The mistake almost all traders make is assuming they can determine a useful probability to plug into their equations. The Black-Scholes-Merton equation uses a normal distribution, which as this video points out is not always the correct assumption. Your risk and reward are actually an unknown quantity in all cases. This is not to say that the risk/reward are bad, just unknown. So, when you size your trades/investments take that into account. In many cases, if you are highly leveraged, the market will eventually find your liquidation. Even if you are a large hedge fund.
Wow - this is an amazingly good comment!!!! Completely agree it's about understanding the risk and modifying what you do to take the limitations of the BS equation into account. If you don't, you can run into problems - which is really just what I'm trying to get across. Thanks for taking the time to comment!!
@@friend2194 Sure, but whether that's a useful probability is still questionable. Your oldest data may not be as useful as your newest data. Markets change over time. That's why a lot of people use time weighted averages. But then that's effectively reducing the size of your model. So the probability of a successful trade is still unknown even with a very large amount of data to look at. Expecting any specific risk/reward based on past data is a bit of a crap shoot.
The quality of your videos is really improving, this is a well done job. I watched the video from Veritasium and I think what they didn't want a too math-heavy video since for people without prior background, learning the idea behind Black-Scholes would have been sufficiently highly informative and I would have been difficult to talk immediately about misbehaviors. These days, there are relatively good tools to estimate volatility distribution (using ML for example), I think exploring fractals would also be a great path to go for.
Thank you very much - nice of you to say so! This channel is quite an experiment to see what I can do and how people would react -- and in the start I was just trying to figure out how even to make a video! I'm of course being a little cheeky in this video and don't really have too much problem with what Veritasium is doing here. But in seriousness also, I have had these conversations with people around the well-known problems of the BS equation, and also been frustrated by people who insist on sticking to it because in their community everyone else does. Most serious traders though know you have to use various compensations with the BS equation to mitigate these. Thanks again!
Day to day movement is largely random, but over the course of weeks and months, there's positive drift (given enough stocks, eg the S&P 500), also (as mentioned) there's often fat tail risk to the downside. So yep, not a normal distribution.
There are four more properties to capture: 1. Skewed tails. Negative extremes movements are generally larger than extreme positive movements. 2. Volatility is highly correlated to trading activity and volume which is partly deterministic (markets are open certain times a day and closed certain days every year, more activity in the beginning of a market session because of market-on-open orders and people waiting for the bell etc). 3. Time between orders is also stochastic. 4. The price jumps themselves for all order ticks are discretized by the limitations of the broker or exchange in question. Is it possible to include these properties in these types of fractal models?
Wow - those are really great points, thanks @AnthonyBerlin. You could try doing analysis on time between orders data or other info to see if they were fractal. I don't know if they would be or not!
came from your recent video, and i think ill stick around! ive always wanted to look into what happens when you start to relax the underlying assumptions in models, so thanks for the additional motivation :D ill definitely be looking into fractals more because of this
This is actually a very insightful video. Came across your channel when I watched your video on chaos theory a while ago and it's good to see you're still posting and somewhat active!
Very interesting video. My advanced derivatives professor taught us this exact same idea but using the Cauchy distribution, I've never heard of Fractal Cascade till now and will definitely be reading some of Mandelbrot's work. Thank you very much!
You're welcome and I'm really glad you found it interesting!! Fractal cascades are probably something that gets talked about more in a geophysical context e.g. for modelling atmospheric phenomena, because they represent a simplistic physical model of what's happening to create these distributions. But I think they are interesting to think about in terms of financial market behavior as well! And they're really useful for generating synthetic data that exhibits a Cauchy distribution.
Thanks! This bugged me a lot when I saw Veritasium's video... I was almost screaming: "No! You cannot predict prices like that! What if someone invents a new type of microchip or there is a war in the only country making 80% of something essential entire world uses??!" ... These models were good enough two centuries ago but the world is way more interconnected and market prices change like weather because it is literally weather-like - the butterfly effect applies here and it's not a nice closed system anymore. And what about money printing? Who knows how many trillion dollars will be printed next week so any kind of "traditional hedging" goes out of the window anyway. And most countries have a wildcard called "emergency state" and can change the very rules of the game arbitrarily, not meaning the rules will be followed - that depends on behavior of the masses which is again... chaotic (deterministic yet unpredictable).
