Quant interview questions

A Derivative questions

Q1 Can you draw the graph for a european call option of the delta as a function of current stock price S(t)?

Q2 Can you write the Black and scholes equation and formula and explain the different terms?

Different parameters:

t is a time in years; with t = 0 generally representing the present year.

r is the annualized risk-free interest rate, continuously compounded (also known as the force of interest).

Asset related:

S ( t ) is the price of the underlying asset at time t, also denoted as S t .

μ is the drift rate of S , annualized.

σ is the standard deviation of the stock's returns. This is the square root of the quadratic variation of the stock's log price process, a measure of its volatility.

Option related:

V ( S , t ) is the price of the option as a function of the underlying asset S at time t, in particular:

C ( S , t ) is the price of a European call option and

P ( S , t ) is the price of a European put option.

T is the time of option expiration.

τ is the time until maturity: τ = T − t .

K is the strike price of the option, also known as the exercise price.

N ( x ) denotes the standard normal cumulative distribution function:

N ( x ) = 1 2 π ∫ − ∞ x e − z 2 / 2 d z .

N ′ ( x ) denotes the standard normal probability density function:

N ′ ( x ) = d N ( x ) d x = 1 2 π e − x 2 / 2 .

The Black–Scholes equation is a parabolic partial differential equation, which describes the price of the option over time. The equation is:

The Black-Scholes formula is the following.

Q3 What is an european call option delta with dividends = 0?

The delta is the first derivative on S based on the Black and Scholes formula.

It has the following formula:

Q4 Explain the N(d1) and N(d2) that appears in Black and Scholes formula, assuming no dividends

An european call formula is the following.

N(d1) is the option's delta, it represents how much the call price changes per unit of change in the price of the underlying.

The second term with N(d2) represents the cost of borrowing (or short bond position).

In other terms we would have the following:

Q5 Considering a European digital option that pays H if the stock price is above the strike X at expiration and zero otherwise. What is the price of this option and how is it related to Black and Schole equation?

H exp (-r(T-t)) N(d2)

H is the payoff if it is in the money at expiry.

Exp (-r(T-t)) represents the discout factor between the initial time and the maturity time.

N(d2) is the probability that option finished in the money.

In other terms it is very close to the second term in the Black and Scholes equation.

A more exhaustive list of Derivative questions can be found here.

B Product questions

Q1 What is a swap?

A swap is a derivative contract through which two parties exchange the cash flows or liabilities from two different financial instruments. Most swaps involve cash flows based on a notional principal amount such as a loan or bond, although the instrument can be almost anything.

Usually, the principal does not change hands. Each cash flow comprises one leg of the swap. One cash flow is generally fixed, while the other is variable and based on a benchmark interest rate, floating currency exchange rate, or index price.

The most common kind of swap is an interest rate swap. Swaps do not trade on exchanges, and retail investors do not generally engage in swaps. Rather, swaps are over-the-counter (OTC) contracts primarily between businesses or financial institutions that are customized to the needs of both parties.

Q2 How the yield cuve is calculated from swaps?

The method is named bootstrapping which consists of implying the rates from the swap price.

Below is an example.

Assume that the 6-month, 12-month, 18-month zero rates are 4%, 4.5%, and 4.8%, respectively.

Suppose we know that the 2-year swap rate is 5%, which implies that a 2-year bond with a semiannual coupon of 5% per annum sells for par:

2.5e−0.04∙0.5+2.5e−0.045∙1.0+2.5e−0.048∙1.5+102.5e−2∙R=100.

Solving for R above gives a 2-year zero rate R of 4.953%. We can keep going to compute the 3-year zero rates, etc.

Q3 What are treasury futures?

Futures allow market participants to take views on future rate movements in an off-balance sheet capacity. There are six futures:

TU (2-year)
FV (5-year)
TY (10-year)
TN (10-year, ultra-long)
US (30-year)
AUL (30-year, ultra-long)

Treasury futures are used routinely in hedging and are incredibly liquid just like the benchmark treasuries themselves are. What this means practically, is that a rates trader will have a large book of inventory. Then a client comes in and wants to buy $100M of 10-year Notes. In order to readjust the risk of the trader's book, then may go into the market and buy some offsetting amount (not necessarily $100M worth!) of 10-year futures.

Q4 If we have a bond trading at 90 with a 10% coupon and it matures next year, then what is the yield to maturity?

To get the YTM for a bond maturing in just one year, then you can use the formula: YTM = (coupons + (face value – current price)) / current price, which gives you: 20/90 or 22.22%.

