Hi everyone, it's MJ and I'm here with Yuri who many of you might remember from the channel which we did a Yeah, when did we do that last interview? December? December slash January. December slash January. The reason why we did it is because Yuri unfortunately decided to abandon Abandon actuarial science and instead you are now pursuing Well, I'm seeing here department of just mathematics and applied mathematics Is that like the new so you know the department of maths at UCT school the department of maths and applied maths The two are technically separate, but one department Okay, okay, and I mean first off how does it feel leaving actuarial science like is they further? To be honest, I think it was the best decision ever the best Although my year was very intense and probably my most intense year since I've been at varsity Okay, it was definitely much more enjoyable. That's so much more fun doing it All right because I mean this this is the thing is some some people find that actuarial science isn't for them and they either move to Accounting data science pure statistics So yeah, it's nice to see another thing that we can jump into if we find actual sciences and for us Yeah, well, I kind of want to go into mathematical finance I just took a route around which would you raise it? It's basically XR Yeah, I mean none of this a or m proff com stuff there fair enough fair. I still didn't think it was this year Okay, so that was great. Yeah, it was the best XR course. I've done at University Well, this is the thing is actual science has got a bit of like a an umbrella to hold bunch of various topics Yes, and I mean the one thing that I really like is I mean mean variance hedging that is subject cm2 What we did is CT 8 point South Africa we refer to it as a 2 and 4 There's there's a lot of different codes for the exact same I don't even know the codes at all. So I'm just like which one is that basically financial engineering Okay, so so yeah And then what I like is I mean and this is why I've included the actual tech logo is you know, this is Yeah, working with is we looking at technology with actual science and I mean neural networks are I don't know I really don't actually understand that's what what I'm hoping to just fine. Let's see if anyone understands Well, maybe maybe maybe give us before we before we get into your whole topic because it's me variance hedging with recurrent Neural networks in incomplete markets. I mean, that's not that's that's like a sentence. That's not like a heading That's like a sentence originally. It was actually longer was it longer because this top the heading now is basically So there's three chapters in this project. Okay, and it's the last chapter Is on this? Okay. Yes. So originally it was I think the mathematics of arbitrage and Mean variance hedging with recurrent neural networks in incomplete markets That's way too much of a mouthful and I think this captures the essence of where I wanted to get to you know The big things are that Everything else is just building up the mathematical intuition behind trading and everything else Okay, so I mean before we get into trading and hedging and incomplete markets Maybe can you just give us a quick word on what exactly is a neural network? Okay, so like in simple language simple language simple language. So it's basically a system of functions Okay, that are universal approximations. So if you have some sort of function You want to approximate neural networks? There's a proof in my thing as well about they can basically Approximate any continuous function. There's there's a lot more functions. They can do approximate but for my case specific I did the continuous function version. Okay, so so correct me if I'm wrong But neural network tries to find the function that links input to output. Yes Okay, cool. Yeah, but a lot of those Looks like spiderwebs with a lot of like Okay, okay, and now coming to I mean mean variants What is what is your opinion on mean variants? Do you see it as this is the framework that defines finance or do you see it simply as a model on which to explore? For this case specific, I just needed some sort of model So the code is actually written in a more general way so that we can add different cost functions at the end Okay, so a lot of papers have been written on mean variance aging Which has a lot of criticize? Criticisms because it wakes losses and gains equally, but yes That's not really good if you try to make something like you don't want to necessarily make profit Do you want to try to get as close to possible to the payoff? But obviously as a company you risk a verse you don't want to take these losses There's a lot of criticisms on it But it was a great framework to at least begin the research Yeah, because yeah, I mean just just for those who don't know mean variants is a Framework for portfolio theory that suggests the two things that investors are most interested in is the return given by mean and Risk given by variance, of course the mean here is isn't an absolute term You know they would say 10% is you know better than say 5% Whereas whereas in finance some people actually prefer getting you know return in a relative Sales as in you know plus three over a benchmark and that benchmark being inflation or something else Because there's no point getting 12% return and then inflation is 15% and you've got a negative three real So the one criticism is that the mean here is too simplistic and the other posture is the variance is As you was referring to it. It's not that we dislike Variants or that we dislike risk. We dislike downside risk Yeah, so if something's got a probability of going below We're unhappy with but there's a probability of it going above We super stuff. Yeah, and this but this framework doesn't doesn't capture that Yeah, and then the other one weakness of the mean variance Theorem is that it requires co-variance is which are normally quite difficult to obtain you using your empirical Data, but with that said and done Very quickly on on hedging. I'll give like a very brief thing and then Chop in if you think that there's there's more to be said essentially with You're gonna come in and explain what this thing is But no, so so hedging. I think a nice example is let's say you you sell a call option Which means you collect a premium and you give somebody the right but not the obligation to purchase a stock at a certain price hedging is when you try and Cover that position and the isling called the covered position where you physically buy the stock But if price goes down then you don't hit that zero more Exactly, and then there's also something known as the naked strategy Which is where you don't do anything and you just assume that additional risk and try and profit from it and then there's the stop loss where you try and buy and sell according to strike price and Then you can also do saying like the Delta hedging and all of that So I want to ask just very quickly. What what type of hedging are we talking about Delta hedging here? Is in like hedging referring just to the price how the price changes? Yeah, so it's more but remember so neural networks are quite black box, but we did compare it to Delta hedge black shawls. Okay, so that's at least the comparison But the neural network itself is a little bit black box. Oh, so it will it will determine what it should hedging what Yeah, it should it buys and sells and you can get a stock price movement and see how it's actually performing Unfortunately, I didn't put it in this actual thesis, but maybe we'll add some pictures or something We can do some little editing. Yeah, so this is some photos. Okay, to put in and then finally You can see it's a it's a big heading Incomplete markets does this mean that you're not assuming that the market is efficient or strong for me so it's not It's not really those so we'll characterize Something so there's two fundamental theorems of asset pricing The first one says that under