Black Scholes Explained - A Mathematical Breakdown

This video breaks down the mathematics behind the Black Scholes options pricing formula.

The Pricing of Options and Corporate Liabilities:

Excel Model

3b1b Normal Distribution Video

Note: Be sure to download the sheet in Excel, as not all formulas will populate in Google Drive.

[Vail]

Thanks for the concise explanation. Very helpful and easier to understand than the one my professor gave.

[holden]

Absolute magnificent. I have been wondering about how options work for years+! Thank you so much for the in-depth walkthrough!

[tarunmathew]

This channel is a hidden gem. Incredibly insightful, explained clearly and perfectly presented!

[SpamMaster-mkbt]

I think your definition of the cumulative normal distribution from 5:26 on is missing a square. It should read N(d_i) = \frac{1}{\sqrt{2 \pi}} \int_{-inf}^{d_i} e^{-x^2/2} dx

[boriscrisp]

notational pedantry: ln is usually used to represent the natural logarithm not the log normal

[namirahasan-vbjm]

what happened to e^1 at 4:12? are we not assuming t=1, so it would be K(e^1)N(d2)?

[RosMyster]

Magnificinet explanation, it was super clear!

[cbe-commerce]

which tools are used for making such a good video?

[davidaloha]

Thanks, this was excellent coverage of Black-Scholes as-is👌

However, put-call parity is BS. This is largely because volatility doesn't have 50/50 correlation with upward/downward price movements (or in other words, it's not entitely independent of price movements). There is often an imbalance in put-option pricing which depends upon the market environment and the expectations of market participants.

[zzprod]

That a beautiful explanation. I was able to reconstruct all the parts and to test it and it works beautifully. However one part didn't work. On 8:20 you are saying that if time to expiration is very short delta is either 1 for itm or 0 for otm. I changed T to 0.0001 and moved the strike above and below the spot and the sheet returned a 0 for both d1 and ď2 and 0.5 for nd1 and nd2. I was hoping to see 1 and 0. Maybe google sheet can't handle these small values.

[waynelast]

3:34 what’s your justification for reducing d2 to d1?

[Kraktur]

Ln is the natural logarithm an ln(1)=0.
You mix things a little first saying it is the « lognormal » function. Lognormal is typically a kind of distribution where if X is normal then ln(X) is lognormal. Moreover you make a common confusion between ln et log. Log(x)=ln(x/10), thus log(10)=1