Stochastic Calculus for Quants | Understanding Geometric Brownian Motion using Itô Calculus

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In this tutorial we will learn the basics of Itô processes and attempt to understand how the dynamics of Geometric Brownian Motion (GBM) can be derived. First we learn what an Itô integral is and how it differs from a regular integral. This leads us to discussing the dynamics of Itô processes and then a special type of calculus for financial mathematics based on Brownian Motion called Itô Calculus.
We will understand why this is different to ordinary calculus in terms of the accumulation of quadratic variation. Also, we discuss how to use Taylor series expansion and using Itô’s Lemma to understand the dynamics of a particular function (valuation) given a defined Itô process, or stochastic differential equation (SDE) that has been defined for the underlying.
We briefly discuss a generic drift diffusion model and the Itô-Doeblin formula for Itô processes. This then leads to a derivation of the dynamics of Geometric Brownian Motion, and it’s explicit formulation which can be used for simulating GBM paths.
00:00 Intro
01:34 Itô Integrals
06:30 Itô processes
09:10 Contract/Valuation Dynamics based on Underlying SDE
12:24 Itô's Lemma
13:35 Itô-Doeblin Formula for Generic Itô Processes
18:04 Geometric Brownian Motion Dynamics
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22 июл 2024

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Комментарии : 66

@mostafaamer3057 2 года назад

Genuinely in love with this channel, keep up the good work!

@panceroti 2 года назад

Very clear explanation. This helped me in my Advanced Statistical Mechanics course I'm doing in my Masters in Physics!

@user-wr4yl7tx3w Год назад

This is really a great channel. Wish I knew about it earlier.

@kostas6915 8 месяцев назад

15:20 -> This second order term actually adds to the drift term, OVER TIME. Exactly, this is the essence of the case. Well depicted!

@thanarttangtanakoon2993 2 года назад

Really love this sorted of content

@user-hi3yc3lq6r Год назад

Nice lecture. The explanation is quite clear and informative.

@luyombojonathan6688 Год назад

Great channel !!! Thanks for the work

@user-oo1bk5dv6c Год назад

watching this is a year 2 bachelor is a painful yet much rewarding experience. thank you, Jonathon

@soumyajitroy6829 2 года назад

Great video with so simple language.

@recsoncanton9965 Год назад

Best video ive seen so far...

@kdpr007 2 года назад

Thank you for the fantastic video. Can you please upload a video on Partial averaging in conditional expectation using fairly basic language. Thank you once again for your videos.

@riccardoformenti4332 2 года назад

keep up this type of content

@omelhordomeualgoritmo 2 года назад

have been in love with quant for many years, but man GBM is the cure for insomnia. God help me :) ... Awesome video btw, thank you!

@mounsifgahit386 2 года назад

what does GBM stand for ?

@QuantPy 2 года назад

Geometric brownian motion

@user-wr4yl7tx3w Год назад

idea. can you consider doing a video on cases where the underlying is interest rate product, such as an interest rate swap?

@TheIllerX 2 года назад

I can really recommend that second volume book of Shreve. That is where I learnt from when I was new to the subject. It explains things cleary, but don't go into allt the more technical and subtle details to prove everything rigourosly.. This is a good start to understand what is happening. After that I recommend a more detailed book. Shreves other book together with Karatzas "Brownian motion and stochastic calculus" is one such option. This is not a light read, but also a good book.

@codyfintech Год назад

Fully agreed, Martin Baxter's Financial Calculus book is also good for developing an intuitive grasp on these concepts.

@ghostwhowalks5623 11 месяцев назад

Great video! What is the context behind the mean square limit....why do we need that particular concept for the stochastic integral? Thanks!

@user-oc8le2ln7o 10 месяцев назад

Pretty good strategy for every beginner trader

@tobiasbergkvist4520 2 года назад

Negative stock prices can be avoided by using exp(brownian motion) instead. Still, I don't think a normal distribution is a good fit, since it severely underestimates the probability of bigger price movements; so called "black swan events", which happen all the time. A distribution with fatter tails is more appropriate. I think the pareto distribution probably has a too fat tail for most stocks though - so a semi-fat tail distribution is likely the best fit for modeling most assets.

@War4Skills 2 года назад

Also the interpretation changes when we use an exp(Brownian motion) doesn't it? Which would complicate things.

