@@Aces314 it’s mostly linear algebra. For risk management, I’d say it’s one of the more applicable areas of math.

Great question. It is e^(-rt) where t=1 and r =0, (interest rate free world). This makes it so that e^(-0*1) = e^0. Anything powered to 0 is equal to 1. So e^0 = 1, and when we see a variable equal to 1, we can ignore it in our simplification.

@@cbe-commerce 😂 glad you liked it. I use Adobe Illustrator to make the puppets and backgrounds and Adobe After Effects to make them move and edit the video. Hope that helps!

@@david0aloha 100%. In the “real world” put call parity doesn’t always shine through the way our model’s theory would suggest. All models are wrong but some models are useful. Black Scholes is often wrong, but still useful. For me, the strength of Black Scholes is in its simplicity, and how well it generalises a complex relationship in so few variables and so little actual math.

Great question! Given our supply demand graph shown in the video - no. Why? Because the supply demand graph in the video makes a few assumptions, such as: perfect availability of the product, perfectly functioning markets, and all agents involved know how much or how little of something there is. In reality, something like what you have mentioned can happen, but its normally because what you have observed is occurring in two (or multiple) distinct markets. Here is an example. In town A, many bananas are grown and there are many banana stands. One day the government puts a tax on bananas, resulting in a higher price per banana, decreasing quantity demanded. But the supply is already fixed, as the amount supplied has all ready been grown. Now we have a surplus of unsold bananas due to the new tax. So, one industrial thinking banana stand entrepreneur goes to town B, that has no bananas, and sells his excess bananas for even more because the supply is so low and demand in town B is strong. Is this a violation of supply and demand? Not really, because town A and B actually represent two different markets. Perhaps there is a geographic or political barrier preventing them from engaging in perfect trade of bananas, but the fact remains, that town B is isolated and therefore its economic model should be considered separate of town A. All this to say, we can observe various prices for things in different parts of the world, not because supply/demand is broken, but because we are observing different markets for the same product due to geopolitical or social differences that cause barriers to trade.

@@financeexplainedgraphics ok thank you very much , one more question , is the given graph restricted due to the base price of the supply , if it altered to the supplier's wish , then how will the graph react ?

​@@vanditstech9081 I might not be understanding your question, but what I think you are imagining is a monopoly, which can dive into a very different analysis. Our model assumes that markets are efficient and costs to purchase are equal everywhere. Another key assumption is that there are many suppliers (competitive market). By asking if the supply can alter with the supplier's wish, then what you are assuming is that a single supplier controls the market's curve - i.e. a monopoly market. In this case, the monopoly creates the curve based on its particular supply curve structure. If the monopoly has a shift in it's supply curve there would be a massive shift in the market, but prices would still ultimately find an equilibrium, however, if the monopoly finds that their bottom line is hurt at that equilibrium, it could charge an inefficient price to maximize profit. The monopoly's actions would be dependent on the shape of the demand curve and its supply cost structure. The only way that equilibrium is not met in competitive markets is if there is a sudden shock to the system exogenously (from the outside). In my last example you had a banana tax, which shocked the system. This resulted in a surplus of unsold bananas. The inverse of this would be if you had had a government sponsored subsidy, then you would have had a shortage of bananas (more people would want to buy the bananas because the government had helped lower the price), but you wouldn't have enough bananas supplied leading to the shortage. After the system is shocked, and the equilibrium is disrupted, the supply curve would then be expected to shift to accommodate this change. All the suppliers in a competitive market would work to meet the lower supply (thought they would do this each on their own and the aggregate change would produce the new supply curve), or in the case of a monopoly the one supplier would alter its production to accommodate (assuming the accommodation aligns with its interests). In the banana tax instance (assuming Village B wasn't there to pick up the extra bananas), the suppliers would produce less bananas next season to meet the new level of lower demand (less demand because the taxes increased the price consumers had to pay). But once again, the equilibrium would be met and the right amount of bananas would be supplied. There really aren't any "restrictions" to the graph. The concept is that there exists, for any product, an equilibrium price that everyone will gravitate towards. That equilibrium is constantly shifting in the real world due to changes in taste (demand) and input prices etc. (supply). However, these shifts in the curves adapt quickly as consumers and producers work to find the new equilibrium. Adam Smith called this concept the "Invisible Hand", because it was a mystery to him as to how all these players (who don't know one another) always find the balance. Today, we call the concept "free markets" where sellers and buyers transact freely to negotiate the right price. When you think of these curves, don't think of one transaction, rather think about the whole economic system for a single good or service. The graph is showing the aggregate for that system. If you look at un-free markets ("command economies" like communist economic systems, where all supply of goods is controlled by the government or central planning authority) you will see massive shortages and surpluses of goods. In the Soviet Union, for example, it wasn't uncommon to have way too much of something or way too little of another thing. This was because what was supplied was based on the government's mandate and not based on businesses working with consumers to find an equilibrium. Moreover, when you have price fixing, where the price someone must pay or must sell at is fixed by law, then the equilibrium cannot adjust to meet the new level of demand or supply, thus causing shortages and surpluses. This is the major economic argument against command economies. Was this helpful in answering your question?

