# Brenner and Subrahmanyam Approximation

The Brenner Approximation, as we can call it for short, is critically important because it allows us to simplify the Black-Scholes model. However, it only works when it is at-the-money (the strike price is the same as the underlying security’s price).

This simple formula is the most basic frame of how we can say an option is priced, using a 0.4 constant as a coefficient for what is otherwise a very dynamic and complex equation. The reason why it can be so simple is that we are phasing out the possibility of volatility skew (differences in IV% at different strikes on the same date). Skew is definitely something that affects options in an important way, but it is also important for our understanding–and also an ability to make quick calculations for an option’s price *by hand if needed*–to see the relationship like this between the underlying, implied volatility, and time.

If we integrate the volatility surface (3D visual of implied volatility compared to price and time) into an oversimplification of everything at-the-money, then we can approximate the price of an option with 0.4 * underlying * volatility * “the square root of time". That last part is where theta (time decay of an option) falls into the equation, and can be explained by how the value of implied volatility is directly proportional to the square root of time. (Sinclair, 2020, p. 108)

This is the Brenner and Subrahmanyam Approximation (1988), which is able to arrive at a constant as a coefficient for the *underlying times volatility* because they are restricting it to the money, which is like a derivative of the volatility surface (removing a the skew dimension).