no arbitrage an equivalent Martin Gale measure exists. Okay, and then In incomplete markets, well, let's just start with a complete market is such that every bounded claim is attainable So you can trade to get your payoff perfectly replicating it But obviously that's not very realistic in real life. You have no idea what the hell is going on and So that's incomplete markets. Not every buck not every claim is attainable. Okay, so how do we minimize our risk? Okay? Yeah, okay. Cool. Well, then let's let's jump into straight this big thing over here First of all, yeah, can you explain what this formula is in words? So they call this discreet Stochastic integral but okay, let's go with simplistic words So the thing on the side obviously that's just notation and the thing on the right. It's basically Phi over there this little guy. Yeah, that Phi represents your position hold in the stock It's a your whole two units and then The XK and XK minus one is the stock movement. So it X the price of XK minus one and XK so You hold a stock and if it goes up Then you're making that gain. So this stochastic integral represents your gains from trading. Okay. Yeah Basically the area and the little curve something like that same like that except now it's the screen time So it jumps Okay, okay, so and I see y'all you've done it now on September 15, so you just handed this thing. Yeah Yeah, I put it was due last week Monday. So I mean could push how many hours did you put into this? In the beginning it was actually relatively easy to like formalize the math behind trading and stuff when I started the code That was a nightmare.  What coding thing did you use like math lab? Or also most of this other plots and stuff as well front-end in Python. Okay, and then I had Lucy move who saw help me do the C++ Yeah, so we tried Python everything turns out tense plus us. Okay, it's very slow Don't want to let's not hate too much Don't want to be tracked down, but yeah, it was it was a little bit too slow, but it We try to use it for things like inbuilt gradients and stuff like that But we ended up just coding the gradient descent ourselves instead Well, I mean that this is the thing I had a friend who did a thesis also from UCT on predicting soccer You know that results and that that thing needed like 18 hours to to calculate the parameters every time This So this one is a dummy version of what I actually want to do didn't get because my computer suck But I just got a new computer actually But yeah, I just got my new laptop and I'm hoping to rerun my code at the more high-level and see Why don't you use it? Yeah, well probably as well, but I want to add all the bonuses foster that Just let's just you know, it's nice to test at least when I'm doing my tests to make sure the code's running fine I'm gonna use a cloud to test my code. So at least that speeds up my testing procedures. Make sure everything's going right But you're a lot. We ran into a lot of errors by mistakes big mistakes They also like I ran my code for like two days waiting for my results and they came back just NANDs errors errors errors errors And that was that was your your bad programming. It was me That's good. You take responsibility. I don't take responsibility, but it's also like when you think of the exponential distribution, right? What is a more natural parameter scale or rate? Guess is a good one. Okay, so Thinking about like arrival times a rate could be great, right? Yeah. So that's more natural, right? So the mistake that I made so I used the involved numpy Exponential distribution thing and they take in a scale parameter And so it was it's the opposite way around one over. Yeah, so the jumps are too much and I mean that it just felt natural to me, but I should have maybe read the documentation a little bit better So we call this model error. We call this modeling error. Yeah, just apply Okay. Yeah, well, let's let's jump into I think you don't need to go through this page or some thingy Talk to me about this is this important for the discussion because I know you said there's some boring math in the beginning And I'm just saying for maybe your community. They might not enjoy too much. That's I mean, so it's important It's just a just an overview, but I think we've spoken about already like I formalized some idea of stocks Then we have the expected square replication. We try to minimize. Okay, I mean variants and then Just the the two neural networks running concurrently. Is this is this the neural network? Yes, so So a lot of times I did see people use N plus one neural networks for each trade. Yeah, each strategy has own neural network so what we did is We have the C parameter here that's Like it's the memory. So it holds some memory. So Believe like so Markovic Markovian processes just need a small dimension of a memory cell. Okay, and something with that you need more Memory for you can increase the size of the memory. So so something like that could You know, like good models mean reversion Momentum stuff, I guess it's something like a memory cell could supersede that Okay idea. Yeah, and then the other neural network is just to predict the trading strategy what position you should hold in the stock Okay, okay. Yeah, so I mean coming coming to that what you're talking about there the mark of because the market of property is that The only information I need in order to predict the future is the present Yes, the fact that the entire past or the entire history is contained in the present Pretty much now is this memory cell saying that's not necessarily the case So, no, no, so if something's Markovian, you don't need a big dimension a memory So she don't need to hold a lot of information. Okay, so that just takes relevant information of like the now You know as small increments possible But if you have some sort of process that has you need a lot of information from the past data Then you can increase the size of this memory cell. Okay, and holds more information And is there I mean, I'm just trying to think what what examples of other information Would you maybe hold like the price the day before or so it depends off of what you want to go Okay, some stop. Maybe let's say you're trying to edge something on Weed forms or something. Maybe it can keep into account the weather Some not always but I mean, that's quite interesting things to think about here. Okay, that opens up quite a lot of modeling possibilities So, okay. Nice thing about neural networks. They're actually pretty flexible. Okay, not too bad Anything else you want to talk about over here? Well, we'll get to it, but so what what allows you to actually Minute like does a parameter thing exist such that it actually minimizes this function. So the first one is The projection theorem, so there is actually a trading strategy such that it minimizes this error and then from the Universal approximation theorem you can further more proximate that strategy with the neural network. Okay. Yeah, that's not confusing Let's go. I must say this is I'm definitely over my over my head. Yeah That's what I'm saying. I'm talking with you through not me Just trying to go through it myself.  