For bonds maturing in two years or more, to calculate this in an interview you would have to use the YTM estimation formula, which is:

If you're looking to test out more variations of this question, you can use a YTM calculator.

Q5 What is repo trading? Where does repo trading fit into the broader trading floor?

When most enter onto the trading floor as a summer analyst, they're drawn to desks like distressed debt or equity derivatives because those are the most talked about in the financial press and seem to involve lots of risk taking.

However, there are plenty of fascinating areas on the trading floor that fly a bit under the radar. This is particularly true in the rates trading area where you generally will have a dedicated money markets desk.

What typifies a money markets desk is that you're dealing with very safe, very short duration assets like commercial paper and repos. But because of their safety, and because they are integral pieces to the plumbing of the financial system, you are often dealing in very large size.

If you're at all interested in exploring how the financial system really works, then you may be intrigured by repo trading. All repo trading involves is an individual who owns a security entering into a contract (a repo contract) where they sell someone the security and agree to buy it back at a specified (higher) price later.

So, a simple example would involve the repo provider giving someone $100 in cash in exchange for a security. Then the next day the repo provider gives back the security and gets $101 in return (making $1 for holding the security for the day). The reason why someone may want to do a repo is because they have lots of assets, but perhaps don't have enough cash at the moment.

Repo trades have been in the news quite a bit lately. This is because, as you can imagine, during good times there's almost no risk involved in repo trades (since they're often just a day in duration, but can stretch longer). However, when the markets become more volatile those taking a security from someone in a repo trade may get a bit spooked.

For example, if you (as a bank) enter into a repo with a counter party with an asset that is currently worth $100, and you're promised to be paid back $101 tomorrow, then what happens if the price of the asset falls to $80 and the counter party no longer is willing to pay back $101? As a repo provider you have their asset, because it was pledged to you as collateral, but you paid out cash to the counter party of $100 and now just have an asset you can sell now for $80 so you're out $20!

Q6 How is a Collateralized Loan Obligation (CLO) structured?

A CLO will be broken down into several tranches. These tranches will have decreasing levels of priority on the underlying cash flows of the levered loans that make up the CLO.

The top tranche, which makes up the majority of the notes issued by the CLO, will often be structured to ensure that it gets a AAA rating from credit agencies (thus making investors feel it is safe). The bottom tranche, called “equity”, will have the lowest payment priority and is not rated and viewed as being quite risky.

C General and Market questions

Q1 When do futures expire? Are the maturities commoditized or organized at all?

For futures, a letter will denote what month the future matures in.

H (March)
M (June)
U (September)
Z (December)

Q2 Putting this all together, if I told you FVZ20 what does that mean?

FV, as noted above, corresponds to the five-year treasury future. Z, as noted above as well, corresponds to December. The 20 simply means the last two digits of the year in which the future matures (so it would correspond to 2020).

Therefore, FVZ20 means the five-year futures contract expiring in December 2020.

Q3What are some important economic indicators to watch out for?

Obviously, this will be somewhat desk dependent. For example, if you're dealing with mortgage backed securities, then you'd keep a careful eye on not only rates, but particularly where mortgage rates are. This is due to how prepayments on mortgages affects the valuation of mortgage backed securities and issues stemming from that like convexity.

In general terms, things you should probably keep your eyes on, even if it’s not viewed as being directly relevant to the product you’re involved with, would include things like:

GDP
CPI (in particular, core CPI)
Changes in 5Y5Y swap rates
Federal reserve speeches
Initial jobless claims
Non-farm payrolls
PMI
Moves in the VIX
High yield and investment grade index trends (as spreads to treasuries)
Moves in the 10-year yield
Moves in Fed Funds
The S&P 500 level (or whatever equity index is most relevant)
P/E ratio (and forward P/E ratio)

As you can see from the above list, things directly related to equity markets make up a pretty small sample of what you're really concerned about on the vast majority of trading desks.

Equities are volatile (as they have a claim on the residual value of the firm) and what moves most asset classes are tied to rates (which in turn are driven by economic and political factors).

Some other things you may want to keep an eye on are FX rates (strong or weak dollar) along with any major spending news coming out of Washington (or whatever political capital is most relevant to where you're applying).

Note: Don't worry about needing to know the exact levels for all of these items. It's generally a good idea to know what GDP and GDP expectations are along with where core CPI is. Everything else you can memorize if you have time, but otherwise shouldn't spend a great deal of time on.

Q4 What is meant by mark-to-market? Why is this important to hedge funds, banks, etc.? Why is it relevant now, more than ever?

Mark-to-market is the process by which securities are recorded on financial statements using current market prices, as opposed to purchase prices or accounting values. As the credit crisis unraveled, many banks and investors were forced to write down assets to current market values.

