Example DRW Interview Questions

Options are financial derivatives that give the holder the right, but not the obligation, to buy or sell an underlying asset at a predetermined price, known as the strike price, within a specified period. There are two primary types of options: American options and European options. The main difference between these two types of options lies in the timing of when they can be exercised.

American Options
An American option can be exercised at any time up until the expiration date. This flexibility allows the holder to capitalize on favorable market movements at any point during the option's life. For example, if the price of the underlying asset moves significantly in favor of the option holder before the expiration date, the holder can exercise the option to capture the profit immediately.

The ability to exercise at any time provides a strategic advantage, particularly in volatile markets or when the underlying asset pays dividends. For instance, an investor holding an American call option on a dividend-paying stock might choose to exercise the option just before the ex-dividend date to receive the dividend payment.

European Options
In contrast, a European option can only be exercised at the expiration date, not before. This restriction means that the holder must wait until the expiration date to exercise the option, regardless of any favorable movements in the price of the underlying asset during the option's life. As a result, European options typically trade at a discount compared to American options, all else being equal, because they offer less flexibility to the option holder.

European options are often used in markets where the underlying asset is less volatile and the need for early exercise is minimal. The pricing of European options is generally simpler due to the fixed exercise date, making them a popular choice for certain financial models and strategies.

The pricing of American and European options also differs due to the exercise flexibility. The Black-Scholes model, for instance, is primarily used for pricing European options and assumes constant volatility and a constant interest rate. American options, however, require more complex models, such as the binomial options pricing model, which can accommodate the possibility of early exercise.

This is a pretty straightforward Markov chain problem. There are 4 states. The transition graph is given in Figure 1.

The problem starts at state 1. As has been explained in the lessons of this course, we use the following equation:
\begin{equation}
s_1 = \sum_{i=0}^{3}p_{1,i}s_i
\end{equation} \begin{equation}
s_2 = \sum_{i=0}^{3}p_{2,i}s_i
\end{equation} Furthermore, $s_0=0$ and $s_3=1$. Then we have
\begin{equation}
s_1 = \frac{1}{3}*0 + \frac{2}{3} * s_2
\end{equation} \begin{equation}
s_2 = \frac{1}{3}* s_1 + \frac{2}{3} * 1
\end{equation} Solving these equations gives us $s_1=4/7$ and $s_2=6/7$. So, starting with 1 dollar, player A has a 4/7 chance of winning.

Proof
If we substitute Equation 4 in Equation 3, we have

\begin{equation}
s_1 = \frac{1}{3}*0 + \frac{2}{3} * (\frac{1}{3}* s_1 + \frac{2}{3} * 1)
\end{equation} \begin{equation}
s_1 = \frac{2}{9} * s_1 + \frac{4}{9}
\end{equation}  \begin{equation}
\frac{7}{9} * s_1 =  \frac{4}{9}
\end{equation}  \begin{equation}
s_1 =  \frac{4}{9} / \frac{7}{9} = \frac{4}{7}
\end{equation}

This is called the coupon collector problem. For a fair n-sided die, the expected number of attempts needed to get all n values is \begin{equation}
n \Sigma_{k=1}^n \frac{1}{k}
\end{equation} which, for large n is approximately n log n.

The time until the first result appears is 1. After that, the random time until a second (different) result appears is geometrically distributed with parameter of success 5/6, hence with mean 6/5 (recall that the mean of a geometrically distributed random variable is the inverse of its parameter). After that, the random time until a third (different) result appears is geometrically distributed with parameter of success 4/6, hence with mean 6/4. And so on, until the random time of appearance of the last and sixth result, which is geometrically distributed with parameter of success 1/6, hence with mean 6/1. This shows that the mean total time to get all six results is \begin{equation}
6 \Sigma_{k=1}^6 \frac{1}{k} = \frac{147}{10} = 14.7
\end{equation}

Exchange-Traded Funds (ETFs) are investment funds that are traded on stock exchanges, much like individual stocks. They hold assets such as stocks, commodities, or bonds and generally operate with an arbitrage mechanism designed to keep trading close to its net asset value, although deviations can occasionally occur. ETFs offer a blend of the characteristics of both mutual funds and individual stocks, providing investors with unique benefits that individual stock investments might not offer.

Diversification
One of the most significant advantages of investing in ETFs is diversification. An ETF typically holds a broad range of securities, often encompassing an entire market index, a specific sector, or even a global market. This diversification reduces the risk associated with investing in a single company. If one company within the ETF performs poorly, it can be offset by the performance of other companies within the same fund. This risk mitigation is particularly appealing to investors seeking to reduce volatility and avoid the pitfalls of poor performance from individual stocks.

Cost Efficiency
ETFs are known for their cost efficiency. They generally have lower expense ratios compared to mutual funds because they are passively managed, tracking an index rather than relying on active management. Additionally, ETFs incur lower transaction costs because they trade on exchanges like individual stocks, and investors can buy or sell them throughout the trading day. This flexibility allows investors to capitalize on market movements and manage their portfolios more actively and cost-effectively.

Liquidity and Flexibility
ETFs offer superior liquidity compared to some mutual funds and individual stocks, especially those that are thinly traded. Since ETFs are traded on major stock exchanges, they can be bought and sold at market prices at any time during trading hours. This liquidity provides investors with the flexibility to enter or exit positions quickly and efficiently, which is particularly beneficial in volatile markets or when rapid investment decisions are necessary.

Transparency
ETFs are highly transparent investment vehicles. The holdings of most ETFs are disclosed daily, allowing investors to see exactly what assets are in the fund at any given time. This transparency is a significant advantage over mutual funds, which typically disclose their holdings quarterly. Knowing the exact composition of the ETF helps investors make informed decisions and align their investments with their financial goals and risk tolerance.

Tax Efficiency
ETFs are generally more tax-efficient than mutual funds. Due to their unique structure and the way they are traded, ETFs typically experience fewer taxable events. Mutual funds often distribute capital gains to investors, which can result in tax liabilities. In contrast, the in-kind creation and redemption process of ETFs minimizes the capital gains distributions, thus reducing the tax burden on investors. This tax efficiency makes ETFs an attractive option for taxable accounts.

For the outcome to be even is the same as the outcome not being odd. For the outcome to be odd, all three dice need to be odd, respectively:

• 1*1*1 = 1 (odd)
• 3*3*3 = 27 (odd)
• 5*5*5 = 125 (odd)

and also all combinations of the odd numbers result in odd outcomes, like:

• 1*1*3 = 3 (odd)
• 3*3*5 = 45 (odd)
• etc.

Basically, multiplying odd numbers with each other results in an odd outcome. If ANY even number comes in play, then we have "a number" multiplying "an even number", which will always result in an even number. So, "odd * even * odd" will also result in even.

So, in any other scenario than multiplying three odd numbers, the product will be even. Therefore, the probability that the product is even is, given that half of the sides of the dice are odd ({1,3,5} vs {2,4,6}) equal to \begin{equation} P(even \; product) = 1 - P(only \; odd \; product) \end{equation} \begin{equation} P(even \; product) = 1 - (\frac{1}{2})^3 = \frac{7}{8} \end{equation}