This is Yuri's project Well, it's yeah, let's discuss it. So, I mean, you know, this is this is your content. I mean stochastic processes Yeah, well, so I'm hoping everyone knows that yeah, we can skip chapters. Yeah, so we can skip I mean, that's basically a random variable that changes throughout time. Yes Martin Gale that is the current value is the best estimator for the future Yeah, intuitively as But in finance only we use like sub Martin girls because we are left with a force of interest in that Yes, because it has the upward So but remember that when you price something You need a Martin girl because it can actually be shown that if you're not using a lot of girl measure There's all the time. Yes, and with the fundamental theorems, as I said, like no arbitrage means they exist a equivalent Martin girl measure and you use the Martin girl measure to price Okay, okay, so it's important. It is important. You don't even know what the Martin girl is. Okay And then so that's chapter zero so we can basically done chapter zero. Yeah, chapter zero Chapter one theory of derivative pricing in discrete time Is this something that you want to spend me some time on like when we get there? Yeah, I think we can spend some time we can skip over binomial model anyone familiar with Stuff and we don't really use binomial model to do anything further. It was just a a nice example of something That's complete. That has no arbitrage You know binomial models often just use just to it. Yes, it's very simplistic The whole idea is that she can either go up or it can go down. Yes, and you kind of Assume jump. Yes, so it's an rate going up and down Yeah, and then that almost all forms the building block of the black skulls Model, yeah, they kind of take that to yeah, they do take it into account slightly obviously with the continuous time Yes, it's can jump anywhere and You know how the sample parts are they crazy? Yes, those those are what those are like ito processes Yeah, those those crazy things Okay And then What does Alice T. M. Sanford I can't really remember but And then that's the recurrent neural networks, yeah, okay cool formulation, okay And we'll yeah, let's jump into and talk about this type of stuff So you can actually click on the thing and just okay You want to say there's a pretty picture then people can Can people access this paper online or is this like it's Like they can go find it on my links, okay, so there's I don't think there's a good quality version So I guess maybe I'll find some way to put it up. Yeah No, look, I mean, that's the thing the views are very small They'll they'll they'll be like hey, this is quite cool. I've been doing something similar or you can improve the chair or you Yeah, that'll be very nice like I would like some feedback on how to improve. Yeah, maybe how to do some extensions because I'm planning to do My Masters in Mathematical Finance next year and I would actually like to extend the sort of thing Yeah, and that's what I'd like to say is I've got the smartest audience on YouTube. Yeah, you think about it Yeah, I mean no one's smarter than that trees So yeah, we got the smartest audience watching here Okay, so like this this is the preamble It's interesting. Yeah, that's scary. Is that that filtration thing? Yeah, so this is a filtration. Oh, yeah, let's just skip it. Should show my tattoos Oh, you've got a tattoo of the filtration being yeah, you know We can do one we can do one glimpsing to it. Hey show some skin on the channel There we go, very cool and a brownie in motion and you're brownie He actually the guy who did that what does he know vine vine or wiener Vina Vina Vina he wrote a book called cybernetics, which I'm trying to get my hands on so yeah And all about your the future and artificial intelligence and it's taking over. Yeah, he was a smart boy smart boy that guy Okay, this is still the preamble preamble. Yes, it's long. Oh gosh. Okay theory of the route of practicing in discrete time Okay Looking nasty talk to me about this so Obviously, we need some sort of Mathematical description these things so obviously something gaining information or changing over time a stochastic process Best best way to go, right? Mm-hmm. So We have some sort of stock dynamics that follow stochastic process and the tuple with The filter probability space and the stock price is called a market. Okay, right? But we just run a price thing. So we simplified a little bit further and introduced discounted prices Yes, because it makes the mats clean, but they're equivalent. Okay makes the mats clean, which is very nice Okay And I mean is that Give me a quick book explain quickly why we're doing discrete time and not continuous time just for simplicity or so also just for More realistic approaches in terms of trading and everything, right? You don't trade every day or every split second. I guess and you don't have a computer that's doing it for you in continuous time Yes, unless you're trading crypto because the crypto market never sleeps But does something trade for you continuously? You can give those little but but I guess you're they're not continuous Yeah, they just discreet at a very yeah fun fun point Okay Keep keep talking here. Okay. So yeah We have a idea of a trading strategy, right? As we spoke about earlier It's just the amount you hold at a certain time up until the next time those are predictable So just means that at time t-minus one You know the trade to make for the period t-minus one to t. Okay Yeah, so obviously you have to make the trade and then either gain or don't gain and What you're trying to do is obviously hedge a specific pay off that only happens at the end And and tell me with with this trading are you also taking in consideration fees trading fees? No, but we were we want to extend it a little bit. We're gonna okay some maybe fees Certain things like that. We did take into account maybe like public holidays But that's just, you know, you take off the amount of trading days. So that was a easy But yeah, we we want to extend it maybe take into account fees But it's not very optimal to compare something like that to the black shawls model Oh, yeah, if you can't any fees or any jumps black shawls just can't keep up Which is a problem.  It's still still like the main thing is it's popular Yeah, I've heard there were there were a lot of mistakes in the first paper that was written about it But you know our life works if you the first one to get something out. Well, the best example is the fact that Merton and skulls Join that hedge fund long-term council management and stuff that big time Big time they were going wall and then what's that one in a million? There was your the Thai currency got devalued the Russian bonds defaulted and they basically got wiped out Yeah Because they kind of thought the market's not They're like that basically Cation, you know, and then they didn't realize panic spreads throughout the whole market and Destroyed them and that's what there's potential yours could maybe take them on in the future Okay, let's see hopefully I can do something big and people of the future will be learning Robots a slash Jordan model because you know, I would have I would have added in my Maybe we can work on something else Okay, so talk to me what's what's happening here. This is just for assumptions coming up I need some sort of bound on the trading strategy Also, just logically you can't borrow Infinite amounts, right? A lot of people just assume you can so it's just also just makes the math scheme Also, it's just intuitive Can't bound like can't borrow infinite amount of money, right? Yeah. Yeah So this is just bounding the trading strategy Yeah, and that final bound is just like right on the bottom of the paper That's just my assumption to make my proofs easier Explain what you mean by that. Is this just just the usual one that they use is the finite credit line We just can't borrow an infinite amount of money. Yeah for mine To make my proofs work. I bounded the whole trading strategy But what so what is so it's just there's just a bound on the trading strategy There's a finite amount that you can either borrow or is that just to make the computational easier That's for the model runs quicker or so in the actual neural network and stuff. That's not necessarily It's just for the proofs of existence Okay, okay I'll betrudge I'll betrudge is basically the idea of making a risk-free profit Yes, or having the probability of making profit in a risk-free. Yeah, so it's you start with nothing Yes, there's a chance of making a gain and then the probability of making a positive gain So is is greater than zero. Yeah, okay, and you are saying here in this section. We still financing Talk to me about the stuff. What's happening? Yeah, this is