Using Volatility Models to Evaluate Quality of Options Data.

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\title{Using the Inverse Black-Scholes Implied Volatility Model and GARCH(1,1) Historic Volatility Model to Evaluate Quality of Data} \author{} \date{}

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\section*{Evaluating Quality of Data}

Due to the nature of Options being an opportunity for investors to Predict The Future', theWisdom of Crowds', for which Aristotle is credited with, rings particularly true in options theory. These predictions will sculpt a Volatility Surface to depict strong behaviours, such as splitting and ridging around catalytic events. This can be used in conjunction with your own sources of news and/or data to see if you have a full picture of the current markets situation, and sometimes even to learn new things from early action of Institutional Investors, who use Options to hedge their downside on large moves.

\subsection*{NIFTY FIFTY}

We have compiled 85,000 different lines of Options values with respect to the NIFTY FIFTY Index on the Indian National Stock Exchange. This large amount of data was unfortunately insufficient to use in combination with an LLM based Sentiment analysis algorithm to calibrate and derive substantial edge. We hope to revisit this project in the near future to take advantage of more time and data.

\section*{Inverse Black-Scholes Model}

The Inverse Black-Scholes model is derived from the following Linear Second order homogeneous Partial Differential Equation:

[ \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0 ]

In the European Model (Exclusively Exercised on Expiration), it is analytically solvable (in the Call case) with the following solutions:

[ C(S,t) = N(d_1)S - N(d_2)Ke^{-rT}, ]

[ d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}, \qquad d_2 = d_1 - \sigma\sqrt{T}. ]

Given sufficient data (in the form of historic options prices, strikes, days till expiry, and underlying valuation), you can solve for your $\sigma$ via Newton--Raphson and graph this on the $z$-axis. These surfaces are called \textit{Implied Volatility Surfaces}. In an ideal world for the Black-Scholes model, these would be exceptionally smooth (Look at one for the S&P 500).

However, this is very rarely the case, and supplies a significant amount of data if you are able to denoise the graphs sufficiently. At the very least, it allows you to evaluate the quality of your data sources.

\section*{GARCH(1,1) Model}

We discovered the GARCH(1,1) Historic Volatility Model could be used in combination with the aforementioned surfaces (by calculating the difference at each strike and DTE) to evaluate the pricing standing of a given set of options. If the historic volatility of an option was disproportionately off from the implied volatility then this would likely be cause for concern. The GARCH model is far simpler, as shown below:

[ \sigma^2_t = \alpha_0 + \alpha_1 \varepsilon^2_{t-1} + \beta_1 \sigma^2_{t-1}, ]

[ \alpha_0 > 0, \quad \alpha_1 \geq 0, \quad \beta_1 \geq 0, \quad \alpha_1 + \beta_1 < 1. ]

With your constants, $\alpha_0, \alpha_1, \beta_1$, usually being determined using regression of some form.

\section*{Volatility Surfaces and Opportunities for Implied Volatility Option Mispricing Strategies}

It is common to see significant spikiness on Implied Volatility Surfaces; this is almost always artifacting with your data, however it can also be an opportunity to capitalise on a market inefficiency (although this was far more common Pre-2008). These spikes can be exposed more visibly by creating a `heat map' of the Implied Volatility Velocity and looking for abnormally large magnitudes on the surface. These indicate sharp breaks from the smooth behaviour expected from a highly efficient market. We have created an example of one with sliders allowing you to fit the data we have sourced and slightly cleaned up in a few different regards. As long as you keep the column titles consistent, any data can be run through our Volatility Surfaces, Heat Maps, and GARCH(1,1) models.

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