Great choice! Asset management firms offer lots of exciting opportunities for quants and Quant Questions can help you get there. Roles can vary from portfolio management, portfolio construction and optimization, transaction cost analysis, finding alpha generating strategies, beta market research or market risk management.

Great choice! Investment banks are the classic place to start your quant career before progressing to more senior ranks within the firm or moving over to the buyside.

Great choice! Hedge funds offer lots of lucrative and intellectually stimulating quant roles and Quant Questions can help you land one of these coveted jobs.

Great choice! Prop trading firms offer lots of lucrative and intellectually stimulating quant roles and Quant Questions can help you practice for these interviews.

Great choice! Financial software firms have lots of modeling opportunities for quants and Quant Questions can help you get there.

Great choice! Commercial banks offers lots of diverse opportunities for quants. You can find roles related to CCAR/DFAST modeling, market risk management, credit risk management, liquidity modeling related to the bank's securities portfolio, asset liability management (ALM), deposit modeling or any other analytics for the bank.

Great choice! Consulting firms offer lots of exciting opportunities for quants and and Quant Questions can help you rock the quant consulting interview. Consultants often advice banks on a host of issues related to stress testing, credit quality, deposit analysis, derivative valuation and market research.

Great choice! FinTech have many great opportunities for very creative quants looking to break new grounds. Here quants can work in crypto currency, in peer-2-peer lending, on crowd sourced trading strategies or on launching new exchanges.

Great choice! Most quants don't think about private equity as a career path. But as this traditionally non-quant industry becomes more quantitative, this is a very viable option. Roles can vary from portfolio construction and optimization, to residential credit modeling and to general analytics.

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We need to remember the Gauss sum equation: where , and Thus, we have

So and
For we have that,

Great choice! BlackRock is the largest asset manager with $6.5 trillion in AUM and offers lots quant opportunities. Quants can find themselves at home in various investment divisions such as Scientific Active Equities or Model-Based Fixed Income, or in roles in Portfolio Analytics or Risk & Quantitative Analytics.

Great choice! State Street is based in Boston and manages $2.5 trillion in AUM. Quants can find interesting roles in groups such as Fixed Income and Currencies, Active Quantitative Equities, Enterprise Risk Management or Model Validation.

Great choice! Pimco is the biggest bond manager based in Newport Beach, California. Quants can find roles within the Client Solution and Analytics or on the Trade Support team on one of their many trading desks.

Great choice! Wellington Management is a $1.1 trillion AUM asset manager. Quants can find technical roles within the Fixed Income, Trading Data & Analytics or Derivative Analytics and Valuations.

Great choice! First Quadrant is a boutique $30 billion AUM investment management firm. They apply systematic approach combining fundamental and economic insights to generate high risk-adjusted returns.

Great choice! Morgan Stanley is a multinational investment bank with over 50,000 employees and over $800 billion in total assets. You can find opportunities as a Desk Strategist, on a modeling team or in the Electronic Market Making Group.

Great choice! JP Morgan is a multinational investment bank based in NYC with over $2.5 trillion in total assets. You can find positions in the Quantitative Research division or in the Data Analytics team.

Great choice! Citi has over 200,000 employees and over $1.9 trillion in total assets. You can find quant opportunities in their trading, asset management or investment banking divisions.

Great choice! Wells Fargo has over $1.8 tillion in total assets and over 250,000 employees. You can find quant opportunities in their trading, asset management or investment banking divisions.

Great choice! Bank of America has over 200,000 employees and over $2.3 trillion in total assets. You can find quant opportunities in their trading, asset management or investment banking divisions.

Great choice! Two Sigma has over $30 billion in AUM and is at the forefront of quantitative investment hiring scientists, academics and engineers. Quants can find intellectually stimulating roles in Data Science & Analytics, Quantitative Research & Modeling or Execution & Trading.

Great choice! AQR is an $80 billion investment management firm based in Greenwich, Connecticut. You can find quant roles within Global Asset Allocation, Global Stock Selection, Risk Management or Trading.

Great choice! Balyasny is based out of Chicago and has over $7 billion in AUM. You can find roles of interest in the following divisions: Financial Engineering, Risk, Quantitative Research or Investment & Trading.

Great choice! D.E. Shaw is based in NYC and has over $30 billion in AUM. You can find quant jobs in their Investment Analysis, Quantitative Strategies or Trading divisions.

Great choice! Citadel is based out of Chicago and has over $30 billion in AUM. You can find quant jobs in their Global Quantitative Strategies, Global Fixed Income or Portfolio Constructions division.

Great choice! Cidatel Securities is the largest market maker in the US. You can find interesting roles in their Information Technology, Engineering or Finance & Accounting teams.

Great choice! Jane Street was founded in 2000 and has about 900 employees. You can find interesting roles in their Quantitative Trading, Quantitative Resarch or Institutional Services divisions.

Great choice! Hudson River Trading is based in NYC and was founded in 2002. They can trade up to 5% of all US stock trading any given day. They have many quant roles in Strategy Development, Software Engineering or Trade Operations.

Great choice! Optiver was founded 1986 and now has over 1,000 employees. Optiver makes markets in options and delta1 products. You can find interesting quant roles in their risk, technology or trading divisions.

Great choice! SIG was founded in 1987 and is based out of Philadelphia. They make markets in equity options and index options. You can find interesting roles in their Trading, Quantitative Research, Machine Learning or Operations teams.