just a nice baby example Just to get it run into this idea of a overdrage, right? Okay, so you're just giving a yeah A little example a two-stock example there woman. Okay, so can we skip that? Yeah, we can skip that We're gonna probably skip a lot of us. Let's not talk about too much Matt. Let's go to the definition of a super Cool, let's go. Okay. Yeah, the probability measure probability measure. So we need this idea of an equivalent martingale measure Okay, something that makes the stocks dynamic a martingale because as we said earlier If you can't find such a Measure you can actually find an arbitrage. So you can make risk-free profit Yeah, so that leads us if you like score maybe like two more pages to oh Jesus all of this Matt's Matt I got you a little limit Yeah, so here comes fundamental theorem of asset pricing one so Under no arbitrage opportunities. Okay, there's a significant martingale measure. So, you know Okay, and then you give this whole proof. Did you come up with this proof yourself? No, I Read a bunch of papers, but I'm learning a lot. It's very interesting. That's great. I Mean, yeah, that just looks yeah, it's a bit messy. I wish I could come up with something Maybe in the future like I'm getting into it. I did come up with some proofs myself in this paper Like I said you combine the Jordan model and I can be your proof guy Okay, definitely. I'll have to make coffee while you try figure out the proof Yeah, but don't you know that's the they call that like the math them they say math and musicians or machines Oh, yeah, we're taking coffee and we output proofs. There we go. Yeah, so I can be the fuel stop. Yeah, exactly Good coffee Okay pricing and replication Anything here that's worth chatting about so it's just a so replicating strategy Okay, so it's something that Replicates a bail, right? So that's a thing you're trying to do. Hey, yes So that's it just getting that whole notion of what we you know, we're building up to this Haging trying to minimize your risk, you know hedge like perfect replication and There's also introducing that notion of a complete market, which is coming just now So in that the complete market is one where every bounded claim is attainable So that means that a replicating strategy exists for each bounded claim. Oh, you see not in real life So incomplete markets in complete markets. Okay. Yeah You've got some definitions. She got another example Yeah, we got to keep it Samples in us. They take up so that's this one. This one is Introducing the notion of what happens if you do have an incomplete market So turns out when you have incomplete markets, your sets of equivalent Martin girl measures is infinite dimension So there's infinitely many equivalent Martin girl measures and then you're no arbitrage price is actually an interval Okay, not one value, which a lot of courses and notes and textbooks assume that You know the law of one price So there's no such thing as the law of one price. Okay. And so how does that then? Influence your your paper So we're gonna be working in an incomplete market where there's some sort of band of prices Won't be talking about the bands, but yeah, so there's there's no unique price. So There's no Replicating strategy, but how how do you hedge in that sense or how do you price something in that sense? When there's this interval that you can actually get no arbitrage prices Yeah. Yeah, so obviously so it's gonna add to the complexity. Yeah So with the with this interval, right? If you if you the seller you want to sell as high as possible But if you the buyer you want to buy as low as possible what where's the middle ground? Mm-hmm. Yeah, so something like that. Okay, okay And then this is still the example. Yeah, it's all examples Oh, here we go. That's what you're talking about a market is complete if every bounded contingent claim is Attainable cool. Does that mean we can skip this? Oh, so he has fundamental theorem of asset pricing to oh, yes, number two Yeah, so this comes with the another characterization of more equivalent things so if Every bounded claim is attainable then it turns out that the equivalent the set of equivalent martingale measures Has cardinality one so there's only exists one equivalent martingale measure and that Results in like a unique price and certain things like that. Yeah, okay, okay And it's because you know, that's the big difference if we had to go Up to the other one who was that one There yeah, so non-empty So that can either have one or infinitely many. Okay, and then that's just the added bonus So it's under no arbitrage The mock is complete if that happens. Yeah Jumped it. Yeah, you jumped it. That's fine. There we go. Okay, cool Talk to me about this This is your proof I made it up So to be honest, I I read papers and I like I couldn't understand anything that was going on So it's like, okay, I'm gonna make that for proof for this mail the proof. Yeah, so I was scared it was gonna be wrong My supervisor said it was fine and just even more So it's a moist nation And then that came into the second matrix and stuff just explaining why this matrix is invert one I just made claims like boom. This matrix is You know, you know, this is like I've died now.  Please if it is like hit me up Put a little comment below Okay, so you actually did this that's impressive. Yeah, so I just took a arbitrary claim ages right you have this trading strategies on the rights and this matrix just represents the linear combinations of these things and Obviously if a unique solution exists the matrix is invertible, okay And the second explanation was just why it's where it's invertible because under No arbitrage and the existence of a unique martingale measure the second equation basically says okay That matrix must be invertible for that solution to be unique. Yeah Okay, okay Let's keep going Going down here. We've got the binomial. Yeah, everyone's bad dream Should we skip it because I do I am looking at the time We've got the formula one soon. I say what should we do this as a part one in a part two video? We might not be able to finish this like like let's let's let's finish this let's finish it. Okay, we should be fine We're gonna do this. Okay Let's get by the model. Okay, let's get to the meat. Let's get to This is the big stuff. This is the big stuff. This is the big stuff Yeah, but we can we've already spoken about a lot of this introductory stuff. Just minimizing this stuff. Okay Time a lot of simplifications. Let's go down So can carry on It's a lot of the the meat. Oh, so so here's one thing That I was talking about earlier just where the training strategy actually exists that minimizes this so we use this theorem from an analysis course, so we've see so this is the projection theorem so This looks cute It's not too bad. I learned this last year already. So it is not too bad Okay, so yeah, it's just basically saying that under this Scaler product that you have with basically that squared expectation thing a Projection exists onto this closed subspace, which is the attainable claims So that's the thing that minimizes the distance between a replicable Hedge and something that's not so you just want to minimize that error and then we go further down Yeah, well, let's carry on keep going down. Yeah, we're gonna talk about the universal approximation theorem. Let's skip skip Still far. It's not too bad. Come on. That's good You know when you see like little weird things above capital letters and all is only so much rotation You can use I know it just get them.  