I wonder why I never heard LTCM co-founded by one of the creators of Black Scholes? That epic fail would tarnish the whole foundation of how options and other derivatives are priced. So it's just not discussed, hidden in plain site. They say never confuse the map aka model with reality!
It's really Mandelbrot's motivations, which are largely about better understanding the nature of these things, and developing a new field of science. But the practical applications are in better understanding the behavior of markets, which in turn can lead to better investment decisions and understanding of the risks. Someone who used these well was Nassim Taleb who wrote the book The Black Swan - by understanding that some derivative contracts were greatly underestimating risk. He had to be patient, but eventually was proven right - from what I understand.
Do you have a nice introductory source for these fractal cascade processes? I couldn't find anything with a quick google search. I was also wondering whether such processes are even martingales and how widely accepted they are as the proper model for price movements.
Thanks very much for your question! I wasn't aware of any really "nice" textbooks on this when I was doing my PhD (some time ago though), and a lot of it just came from the scientific literature. That's actually why I started making these videos, because I can't believe this stuff still isn't widely known - I suspect big data and AI have replaced chaos theory as the "sexy" topics so don't get talked about that much any more. What's really important to understand is that these (multi)fractal properties are a characteristic of many complex, high-dimensional chaotic systems, like weather, and so the sources on this might be talking about any of these systems. There's a book: Lovejoy, S., and D. Schertzer (2013), The Weather and Climate: Emergent Laws and Multifractal Cascades which is an example of this. I've (tried to) read an older version of it and found it pretty dense. If you're interested specifically in fractals and financial markets, then read Mandelbrot's books and papers, and anyone who references these - those Mandelbrot's work can be a bit cryptic at times! There was a journal of Econophysics at one point (not sure if it's still going). Hope that helps!
Correct me if I’m wrong because I’m confused, I thought the black scholes Merton equation factors in the underlying movement of the market based on past and current events through drift, isn’t the drift aspect of the equation stating that current and past trends can effect the pricing of contracts.
I think you're right that drift can be taken into account, if by that you mean underlying trend. What I am trying to highlight is the importance of extreme, unexpected events, which I think are different. You need fat-tailed probability distributions to describe these.
Just stumbled across your channel after discovering Benoit Mandlebrot and his talks. Really enjoy your content. One thing I was wondering is there a huge difference between the distribution implied by the fractal model vs a distribution created by fatter tails ie Cauchy/Pareto etc?
Thank you for your comment!! Really appreciate it. Fractal models of financial markets would generally have fat tails, and you can adjust them to make them have fatter tails. Fractal models are ultimately also just another abstraction of reality - so you're really just trying to fit the best model to the data. Depending on what you are trying to do, you can just focus on the parameters for the probability distribution and not worry about making a fractal to generate synthetic "price action". If you're comparing the difference between a fractal model and one based on a normal distribution, most of the time there isn't much difference. It's just the failure to be able to properly capture the likelihood of extreme values, which can be critical in a market shock. Hope that fits what you're asking!
any good idea has its own limit - this is my own citation from my interview for Facultimedia, a while ago. That is why veratisum video is only good as populism of math science, and... proving that Nobel prize committee has no clue what they award. Thank you for honesty and clear explanations.
Thank you - absolutely right and a good point - the BSM equation is obvious a brilliant piece of work, and it still gets used despite its flaws but usually with some fudge factors applied. What surprises me is how little though Mandelbrot's work seems to get attention (maybe that's a false impression on my part?), which I think discovered something much more interesting than that markets can behave approximately like a random walk. Thanks again!!
Wow. Is BSM Modelling still mostly relied on by majority of people? So does that mean institutions and traders still havent learnt from past events like LTCM black swan events??
I've definitely come across people who are convinced that the BSM model is fine. But I think smarter traders know that there are problems with it and have fudge factors they use to deal with this, particularly in terms of how they cope with the possibility of more extreme market movements. I think this would make a good subject for future videos!! Thanks again!!
Good video, however the Fed did not bail out LTCM nor their investors. And over time, LTCMs investments turned out to be profitable, so in the end their models were correct.