Great choice! Bloomberg was founded in 1981 and now has over 19,000 employees. Bloomberg has great opportunities for quants in Multi-Asset Risk System (MARS), Enterprise Data, Quant Analytics and Desktop Build Group (DBG).

Great choice! Moody's Analytics was founded in 2007 and now has over 10,000 employees. They offers opportunities for quant in their Enterprise Risk Solutions, Credit Analysis & Research or Strategy & Analytics groups.

Great choice! MSCI is based out of NYC and has over 3,000 employees. They have opportunities in Multi-Asset Class Research, Risk Analytical QA and Analytics Platform Development teams.

Great choice! Numerix was founded in 1996 and is based out of NYC. You can find interesting quant roles in their Financial Engineering, Client Solutions Group or Global Operations groups.

Great choice! HSBC is headquartered in London and has over 230,000 employees. They have quant opportunities in Traded Risk Management, Risk Analytics and Modeling or Data Analytics teams.

Great choice! Santander is headquartered in Santander, Spain and has over 200,000 employees worldwide. They offer quant opportunities in Risk Modeling, Operational Risk and Technology divisions.

Great choice! US Bank is the 5th largest bank in the US. They have quantitative positions within their Enterprise Research and Insights teams, Data Analytics and AI/ML & Data Platform teams.

Great choice! CIT Group has over $50 billion in total assets. CIT has quant opportunities in the Quantitative Strategies (QS) or Analytics teams within Audit or Information Technology.

Great choice! Deloitte has over is headquartered in London and has over 280,000 employees world wide. Deloitte has quant roles in their Analytics & Cognitive, Managed Analytics or Risk and Financial Advisory (RFA) groups.

Great choice! PwC is headquartered in London and has over 250,000 employees. They have opportunities in their Valuations or Financial Markets Business Advisory groups.

Great choice! EY was founded in 1989 and is headquartered in London. They has over 270,000 employees and you can find roles in the Quantitative Advisory Services (QAS) or Risk Technology teams.

Great choice! Bain is headquartered in Boston and has over 8,000 employees. Bain has lots quant opportunities within their Advanced Analytics Group (AAG), including data science, engineering, marketing science, operations research.

Great choice! KPMG has over was founded in 1987 and now has over 200,000 employees. has quant positions in Economic and Valuation Services, groups, Technology Enablement, Risk Strategy & Compliance and Data & Analytics groups

Great choice! Lending Club was founded in 2007 and is based in San Francisco. You can find quant roles in their Data Technology, Risk or Engineering teams.

Great choice! Affirm was founded in 2012 and is based out of San Francisco. They offer quant positions in their Strategy & Analytics, Data/Risk and Finance divisions.

Great choice! Square is headquartered in San Francisco and was founded in 2009. Square has quantitative opporuntities in Marketing Analytics, Data Engineering, Data Science and Product Analytics.

Great choice! Coinbase is a digital currency exchange based out of San Francisco. Coinbase has quant opportunities in Operations & Technology, Data and Engineering.

Great choice! You may have heard of IEX Group from Michael Lewis's book Flash Boys. IEX is based in NYC and you can find interesting roles in Quantitative Research, Development or Finance teams.

Great choice! The Carlyle Group is based out of Washington D.C. and has over $220 billion in AUM. You can find quantitative roles on the Enterprise Data team or Analytics team.

Great choice! Apollo is based out of NYC and has over $303 billion in AUM. You can find quant roles on the Risk Management and Quantitative Business Modeling team.

Great choice! Blackstone is based out of NYC and has over $450 billion in AUM. You can find quant jobs as part of the Blackstone Innovations, Blackstone Alternative Asset Management or Portfolio Management Program teams.

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Let's label each horse 1 - 25. You first need to race each horse: (1, 2, 3, 4, 5), (6, 7, 8, 9, 10), (11, 12, 13, 14, 15), (16, 17, 18, 19, 20) and (21, 22, 23, 24, 25). That is 5 races. Then we take the winner of each heat. Let's say that is the lowest horse number in each heat and race them (1, 6, 11, 16, 21). This guarantees that #1 is the fastest horse. This is the 6th race. Are #6 and #11 necessarily the 2nd and 3rd fastest? No, because it is possible that #2 and #3 just lost in the first heat to the fastest horse (#1). We need a final 7th race between (6, 11, 2, 3, x) where x in any horse.

The delta of an option is the 1st derivative of the value of the option with respect to the price of the stock. As stock price increases, the value of a put option decreases thus we know the delta must be negative. The delta for a put can't be lower than -1 because as delta gets closer to -1, gamma (2nd derivative w.r.t. stock price) will go to 0. Also, delta can be interpreted as the probability the option ends up in-the-money, thus a value less than -1 would not make sense.

Use the binomial distribution. Since p = 0.50 and N = 100, the mean is 50, the variance is 25 and the standard deviation is 5. So 40 corresponds to a one sided 2 std move. A two sided 2 std move contains 95% of the area, so a 1 sided 2 std move contains 95% + 2.5% = 97.5%.

The opposite corners of a chess board have the same color. Placing a domino on the board requires covering both a white and black square. So, we will not be able to cover the chessboard.

We have that , and . We recall that the variance of a portfolio with 2 assets is

To find a local minimum, we need to differentiate, set equal to zero and solve for our parameter . Once you work out the math, you will get that

We evaluate the expression with the given inputs. The weight of stock X is 93.75%.

The normal errors assumption is only required when doing OLS on small samples. In most financial applications of OLS, you should have sufficient data so won't need worry about having normal errors.