People have been trading options for a very long time, but there wasn’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 $σ$—the market’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—“volatility”

## Black-Scholes Formula for an European “Call” 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’s dissect this a little.

The first term:

\begin{equation} S_0\mathcal{N}(d_{1}) \end{equation}

is the “current” 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 “price” 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’s comparable to \(S_0\).) As \(r\) is the risk free interest rate, we are essentially saying: “in a perfectly functional market, over the next \(T\) days, how will our asset grow?”

This is again weighted by the probability of you being willing to exercise it—through modified slightly differently.

Therefore, subtracting the two terms, we get the actual value of the option—the money you would gain by exercising it, then immediately selling the stock, weighted by how willing you are actually to excercise it.

Let’s now take a look at those “probabilities” \(d_{\{1,2\}}\). These factors essentially provide quantification of the statement that: “the higher our current price is ABOVE the excrecise price—accounting for volatility—the more willing we are to excercise the option.”

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 “pay” (price) is, but your option—given its constant strike price—provides the certainty in gain.