volatility. This phenomenon is called the **Volatility smile** (refer to the chart below),
- the model assumes constant risk-free rate of return (something difficult to anticipate as the global economy changes every day),
- the model does not account for liquidity risk and additional fees/charges,
- it assumes that stock returns follow lognormal distribution (meaning that the model ignores the possibility of large price deviations, surge or drop, something that can easily be observed in real trading),
- assumes no dividend payout. Dividend payouts change the current valuation of the stock, which in turn should change the options price,
- the model is applicable only for European options.
The fact that the aforementioned assumptions can rarely be met in real-life is exactly why anomalies can be observed. This, in turn, creates a lot of opportunities we can explore and exploit with machine/deep learning, such as arbitrage trading.
```python
"""
===== IMPORTANT:
Execute the following code cells (below) first in order to instantiate the 'options_df' dataframe
"""
curr_price_ = options_df.iloc[0,0]
plt.figure(figsize=(10, 6))
plt.scatter(options_df.Strike, options_df.IVAsk, label='IV to Strike', \
c='royalblue')
plt.vlines(curr_price_, options_df.IVAsk.min(), \
options_df.IVAsk.max(), \
label='Current GS price - ${}'.format(curr_price_), \
linestyles='--', colors='gray')
plt.xlabel('Strike price - $')
plt.ylabel('IV')
plt.title('Options Volatility smile')
plt.legend(fontsize='large')
plt.savefig('imgs/Options/volatilitysmile.jpg', dpi=100)
plt.show()
```
_Not a perfect smile at this example, but we only used data for one day._
The difficulty in options valuation creates a lot of opportunities for arbitrage (something we will explore in another post). So let's go back to our original idea - predicting stock prices. As we already mentioned, we want to use as many different types of features in order to find, and account for, eExcerpt of 22,825 characters
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matched fp:7fe2c72ca9ed0f03, llm:description: 'In this notebook we will explore a machine learning approach to find anomalies in stock options pricing.' Language: Jupyter Notebook
matched fp:7fe2c72ca9ed0f03, llm:description: 'In this notebook we will explore a machine learning approach to find anomalies in stock options pricing.' Language: Jupyter Notebook
matched fp:7fe2c72ca9ed0f03, llm:description: 'In this notebook we will explore a machine learning approach to find anomalies in stock options pricing.' Language: Jupyter Notebook