Thanks Tobias - that's a really good point. The Fed just orchestrated the rescue of the position and I think my video is a bit unclear on this - though most references I see to this event do colloquially describe it as a bailout. I don't want to create a false impression, so I've decided to edit out reference to the Fed - and again thank you for raising this. I'm not sure that I agree that the fact that LTCM's investments eventually became profitable proves the models were correct - mainly because you can demonstrate (and Mandelbrot did a lot of work proving this) that financial market statistics are inconsistent with theories based on statistical independence and normal distributions. LTCM clearly got something wrong if it required this intervention, and the nature of the rescue created controversy. But that doesn't mean that the BSM equation is useless either, you can make adjustments to better deal with the likelihood of extreme events. Of course, these are just my opinions. Thanks again Tobias
So l don't get exactly how BSM is fundamentally wrong. Are you saying that events classed as low probability in BSM are actually high probability irl? Just wondering.
It's more a case of how it handles rare events with large consequences. Like Nassim Taleb points out, such events are not as rare as we typically regard them to be, and that's where you can run into trouble with the BSM equation. Generally these days people who know what they are doing with the BSM model have means to take these better into account, but it's difficult to handle these sorts of events. The classic BSM model thought it could handle these, but by assuming price fluctuations behave like a random walk, they underestimate them. Hope that explains it!
@@fractalmanhattan Thanks for responding. l must say that it sounds to me like what you are saying has more to do with how people classify risk or rare events and types of random walks; simple random walks and random walks with drift.
Thanks so much again Brumor - really nice to get your messages. Glad you enjoyed it!! I probably overdid some of the sound effects and edited out some of the B roll and accompanying humor as I think some viewers didn't really like this. It's great though to experiment and learn some lessons from that! All the best and thanks again!
@@fractalmanhattan No worries! I think the sound effects and humor were a great idea. Can't please everyone, I suppose. If you decide to keep this style in future videos, I'm sure you'll improve over time. Either way, you're already doing great and are bound to do awesome in the future also! :)
I'll join the other commenter in saying the video would be improved three fold by replacing the stock footage by nothing. Otherwise an okay video, and potential for a great channel.
Yeah - some of it is a bit random for sure - perhaps too much of an in-joke in some bits, and partly because I'm experimenting. Your comment is appreciated -thanks!
Thanks for the feedback, it's always good to get. Yeah, I can definitely see what you mean here - probably experimenting a bit much! Also, I didn't want to seem too serious in this one, as if I'm offering criticism I want to seem like I am poking fun rather than being too intense :). It's a good lesson for next time though. Thanks again - great to get your comment.
@@fractalmanhattanyou honestly put more information into 8 minutes compared to other peoples hour long courses. The visual representation isn't too bad, but the sound effects really are a bit much
@@fractalmanhattanSeth Godin once said that the reason He has a blog is to not spend time learning TH-cam and only focus on his skill. Therefore, I'm grateful for the work you put in your videos to spread that gems, and I don't care about that other things. You are so smart! Thank you!
Yes indeed there are approaches which have subsequently been developed that you can use to take into account changing volatility - topics for the future!
The mistake almost all traders make is assuming they can determine a useful probability to plug into their equations. The Black-Scholes-Merton equation uses a normal distribution, which as this video points out is not always the correct assumption. Your risk and reward are actually an unknown quantity in all cases.
This is not to say that the risk/reward are bad, just unknown. So, when you size your trades/investments take that into account. In many cases, if you are highly leveraged, the market will eventually find your liquidation. Even if you are a large hedge fund.
Wow - this is an amazingly good comment!!!! Completely agree it's about understanding the risk and modifying what you do to take the limitations of the BS equation into account. If you don't, you can run into problems - which is really just what I'm trying to get across. Thanks for taking the time to comment!!
With a large enough sample size, it'd be distributed normally, according to central limit theorem
But then assumption of Independence is mostly untrue lol
@@fractalmanhattanwhats BS equation?
@@friend2194 Sure, but whether that's a useful probability is still questionable. Your oldest data may not be as useful as your newest data. Markets change over time. That's why a lot of people use time weighted averages. But then that's effectively reducing the size of your model. So the probability of a successful trade is still unknown even with a very large amount of data to look at. Expecting any specific risk/reward based on past data is a bit of a crap shoot.
The quality of your videos is really improving, this is a well done job. I watched the video from Veritasium and I think what they didn't want a too math-heavy video since for people without prior background, learning the idea behind Black-Scholes would have been sufficiently highly informative and I would have been difficult to talk immediately about misbehaviors.
