Itos Lemma Explained

Video on Taylor-Explansion: • Taylor series | Chapte...
This is part 3 of my series on "Understanding Black Scholes".
Ito's Lemma is a key mathematical lemma used in the derivation of Black-Scholes - so here is my understanding.
I know that this is mathematically not super rigorous. This video is here only to give intuition.

[@kimchi_taco]

Best ito lemma explanation I've ever listened.

[@augustjung5018]

Honestly, you’re videos have been a huge help for me conceptualizing and understanding the topics at hand. Thank you!

[@joshmccraney4020]

Underrated video! Very well done

[@jadeddib]

Thank you for your outstanding explanation!

[@tsurotafadzwachikari4039]

Thank you for the video. I would like to understand the Itos Lemma and Brownian Motion. Any videos that can take me through, its for the financial engineering class under stochastic and optimisation in finance so would @ like something finance related but handy

[@nakshathra5149]

good video, thanks for explaining

[@pritshah3119]

Link of the video to be watched before?

[@stochasticNerd]

The taylor expansion video you linked is about approximating a function near a point in domain. But what you use initially in the video is about approximating the differential of the function. I tried but couldn't derive it from the logic followed in the linked video. Could you please explain it?

[@FinAndEcon]

Honestly, my de4rivation is extremely hand wavy and in no way mathematically correct. I just want to give an intuition. So when thinking about Ito's lemma, you can think about Brownian motion being approximated by a second order term. The similarity is in the concepts. Hope that helps

[@roshannanayakkara5865]

Excellent

[@Ash-vu5vo]

At 2:56, “doesn’t depend on an ordinary variable x but Brownian motion Zt”. Umm… whyyy?? Feels like there is a conceptual leap between df(x) and df(Zt), that needs more context - what is this new f? Does it transform the stochastic process into a new stochastic process? If so, what’s the point of doing that, what purpose does it serve? Determinism, class of random variables, differentiability, so much conceptual context needed. Not to mention you haven’t actually stated what the lemma is. But ofc, no one questions any of this.

[@MLDawn]

But why in a random process, the second term of the Taylor expansion cannot be ignored? In your example, the result of keeping the second term is dz**2=dt. Could you please elucidate this a little?

[@FinAndEcon]

There are books about this. In the end, this is about the theory of stochastic calculus. Within the rules of stochastic calculus, it clearly follows that this must be the case - however, for "traditional" functions, it clearly follows that the second term can be ignored. So in the end, if you really want to understand this, you need to go through a multitude of theorems and proofs from stochastic calculus

[@JuhoKim-qg1tk]

my professor needs to see this video

[@djinnhe4rt340]

why can we ignore the terms higher than 2nd order in stochastic calculus if dzt is non neglible? wouldn't the terms become more and more weighted as the order increases? i know you didn't mean to do a rigorous proof, but i was just wondering

[@FinAndEcon]

I mean, there is really no intuition here - it is just a mathematical property of Brownian motion : (. I could only send you the proof if you want to :)

[@seannguyen3260]

Thanks for your video, and the chart shuold be with out drift because of EX = 0.

[@FinAndEcon]

You are right. I should probably not use a drift. However, a representation of Brownian motion can look like it has a drift although EX = 0

[@user-ff8qw2ry6u]

Ito san no lemma.

[@hakimbeldjoudi2818]

Totally wrong, you confused Taylor's development with calculating the differential of a function over a domain? df(x) = f'(t).dx + (1/2)f''(t)(dx^2) ?? that's not even remotely close to being correct: df(x) = f'(t).d(x) and that's it, it's also just writing system equivalence that holds true by definition, you're confusing more than helping with this video.

[@FinAndEcon]

Thanks for the comment. I know that the formulas spelled out are correct - they are taken from popular finance textbooks and the MIT Lecture on the topic. And the Taylor expansion analogy is from my finance professor who has a math degree. So I am quite convinced this is not totally incorrect.