Option Price Engine Implied Volatility Analysis

Initial results when reconstructing the implied volatilities for one timestamp of APPLE options from Yahoo Finance data shows significant discrepancies between what Yahoo’s data claim the implied volatility to be. The figure shows these discrepancies with several outliers. The blue scatter points were computed by taking raw Yahoo Finance option prices and reconstructing their implied volatilty using a Black Scholes pricing module for European Vanilla options based on the closed BS formula in the C++ qengine by David Doebel. This was done with an iterative solver for non-linear equations. In particular,

$${\sigma }_{n+1}={\sigma }_n-\frac{C_{BS}({\sigma }_n)-C_{market}}{Vega({\sigma }_n)}$$

This method is known as the Newton-Raphson method and it converges quadratically. $C_{BS}$ denotes the price computed via the Black Scholes formula and $C_{market}$ is the price of the option that we extract from the real-world data. Furthermore, we call $Vega(\sigma ):=\frac{\partial C}{\partial \sigma }$. Despite the fast convergence, the Newton-Raphson method has certain limitations when it comes to stability. Which is why, we use a different iterative method called Brent’s method, which is a hybrid approach of bisection and secant method. Therefore it converges slower but enjoys better stability.

Apparent issues

• Extreme IV spikes for deep OTM options
• Instability for low-priced contracts
• Using Black Scholes on American Option Contracts
• Constant risk-free rate $r$
• Using the mid price and not considering the bid-ask spread

Future roadmap

• Outlier detection
• Filtering based on liquidity to avoid including illiquid or low-priced options
• Select the price better (i.e. consider last price vs mid)