These days, there are relatively good tools to estimate volatility distribution (using ML for example), I think exploring fractals would also be a great path to go for.
Thank you very much - nice of you to say so! This channel is quite an experiment to see what I can do and how people would react -- and in the start I was just trying to figure out how even to make a video! I'm of course being a little cheeky in this video and don't really have too much problem with what Veritasium is doing here. But in seriousness also, I have had these conversations with people around the well-known problems of the BS equation, and also been frustrated by people who insist on sticking to it because in their community everyone else does. Most serious traders though know you have to use various compensations with the BS equation to mitigate these. Thanks again!
Day to day movement is largely random, but over the course of weeks and months, there's positive drift (given enough stocks, eg the S&P 500), also (as mentioned) there's often fat tail risk to the downside. So yep, not a normal distribution.
In general stocks are lognormally distributed with a positive forward value. Right?
There are four more properties to capture:
1. Skewed tails. Negative extremes movements are generally larger than extreme positive movements.
2. Volatility is highly correlated to trading activity and volume which is partly deterministic (markets are open certain times a day and closed certain days every year, more activity in the beginning of a market session because of market-on-open orders and people waiting for the bell etc).
3. Time between orders is also stochastic.
4. The price jumps themselves for all order ticks are discretized by the limitations of the broker or exchange in question.
Is it possible to include these properties in these types of fractal models?
Wow - those are really great points, thanks @AnthonyBerlin. You could try doing analysis on time between orders data or other info to see if they were fractal. I don't know if they would be or not!
iirc they mention the weaknesses of the model, ie being based on CAPM & normal distiributions.. volatility that isn’t dynamically changing… etc etc.
came from your recent video, and i think ill stick around! ive always wanted to look into what happens when you start to relax the underlying assumptions in models, so thanks for the additional motivation :D
ill definitely be looking into fractals more because of this
Thank you!
This is actually a very insightful video. Came across your channel when I watched your video on chaos theory a while ago and it's good to see you're still posting and somewhat active!
Thanks - I really appreciate you taking the time to say so!! I'll try to get back to making some more videos soon.
Very interesting video. My advanced derivatives professor taught us this exact same idea but using the Cauchy distribution, I've never heard of Fractal Cascade till now and will definitely be reading some of Mandelbrot's work. Thank you very much!
You're welcome and I'm really glad you found it interesting!! Fractal cascades are probably something that gets talked about more in a geophysical context e.g. for modelling atmospheric phenomena, because they represent a simplistic physical model of what's happening to create these distributions. But I think they are interesting to think about in terms of financial market behavior as well! And they're really useful for generating synthetic data that exhibits a Cauchy distribution.
Good video, but the random, loud sound effects are not.
Hee hee ... a few people have said that!
Thanks! This bugged me a lot when I saw Veritasium's video... I was almost screaming: "No! You cannot predict prices like that! What if someone invents a new type of microchip or there is a war in the only country making 80% of something essential entire world uses??!" ... These models were good enough two centuries ago but the world is way more interconnected and market prices change like weather because it is literally weather-like - the butterfly effect applies here and it's not a nice closed system anymore. And what about money printing? Who knows how many trillion dollars will be printed next week so any kind of "traditional hedging" goes out of the window anyway. And most countries have a wildcard called "emergency state" and can change the very rules of the game arbitrarily, not meaning the rules will be followed - that depends on behavior of the masses which is again... chaotic (deterministic yet unpredictable).
Thank you! Really appreciate your interesting comments and insights!
Wow Im glad, that I found your channel
Thanks again que bono! I'm glad you enjoyed the content, and I really appreciate you taking the time to say so!!
I wonder why I never heard LTCM co-founded by one of the creators of Black Scholes? That epic fail would tarnish the whole foundation of how options and other derivatives are priced. So it's just not discussed, hidden in plain site. They say never confuse the map aka model with reality!
What are some motivations for your proposed fractal cascade model vs replacing the gaussian distribution with a heavier tails one like laplacian?
It's really Mandelbrot's motivations, which are largely about better understanding the nature of these things, and developing a new field of science. But the practical applications are in better understanding the behavior of markets, which in turn can lead to better investment decisions and understanding of the risks. Someone who used these well was Nassim Taleb who wrote the book The Black Swan - by understanding that some derivative contracts were greatly underestimating risk. He had to be patient, but eventually was proven right - from what I understand.
