The Easiest Way to Derive the Black-Scholes Model

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In this video, we are going to derive the Black-Scholes formula via a delta-hedging argument. We'll construct a portfolio consisting of one option and some underlying shares and try to make the portfolio risk-free by eliminating the option's risk.
In the video, I'm using such concepts as the lognormal random walk, Ito's lemma and stochastic calculus.
If you have any questions or suggestions, feel free to let me know. Thank you for watching!

[@PerfilievFinancialTraining]

Hello friends! Thank you so much for watching! I’ve only recently started on KZbin, and this is one of my first videos. I really hope you’ll find it interesting and somewhat entertaining. Please, please do subscribe to the channel - at this early stage, your support has a HUGE impact, and absolutely every person counts. I am doing this full-time now, and if you want to see how it goes, it would be great to have you on board! As always, feel free to reach out for any feedback, questions and suggestions. You can ping me on Twitter or via email in the channel description. Thank you for your help and support!

[@zkkrhfhska]

Have you seen the derivation via Wang's transform? I come from an insurance maths background and I found that the "easiest" for me. I've also see a good explanation based on option price : probability duality which was very intuitive

[@anandkulkarni2111]

The proof is very intuitive. I recommend that you discuss why those terms like dt*dt and dt*dw tend to zero since they are infinitesimally small. It just helps people from non financial math background a bit more.

[@steez54321]

This is actually the best explained derivation I've found on KZbin so far. Thank you so much!

[@Zzzexie]

This is absolutely this best video for BSM! And the explanation is much easier to understand than the green book Thx

[@richardxue1506]

insanely high efficiency. Thank you for the great work

[@scentilatingone2148]

Brings back memories of Dif EQ class! I would have learned alot more with you as a professor.

[@PerfilievFinancialTraining]

Much appreciated, thank you for your kind words!

[@sdsa007]

I like that this video was a concise overview! It made everything connect! It complements the other videos that I saw where I got stuck in the weeds... which means I have a good understanding of the high-speed sections, but I still needed this overview to confirm all the math substitutions! Thanks!

[@burnoutparidise1]

"This is left as an exercise for the reader". Oh lord. It's my math classes all over again.

[@PerfilievFinancialTraining]

Hahaha, sorry for the flashback :) I also couldn't stand it when math textbooks did that!

[@jon5532]

This is awesome. I'm probably going to watch it a few times to get comfortable with all the material. Thanks for the knowledge!

[@PerfilievFinancialTraining]

Thank you for watching! If anything's unclear or confusing, feel free to let me know - would be happy to help out!

[@Tweeteketje]

Great content, super clear! I hope you will make more videos! Why I think this is fantastic, is that it is intuitive, clear, step-for-step and yet concise.

[@Lexis.options]

This was the best math lesson! You have a love for teaching and made it easy to understand. Looking forward to more of the same!

[@PerfilievFinancialTraining]

Hi Lexi, thank you so much for your kind words! I'm really glad to hear it was easy to understand :) Thank you!

[@Jenna-iu2lx]

The explanations are so clear, thank you so much for this video!

[@alovyachowdhury9143]

This is fantastic! Thanks for the clear indications about the assumptions on delta hedging and portfolio growth at risk-free rate, it made for a really easy to follow derivation

[@salimrhmaritlemcani1936]

Amazing strating point thanks a lot. clarifies a lot ! The only thing that I think would be relevant to point out is that this is the Black Scholes Merton differential equation, not the Black Scholes formula: they are similar but serve different purposes. The black sholes formula is a closed-form solution derived from the BSM differential equation. Black Scholes Merton differential equation is used to calculate the fair value of European-style options and to determine the option's sensitivity to changes in various factors, such as the underlying asset price and time, while the Black Scholes formula provides a mathematical formula for calculating the theoretical price of a European-style call or put option. Thanks for the content!!

[@harryj1081]

Great video as always. I’d be thrilled if you could elaborate more on option trading strategies that the cornwall capital turned 110k to 80MM from the big short. Apparently they relied heavily on the models and maximize the Convexity of option.

[@PerfilievFinancialTraining]

Hi Harry, that's an interesting story! Haven't heard of it, thank you. If time permits, I'll try to check it out! Thanks for watching! :)

[@miquelmalaga]

Sir, what are you selling and how can I buy it?