This had a dramatic impact for a multitude of reasons. For example, many banks and investors used these assets as collateral to borrow against, in order to make other investments. As the value of the assets declined, so did the amount these investors and banks could borrow. Secondly, as assets are marked down, investors are often likely to sell (and/or forced to sell) their assets so that they don't keep losing value. Selling naturally depresses market values further, causing a ripple effect: As more and more is sold, more and more value is lost.

As we saw in late 2008, this can cause a panic, where people seek to move large amounts of personal investments into cash and are willing to sell at any cost.

Q5 What is meant by recovery value?

Recovery value is the amount an investor receives in bankruptcy liquidation from his investment in a particular financial instrument (and recovery rate is the associated percentage). For example, if an investor received 40 cents on the dollar for every bond that she purchased in a particular company, the recovery value would be 0.40 and the recovery rate would be 40 percent. Distressed credit investors are particularly concerned with recovery values. It is also important to note that the more "senior" the investment an investor makes in an company's capital structure (debt vs. equity), the more likely the investor will recover his investment in bankruptcy.

Q6 What are some examples of defensive stocks?

In general, a defensive stock is one that performs well during a period of economic slowdown, such as that of a basic goods company or even a discount retailer. These stocks do not generally outperform the market during periods of rapid economic growth, and thus they typically trade at lower P/E ratios than many competitors, as well as have significantly lower volatility. Defensive stocks are often mature, dividend-paying stocks.

Q7 What are credit spreads? What are they really telling you?

Credit spreads simply refer to the difference between where corporate debt is trading vs. the underlying treasury of the same duration.

So, for example, if the five-year bond of a certain company (say Ford Motors) is trading at T + 350, all that means is that it is trading at the five-year treasury rate plus 350 basis points (3.5%).

Remember that we view treasuries as having no credit risk since the U.S. dollar is the global reserve currency and (hypothetically) the U.S. can always meet its debt obligations given that it can print whatever level of currency it desires.

So what the 350 basis points (in this example) is showing us is the credit risk of Ford. This can then be compared to other five-year bonds issued by other automotive companies and we can infer the markets general view on the credit risk of Ford (although this is all a bit simplified, but sufficient for an interview).

Now what do we mean by credit risk? What is this 350 basis points really representing? What it is representing is the probability of Ford potentially not being able to meet its obligations and defaulting. In other words, the market is demanding 350 basis points to compensate for the risk embedded in Ford in this part of the capital structure for this level of duration.

So hypothetically, if Ford went into the market and tried to sell a new five-year bond at T + 50 we would expect very little demand from the market because the level of compensation (just 0.50% above the underlying treasury rate) isn't sufficient compensation for the risk of lending to Ford.

Q8 What do we mean by curve trades in rates trading?

When we talk about a rates trader’s book being properly positioned and taking advantage of the market, this obviously doesn’t mean that if 2yr bonds are overvalued he or she just won’t have any!

Rates traders have such large books that they will always have millions of dollars’ worth of bonds that they may or may not like in order to facilitate client flow and generally be in the market.

Instead the way that a rates trader positions their book is by taking advantage of changes in the yield curve. For example, there’s nothing you can do about owning a lot of 2s. However, maybe you think that 2s will go down more than 5s. This would create a flattening of the yield curve between those two points in the yield curve.

Curve trades involve taking advantage of the relative changes in one bond compared to another along the yield curve by using treasuries and treasury futures to be relatively overweight or underweight certain areas of the curve.

It's generally a good idea to know if the yield curve has been steepening or flattening over the past few months prior to your interview. While this may not be something that comes up in every interview, if you express an interest in any area of rates trading (broadly defined, so this would include MBS trading) then you should know this.

Q9 Can you give me a simple example (with numbers) of how delta hedging works?

As a derivatives trader within sales and trading you're always going to be looking to hedge out of your position to lock in profits on every trade that you do.

But how do you do this? How do you know what to hedge and how much of it to hedge? Well, like with many things in derivatives, it quickly gets very complicated, very fast.

This can be especially true for more esoteric trades clients want to do with may require hours of work to properly give a price on.

However, if we try to simplify things down we know that delta of an option simply measures the change in the value of the option for a change in the underlying (with everything else held constant).

So if we're looking at a delta of 0.5 on a trade - garnered from our model - then if we (as a trader on the sell-side) sell calls on a certain number of shares we will then be going and buying half that number of shares to hedge out the risk. This is because, if the underlying moves up in value given that we sold calls we will be losing money (the client making money), but our loss in the value of the call options we wrote will be negated by the long equity position we put on as a hedge.