Please make more videos on maths and trading
Absolutely love this channel, hope you release more videos in the future 🙏🏻
Thank you!! I hope to get back to this soon - I've just been super busy.
Do you have a nice introductory source for these fractal cascade processes? I couldn't find anything with a quick google search. I was also wondering whether such processes are even martingales and how widely accepted they are as the proper model for price movements.
Thanks very much for your question! I wasn't aware of any really "nice" textbooks on this when I was doing my PhD (some time ago though), and a lot of it just came from the scientific literature. That's actually why I started making these videos, because I can't believe this stuff still isn't widely known - I suspect big data and AI have replaced chaos theory as the "sexy" topics so don't get talked about that much any more. What's really important to understand is that these (multi)fractal properties are a characteristic of many complex, high-dimensional chaotic systems, like weather, and so the sources on this might be talking about any of these systems. There's a book: Lovejoy, S., and D. Schertzer (2013), The Weather and Climate: Emergent Laws and Multifractal Cascades which is an example of this. I've (tried to) read an older version of it and found it pretty dense. If you're interested specifically in fractals and financial markets, then read Mandelbrot's books and papers, and anyone who references these - those Mandelbrot's work can be a bit cryptic at times! There was a journal of Econophysics at one point (not sure if it's still going). Hope that helps!
@@fractalmanhattan Thanks.
Correct me if I’m wrong because I’m confused, I thought the black scholes Merton equation factors in the underlying movement of the market based on past and current events through drift, isn’t the drift aspect of the equation stating that current and past trends can effect the pricing of contracts.
I think you're right that drift can be taken into account, if by that you mean underlying trend. What I am trying to highlight is the importance of extreme, unexpected events, which I think are different. You need fat-tailed probability distributions to describe these.
Just stumbled across your channel after discovering Benoit Mandlebrot and his talks. Really enjoy your content.
One thing I was wondering is there a huge difference between the distribution implied by the fractal model vs a distribution created by fatter tails ie Cauchy/Pareto etc?
Thank you for your comment!! Really appreciate it. Fractal models of financial markets would generally have fat tails, and you can adjust them to make them have fatter tails. Fractal models are ultimately also just another abstraction of reality - so you're really just trying to fit the best model to the data. Depending on what you are trying to do, you can just focus on the parameters for the probability distribution and not worry about making a fractal to generate synthetic "price action". If you're comparing the difference between a fractal model and one based on a normal distribution, most of the time there isn't much difference. It's just the failure to be able to properly capture the likelihood of extreme values, which can be critical in a market shock. Hope that fits what you're asking!
any good idea has its own limit - this is my own citation from my interview for Facultimedia, a while ago. That is why veratisum video is only good as populism of math science, and... proving that Nobel prize committee has no clue what they award. Thank you for honesty and clear explanations.
Thank you - absolutely right and a good point - the BSM equation is obvious a brilliant piece of work, and it still gets used despite its flaws but usually with some fudge factors applied. What surprises me is how little though Mandelbrot's work seems to get attention (maybe that's a false impression on my part?), which I think discovered something much more interesting than that markets can behave approximately like a random walk. Thanks again!!
Wow. Is BSM Modelling still mostly relied on by majority of people? So does that mean institutions and traders still havent learnt from past events like LTCM black swan events??
I've definitely come across people who are convinced that the BSM model is fine. But I think smarter traders know that there are problems with it and have fudge factors they use to deal with this, particularly in terms of how they cope with the possibility of more extreme market movements. I think this would make a good subject for future videos!! Thanks again!!
How to trade using fractal
Good video, however the Fed did not bail out LTCM nor their investors. And over time, LTCMs investments turned out to be profitable, so in the end their models were correct.
