DRW Quantitative Trading Internship Round 1 Tech Phone Screen Summary

Q1 a): Given Z standard normal. Y_1 = 1 if abs(Z) < 1 else 0; Y_2 = Z^2. Make a market for Y1, Y2. A: First calculate Y1. The meaning of abs(Z) < 1 is that Z is within 1 standard normal. Therefore, we

Q1 a): Given Z standard normal. Y_1 = 1 if abs(Z) < 1 else 0; Y_2 = Z^2. Make a market for Y1, Y2. A: First calculate Y1. The meaning of abs(Z) < 1 is that Z is within 1 standard normal. Therefore, we can know from the statistics that the probability is 68% (I initially misremembered and said 65%, which was corrected). Thus, we can directly calculate the fair price of Y1 as 0.68 using the expectation. Then I struggled with how to calculate the spread. I asked the interviewer for help, and he said it can be understood as how confident we are about the price and how risky it might be. If our pricing for an asset is not very certain, we should have a larger spread. I listened to this, but didn't quite grasp the hint. He said, "For simplicity, I'll price it at [email protected]." This ensures symmetry and avoids overly complex calculations. (During my debriefing, I thought the spread might be determined by standard deviation (std), which means calculating the Var of Y1 and then choosing one standard deviation. The problem is that Y1 should be a Bernoulli distribution, but its p is difficult to calculate, and I didn't have a calculator on hand. I wonder if any experienced people with M&A game experience could advise on how the spread is usually calculated in this scenario.) Next, I calculated Y2. Following the previous approach, I first found E[Y_2] = E[Z^2] = Var[Z] - E[Z]^2 = Var[Z] = 1. Here I made a small blunder. When calculating E[Z^2], my first thought was that it was independent and could be decomposed into products of Z. After the interviewer reminded me and asked if I was sure it was independent, I realized that I had used variance to calculate the fair price. Again, I wasn't sure how to assign the spread (Var[Y_2] = E[Y2^2] - E[Y_2]^2 = E[Z^4] - 1 = 2? But I was confused because in this case, sqrt(2) > 1, and the minimum bid should be 0, so even +- 1 std cannot satisfy the spread requirement). I just randomly assigned [email protected]. Continuing to maintain the symmetric characteristic. The interviewer followed up by asking why I considered giving Y2 a wider spread and Y1 a narrower one. My brain went blank here; I didn't think that much about it. But considering he mentioned the concept of "the more risky, the wider the spread" when explaining how to assign the spread, I forced a connection and said that Y2 might have greater volatility (which is actually correct, because Var(Y2) happens to be greater than Var(Y1), so we should be less confident on the price of Y2). I didn't mention Var at the time, but the interviewer still let me pass. The question continued: How are Y1 and Y2 correlated—positively, negatively, or not correlated? I chose to draw a graph here, as I wasn't entirely confident about the probability at the time… I looked at Y2; it's a parabolic function, and Y1 is a discrete function with the same step function. We can clearly see that Y1 increases when Z approaches 0, and remains at 0 outside of 1. Conversely, Y2's value increases as Z increases, so I concluded it's negatively correlated. After receiving confirmation, we moved on to the formal market making part. Q1 b): First, the interviewer confirmed my understanding of mm game, introducing basic terminology like x@y. Then, he asked me to quote prices for Y1 and Y2. He hit my Y1 bid (0.5) and asked me to provide new prices for Y1 and Y2. A: I explained the situation, saying that assuming my counterparty is rational, their willingness to complete the transaction at my bid indicates they still have profit potential, therefore they considered our pricing too high. Therefore, I would consider changing the fair value of Y1 to its transaction price of 0.5. Also, because its offer provides me with new information (information about Z that I didn't know before), we can reasonably narrow our spread. Here, I'm referring to Y1 as [email protected]. The specific logic is: if the opponent considers Y1 to be of low value, it means the absolute value of Z might actually be greater than 1 more likely than we expected, making Y1 more likely to be worthless. Under this premise, since we just assumed Y1 and Y2 are negatively correlated, the price of Y2 will inevitably rise. However, I don't know exactly how much it will rise; intuitively, I'll put the midpoint at 1.2. So, Y2 is [email protected]. I'm actually quite uncertain about the pricing part; if there's a complete process and calculation method, I hope an expert can provide some guidance. Q2 a): They asked if I watch football, saying we need to price some football-related things. The first is the total number of goals scored in La Liga last year, and they asked me to bid for it. A: I'm a bit confused because I don't watch much football. First, they said we need to do some assumptions and mathematical modeling. After confirming with the interviewer that La Liga has 20 teams, each playing one home and one away game per season, we calculated 19210 = 380 games per year. I mentioned I vaguely remembered the Golden Boot winner scoring around 30 goals last year. Since teams vary in skill level, some might score fewer goals, while stronger teams would generally score more. Based on my expectation of La Liga's style, I estimated an average of 3 goals per game. This would give a total of 380 * 3 = 1140 goals per year. I thought this number seemed a bit high, and confirmed with the interviewer that it was reasonable. He agreed, suggesting we lower it to 1000 as our midpoint. Because I had relatively low confidence in the total number of goals, I considered a wide spread, and impulsively suggested 600@1400 (the interviewer then told me this was meaningless and asked for a 200-width spread, but with a skew). Then, based on our discussion, I said I thought weaker teams might score fewer goals, so I chose 760@960 as my bid for the team's total goals. Q2 b): I was asked to give a bid for Real Betis's goal tally for the season. A: I asked if he had any additional information, such as the team's approximate level (at this point, I completely forgot about this team). The interviewer said he wouldn't provide any additional information. Here, I first calculated based on the midpoint of the goal tally, which is 860: 860/20 = 43, indicating that the average number of goals for a La Liga team is 43. Without additional information, I assumed Real Betis was a mid-table team, so I used 43 as the midpoint. He then asked me to give a bid with a width of 10. I said that mid-table teams might not be as attack-oriented overall, so I thought it was more likely they would score fewer goals than average. Based on this, I would give a skewed bid. Finally, I gave 36@46. Q2 c): I was asked to give a bid for Real Betis's ranking last season, from 1-20. The requirement was a width of 4. A: I didn't think much about this part. Based on the above assumptions, it's a mid-table team, and I didn't know much about them, so I directly gave them 12@16. Q2 d): In the formal mm game, his requirement was for me to quickly change the pricing of all three assets. He would tell me what he bought/sold each time, and I needed to adjust the pricing based on his actions (as quickly as possible, so not much explanation is needed) while ensuring the spread remained unchanged. A: I didn't adapt well; I still habitually relied on logic, which might be a point deduction. Round 1: Buy team rank. This indicates the actual ranking is higher. Team: 10@14 —> This indicates the number of goals may have increased; 40@50 —> But the number of goals is unrelated to the total league goals: 760@960 Round 2: Buy team goal count: This indicates the number of goals is low, but I felt I couldn't significantly increase the expected number of goals, so goal count: 42@52 —> The team ranking continues to rise: 8@12 —> Same logic, the total league goals remain unchanged: 760@960. Round 3: Bet on league goals: Similarly, continue to increase, but don't want the increase to be too large. Considering that the interviewer thought 1140 was too high when calculated, so: 800@1000 —> The team's average number of goals increases. Assuming each team increases the same number of goals: 44@54 —> However, if all teams increase, it won't affect the ranking: 8@12 Round 4: Bet on team ranking: 6@10 —> It seems that the number of goals shouldn't be increased significantly, as the difference might not be large, so: 45@55 —> Maintaining the logic of Round 1, league goals shouldn't be affected: 800@1000. Game over.