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"Quantitative Researcher" or "Trader" are jobs where even young, inexperienced STEM-related students can earn a ton of money, at least relative to most other opportunities. This is true, this is how most people perceive those jobs and this is why the majority of maths/IT-minded, in particular the majority of maths/IT-olympiad, people consider doing them. At least for a bit at some point in their lives. Most probably, you are not an exception :)

And it is **totally fine** to care a lot about the salary aspect when applying for jobs, no matter what many people and books say! It is important, given how our world operates. So it makes 100% sense to think of it is as "a significant factor". Plus, it should be a relief for many people to know that there is very decent money in a possible future of a mathematician.

But what are those jobs actually about? Maybe it is just not what you want to do with your life. Let's read the main descriptions from a few of the largest and most successful *trading firms* that you could actually end up in:

- Citadel Securities: "We provide liquidity to ensure institutions and retail investors can tradewhat they want..."
- Jane Street: "We are a global liquidity provider and trading firm that uses sophisticatedquantitative analysis and..."
- Optiver: "We have over 1,900 employees in offices around the world, united in their commitment to providing liquidity, competitive pricing, ..."
- XTX Markets: "We partner with institutional clients, counterparties and trading venues globally to deliver liquidity in the...."
- Hudson River Trading: "We are a multi-asset class quantitative trading firm that provides liquidity on global markets..."

One of the reasonable comments on this could be "Lol". More seriously, it probably does not tell you anything about those jobs. To be completely fair, that "providing liquidity" phrase mentioned in the descriptions sounds plain silly in the first place... It is actually a topic for jokes between many of the finance people :) Even despite the importance of "market liquidity" on a global scale that they are well aware off...

Though not every finance firm is about "providing liquidity" (whatever it means). Some of the other ones, e.g so-called *hedge funds*, are about making a bunch of very rich people even more obscenely rich. It does not have to be a bad thing by the way: some millionaires are great people. Then there is so-called *investment banking*, which is mainly about providing financial advice (or other services) for corporations and governments. Sometimes among those governments there are totalitarian ones. Often not. But sometimes there are. I hope that not that often though.. But it happens. Sadly.

Anyway, if you do join a decent finance company, it is very possible that you will find yourself in a position of doing interesting (from maths perspective) work that does not significantly impact a good cause – nevertheless, you will be utilising your maths & IT skills, while being very well-compensated. In particular, such a career path is NOT necessarily inherently evil. It is likely to be rather neutral if anything, at least up until you use the extra earnings you got for good.

To be again fair, just like this entire course will be, the details of what people in those companies do are secret. So the job descriptions you get, especially from their websites, are often very vague. Being even more honest, those details do not matter as much actually, definitely not as much as knowing what set of maths & programming skills they need. And that information is not a secret! But for the sake of completeness, let me share a description for the three most common roles for those companies anyway:

**Trader****.**

This role is more commonly known than "quantitative researcher," and sounds simpler. However, it technically doesn't even exist in many trading companies, especially smaller ones! This job often involves sitting in front of one or more computers, monitoring price changes, and following a trading algorithm designed by someone else. You'll work with data (sometimes in Excel), think quickly, and need a basic understanding of probability theory. You must be ready to modify the algorithm or deviate from its suggestions too. Here is a toy example: if the algorithm instructs you to buy 10,000 shares immediately, but some ... stuff went down recently, like a major trading company has just collapsed causing market turmoil, you might decide to buy only 1,000 shares instead. Also, in some companies, you get to update and improve the model itself, but it is more of a rare case (as this is what another role is for);**Software Engineer****.**

Most finance companies automate trading where possible. "Automated" means a team has developed an algorithm to trade based on predetermined rules. These rules might be as simple as "if the price variance in the last hour is high enough, we pause trading for the next 10 seconds." Coding the algorithm involves ensuring it functions correctly as well as making sure it is fast and that it can handle system collapses. This is where more advanced software engineering comes into play, requiring significant study to master. Though AI might replace a bunch of those in a year, who knows;**Quantitative Researcher or Quant, for short****.**

In some companies, this role is also called "Quantitative Analysts". Anyway, these individuals search for market opportunities to make money, and they are the ones creating trading models. In smaller companies, there often just Quants and Software Engineers, no one else really (maybe a legal-advisor or an HR as well). Quants may also code algorithms and take on tasks typically assigned to Software Engineers. This role is usually the most mathematical of all, and a solid understanding of probability (usually stats as well) is essential here. Many companies prefer candidates with extensive research experience and often value those with PhDs, even if the degree is not in mathematics. But far not always PhD is a must, so don't worry. Additionally, these roles almost always require at least basic knowledge of programming languages like Python or C++, as you might need to analyse datasets containing billions of numbers.