Oh no! I think you are right. I went back into the production file and the ^2 was there but I had hid the layer 😢. Thank you for catching it. Yes, it should be {-x^2/2}

You are right! Good catch …. I’ll be having a word with my script writer later 😉 Thank you for clarifying!

Sorry. I had a mistake with the denominator. Fixed it. Perfect

Great! I was going to reply to this earlier but needed access to my computer to check, however you beat me to it. Glad it is working for you now!

d2 is not reduced to d1, there is still a negative sign in d2 that is not in d1. However, the reason I reduce it down from the original long form of d2 = (d1 - sig(T)) is because it is a little easier to see visually, and it highlights that the difference between d1 and d2 are their inverse relationships with regard to volatility (sigma). I show the long form initially because when you see Black-Scholes, much of the time d2 is shown in the long form.

@@financeexplainedgraphics I see thanks

That’s incredibly kind! Stay tuned for more upcoming videos

You can calculate in Excel or any software you prefer. The linked Excel doc shows how this is done. Standard calculators do not perform these operations, and doing it by hand is very impractical.

It's not a matter of courage, it is a matter of ensuring the "right" audience is viewing the video. I can see in the metrics a major drop after the warning is given - this is good, as it prevents viewers who don't want to watch the video seeing it and then giving it a "dislike", which harms my content. As for not wanting to deal with the math - you'd probably be surprised by the number of financial professionals who find mathematics "boring" and "tedious".

It does! Any asset can be modelled this way and added to the model

On the way!

Haha, wow! You are right, I did. I guess we can just pretend the indexes of w match r 😉 Great catch.

Impressive that you want to work on this while still in high school. Looking at the 2:46 time stamp, think of it this way: You have assets (stocks, bonds, whatever) and each are called A, B, and C. The vectors (vectors are essentially single column "matrices") are the historical returns of the asset. For instance, you might have a stock A that has a return vector of [+0.2%, -1.1%, +1.3%, +0.7%, -0.8%] over a given 5 day period. Now you do this for each of your assets and find what their returns were each day (or month or whatever you like, just be consistent and make sure all assets have the same number of observations). Then, you are going to create a matrix. A matrix has rows and columns, and a covariance matrix, like the one in our video, is ALWAYS square (number of rows = number of columns). HOWEVER, the size of the matrix will depend on the number of assets. We have three assets, so the matrix will be 3x3. If we have 10 assets, the matrix would be 10x10. So, assuming we have 3 assets, we will have a 3x3 matrix, which means our matrix will have 9 data points. Each data point is the covariance of two of our vectors, which equals a single number. A covariance is a statistical term and you can read up on in a number of places (Wikipedia is great for this). If you are doing this in Excel (which I recommend as a great first step to financial modeling), you can use a function covariance.s( ). Inside the parenthesis you will put the range (your vectors) of returns for the assets separated by a comma. Make sure you have the same number of observations in each range (vector). You can see this in the Excel document in the video description. Now, you have your matrix, which will shown (in this case) 9 covariances. However, there will be something interesting about this matrix. You will see that it is special, because it is a symetric matrix. If your matrix is not symetric, you have messed up. Why is this a symetric matrix? Because cov(A,B) is the same as cov(B,A), etc. Here is something else that is interesting, if you found the variance of any single vector (A, B, or C) that variance would be the same as the cov(A,A) for variance A, and cov(B,B) for variance B, and so on. So what you have in your matrix is the variances of A, B, and C running along the diagonal from the top left to the bottom right, and in the fields you have a mirror of values which are the covariances of A,B B,A A,C C,A B,C and C,B. In other words, our 3x3 matrix looks something like this in variable terms: | varA X Y | | X varB Z | | Y Z varC | That's probably more than you asked for. I highly recommend you play around with the excel file. This math is well beyond what most people are taught in high school. If you are interested in learning more about this type of math, you should explore classes on Linear Algebra, which is the specific mathematics applied to matrix and vector relationships (just like the ones in the video). I highly recommend the MIT Classes that are free on KZbin for Linear Algebra (Professor Strang is a legend in this field and he recently retired but all his classes are available on their channel). You don't need to understand calculus at all to understand Linear Algebra, but I would highly recommend you familiarize yourself with the concept of Vectors and Matrices if this type of math and financial analysis interests you. Let me know if you have any more questions.

