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:
www.cs.princeton.edu/courses/...
Excel Model
docs.google.com/spreadsheets/...
3b1b Normal Distribution Video
• Why π is in the normal...
Note: Be sure to download the sheet in Excel, as not all formulas will populate in Google Drive.

[@tarunmathew2801]

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

[@financeexplainedgraphics]

That’s incredibly kind! Stay tuned for more upcoming videos

[@drumline75]

LOVE THE CONTENT KING

[@Vail88966]

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

[@holden1886]

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

[@jamesmanning5159]

No YOU have a great day

[@boriscrisp518]

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

[@financeexplainedgraphics]

You are right! Good catch …. I’ll be having a word with my script writer later 😉 Thank you for clarifying!

[@SpamMaster-mk8bt]

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

[@financeexplainedgraphics]

Oh no! I think you are right. I went back into the production file and the ^2 was there but I had hid the layer 😢. Thank you for catching it. Yes, it should be {-x^2/2}

[@zzprod1]

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.

[@zzprod1]

Sorry. I had a mistake with the denominator. Fixed it. Perfect

[@financeexplainedgraphics]

Great! I was going to reply to this earlier but needed access to my computer to check, however you beat me to it. Glad it is working for you now!

[@waynelast1685]

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

[@financeexplainedgraphics]

d2 is not reduced to d1, there is still a negative sign in d2 that is not in d1. However, the reason I reduce it down from the original long form of d2 = (d1 - sig(T)) is because it is a little easier to see visually, and it highlights that the difference between d1 and d2 are their inverse relationships with regard to volatility (sigma). I show the long form initially because when you see Black-Scholes, much of the time d2 is shown in the long form.

[@waynelast1685]

@@financeexplainedgraphics I see thanks