# BSM (Black-Scholes-Merton Model)

## Basic Points

- The Black-Scholes model was published in 1973 and became the basis of how options are priced. They won the Nobel Prize for this in 1997.
- To this day, we still use this model to speak of the major components of option dynamics conversationally, using the Greeks. However, brokers and market makers will use more mathematically-complex and computationally-intensive approaches such as the binomial model, but these offer little conceptual help compared to breaking the dynamics down into Greek objects like the BSM does.
- One of Merton's main contributions to the original model was helping to price in dividends. These commonsense improvements evolved into the BSM (Black-Scholes-Merton) model.
- There are some limitations of this model, but if understood well, these limitations can be used to uncover edges (competitive advantages). For example, the BSM model assumes that volatility (the one-year expected range in percentage terms at 68.3% confidence) remains constant, which practically speaking is never going to happen.

## Intermediate: Greek Implications

The rest of the option chain can be expected to have accurate readings for delta (directional exposure), gamma (acceleration of directional exposure), and theta (time decay). However, much productive work can go into forecasting changes in IV (implied volatility - volatility measured by relative option pricing) in order to anticipate—with reasonable likelihood—what will happen to the price of options. As Sheldon Natenberg breaks it down in *Options Pricing and Volatility*: “Black and Scholes tried to answer this question: if the stock price moves randomly over time, but in a manner that is consistent with a constant interest rate and volatility, what must be the option price after each moment in time such that an option position that is correctly hedged will just break even?” (2015, p. 339). Our task as traders is made easier by how we only need to test and examine the volatility input, and can trust brokers to be accurate with the rest. Therefore, if an option seems over or underpriced according to a theoretical pricing model, then this is the market pricing in either weaker or stronger volatility. At SpotGamma, we are studying shifts in volatility every day because that is where the most inefficiency is, and where there is the most edge to discover for boosting our trading.

## Review

This model is a lot to take in and so it is worth a review of what is going on here. Part of how we can simplify its formula in our minds is to realize that only the stock price and time are the variables (what is changing). In contrast, volatility and rho (interest rate exposure on options) are inputs. Rho, pertaining to proportionate pricing increases of interest rates increase, is usually the least impactful part of the model, and so this isolates volatility as the key input. The reason why volatility is an input and not a variable is because the BSM model accepts the knowingly-naïve assumption that volatility will remain constant throughout the life of the options contract, which it never does. This is mainly where we dig in as traders and look for trading edges.

To summarize what the BSM is trying to do, “it is just a mathematician’s way of expressing how changes in one set of variables—stock price *S* and time *t*—affect the value of something else, a call *C*” (p. 339). As a note, this formula sometimes uses *V *instead of *C *to represent the value of the option as opposed to the call price.