Optimizers can do it for you, but as a beginner I strongly believe it is important to learn the fundamentals of the theory (risk vs. reward and volatility as risk). The math is secondary and for someone without knowledge in linear algebra it could be more challenging. If you use an optimizer it can be convenient, but it does little to help you understand what is going on and why it is determining what it has calculated. One thing investment apps don't do is calculate the expected return, which is necessary for computing the optimization. So developing your own theories and investment strategies is still key.

@@financeexplainedgraphics thanks

Figure out/eQasian Q..uestion works , it is mathematic B and A level. You go in a mathematic web site and when you see there your skills, than you need to start reading from year 5, so you make sure that you need to connect the past knowledge with new formulas. Ask more Q..uestions , dont bother to ask somethink clever.

Hi Haneul, and thanks for the questions. 1. A portfolio's volatility cannot be found by summing standard deviation, or even using a dot product (sumproduct) for the values. The reason for this is that there are covariations which exist between these assets. Let's say that you have two equally volatile positions A and B. However, A and B have inverse volatilities (when one is up the other is down). If you only take their respective standard deviations and apply to their respective sizes, your volatility will be the standard deviation of A and B. However, because when A is up and B is down, the portfolio actually has lower risk, likely near 0 (hypothetically) because the two cancel each other out (perfectly in our scenario which is not realistic but useful as an example). This is why diversification among uncorrelated assets is important! It can actually reduce risk. 2. This is probably incorrect. One issue that you might be facing is that you have not annualized the data. You have taken daily returns, and therefore you must multiply the standard deviation value by the SQRT(252) <- 252 is the number of average trading days in a year. Try that and see if that doesn't help fix the issue.

@haneulkim4902 The other issue might be that you are taking just the closing prices. You need to take the percentage change between days. This is also likely your issue. Take the (current day - previous day)/previous day for each day to get the daily percent change.

@@financeexplainedgraphics Thank you! Why do we need to annualize the standard deviation? and I did use return (= price_t -1/ price_t), not closing price. Lastly please post more videos~ your videos are very well structured and informative :)

@@haneulkim4902 Glad I could help! The reason you want to annualize is so that it "looks" right. You don't have to annualize, but it is best practice. Most times when professionals quote volatility they do so on an annualized basis. When you say a security has an volatility of 20% annualized, it is more familiar to investors than saying your volatility is 1.26% - so when you said your vol was extremely low, I thought it might not have been unannualized (but 5.9*10^-11 is WAY to low 😂 even for daily vol). As for the videos, I am trying. They take a very long time to make and I am a bit of a perfectionist. I also have a day job which takes up a lot of the energy I would otherwise put into making videos. However, I am so glad you liked it! Still working on Black Scholes, which is a beast of an equation to explain in ~15 minutes.

@@financeexplainedgraphicsI see, the reason it was so low was because for calculating return I was using (Return = price at t / price at t-1) instead of (Return = price_at -t - price_at_t-1) / price at t-1. Wow already excited for the next video, thanks again for an amazing video and answering my questions!