So let's see you can carry on carry on. Yeah Examples of activation. I mean this this is like first-year XR, you know, everyone should Everyone should be fine. Okay. Yeah on the next page starts Just at the start to the page there. So my mouse is like You can see it's this just the PDF is giving my computer this little ring So the universal approximation theorems for neural networks This is basically just saying can we approximate that training strategy that now we've shown exists, okay? And spoiler alert we can we can we can Okay, unfortunately, I couldn't But I think with my bad computer, but I can you can in theory, okay? Yeah, so it's basically just saying that neural networks are Dancing the space of continuous functions so So dance just means that the closure of that space is equal to the continuous functions So it means that If you choose a continuous function, yeah, you can find with arbitrary Precision a neural network that approximates it So you get as close as possible to this continuous function using a neural network. Okay? I'm gonna pretend understood that This is this I feel like you're my brain is getting like worked over time Hey, I think Too much for you man. It's a lot to take in first one. We go to the juicy stuff Let's let's carry on. Let's go see some pretty pictures. Okay, so silly. So let's go to just we'll maybe just do geometric Brownian motion Okay, let's do yeah, we'll leave it written model. We'll just keep it nice and simple So we can always we can always do part two, you know, we can be one of the people if this video gets a thousand likes We're gonna do part two Maybe then maybe I can re-run the model by then. Yeah, maybe not Kind of feel like starting new research now as well as those was No, you're you're on to something you've got to You could sell this to some hedge fund manager for billions I Don't want to make money. I just want I want to do Obviously everyone wants to make money I do something but I still I first one I'll do you want to make money while you make good stuff Yes, yes, that's that's the truth. That's the dream. Okay. This is something that I have fortunately seen before This is nice Done a video on this I should make a video I should make a video kind of explaining. Let's see if I'm right This is your drift parameter and that is your diffusion parameter. Yeah, so the volatile over other you've called it drift and volatility And yes, yes, I know that that's cool. You get this through. This is I toes Lemo, yeah, she to this thing. So you solve that stochastic differential equation. Yes, which And then you get this nice formula, which keeps the stock prices above zero. Yes Do you know I keep it by zero because you're taking the limb? Yeah, well, and then the little conquer. Oh, I say Lin. How do you say that? How do you say the natural is it? Yeah, okay, just I don't know people just check just check. Okay Talk to me about the pretty pictures. Okay, so this one So this one is just 30 simulated Yeah, so this is nice and clean obviously for the data nice like 25,000 But we didn't we don't want to picture up 25,000 balls like this. Okay, but the big thing here is we just try to make it relevant right so the stock price of 100 Cool The time period of three months. Okay, so we said they're 80 trading dates So that was taken to account public holiday. We can trade volatility of nor comma naught six and a drift of nor comma naught three So this is just a pretty picture of some stuff. What could have some go up some go down So sometimes this purple one did really well. Yeah, and then this red one didn't do to do to lack So obviously now you're trying to hedge a payoff on this But you don't know if it's gonna go up or down. So can you try you've got this little final doubt? Yeah, so we have there on the right some call call payoffs on These stock parts so you can see some white zero some goods like 110. So obviously the stock price Escalade, yeah, very far. Yeah, that's quite interesting that distribution. Did you expect that shape? Hey, not really. I didn't expect anything. I was just making cool plots and just seeing how things look should have not have been somewhat normal Because remember this is First is log normal. So maybe maybe it has some sort of thing there. Also, there's a bound at the bottom So once things go down, they just hit zero, right? Okay. Yeah, so it's a bit of a shift in there in that Okay, yeah, and then so this is just explaining what I'm just saying And then oh, this looks like I didn't actually change. There's a typo there. Let's not go back up Well, let's see if any of the viewers can identify the top. Okay, so it can go down Okay, what is this validation error thing? So this was a function I made to Obviously now you want to choose hyperparameters. So you want to run different models Just to see which one is the best So around a few parameters, which you've listed there at the top. So changing the seed I mentioned That's that memory cell. So one three or five and We could only run on my terrible computer neurons of 25 25 on each Which will maybe try and make it a little bit wider and then three activation functions So these are the best Parameters, okay, I got and that three sigmoid tan H 25 25, you know that that was the best parameters. So what you do is You train your set, right? You have one validation sets. We just check the error on that validation set it then Checked all of these parameters plotted the middle one and passes those the best parameters through to another training session after it's checked that And then trains it some more. Okay, so that's how I built this System here going obviously don't see the parameters then come back then rerun it on the best parameters So just had this continuous flow checked it plotted it pass those best parameters through to the next one And then trade it some more, okay Yeah, it was quite a little time save with it. Yeah, well, it's still runs for like two days, but Time's a bit. Okay, and then talk to me about what's happening here So that you let me the black skulls did a better job.  Okay, I'll try rerun it sometime and then we'll see I just have a lot of things coming up I'm hoping I'm like hoping to beat it Just couldn't do it here.  Oh, yeah But it does beat it for Yeah, so we didn't beat it on geometric Brownian motion. So thing is black souls is Derived from this notion of the stocks following a geometric Brownian motion Also knows the exact parameters But in theory we could maybe beat it because of this discreet timeness, right? So now black souls is a perfect page in continuous time But in discreet time, you can see it actually has a distribution around zero. It's not it's not perfect Okay, so it was perfect then that pink ball would just be at zero down. Okay. Yeah, so this is a profit and loss So you all makes quite a lot of love But it was a badly trained well But it's more about the theory. Yeah, it's about the theory and I'm can it actually do it Yeah, and it actually to trade. Yeah, I mean it's not that's not too bad. So something I didn't train that Yeah, yeah, that's actually pretty good. But like I said, the formula one's coming out Yeah, well, we can actually just end it. We don't need to talk about this Do we we don't need to talk about the merchants and all these other things exactly the exact same thing just And you actually graph and you beat it this time. Yes, I'm beat at that time. Okay, so maybe just the extensions Just a quick talk to me if you want talk to me very quickly about the Yeah, so in the actual project, obviously we only did a call option and it was European so the payoffs at the end and has the same sort of structure, but You can obviously extend this to different types of Options so we want to do that. So big thing was American and the muta options Can we change the neural network to exercise? Tell us quickly what so American you can exercise any time before the expiry date European you can do it on the date The muta is basically in between European and American. Okay, so what it is is that you can only trade on specific dates So let's say maybe every three days or something. So Yeah, that's you can't trade any time before the thing it has these jumps So in those three days the stock price could plummet and come back up, but you so you can only trade on those dates, so Okay, so it's like a little bit. So you want to get it for I mean, but this I mean