Weekly Lesson Wednesday📝 

Black-Scholes Model: What It Is, How It Works 🎯

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What Is the Black-Scholes Model?

How are you boys,
glad to see you are back keeping up with the lessons, and to any new members, welcome. We do this shit every Wednesday.

Today’s lesson is the Black-Scholes model. also known as the Black-Scholes-Merton (BSM) model. It is one of the most important concepts in modern financial theory. Just a heads up there will be a few mathsy words and concepts thrown around, seeing as it’s a mathematical formula. Fortunately, you don't need to know or even understand the math to use Black-Scholes modeling in your own strategies because of a variety of online options calculators to help.

Essentially in the simplest words The Black-Scholes model is a mathematical formula used to estimate the price of financial options. The model requires 5 inputs the strike price of an option, the current stock price, the time to expiration, the risk-free rate, and the volatility.

The standard model can only be used for pricing European-style options, which can only be exercised at expiration. This is important as as it does not take into account that American options could be exercised before the expiration date. For American options a similar model called the Bjerksund-Stensland model is often used.

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How it works ❓️ 

OK, so you know what it does, now how does it work ?

Well, like I said before it operates on 5 variables. I’ll run you through them if you are unfamiliar.

⭐THE INPUTS ⭐

1. The Current Stock Price (S)
Easy stuff first. I’m sure you all know. This is The current market price of the underlying asset

2. The Strike Price (K)
Might be unfamiliar if you don’t deal in options. The strike price, also known as the exercise price, represents the price at which the option holder can buy (in the case of a call option) or sell (in the case of a put option) the underlying asset.

3. Time to Expiration (T)
Again another options term. Time is a critical component of option pricing. The longer the time until expiration, the more valuable the option, as it allows for more price movement in the underlying asset. This is time placed on a given option.

4. Volatility (σ)
Volatility measures the degree of price fluctuations in the underlying asset. A higher volatility implies a greater potential for price swings, which generally increases the option's value.

5. Risk-Free Interest Rate (r)
The risk-free interest rate is the return an investor could earn on a risk-free investment, such as a government bond. It's used to discount the future cash flows of the option back to their present value. Easily found online.

Armed with these five components, the Black-Scholes Model calculates the theoretical price of an option. So, you can determine if an option is overvalued or undervalued compared to the market price and profit off the arbitrage.

AN EXAMPLE ⚡

Imagine you are considering buying a European call option on XYZ Corporation. The current stock price (S) is $100, the strike price (K) is $110, the time to expiration (T) is 90 days, the volatility (σ) is 0.20 (or 20%), and the risk-free interest rate (r) is 5%.
With this you can plug it into your online calculator or do the equation as shown down below for the hard core nerds. But, one way or another you will get the theoretical value of the call option, which is approximately $6.64. This means that, according to the Black-Scholes Model, the fair market price of this option should be around $6.64.

Now, as always the model has assumptions, which are the following 👇

• No dividends are paid out during the life of the option.

• Markets are random (i.e., market movements cannot be predicted).

• There are no transaction costs in buying the option.

• The risk-free rate and volatility of the underlying asset are known and constant.

• The returns of the underlying asset are normally distributed.

• The option is European and can only be exercised at expiration.

These assumptions also pave the ways for the potential pitfalls of this model.

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It’s not your lucky lotto ticket 🎟

Like always with most of these models and lessons they are guidelines and well LESSONS ! If this thing was accurate and perfect we would all be fucking millionaires. The Black-Scholes was the the first of the arbitrage-free models that show how to relate the price to hedging costs - under some assumptions. It has been used and manipulated since then to work for people. Now I’m not saying don’t use it. At least it is mathematically based and it’s sound, and it’s definitely better than some of the shit out there that’s basically guess work. However, you should know why it is has limitations. 

LIMITATIONS 📢

1. Assumptions:
The model relies on certain simplifying assumptions, such as constant volatility and a risk-free interest rate, which may not always hold true in real-world markets.

2. Market Dynamics:
The model doesn't account for sudden market shocks or extreme events, which can lead to significant deviations between theoretical and actual option prices.

3. Dividends:
It assumes that the underlying asset doesn't pay dividends, which may not hold true for certain stocks.

4. Volatility Estimation:
Accurate estimation of volatility can be challenging, and small errors in volatility inputs can lead to significant discrepancies in option pricing.

Finally, I don’t want to leave you thinking this formula isn’t worth your time. So I will list the many advantages of it quickly, because I know this has been long piece.

1. Standardization:
The model provides a standardized framework for pricing options, making it easier for investors and traders to assess their value consistently

2. Transparency:
The mathematical nature of the Black-Scholes Model offers transparency in option pricing, allowing market participants to understand and compare valuations easily.

3. Speed:
It takes two fucking seconds. Why not use it.

4. Risk Management:
By knowing the theoretical value of options, investors can better manage their risk exposure and make informed trading decisions.

^_NEWS
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Thank you
That’s All Folks

Thank you for reading. I know it was a longer one today, but I hope it will do you some good.
Like usual. Not financial advice. Full disclaimer here

Cheers,