Black-Scholes-Merton (BSM)

Brian Lee
Aug 12, 2025
Part of: {{linktitle(“v2_option”)}}

The Black-Scholes-Merton (BSM) model is a mathematical framework used to estimate the price of European-style options. It helps traders understand how the price of an option is affected by Time To Expiration, Implied Volatility (\(IV\)), Risk-free Rate, and the underlying’s Spot Price (\(S_{spot}\)). One of its key assumptions is that prices follow a normal distribution, meaning most price changes are small and extreme changes are rare.

A call option price under BSM is:¹

\[ C = S_0 N(d_1) - K e^{-rT} N(d_2) \]

The corresponding put option price is:

\[ P = K e^{-rT} N(-d_2) - S_0 N(-d_1) \]

where

\[ \begin{aligned} d_1 &= \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma \sqrt{T}} \\ d_2 &= d_1 - \sigma \sqrt{T} \end{aligned} \]

• \(C\) is the price of a European Call Option
• \(P\) is the price of a European Put Option
• \(S_0\) is the current Spot Price (\(S_{spot}\)) of the underlying asset
• \(K\) is the option’s Option Strike Price
• \(r\) is the Risk-free Rate
• \(T\) is the Time To Expiration in years
• \(\sigma\) is the Implied Volatility (\(IV\)) of the underlying asset
• \(N(\cdot)\) is the cumulative distribution function of the standard normal distribution
• \(d_1\) and \(d_2\) are intermediate terms defined above

BSM can also be written as a partial differential equation for the option price \(V(S, t)\):

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

Solving this PDE with the appropriate terminal payoff and boundary conditions yields the closed-form call and put pricing formulas above. The model also underpins the Option Greeks, which measure how option prices respond to shifts in these inputs².

I believe it’s worth learning the Black-Scholes-Merton model, especially if you’re interested in how options are priced. While I haven’t studied it deeply, I do know that it assumes stock prices follow a normal distribution. That idea gave me the inspiration to try a simpler approach in my own strategy.