@drdca8263 Год назад

[note:I don’t know what I’m talking about when it comes to finance.] What about the Martingale assumption? Would that being violated contradict the Efficient Market Hypothesis or something? Or... Well, I guess momentum trading working (iirc?) sorta maybe implies violating the martingale assumption? Hmm... now I’m wondering how one would measure how well a model is doing. I think maybe you could look at like, the surprisal of the observations under the model, minus the entropy of the model? Or, uh, Hm, well, if things at separate intervals are independent (or across different stocks? Or something that would let you take an empirical average), you could maybe take a KL divergence? Of course, the real test in practice would be “does it make you money”, but pretending that wasn’t the goal and you just wanted an “accurate” model for its own sake, how would you measure its “accuracy” (when it is all stochastic like this)

@ingolfura.4327 Год назад

This is actually true, and priced in the market aswell I highly recommend the video "the volatility smile" by professor patrick boyle here on youtube, he talks about how this is dealt with as the volatility is actually changing all the time

@emmanuelameyaw9735 2 года назад

Is there a software to do this? Or this is just theory stuff for intuition?

@kdpr007 2 года назад

Thank you for the fantastic video. Can anyone please help me understand one basic concept. Why do we have break the [0,t] into smaller intervals. What do we miss by not doing so? Does it eliminate any unnecessary /overlapping factors? Thank you

@csaracho2009 Год назад

That is why it is called infinitesimal...

@FoobsTon 8 месяцев назад

What a brilliant explanation. Well done.

@shahbajsingh4228 2 года назад

I fucking love this channel

@joemcdanel3708 2 года назад

Great video! I do have a question. In the first 30ish seconds you mention that stocks can’t go negative. Why is this so? WTI crude went negative in the year 2020.

@iceman5321 2 года назад

Crude oil are futures and not stocks. The residual claim right of stocks on earnings just stops at $0 because the company is limited and does not need to come up for additional capital if it files for bankruptcy because it is legally limited. For the future crude oil however the future contracts expired, which means the oil barrels are delivered and because of the storage problem at the time, you did not pay for the oil, but you where paid to store the oil. As you can see very different issues, and barely comparable.

@QuantPy 2 года назад

Unlike the equity market, commodity markets have quite inelastic demand. The extreme example of this is in the power market where instantaneous power cannot (on a large scale) be stored. In commodity markets it is essential to have negative price signals. As in the comment above, companies are entities that have limited liabilities.

@davidlenir7517 Год назад

Can I read volume 2 without volume 1? I have a high energy theoretical physics background.

@johnhudson9248 Год назад

Nice lecture, but in the intro for Wiener stochastic processes, though I'm not a professional quant, I always thought that the stochastic variable was the log of the asset value rather than the asset value itself. This is what it says in Wikipedia too, and using the log has two effects (1) the variable can't go negative and (2) the statistics look similar whatever the lever of the asset, whether its $1 or $1000.

@QuantPy Год назад

Thanks John, this lecture starts with the Wiener process and progresses to the Geometric Brownian Process where assets prices have the log of asset price as you’ve mentioned

@fernandojackson7207 6 месяцев назад

Excellent Lecture. In 21:30, we know Ln(St) is Normal. How about St itself, is it Normal too? Not sure what the distribution of e^X is when X is Normal.

@xRedNationx 2 года назад

So why is it an adaptive process into the filtration? Why can’t it be static?

@STheBlackDragonS 2 года назад

u can be perfectly a teacher xd

@Gerard91999 2 года назад

For quantitative finance which could be better? CFA or MBa?

@QuantPy 2 года назад

Just my opinion neither, financial math would be a good choice 👍 CFA & MBA are programs not specific to quantitative finance in terms of derivative pricing theory ect. Great if you want a job as an analyst in the financial industry though.

@Gerard91999 2 года назад

@@QuantPy The next year I’ll finished my bach in Financial Engineering and now I'm looking for the CFA or Quantitative Finance MSc in Glasgow. Which MSc could be better for the Quant path? Or in your opinion, what could be the next step?

@siddharthchaudhary1266 2 года назад

@@Gerard91999 Don't worry about CFA or finance at all. If you are serious about Quantitative Finance then do MSc either in Quantitative Finance or Mathematics (better if focused in PDE or probability theory) and do lot of good quality computer science projects either in Python, C++ or JAVA. With good skills in Computer Science and Math you could easily crack it.

@Gerard91999 2 года назад

@@siddharthchaudhary1266 thank you m8!!!

@sueton1797 2 года назад

Neither. But the CFA is cheap compared to the MBA, so if you can, take that. For a quant career, the best is a PHD, where it doesn't matter what you choose to study. Whether phyisics, mathematics or computer science. It depends on your focus and your research interests.

@2late4coffee 2 года назад

I am an undergrad and taking intro to stochastic modelling class, how can I go deeper to understand these topics, what should I begin with?

@QuantPy 2 года назад

I suggest buying Stochastic Calculus for Finance II by Steven Shreve

@patbateman69420 2 года назад

Try out Stochastic Calculus by Calin. It should be simple for you based on what you say your knowledge is, but it should prepare you for further study.