[@PerfilievFinancialTraining]

Hi Miquel! Thank you so much for your support! At the moment, I don't have much to offer, but I will let you know as soon as I do :)

[@kurian0_0]

So easy and clear to understand

[@RishabhKhare]

Really good explanation. Thanks for doing this!

[@PerfilievFinancialTraining]

My pleasure! Thanks for watching.

[@774471jr]

9:20 "annnd, that's pretty much all you have to do" lol Loving your channel!! I know the basics of options, but you definitely make it easier to understand all these complicated things.

[@PerfilievFinancialTraining]

Hi, thank you so much for watching the video! Glad it was useful :)

[@nikolaykogut7546]

Thanks Sergei... I really liked the video

[@PerfilievFinancialTraining]

Hi Nikolay, thank you so much!

[@federicocremonini4741]

Thank you so much!

[@PerfilievFinancialTraining]

No worries, thank you!

[@bomfim04]

Amazing Chanel! Hello from Brazil!

[@PerfilievFinancialTraining]

Hey, Diego! Thank you for subscribing! Hello to you too! :)

[@harshvardhanranvirsingh9473]

Perfect!! looking for more content like this!

[@PerfilievFinancialTraining]

Thank you! :) Will do my best.

[@aj_actuarial_ca]

Very well explained. Are you an actuary?

[@dylan-kt7kd]

Amazing video, thank you

[@UniversalDegen]

Damn this is like Sheldon Cooper level shit. Awesome explanation 😃

[@PerfilievFinancialTraining]

Hahaha, thank you so much for watching and for your feedback! :) Really glad it was useful :) All the best!

[@hassamkhan7861]

Video is short and To the point and i really like it though, looking for more practical content . But there certain topic which people should be comfortable with, stochastic calculus and stats , . Im sure these prerequisite would be handy . For further topics .

[@leoafrifanus]

Most intuitive trading view is the one you presented ! Some people like to say the discounted option price is a martingale, then apply Ito to it and say the drift is 0, but that’s too abstract vs this one ! Great content man !

[@PerfilievFinancialTraining]

Hi Mohd, thank you. Yeah, I see what you mean - it is indeed just theory. I'll probably do a few of these at the start, as I experiment with different topics/subjects. Hopefully will have more practical stuff later on too! Thanks for watching! :)

[@PerfilievFinancialTraining]

@@leoafrifanus Thank you, glad you like it! :) Yeah, hahaha, I know that derivation and it is too abstract indeed (especially if one doesn't know much about martingales or risk-neutral expectations etc)...

[@raulzevallos3399]

AMAIZING EXPLANATION FINALLY I UNDERSTAND IT. THANKS

[@choicedeals5041]

Hi there! This is great content and you have made it really easy to understand complex concepts. Could you make an episode to explain and demystify what exactly is a Partial Differential Equation (PDE) and how this is different from other types of models e.g. trees, monte carlo Thank you!

[@PerfilievFinancialTraining]

Hello, yes, of course, it's certainly something I could do. I am a bit short on time at the moment to film/present everything I want, but I've noted your request and will do my best. Thank you for watching this video!

[@CAfinalspeedruns]

I know it's been 2 years so you probably figured it out by now, but Khan Academy has an excellent playlist on multivariate calculus where you can find the relevant video for what a partial derivative is

[@patrickaungier3197]

Awesome work, thank you !

[@riccardoformenti4332]

Keep it going, loving the content

[@PerfilievFinancialTraining]

Hey Riccardo, thanks so much! Glad you're enjoying it :)

[@sergeyyatskevitch3617]

Hmmm..... d2S technically is not a "square" of the dS, but rather a second differential, thus the issue here is how to treat a differential of the stochastic process W. I understand that you tried to simplify the process, but omitting several important math steps in understanding this equation, led LTCM to its demise. But I enjoyed your way of presenting this very important, but complex equation. BTW, this equation is very well known in Theoretical Physics as the Fokker-Plank equation. Cheers!

[@PerfilievFinancialTraining]

Hi Sergey, thank you for your feedback and insights! Indeed, I haven't noticed I called the second derivative a "square" :) That would be a video-level typo :) Thank you!