this like Russian is Asian There's all very so you're so Asian everything and so you want to get model so that it can do all of these Yeah, okay talk to me about These are the two extensions here. Okay, so applications only provided for aging. Okay But there are other applications in the financial interest industry like forecasting and filtering and trading This one's an interesting one. Yeah, so how many how much profit? Yeah, why don't you make it about this? Do you know how we would have got like a million views if you've made about trading? Because do you know people don't like hedging? Hedging just means less money if you're right. Yeah Pretty much like you want to make a profit. Yeah, so I haven't but I haven't done any of these extensions yet So okay, we'll see how it goes, but they are these applications and they are there's some code What why would you want to do it for filtering? What's the whole point of doing that? I was just thinking about the other things So finding the true underlying stock pattern, which is pretty much I guess for forecasting and trading It's like a middle ground. Yeah, can you get this true pattern of the stocks instead of this Disgusting view of everything long-term trends things like that. Yeah, okay. That could actually be very interesting. Yeah, okay, and then Finally talk to me about This lots extension. Yeah, so this is just obviously we only did Monte Carlo simulation so we compared the stuff to the black I mean so geometric Brownian motion model But can we train it on these two models and actually trade stocks?  I'll give you two hundred bucks to go behind this model Just don't try when I lose it Okay, just want to see how it performs on real days I often perform and then I choose what one one last question. We've got five minutes until the wrong pre-start One last question of every one that you read in your thing of a jig here. I like how you're referencing yourself. That's pretty cool Is that Mario? Yeah, just a quick basic quick little thing there Where's your stuff? Yeah, where's where's my stuff you didn't you didn't reference? I've got one one article out It's got one citation only So you need it you decided to get get that little thing, but anyway coming back to the question of all the people you read Who would you recommend or it's worth reading more of just their work? Like you saw this person you like one discuss like a genius. Okay, so in terms of trading and financial math and everything Watson schweizer number 12 number 12 month So he's pretty good at the stuff and sure I have in general So things about probability theory Finance and stuff like that. I think I'm good at textbook by this guy. Yeah. Yeah, you're great Yeah, I was just thinking I know that and then biggest inspiration is Melusi my whistle Six seven and eight just different things. So here's my lecture and my supervisor. Okay So we're checking those guys out and then you're like I said, we do have the formula one coming up You know do it the clerk. Let's hope you win Singapore. Did he Position go Patrick he's gonna go for it three three three four three yeah So so let's go watch that and just to everybody watching if you've also written a really cool paper and You would like to come and discuss it on the channel hit me up and Yeah, cuz we're gonna even do this as a series that could be and then you can you must join as well You can like criticize the papers as people talk about no joking Okay, cool. Anyhow, I think that further ado Cheers everyone and have a great weekend or week whenever you're watching this


That's what I did, 1 by z, so when you subtract it you get 2 y sizes, okay, into sine to the power 4, if I am not mistaken it will be z plus 1 by z to the power 4 and again 1 by 16 z minus 1 by z to the power 4, please note i to the power 4 becomes 1, so you realize integration, I will be already, so it will be either 1, so 16 into 16 to 56, can I write this as z square minus 1 by z square to the power 4, I am just writing the final terms, will you get 6 plus 1 by z to the power 6, correct, then plus 6 z to the power 4 plus 1 by z to the power 4, z square plus, this term, if I raise it to 16 i to the power 4 which is 1 by 16, correct, z minus 1 by z to the power 2, now expanding this multiplying z minus 1 by z and z plus 1 by z I get z square minus 1 by z square and both of this, so that you will understand it easily, so z to the power 8, next term would be what 2, minus 4 z to the power 6 into 1 by z, next term would be what 6, 6 z to the power 4, next term would be plus z to the power 4 and this is minus 4 minus 8 cos 4 x and 6, so instead of sum which is much easier to integrate, so this will become 1 by 256, this will become 2 sin 8 x by 8, this will become, this will become 6 x plus c, don't start up by, by how we are going to do this, which is case number 4 minus 1 by 3, you get minus 12, which is a, so in this case, by sin to the power 11 by 3, when you do that you will get x, we get tan, put tan x not k, I already use k, let's put it as t, so what will happen, 6 square x d x square which is 1 plus t square, which you can further write it as t to the power minus 8 by 3 into minus 3 by 8 and minus 3 by 2 tan x to the power, it becomes very easy to separate a c square from it and the beginning that's the beauty of a c term, so c square should be separated out to facilitate becoming d t part, this part becomes, so we can take sin to the power k, so we get cos c can do that also, cos c part also is equal to, it will give you this, so first of all if you divide this by cos square, so ultimately you are, so c can square x d x will become d t, so this integral converts to d t by t to the power 7 by 2, which is t to the power 2 into minus 2 by 2 minus c, which is minus 2 by 5 tan x to the power of minus 5 by 2 plus, does it help? Sir, the special


The National Archives and Records Administration established a state-of-the-art digitization center at the National Archives at College Park, Maryland to support the agency's continuing commitment toward meeting our strategic plan goal of increasing public access to records through digitization and online delivery. The new Digitization Center enables the digitization of large volumes of textual, photographic, and microfilm archival records. Once digitized, millions of pages of archival records will be made available in the National Archives catalog for anyone to view anywhere in the world. The Digitization Center houses a fleet of top-of-the-line imaging equipment, giving the National Archives the capability to scan the wide variety of materials within its holdings. These are our three new IBML high-speed conveyor belt scanners. They are a very significant purchase for us here, and they will allow us to greatly increase our digitization. They can produce up to 250 images per minute. We're very excited to get these into full production over the next couple of months, and I know our staff are looking forward to working with them. The Digitization Center has recently received additional enhancements and upgrades. This is our new oversized setup with a Phase 1 camera, with cinema lights and pantographs to move the lights up and down, and it's going to give us the ability to shoot things at 450 to 500 dpi, which is actually a significant improvement, almost double of what we could do before. We can take some of our holdings that previously we were unable to digitize, and we'll be able to digitize them, get them to the catalog, and out to everybody who's interested. Today, the American public expects to interact with their government, conduct business, and retrieve information online. Connecting the public with our records is core to the National Archives' mission. Over the last two decades, we have made more and more of our records digitally available on our online platforms. With the establishment of the National Archives' new Digitization Center, we are more than ready to meet the expectations of the public we serve.


