Black-Scholes Model Explained: The Math Behind Options Pricing

Añadido: 2026-05-30

[00:00:00]In 1997, Myron Scholes and Robert Merin received the Nobel Memorial Prize in Economics for work that helped answer one of Wall Street's hardest questions.

[00:00:10]What is an option worth? At first glance, that sounds simple. A call option gives you the right to buy a stock later at a fixed price. But once the stock price is uncertain, time is passing, interest rates matter, and volatility changes everything, that right becomes surprisingly hard to value. This is the blacksholes model.

[00:00:31]One of the core frameworks that turns uncertainty into a theoretical options price. By the end of this video, the goal is not just to recognize the formula. It is to understand what the equation is doing, why bell curves show up, and why the assumptions matter.

[00:00:47]Before 1973, options pricing was far less standardized. If you wanted to buy the right to purchase a stock at a certain price, how much should you pay for that right today? Traders had no widely accepted consistent way to answer that question. Then Fischer Black, Myron Scholes, and Robert Merin came along with the breakthrough. They showed that under specific assumptions, an option could be replicated by continuously adjusting a hedge made of the stock and cash. And that is the economic insight.

[00:01:17]If two positions can produce the same future payoff, they should have the same price today. Otherwise, traders could create an arbitrage. The result was a formula that gives a theoretical no arbitrage price for an option based on five main inputs. For this video, we will focus on the basic black shells formula for a European call option on a non-ividend paying stock. At expiration, the payoff is straightforward. If the stock finishes below the strike price, the call expires worthless. If the stock finishes above the strike price, the call is worth the stock value minus the strike price. So, the payoff is easy to describe at expiration. The hard part is figuring out what that uncertain future payoff is worth today. That means the model has to answer three questions at once. How likely the option is to finish in the money, how much the stock can move, and how future cash flows should be discounted back to today. Here is the black trolls call option formula. C is the theoretical call option price. S is the current stock price. K is the strike price. T is the time to expiration measured in years. R is the risk-free interest rate. Sigma is volatility. And capital N is the cumulative normal distribution, which we will unpack visually in a minute. The most important intuition is that the formula has two big pieces. A call option is valuable because it gives you stock upside if the option finishes in the money. But to get that stock you have to pay the strike price. So the model is roughly value of the stock side benefit minus the present value of the strike side cost. So this is not just two pieces glued together.

[00:03:04]It is the call payoff logic translated into today's dollars under the model's assumptions. D1 and D2 are not new market inputs. They are intermediate values the model calculates from the five inputs. Here is D1. You do not need to derive it to understand the model, but the pieces are useful. The ln of S over K is the moneyiness term. It compares the stock price to the strike price. The R + sigma squar over two term comes from the model's log normal pricing setup. and the denominator sigma * the square root of t scales everything by the total volatility over the life of the option. d2 is simpler. d2 is just d1 minus sigma * the square root of t. In other words, d2 is d1 shifted down by the total volatility over the options life. The key idea is that d1 and d2 turn the option setup into locations on a standardized bell curve. Once the model has those locations, it can convert them into areas under the curve.

[00:04:12]Why is there a bell curve in an options formula? Because black shores assumes the stock follows a process where log returns are normally distributed. The stock price itself is modeled as logn normal, but the log returns are what produce the bell curve. The model then standardizes the problem. So d1 and d2 can be read as positions on a standard normal curve. A bell curve measured in standard deviations. Capital N means the cumulative standard normal distribution.

[00:04:43]Put simply, N of X is the area under the bell curve to the left of X. It is an area, not the height of the curve. If you pick any point on the curve, call it D, then N of D is the area to the left of that point. Because it is an area, it is a number between 0 and 1. If D is positive, the area to the left is more than 50%. For example, N of positive.5 is about69.

[00:05:11]If D is negative, the area to the left is less than 50%. N of.5 is about 31. Because the standard normal curve is symmetric, the negative version is the complement of the positive version. So N ofD equals 1 - N of D.

[00:05:30]That is why positive and negative versions of D1 and D2 are closely related. For calls, the standard formula uses N of D1 and N of D2. For puts, the standard formula uses N of negative D1 and N ofD2.

[00:05:47]Same bell curve, different side of the option payoff. Now we can connect this directly to the option. In the black shaws call formula, n ofd2 is the model implied probability that the option finishes in the money. More precisely, it is a riskneutral probability. That does not mean investors are actually risk neutral. It means the model uses a pricing framework made possible by the hedging argument. If the option can be replicated by a continuously adjusted hedge, the model does not need your personal expected return for the stock.

[00:06:20]It prices the option using the risk-free rate and riskneutral probabilities. This can feel a little backwards at first because a call finishes in the money when the stock is above the strike, but after the model standardizes and rearranges the problem, that event maps to the area to the left of D2 on the standard normal bell curve. N of D1 is different. In the basic black shells call formula, N of D1 is the call options delta. Delta tells you how much the option price tends to move for a small move in the stock price. It is also the hedge ratio. The amount of stock exposure used to hedge the option in the model. So N of D1 and N of D2 are related but they are not the same thing.

[00:07:04]N of D1 is tied to delta. N of D2 is tied to the riskneutral probability of exercise. Now the formula should feel less arbitrary. The first term S * N of D1 is the stock side value. The second term K * E to ther R * T * N of D2 is the strike side cost. E to ther R * T is the continuous compounding discount factor. It brings the strike price back to today's dollars using the risk-free rate. Let's use a simple example.

[00:07:38]Suppose the stock price is 100, the strike price is 105, time to expiration is 6 months, the risk-free rate is 5%.

[00:07:46]And volatility is 20%. When you plug those into D1 and D2 formulas, you get D1 of about.1 and D2 of.24.

[00:07:57]The bell curve then converts those into N of D1 of461 and N of D2 of about.46.

[00:08:06]Now substitute those values into the formula. The stock side value is about 100 *461 or 46.1.

[00:08:16]The discounted strike is 105 * e to the.025 or about 102.41.

[00:08:24]Multiply that by.46.

[00:08:27]Subtract it from the stock side value and the model price comes out to about $4.58.

[00:08:34]That is not a promise that the option should trade at exactly that price in the real world. It is the theoretical price the model gives you when those inputs and assumptions are used. This also explains why you sometimes see N of D1 and N ofD2.

[00:08:50]Those appear in the standard put option formula. A call payoff focuses on the upside above the strike. A put payoff focuses on the downside below the strike. So the put formula uses the complimentary areas of the same bell curve. For example, if n of d2 is about46, then n of negative d2 is about.594.

[00:09:12]Same curve, same model, different side of the payoff. Now that the formula makes more sense, the assumptions matter even more. Black shells is powerful because the assumptions make the math work. But those assumptions are simplified. The basic version assumes the option is European style, meaning it can only be exercised at expiration. It assumes no dividends. It assumes volatility stays constant. It assumes you can trade continuously. It assumes no transaction costs and it assumes stock prices move smoothly according to a lognormal process. Real markets are messier. Prices can jump. Volatility changes over time. Trading costs money and different strike prices often trade with different implied volatilities.

[00:09:55]That is why traders often treat black trolls as a baseline rather than a perfect description of reality. It gives a disciplined starting point, but real options markets reflect volatility smiles, skews, jumps, and fat tails that the basic model does not fully capture.

[00:10:12]So, black shles is not just a formula to memorize. It is a way of translating a call options payoff into today's dollars using payoff logic, bell curve probabilities, discounting, and no arbitrage reasoning. Used carefully, it is an incredibly useful framework, but the price it gives is only as good as the assumptions behind it. Subscribe for more finance videos just like this