[@joaoricardosimas2036]

This is awesone! Thank you Perfiliev!

[@mindingthedata4218]

Another incredible video! Cannot wait for the next one :)

[@PerfilievFinancialTraining]

Thanks so much for watching! Glad you're finding it useful :)

[@vvardhan14]

Thanks a lot buddy !!

[@PerfilievFinancialTraining]

Thank you for watching!

[@jonathanseo6728]

My major is civil engineering, and I also learned about stokes theorem. It’s look similar as Brownian motion is that I first impressive part. And second is that I listened your lecture from 0 to end. But I don’t have any idea about how to treat my stock portfolio 😅😅

[@Vijaykumar1614SK]

fantastic

[@PerfilievFinancialTraining]

Thanks, Vijay!

[@ttwtrader]

Hi Sergey, great video. Thank you. Is there any way to visualize the formula in terms of graphs? So, to "play" around with different "parameters" and see the graphical output? Could you add or do that as sequel of this video using mathematica for example?

[@PerfilievFinancialTraining]

In this video, I'm discussing the Black-Scholes equation, which still needs to be solved to get an options' pricing formula. The Black-Scholes equation can be written in terms of Greeks: Theta + 1/2 * vol^2 * spot^2 * Gamma + rS * Delta - rV = 0. And you can visualise the Greeks via a simple Black Scholes calculator. Unfortunately, I can't give you a link, since KZbin hides comments with links, but google "perfiliev financial black scholes" and check out the first link!

[@ttwtrader]

@@PerfilievFinancialTraining Thanks a lot.

[@ttwtrader]

Btw, it can be done in Mathematica as well.

[@PerfilievFinancialTraining]

@@ttwtrader Definitely! To be honest, even a simple Excel sheet can do :)

[@TrungPham1310xx]

You're much better than my lecturer and I have to pay for it.

[@Timbel26]

Hello,I need help on linear fractional black-scholes model.

[@bongiwelanga1706]

Thank you!!

[@lebleb8603]

Awsome! Can you explain how Cem get his levels?

[@PerfilievFinancialTraining]

Thank you for watching the videos! I'd have to dig into that a bit more, to come up with a good explanation. If time allows, I'll try to get into it. Thank you!

[@lebleb8603]

@@PerfilievFinancialTraining thank you for your reply!

[@christopherrose5554]

Hey found you on Twitter and love the videos Will you provide any practical examples in the future? EG Using these equations to construct a hypothetical neutral portfolio of apple

[@PerfilievFinancialTraining]

Hey Christopher, thank you for watching the videos! Glad you liked them :) Yeah, this one was a purely theoretical video - I'll try to do more practical stuff in the future if time allows. Thanks again!

[@ddyms]

Awesome video. Thanks :)

[@PerfilievFinancialTraining]

My pleasure, thank you!

[@HaigInstruments]

If it is risk free, then how did people use this knowledge to beat the markets?

[@freqh]

"Don't try to measure the water depth with both legs" is a old saying... It's base on fair price, but reality is different in the market... Market might be overvalued or undervalued most of the times because of greed, fear, emotions, natural calamities, war etc.

[@user-cz8lv7lw5u]

Thanks bro.

[@paulmalliga9996]

Thank you for your work!! :D

[@giovanniberardi4134]

Hi Sergei! I have a question. Is there an intuitive reason explaining why dt*dw=0 dt*dt=0 and dW*dW=dt? Why the uncertain factor in the price model is dropped when it is plugged in the BS model? Is it the direct consequence of the hedging? Thank you very much

[@lukedoyle7802]

This is the Magnum Opus of Black Scholes explanation videos

[@nemachtianimx343]

Gracias!

[@xntumrfo9ivrnwf]

Hey, ya silverback s twitter: great video! Some 'comments' (lol), or rather tiny points that *maybe* might help: - title of the video will turn some people off. Why not something like "... intuition behind the BS formula" etc. 'Derive' is scary for some people... actually thinking about this now, unless you are splitting your content between more technical and more intuitive, etc. --> in that case cool, you can just put relevant videos in a playlist - have you / will you do a binomial option pricing? if you have, apologies, been swamped so am catching up on good channels - ^^ same for no-arbitrage bounds --> that's a really easy intro to all of this; P-C parity? - Idea: create a quick XLS template to calculate this and let people follow along?