I'll go back to another one if you're new to the channel. I am Goldpony Edgy, new car truck SUV reviews on YouTube and today we're in the brand new 2024 Lexus IS 350 courtesy of Bobby Ray Hall Lexus in Mechanicsburg PA for more information on their inventory. Please feel free to check out the link in the description box below. So we're in this one today because there is a new color for the 2024 IS and we do have it with us here today of course so gotta love that not only that you have stellar reliability with the IS that has been known for forever now I feel like just check out a Consumer Reports magazine that will substantiate that and I can already tell you the steering feel is absolutely amazing as well so ultimately in this video we will be testing out and going over everything about this one from acceleration to braking, steering feel, ride quality, sound system, exhaust clip, all that fun stuff so having said all that what do you guys say let's just go ahead and jump right into it and as always let's start with pricing it says you can imagine there are a couple different configurations for the 2024 IS 350 you have the F-sport design starting at $44,410 and then the F-sport starting at $45,660 that was pricing for the rear-wheel drive configuration if you wanted to add all-wheel drive you can do that simply add $2,000 to either of those prices but regardless of trim level that you go with the power plant is going to be the same powering the beast is a 3.5 liter naturally aspirated reliable v6 putting out 311 horsepower at 6,600 rpm 280 pound feet of torque coming in at 4,800 rpm that power being sent to rear wheels or all-wheels through a six-speed automatic with paddle shifters for the all-wheel drive but then an eight-speed automatic for the rear-wheel drive so that's really where the difference comes in with the powertrain 0 to 60 time approximately 5.6 seconds that's plenty respectable there top speed 143 miles per hour that's pretty good as well fpg numbers then coming in at 20 in the city 28 on the highway for the rear-wheel drive 19 city 26 then on the highway for the all-wheel drive taking premium unleaded fuel and so before we do any kind of fun acceleration test here in the IS did want to mention to you guys the drive modes drive mode is a circular dial located just to the right of the shifter there those driving modes will include eco normal sports sport plus snow and custom adjusting things like the shift points of throttle response steering sensitivity the all-wheel drive system engagement and actually suspension settings as well that's a ton of adjustments but now have you got all of that out of the way what do you guys say let's go ahead and find a straightaway let's put the paddle shifters here to the test first and let's see how quickly these paddle shifters are going to react for us here all right we are in sport driving mode in three two one go it's not bad not as quick as the CA Corvette I just got done driving but honestly they're not bad paddle shifters are relatively quick and there is a full manual shift mode as well so it didn't actually do any of the shifting for me like some paddle shifters will do if you don't hit anything so that's pretty cool so I just slid the shifter all the way to the back and to the left that's full manual shift mode I had in the sport driving mode paddle shifters are respectable I got no problem with them whatsoever so I kind of like them but anyway so let's now go ahead and find one more straightaway here let's get back full control to the IS 350 and let's see how quickly the car now is going to get us here up to speed all right in three two one go oh you gotta love that sound I'll get more to the sound in a second whoa baby I love that man there is nothing wrong with that acceleration 0 to 60 and what would I say 5.6 seconds that is plenty respectable guys so 100% you are not going to have any issues emerging on to the highway that was fun man I love that acceleration itself speaking of the sound there's a AC button just by my left knee and you can turn that up or turn that down essentially what that is is it pumps the engine noise into the cabin which makes it sound that much cooler in my personal opinion and I know it's kind of just simulated engine sound but it's sticky cold man I really like that but yeah you are not going to have any issues emerging on to the highway that was fun but anyways to go along with that acceleration as always braking is equally important so up front you will find 13.1 inch ventilated front discs in the back 11.7 inch ventilated rear discs as far as that six easier stopping distance goes that comes in at 110 feet and just to put it to the test here it's actually a little bit on the softer side of things but I will say that 110 foot number that is sports sedan go without a doubt usually sports sedan number is in the one teams so 110 feet that is plenty respectable but to my surprise it is a little bit of a softer braking feel which is more on the luxury side of things so that is interesting I kind of expected a little bit more firm bite to the braking but again that number is plenty respectable but anyways then touching on suspension and handling up front you're gonna get a double-wishman type front suspension in the back independent multi-link rear suspension but for the F Sport you will find an F Sport tuned suspension of course get front and rear stabilizer bars and then for the rear wheel drive configuration you got a torsion limited slip differential then as well as far as steering feel goes like I said previously depends upon the drive mode that you put it in if you put it in that sport driving mode it is a noticeably heavier steering feel 100% instantly points you in the direction that you want to go and definitely one of the heavier steering feels that I am used to feeling so I am a huge fan of that and if you take it out of that sport driving mode it does loosen up a good bit but still on the heavier side of things but it is a 100% noticeable difference in that steering feel dependable on the drive mode dang anyways then touching on cabin noise we were going 23 miles per hour right now so I'll let you guys be the judge of that and the other thing is with the ASC button if you wanted more engine noise you got it if you want it so that's kind of cool so as ride quality goes that again has been perfectly fine in my short little test drive here today so I personally don't have any issues there then touching on rear visibility that is something you definitely are not going to have any issues because of the shape of this and you can see perfectly fine out of my rear view mirror here and also touching on forward visibility rain sensing windshield wipers actually do come standard on the is 350 so that essentially is going to turn those on whenever it detects any kind of mist or rainfall so just one less thing you got to worry about it's kind of like automatic headlight so that's pretty convenient too so you can better focus more of your attention on actually enjoying the drive but that pretty much rounds out the performance segment of this review guys let's now go ahead and take a look at the exterior of our brand new with our brand new color 2024 Lexus is 350 all right so here she is you guys the new 2024 Lexus is 350 finished in incognito yes that is the new color for the 2024 is let me know what you guys think of it in the comment section below but as always let's go ahead and start with where this one is made take a look at the bin first character is the letter j indicating that the is 350 is built and assembled in japan you got to love that but up front when it comes to the headlight setup the standard setup is by led headlights with led daytime running lights however there is an optional triple beam led headlights we do have that option with us here today that goes for $1,250 for a little better added illumination at night either way there's still leds you're going to be able to see plenty fine but anyways automatic feature does come standard on the is 350 along with automatic high beams so when you have your high beams on at night and it senses a vehicle coming in the opposite direction it's going to automatically dim them back to low beams and when that vehicle is gone it's going to automatically bounce it back up to high beams for you there Lexus spindle front grille of course in typical Lexus fashion there do you get a little bit of front air curtains then down to the sides help a direct air around the wheel entire combination so a little better aerodynamics there as well but that pretty much rounds out the front end of the is let's now go ahead and make our way to the side right and something else since we are around to the side of this one chrome windows surrounds do come standard take a look at the side mirrors they are body colored power adjustable side mirrors they will be heated with led integrated turd signals then as well you're going to find that f-sport badging found on the front fenders of course because we have the f-sport f-sport specific side skirts as well this is all coming standard for the is 350 and I'm just saying this because if you were to compare it to like an is 250 or 300 so it's going to have specific side skirts for the s-sport trim levels but then take a look down at the wheel setup 19 inch split 5 spoke alloys do come standard but then there are 19 inch forged bbs alloys available in case you wanted those but that pretty much rounds out the side profile let's now go ahead and make our way to the back and something else since we are around to the back of this one all the way to the top you will find a body colored shark fin antenna just below that you will find a rear spoiler there is a carbon fiber rear spoilers available as well today's rear spoiler is going to be finished in gloss black in case you