Thanks Tobias - that's a really good point. The Fed just orchestrated the rescue of the position and I think my video is a bit unclear on this - though most references I see to this event do colloquially describe it as a bailout. I don't want to create a false impression, so I've decided to edit out reference to the Fed - and again thank you for raising this. I'm not sure that I agree that the fact that LTCM's investments eventually became profitable proves the models were correct - mainly because you can demonstrate (and Mandelbrot did a lot of work proving this) that financial market statistics are inconsistent with theories based on statistical independence and normal distributions. LTCM clearly got something wrong if it required this intervention, and the nature of the rescue created controversy. But that doesn't mean that the BSM equation is useless either, you can make adjustments to better deal with the likelihood of extreme events. Of course, these are just my opinions. Thanks again Tobias
@@fractalmanhattan thanks for your thoughtful answers. I understand that it's not always possible to discuss everything in a video.
I saw the center of a fractal and the universe opened up to me.
April 1st 11 pm.
That is so true!!🤣
Amazing channel please please make more videos on maths of stocks or something related to that
Thanks for that!! I'm definitely planning to make more videos on this subject - just been really busy lately! Thanks again!!
Very good explained. Thanks a lot!
Thank you - glad you liked it!!
So l don't get exactly how BSM is fundamentally wrong. Are you saying that events classed as low probability in BSM are actually high probability irl? Just wondering.
It's more a case of how it handles rare events with large consequences. Like Nassim Taleb points out, such events are not as rare as we typically regard them to be, and that's where you can run into trouble with the BSM equation. Generally these days people who know what they are doing with the BSM model have means to take these better into account, but it's difficult to handle these sorts of events. The classic BSM model thought it could handle these, but by assuming price fluctuations behave like a random walk, they underestimate them. Hope that explains it!
@@fractalmanhattan Thanks for responding. l must say that it sounds to me like what you are saying has more to do with how people classify risk or rare events and types of random walks; simple random walks and random walks with drift.
Thanks! - please do feel free to elaborate. I'm working on ideas to make some more videos so hopefully I can expand on some of these themes soon.
The use of surprise give’s humans a tremendous edge, and may save us from AI runaway. Thinking outside the box is our forte.
Let's hope so!! :)
Great video! Growing fast :)
Thanks so much again Brumor - really nice to get your messages. Glad you enjoyed it!! I probably overdid some of the sound effects and edited out some of the B roll and accompanying humor as I think some viewers didn't really like this. It's great though to experiment and learn some lessons from that! All the best and thanks again!
@@fractalmanhattan No worries! I think the sound effects and humor were a great idea. Can't please everyone, I suppose. If you decide to keep this style in future videos, I'm sure you'll improve over time. Either way, you're already doing great and are bound to do awesome in the future also! :)
Thanks again Brumor - great feedback to get!
great video, keep it up
Thank you!!
Thank You
Thank you!! Really appreciate you taking the time to comment
I'll join the other commenter in saying the video would be improved three fold by replacing the stock footage by nothing. Otherwise an okay video, and potential for a great channel.
Yeah - some of it is a bit random for sure - perhaps too much of an in-joke in some bits, and partly because I'm experimenting. Your comment is appreciated -thanks!
Im also a huge fan of Mandelbrot.
Great channel
Thank you!!
Love this
Thank you!
4:11 ok you got a new sub
Thanks!! Really appreciate that!
Gosh, these are good
Thank you!
Black-Scholes is only accurate for European style options, not American style options that can be assigned anytime.
Thanks :)
Great, but the random stock footage can be cut down, and the random sound effects are extremely annoying.
Thanks for the feedback, it's always good to get. Yeah, I can definitely see what you mean here - probably experimenting a bit much! Also, I didn't want to seem too serious in this one, as if I'm offering criticism I want to seem like I am poking fun rather than being too intense :). It's a good lesson for next time though. Thanks again - great to get your comment.
@@fractalmanhattanyou honestly put more information into 8 minutes compared to other peoples hour long courses.
The visual representation isn't too bad, but the sound effects really are a bit much
@@GHOST25938 Thanks for that feedback - really useful.
@@fractalmanhattanSeth Godin once said that the reason He has a blog is to not spend time learning TH-cam and only focus on his skill. Therefore, I'm grateful for the work you put in your videos to spread that gems, and I don't care about that other things. You are so smart! Thank you!
@@marujo4206 thank you so much - that's really appreciated!! I think Seth has a point about blogs being less work - hee hee
like this comment if you know about LTCM from the Swedish investor.
Heston models deals with changing vol though!
Yes indeed there are approaches which have subsequently been developed that you can use to take into account changing volatility - topics for the future!
Please make more videos on maths and trading
I will try!! Thank you!