Risk Latte - Interview Questions # 6: Job Interview of a trainee Trader

Interview Questions # 6: Job Interview of a trainee Trader

Team Latte
May 25, 2007


Here is a partial transcript of a job interview of a trainee rates trader. The interviewer was the head of fixed income trading of a large European bank and the interviewee was an MBA with one year of work experience in the area of risk management.

Interviewer's Question :

A customer of a bank knows very well what a call option by definition (that on maturity of the option if the spot is greater than the strike he gets paid the difference). However, he has no idea or knowledge of Black-Scholes model, any other option pricing models or methodologies. He is asked by his banker to price an option on a stock. The stock is currently trading at $100. The banker further tells him that according to their research in a month's time the stock can either go up to $110 or go down to $90 with equal probability of 50%. The banker then asks him to price a one month at the money (the strike is the same as the current stock price) option on this stock (ignore the discounting factor).
The customer immediately answers: zero. His rationale is simply to take the average of the stock price in both the states (up state and down state), which is $100, as the value of the terminal value of the spot and then apply the condition of call option, i.e. if spot at maturity is greater than the strike then get the difference or else nothing.
The banker tells him that he is wrong. The correct price of the option (ignoring discounting) is $5 and not zero. What is wrong with the customer's analysis and rationale?

Candidate's Answer :

N/A

Our Comment :

The issue here is that of Jensen's inequality and the problem of convexity in options. Without going into any detailed math, it can be explained to the customer that if we have a convex function then it can be shown that: . In other words, the expectation of a convex function is almost always greater than the function of the expected value. The payoff of an option is a convex function - as demonstrated by the two state probabilistic existence - and hence if we take the expectation of the variable, (which is the stock price in this example) and then calculate the value of the function (which is the option payoff function) it will be always less than if we were to first calculate the option payoff function for each state, the up state and the down state, and then take the expectation (the probability weighted average). If you first take the payoff of the option at each state and then take the average, you'll get $5 as the value of the option.
We will not go into any detailed analysis of Jensen's inequality here, but suffice it to say that it cuts through to the heart of derivatives modeling, i.e. modeling the convexity in derivatives.


Interviewer's Question :

So, you want to join our bank as a trader? That's great, we are flattered. Now you look around you and I'll introduce you to five other traders who work with me and in the last two years they have all made pots of money. Will this make you think that you would also be able to make money when you join us?

Candidate's Answer :

N/A

Our Comment :

This trader, of course, embraces the Nassim Taleb school of thought; We are not going to go into the entire argument about a rare event and "black swan" but will only mention (paraphrasing David Hume): no amount of observation of successful traders will make us believe that all traders are successful.)


Interviewer's Question :

In a volatility matrix of an FX rate with ATM, and 25 delta and 10 delta call and put on either side what is the symmetric part and what is the skew? Can you retrieve the symmetric part and the skew part of the matrix from the ATM?

Candidate's Answer :

N/A

Our Comment :

ATM is the centre of the symmetry. The symmetric part of the function is called the "butterfly" and the skew part is the "risk-reversal". A real function can be decomposed into:
and the skew and the symmetric part can be retrieved using:

For further reference on this subject see Uwe Wystup's excellent paper: "Volatility Management".


Interviewer's Question :

What is the put-call parity relationship for a coupon paying bond (i.e. call and a put option on a coupon paying bond).

Candidate's Answer :

N/A

Our Comment :

The put-call parity for bond options is:
Put Price = Cal Price + PV of Strike + PV of Coupons íV Bond Price


Interviewer's Question :

What do you understand by the "gapping" risk in the case of the barrier option?

Candidate's Answer :

N/A

Our Comment :

At the barrier the delta becomes discontinuous for a KI or KO option. If the spot moves significantly as and when the barrier is triggered the trader will have a problem of hedging it continuously across the barrier. This is the gapping risk.



Reference: Uwe Uystup's "Volatility Management" is a good reference for mathematical exposition on risk reversal, butterflies and ATM vols.


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