were curious so a couple different options there led tail lights do come standard with led license plate lighting as well I don't want to forget to mention that you will find a gloss black rear diffuser down towards the bottom of course you got the all-wheel drive badging if the is that you have is equipped with all-wheel drive and to the sides there you will find dual exhaust outlets with satin chrome tips so having said that I do believe you guys know what we have to do next here as always here is that exhaust clip so now since we are still around to the back here when it comes to opening that rear trunk there is a button on the trunk itself of course but there is a button on the key fob though as well but once opened up cargo capacity comes in at 10.8 cubic feet if that was not enough space there is a 6040 split meaning the rear seats do fall down for quite a bit of extra space then if you needed it there is a first aid kit back there in typical lexus fashion they always do there's some tie-down anchors grocery bag hooks there's cargo lighting as well then if you were to lift up underneath of that cargo floor you are going to find a spare tire as opposed to the fix-a-flat so I personally like that but then make your way up to the rear legroom that comes in at 32.2 inches for reference I mean even six feet tall this is how much space I have back there rear center armrest with cup holders does come standard but this is the best part rear ventilation coming standard that typically you won't find that even in luxury cars when it's a vehicle of this size with only 32 inches in the rear seats so that's pretty cool I love that power rear sunshade is available for an additional $250 but the one thing I would wish that Lexus would have added to the rear passengers is rear charging ports so would have loved to have seen that I know my kids love their tablets and they probably would definitely want to stay charged up back there but then make our way up to the front seats 10-way power driver seat does come standard 8-way power adjustable passenger seat though also coming standard leather seating will come standard heated and ventilated front seats also coming standard f-sport bucket seats you gotta love that man these are the most comfortable seats in existence I'm telling you guys it just like hugs you not even around the turns it just constantly is hugging you everywhere you go and the best part about this is I should have mentioned it during the driving portion but the is has this really nice leg rest that you just rest your leg on when you're actually driving the leg that you used to hit the pedals basically so it's something that no other vehicle that I test drive has and it's so comfortable this is it makes it such a luxurious drive it's probably something that they're just trying to hide but I'll actually love that it's there because it's so comfortable but back to the seats memory settings are optional we do have them for up to three different drivers but overall these are still to this day my very favorite seats in existence out of the last 800 plus cars that I've driven I don't even know how many at this point so I love the seat comfort here but then take a look at the steering wheel it is of course tilt and telescoping it is leather wrapped ash wood accents are available I've driven that one before that's pretty darn cool f-sport badging at the bottom of course and it is heated actually as well so then make your way to the startup let me start by showing you guys the key you got your Lexus logo on the one side when you flip it over lock unlock and a button to pop the rear trunk there but it is all keyless entry with a push button starts all I'm going to do here is simply put my front of the break and press that silver engine start button located just to the left of that infotainment screen and so once started up when it comes to the gauges these are Lexus LFA inspired gauges and I love them this may be my favorite gauge cluster as simplistic as it is because there is this cool button where you can slide that kind of circle to the left and to the right or I should say to the center and to the right completely changing the look just like in the Lexus LFA it's kind of like a digital slash analog style gauge cluster because you can move it and all that fun stuff and that has this cool circle in the middle of it but anyways I love it of course it gives you outside temperature how many miles you have left until you hit empty there's a digital speedometer trip a trip b so everything you could possibly want up there so well done Lexus yet again but now let's go ahead and make our way to overall interior quality power moonroof is going to be available we got that one dual zoom climate control aluminum foot pedals you're gonna find an analog clock I always love seeing those in luxury vehicles ashwood trim is going to be available to go along with the steering wheel of course just behind the shifter you have a couple cup holders you have your touch pad controller and buttons for the infotainment screen I'll get more to that in a second but within the center armrest there's a okay amount of storage there you got a couple usb charging ports auxiliary port and a 12 volt power outlet actually in there as well I like the little details on the side doors although that does feel plastic but there's nothing wrong with that let's check out the glove box here not a ton of space but it'll certainly get the job done I like the contrast stitching just above that got homely controls throughout the three different garage doors I almost forgot to mention that as well so that is pretty cool overall like I said it's finished extremely well but it is extremely comfortable that is really where it gets me with the seats and the little leg rest here as well but anyways now let's go ahead and take a look at the infotainment screen it is an eight inch color touch screen display that comes standard however there is a 10.3 inch color touch screen display that is available and yes it is touch screen but you do have the functionality of the touch pad controller and buttons as well so of course bluetooth and audio streaming does come standard android auto apple car play as well you can actually check out your weather information up there that is pretty cold there's some climate control settings up there as well check out your driving statistics of course there's a factor in navigation system and of course your radio information and so when it comes to the sound systems there are two of them there is a 10 speaker sound system that does come standard however there is an optional 17 speaker mark levinson sound system that goes for $1080 that comes with 1800 watts and that is the sound system that we have with us here today so having said that what do you guys say let's go ahead and turn on the radio see what we got playing today and let's test out the clarity of this one these sound systems these days in cars are absolutely ridiculous how is madness man like the clarity was crystal clear base was insane as well 1800 watts in this vehicle the size of this is 100 overkill without a doubt i love that sound system i would have no problem with that for sure but anyways last thing i want to mention to you guys on the infotainment screen at least is when you do put the is in reverse we'll find a rear view camera coming standard across the board and by the way there is a panoramic view monitor available for $1365 giving you the bird's eye view letting you know who or what is behind you which is always is going to lead us into safety and so IIHS top safety pick plus which is the very highest rating given by IIHS so that's a great start right there front side side current airbags do come standard you got a driver's knee airbag up front as well in the back you're going to have latch aka lower anchors to others children for the rear car seats rear child door locks temperature monitoring system but also coming standard pre-collision system with pedestrian detection lane tracing assist lane departure alert restraining assist road sign assist adaptive cruise control any blind spot monitoring system with rear cross traffic alerts so overall what it comes to my final thoughts here of the IS 350 excellent design i love the look of it personally great safety as well you can't beat an IIHS top safety pick plus that's brilliant brilliant steering feel as well especially in the sport driving mode it's such a heavy weight to it instantly points you in the direction that you want to go excellent reliability i love the LFA inspired gauges as well it's amazing great sound system here especially the mark leavenson that we have today i probably could keep going because i could say something about the seats being the most comfortable ever i like the little leg rest as well so comfort is 100 on point i think there are two things missing for this car i think one thing is the rear charging ports i would have loved to have seen those and the other thing is i didn't know lexas could do it because they're brilliant at this is a hybrid configuration i would have loved to have seen maybe this car mixed with one or two electric motors giving it excellent mpgs because i love this car but what what if you got a commute to work and you don't want to spend a ton on gas money that i know somebody's gonna say well just get the ux well yeah i guess that's a possibility so anyways that about rounds out this when you guys thank you so much for watching for free to follow me on social media for different spy shots of these vehicles before they actually get to youtube be sure to hit the subscribe and the bell notification button if you're in any new car reviews because that's what we do here on this channel after all do appreciate you guys watching more than you know and i will see you guys all in the next video stay gold