Options and the Black-Scholes formula (part 1)

Part 1

In this series of articles, I will derive the Black-Scholes equation for valuing a European call option in the classical way.

DISCLAIMER: in these articles I try to give an intuition of what is happening in general, so in many places I deliberately omit mathematical formalities so as not to complicate the understanding.

The previous article discussed what options are and how they work. Now let's derive a formula for estimating the value of a European call option.

Don’t be alarmed by all the terms throughout the article, I will try to explain each term clearly.

A small retreat

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Contents of articles (both parts)

1. Naive representation of price movement in the market

2. Adding randomness – price movement as a random process

3. Normal and log-normal distribution

4. Equation of price movement when it is distributed log-normally

5. Problems with the resulting equation

6. Girsanov's theorem and risk-neutral measure

7. Got rid of unknown drift. What's next?

Naive representation of price movement in the market

Let's imagine that we have some risky asset that is traded on the stock exchange. For example, the cost of ether depends on time.

Let's assume that we are looking for a function that calculates the relative cost (in fact our S