Black-Scholes-Merton Model

The Black-Scholes-Merton model is a Nobel Prize winning mathematical contribution used to determine the fair price of a financial option. At its core, the BSM model is a partial differential equation, specifically a diffusion equation. It models how the probability of an option's value diffuses through the market as it approaches expiration, linking it to the concept of geometric Brownian motion.

The model assumes that stock prices take a continuous "random walk" over time. By mapping this random spread, it calculates the probability that an option will cross its target threshold before the clock runs out. Some important concepts involved are:

• Current Price versus Strike Price: Where the asset is currently valued versus the target threshold it needs to hit.
• Time to Expiration: How much time is left before the contract expires.
• Volatility: This acts exactly like thermal conductivity in a heat transfer problem. Higher volatility means the probability "heat" spreads out much faster and wider, which increases the theoretical value of the option. This can be thought of as how erratically a stock has behaved in the past.

By comparing the theoretical value generated by this diffusion model against the actual cost to buy the option, you can estimate your expected profit or loss. This is all a massive simplification of a topic that is still challenging to me, and it doesn't even touch on the "Greeks" or the advanced statistical option analyses out there. Please look at the references for a more comprehensive discussion, as this was just an exercise for me to further explore financial engineering.