This list is not exhaustive, plus the exact descriptions of those roles vary from company to company. But once again, all of it is much less important than having the skills those companies seek in people! Once you get in the company, they will explain you everything there. So "getting into a company" is a clear priority here, and thus this course revolves around this rather than teaching you a potentially successful, but almost surely outdated, trading strategies.

Now, let's finally get down to increasing the chances of getting into those companies. This course is for young and unexperienced people, so we will start preparing for getting an offer for an *internship* (which is similar to the process of getting into a company on a Junior position anyway). Just to confirm, throughout this course** we will be focused on the roles of Traders, Quants and similar ones, i.e more mathematical roles**. **We will not be focusing on Software Engineer's path at all.**

First things first, the application process is almost straightforward: either you have a friend who recommends you to the company from within the company (and then tells you what to do), or you google a "[*Name of The Company*] Internship" and follow the instructions from their website. You will need to attach a CV (if you do not know what this is: time to google it and create it), answer a few more general questions and then click "submit". That is all, it really is that easy in 90% cases :)

If you are lucky (yeah, luck is involved), and your CV is good enough, you will be invited to do an assignment or to do an interview over a phone/e-meeting. If you succeed, you will move to the next round, and so on.. What is interesting here, is that for many companies there are mostly brainteasers and quite basic questions on probability at the initial rounds. At least in the maths part of them, there might also be a coding assignment (but it is often quite easy, we will get to it briefly). Therefore being ready for those is absolutely crucial, and luckily you can already do a decent amount here! All of this applies both for trader and quant roles by the way. And so those brainteasers and basic probability questions is what we will focus on for the rest of this lesson, i.e we are finally getting to doing maths 💪

Exercise 1.

What is the smallest integer whose digits multiply together to 10000?

Explanation and comments

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The answer is 255558. It is a non-difficult exercise on making use of the greedy algorithm (in particular, not forgetting to sort the digits in the non-decreasing order).

And yes, this is for real! You can even google this particular question if you do not believe it, it was a question at an interview to get into Jane Street.

Exercise 2.

You are playing a game, in which you roll a fair, 6-sided die. Whatever you roll, this number is the number of dollars you will get. How much would you pay for this game?

Explanation and comments

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You could say "I am not gonna pay a penny for this stupid game", and this would be technically an answer. You do not have to take any risks, it is fine and understandable.

However, as an interview question, this problem is designed to assess the math and reasoning you employ. Even though it's an usual problem, the mathematics and logic required to derive a reasonable answer — which by the way is greater than 0, even 0.1 — are the types you will use a lot in a real job. The problem is also somewhat open-ended, thus not being 100% perfectly rigorous in your explanation is fine (unless the interviewer specifically asks for it).

So, let us share an answer that would be accepted. It is "something less than 3.5, it could be even 3.49". You may recognise the 3.5 as the expected value of a die roll and this is exactly why it appears here! Especially if you can play this game many times, with high probability you will be winning money, even if you pay 3.49 each time to simply play that silly game. So appealing to the maths expectation is how you should go about explaining that answer of 3.49.**Remark:** The way the question is formulated is exactly how it was asked at the interview. So be ready for hearing an unusual question.

Exercise 3.

**a)** Let's update the rules of the game from above: you are now allowed to reroll the fair die if you’re not satisfied with the first roll, but that you must accept the result of the second roll. How much would you pay for this game?**b)** What if you can reroll up to two times? How much would you pay?

Explanation and comments

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**a)** We will again use mathematical expectation of something as our "cut off" point. However, there is this "you can reroll if you want to" part, which complicates things. When should you "want to reroll"? This is a question of finding the "most optimal" strategy, and luckily it is not difficult for this problem.

Indeed, it only makes sense to reroll when our first roll is 1, 2 or 3, cause it is only then the expected value for what we will get if we reroll, i.e simply 3.5, is larger than what we just rolled. Yes, it is more "risky", but "on average" we will be better of that way. Especially, if we can play this game many many times.

Finally, we are left with calculating the expected value of the final result if we adhere to the strategy just described. It is not difficult: it is \[ 1/2 \cdot 3.5 + 1/2 \cdot \frac{4+5+6}{3} = 4.25 \] where the first addend is equal to the expected value of the result conditioned on seeing 1, 2 or 3 on the first roll (which happens with probability 1/2, and in this case we reroll), and the second addend is equal to the expected value of the result conditioned on seeing 4, 5 or 6 on the first roll (which happens with probability 1/2, and this is the case when we stick to it, i.e no rerolling). Thus the answer to the original question is "I would pay smth less than 4.25, e.g even 4.24".**b)** This problem is very similar to the one we just solved, we simply need to take it one step further. Below is the formula for the expected value of the result for the optimal strategy, you should be able to understand where it comes from: \[ 4/6 \cdot 4.25 + 2/6 \cdot \frac{5 + 6}{2} = 4\frac{2}{3}\] therefore the answer to the original question is "I could even pay 4.66".

