---
title: "Black-Scholes Formula"
source: https://www.jemoka.com/posts/kbhblack_scholes_formula/
---

People have been trading options for a very long time, but there wasn&rsquo;t a good way of quantify the value of an option.
There are two main types of uses for Black-Scholes Formula
you can use all variables and determine the value of options you can get the price of options being traded, then compute the $σ$&mdash;the market&rsquo;s estimation of volatility (how much they want the insurance policy that is the options) constituents \(S_0\): stock price \(X\): exercise price \(r\): risk-free interest rate \(T\): maturity time \(\sigma\): standard-deviation of log returns&mdash;&ldquo;volatility&rdquo; Black-Scholes Formula for an European &ldquo;Call&rdquo; Option Here is the scary formula:
\begin{equation} C_0 = S_0 \mathcal{N}(d_{1})-Xe^{-rT}\mathcal{N}(d_{2}) \end{equation}
where, the variables are defined above, and:
\begin{equation} \begin{cases} d_1 = \frac{\ln\qty(\frac{S_0}{X})+\qty(r+\frac{\sigma^{2}}{2})T}{\sigma \sqrt{t}}\\ d_2 = \frac{\ln\qty(\frac{S_0}{X})+\qty(r-\frac{\sigma^{2}}{2})T}{\sigma \sqrt{t}} \end{cases} \end{equation}
and \(\mathcal{N}\) is the area at point under the standard normal distribution.
oh god So let&rsquo;s dissect this a little.
The first term:
\begin{equation} S_0\mathcal{N}(d_{1}) \end{equation}
is the &ldquo;current&rdquo; stock price, weighted by the probability of you being willing to exercise it.
and the second term:
\begin{equation} Xe^{-rT}\mathcal{N}(d_{2}) \end{equation}
is the &ldquo;price&rdquo; of the exercise (what you need to pay, if exercising the option, to get the stock.)
This strike price \(X\) is discounted by \(e^{-rT}\), which is like a time machine that rolls that strike price back to what it would be today (so that it&rsquo;s comparable to \(S_0\).) As \(r\) is the risk free interest rate, we are essentially saying: &ldquo;in a perfectly functional market, over the next \(T\) days, how will our asset grow?&rdquo;
This is again weighted by the probability of you being willing to exercise it&mdash;through modified slightly differently.
Therefore, subtracting the two terms, we get the actual value of the option&mdash;the money you would gain by exercising it, then immediately selling the stock, weighted by how willing you are actually to excercise it.
Let&rsquo;s now take a look at those &ldquo;probabilities&rdquo; \(d_{\{1,2\}}\). These factors essentially provide quantification of the statement that: &ldquo;the higher our current price is ABOVE the excrecise price&mdash;accounting for volatility&mdash;the more willing we are to excercise the option.&rdquo;
Note then, \(\ln\qty(\frac{S_{0}}{X})\) form the top of both expressions. That essentially measures how high the current price \(S_0\) deviates from the strike price \(X\).
Now, as volatility \(\sigma\) increases, \(d_1\) increases and \(d_2\) decreases (as \(\frac{\sigma^{2}}{2}\) is being added in \(d_1\) and subtracted in \(d_2\)). This is because, as volatility increase, you are less certain about what the actual &ldquo;pay&rdquo; (price) is, but your option&mdash;given its constant strike price&mdash;provides the certainty in gain.