[@PerfilievFinancialTraining]

Hey Nick! Great to see you here! :) These are some great suggestions, thank you so much! - Title + thumbnails - I am trying to improve on those and make them less "scary". - Playlist - yes, as soon as I add a few more videos in a similar genre, I'll start grouping them into playlists. - Binomial model - funny you should mention it, as I did it alongside this video. Thanks for the other suggestions! Much appreciated!

[@amirulfadlan1718]

OMG HAHAHAHAHAHAHAA thank you so so much for this simpler equation. Its so hard to understand the one from the textbook. thank you sir! 😁

[@leedunkelberger9768]

great stuff " )

[@sanjithramanmohan8971]

That was some reallyy dope explanation ! Can you derive it into the formulae which they give for N(d1) and N(d2) .It'd really help if you put more videos on stochastic calculus and stuff too ! Thank you so much !

[@Jupiter1423]

Ill just use a bs calc thx

[@AaronLloyd-Jones]

The best way to derive a premium formula for an option is how I do it (and recommend others do also), and this is not the Black-Scholes formula: The Black and Scholes equation is wrong: The Black and Scholes (risk-neutral) premium is the first moment of the option expiry for an asset that has all risk and no market return (the risk-neutral measure), that which has been debased of market return (by holding portfolio returns fixed flat at r). This idiotic asset (the risk-neutral measure) is stochastically dominated by bonds in that bonds have the same return (r) but without the risk whilst it is stochastically dominated by stocks since stocks earn market return for the equivalent amount of risk: bonds have LOWER RISK for the SAME RETURN as the debased market asset (the risk-neutral measure) whilst stocks have HIGHER RETURN for the SAME RISK as the debased market asset (the risk-neutral measure) Either way, the 'risk-neutral measure' is totally idiotic and stochastically dominated by all non-redundant asset classes. It is not deep and it is not abstract. All it is is the market asset without return (which is then used to price the derivative and so is wrong and inaccurate). If a trader wants an option, then he must not take an offsetting position that nullifies the option position. There is nothing risk-neutral about that. An option premium must have a mean mu in the drift term, otherwise it is wrong... wrong for derivatives and wrong for efficient and non-communist finance. nb: I had to say 'no risk' when I sat several of the courses in undergraduate (almost two decades ago). It was clear as day to me then that it was inaccurate (and proved by me definitively now more than one decade ago). I debunk Black and Scholes fully here: drive.google.com/file/d/1drOy89roxTawddpbFv03MEgrNSRwPRab/view?usp=drive_link here is new theory for markets (crystal ball formula): drive.google.com/file/d/1POgaFZxaXpGPbxDh8p9IHP_Kr2-VXok5/view?usp=drive_link PhD examiner report 3: drive.google.com/file/d/1z2Cflnp1uQ059GIonv2lzfqOj0EcMXrv/view?usp=drive_link PhD examiner report 2: drive.google.com/file/d/1K07G377R0ZSUs9ax6EXAzYealrjbo2vS/view?usp=drive_link PhD examiner report 1: drive.google.com/file/d/1BXwbk-uFrQDH_es_T5FiIJOnJ_42oA0q/view?usp=drive_link

[@ffust3740]

Please!! We want the martingale approach!!

[@PerfilievFinancialTraining]

Hi Ferran, hahaha :) I think that would take us all the way to the solution and not just the BS PDE, right?

[@apundude]

Great video.. Can u just give an example with market datas on how it actually works as a risk free model by choosing and stock and it's underlying option?

[@kennethamoahnyame4678]

"easiest"

[@PerfilievFinancialTraining]

Hahaha, as easy as it can be :) But yes, I agree, even this method is based on some relatively complex mathematical concepts.

[@quant-prep2843]

Guys are we in heaven?

[@PerfilievFinancialTraining]

:)

[@tsunningwah3471]

zhins

[@chihuahuafink3644]

I knew I’m a 100% nerd when I thoroughly enjoyed going through all the math 🧮