**Remarks about the two previous exercises**: **1)** One could ask if relying so much on the expected value is reasonable in the first place. Or as the very least, why does it necessarily produce "the most optimal strategy"? And the answer is "it may not". It is merely reasonable to look at it, especially if we add the assumption that we are playing this game many times (formally because of the so-called "Law of Large Numbers"), but it does not have to lead to "the best answer". There are many problems, where it seems like the expected value should lead to a great answer, but instead it leads to something weird. If you mention all this during the interview, you would not be wrong;**2)** I have mentioned the "if we add the assumption that we can play this game many times", even though it is not written in the problem statement. How come, who let me do assumptions? Well, it is an interview question that is supposed to test your reasoning skills. In particular, making certain reasonable assumptions is actually normal, sometimes it is even anticipated from you since the original question is unsolvable! So as long as you clarify what assumption you are making to the interviewer, you are fine. He or she can then accept your assumption, or talk to you about it, e.g clarify that "no extra assumptions are needed".

Exercise 4.

There are two coins: one is fair, another has Heads on both sides. One of the coins is picked at random, and then tossed 3 times. You do not see which coin has been picked, but you are told that it was three heads. What is the probability that the 4th flip will also be heads?

Explanation and comments

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If you got the answer of 3/4, please think again. It is clearly wrong, as it is simply $1/2 \cdot 1/2 + 1/2 \cdot 1$, i.e it is the probability of seeing heads when you pick one of the two coins at random not having any information about the previous tosses. It should be clear that if the first three tosses were all Heads, then it is much more likely that the coin that is being tossed is the one that has Heads on both sides.

Moving on to the correct solution. It is a basic exercise on Bayes' theorem. If you are not familiar with it, or if you struggle with it: it is time to change that. That theorem is not difficult at all, it is also intuitive and happens to be an unbelievably applicable result. Anyway, the problem is really about calculating \[\mathbb{P}(\text{fair coin picked} \, | \, \text{info about 3 heads} ) \] since once we have this number, call it $p$, the final answer will be $p \cdot 1/2 + (1-p) \cdot 1$. In order to find that probability, one could use common sense (and relatively quickly understand that it is \frac{1}{9}), or, and we expect you to do it this way at this stage, apply the Bayes' formula:

\[ \mathbb{P}(\text{fair coin} \, | \, \text{info} ) = \frac{\mathbb{P}(\text{info} \, | \, \text{fair coin}) \cdot \mathbb{P}(\text{fair coin}) }{\mathbb{P}(\text{info} \, | \, \text{fair coin}) \cdot \mathbb{P}(\text{fair coin}) + \mathbb{P}(\text{info} \, | \, \text{UNfair coin}) \cdot \mathbb{P}(\text{UNfair coin})} \]

where "fair coin" = "fair coin picked", "info" = "info about 3 heads" (just to make the formula look shorter). The terms on the RHS are all trivial, $\mathbb{P}(\text{fair}) = 1/2$, $ \mathbb{P}(\text{info} \, | \, \text{fair coin}) = 1/8 $, etc... It is an exercise for you to fill in the details here to obtain \[ \mathbb{P}(\text{fair coin} \, | \, \text{info} = \frac{1}{9} \] which will lead to final answer being \[ \frac{1}{9} \cdot 1/2 + (1-\frac{1}{9}) \cdot 1 = \frac{17}{18}. \]

Finally, just to be clear, we do not want you to apply the Bayes' formula blindly. For anyone doing this course, as the very least we expect you to have been exposed to that formula before, and in particular it should make sense to you. It is fine if you have not used it a lot in life, but it should not be school-book random bunch of letters to you.

This is just the beginning, the exercises above are an absolute classic. Also, apologies for a lot of bla-bla kind of text for this lesson, it is an introductory classwork after all...

From maths world, we had a look at a brainteaser and two problems on basic probability theory: one on expected value, one on Bayes' theorem. Moreover, the law of total probability and general understanding of the basics of discrete probability were needed. All of these is absolutely crucial knowledge, and it is a prerequisite for this course. If you are not sure whether you know those things well, you do need to work on those. If you struggle a lot and feel lost, feel free to email us to ask for some guidance (maybe doing another course is more reasonable in you case).

The next step is proceeding to the Problem Set for this lesson. There you will see a lot of problems, neither of which goes beyond the exercises below when it comes to the theory required. Even though, some of them are more difficult. All of them are from the interviews that really did take place!