Here are some sample questions I got from recent entry level interview.  Are these typical questions for entry level?  If so, then I am in trouble with me and my stupid pure math PhD On the Existence of Algebraic Oriented Equivariant Cohomology Theories!!!!!!!  Assume all stocks in question satisfy:

dS = rS dt  +  \sigma S dW_t along with a cash dividend being paid.

1) What is the value today for a contract that at maturity pays the inverse of the stock price at maturity.

a) How would this be hedged and what drawbacks would there be to this strategy?  What additional piece of info would dictate whether you would recommend this trade to the trader?

2) Suppose we have a contract that at maturity pays the maximum of two European calls C1 and C2 expiring at the same maturity on stocks S1 and S2 assuming the stocks satisfy the same SDE and are correlated.

a) What is the PDE for this contract? Express the analytical form of the price in terms of time to expiry.  Describe your approach if all options in question are American.  Also try the same for 5 underlyings and five calls.

3) Assume that call options are traded in the market, all of them with matuity T.  Assume that the strikes form a continuum.

a) Obtain the pdf of the unerlying at maturity. Assume that the implied volatility of the options follow the rule sigma(K) = sigma(0) - epsilon log(S_0/K) where S_) is a reference spot value, epsilon a constant, and K strike.  What does the pdf look like? Is the above form possible in the market?

4) There are 3 dividend regimes: yield, discrete proportional, discrete cash.

a) Write the risk neutral stock process under BS' GBM assumption with continuous dividend yield.  Discrete proportional and discrete cash dividend assumptions affect stock price as follows:

At ex-dividend time t, S(t) = S(t-)(1-p(t)) for proportional dividends and S(t) = S(t-) - c(t) for cash dividends.  Write out the strong solution for the stock process for both assumptions given a stream of proportional dividends, p(t_i) for one and a stream of cash dividends, c(t_i) for the other.  How do the prices of a European call priced with a continuous yield, discrete proportional dividends, and discrete cash dividend assumption compare? why?  What happens to the European call prices above if the volatility of the GBM between dividend dates are time dependent?

5) Describe the OO concept of an "object factory."  Google - yay!  Discuss how an object factory could be used in the design of an equity derivative pricing library.

6)  Suppose that we believe that there are similarities between the implementation of a barrier pricing engine and a convertible bond (assume that the convertible contains an early exercise provision and that the barriers and convertible barriers are NOT constant in time):

a) WHat numerical method would you use? Give an object orientated ovierview of how code could be shared between the models, with some description of the inputs and outputs for the model. What considerations need to be thought about when sharing the same numerical technique between a convertible bond and a barrier?

7) What is the value today (t) of a contract at some maturity (T) pays the squared natural log of the stock price at maturity? (log S)^2

a) How would this be hedged and what would the drawbacks be to this hedge?

Consider and American call option in the money.  Under what circumstances would the option be exercised before maturity? (Easiest question by far)  Would you be mroe likely to have early exercise if volatility is higher, interst rates are higher, or dividends are higher?  Explain as if you were talking to a trader.  Now consider a convertible bond with a conversion at the same strike as above.  How might your exercise decision differ from the call?

9) You are debugging a problem with a Variance Swap model and upon inspection you isolate the problem to the integrand that is:

double VarSwapIntegrand(double x)

{   double result, vol, putCall, vol1;

     double strike = exp(x)*cStrikeBreakPoint;

     if (x<=0)

                putCall = -1.0;

     else putCall = 1.0;

     vol = CalcVolByStrike(strike, cExpiry);

     result = BlackScholesValue(putCall, Spot, strike, cExpiry, vol, Rate0, Rate1)/(strike*strike);

return result;

}

The calling procedure integrates the integrand from -10000 to 10000.  Assuming that all the variable used are valid and that the integration routine is convergent, can you:

a) Identify the problem with the Var Swap implementation? Correct the code and derive the correct form?

10) Please pose and answer a question you would ask a candidate.

Does anyone have an idea what the answer is for Question 9? I am completely stumped.

I should remark that  the bank that gave me  this (PM me for details) usually gives the exam to screen people before on site interviews.  I was in town already, so they were willing to meet me.  Some of the questions on this were asked live.  I did not know some things, but they seemed understanding based on my background and would steer me a bit.  I was fortunate in that each time they made a suggestion, I was able to run with it.  As far as hedging some of those exotic contracts, I had no idea - the guy mentioned a paper by Breeden & Litzenberger (197 that would be interesting and helpful.  I will get it and work through this.  I am sure I can do this with a little research - the guy said to take my time because he knew I had things to investigate.

That said,

Yes....#9 is tough...I am confused on it myself... 

I just wonder why none of the books I have discuss Breeden and Litzenberger's apparently standard result....and they also do not discuss variance swaps - though my interviewers were kind enough to teach me a bit about variance swaps and the general results of Breeden and Litzenberger.

The B&L paper is in this thread.

By the way, I believe the question set you have given would be only applicable to somebody recently graduated from the rigours of formal Math Finance education. For somebody who never did this, and for somebody who did it more than a year or so ago (unless they have been keeping up with their gym work), I don't think it is especially attainable.

I think that for the latter people the more relevant skill is knowing where to find answers, and what might be wrong with those answers, than the actual formal answers themselves.

The trouble with #9 is the following:

The formula in the line for "result" is correct except for the fact that they made a u-substitution in the declaration of strike by exponentiating.  You therefore have to realize that du = strike * dx....so, in the line for reult you need to divide by only one "strike" and not two!  It seems liek a nice numerical technique to get a lot of accuracy with far fewer calculations.

I dunno Jumpstart, but I think a major mathematical blunder counts for more than semantics!  The code certainly has the blunder I indicated.  Usually, you would integrate from 0 to infinity the expression they have for result in the code below - and you might approximate that by integrating from exp(-10000) to exp(10000).  So they made a u-sub as I indicated, but forgot to consider that du = strike*dx.  That is very significant of course, but could easily be overlooked when looking at it in code form as opposed to looking at it as a math formula that you are more accustomed to...it may converge, but not to the correct answer.  They also said to assume that all variables are fine.  This of course is a logic error - much harder to figure those out than a syntax error because it will compile and run quite well!

I thought that maybe they had a different formula for variance swaps than what I was finding by scanning the web which casued me trouble...

Bear in mind that this question pissed me off for a long time too.  Then I spoke to the bank - and they mentioned that the formula I found on the web was correct!  Then the little light bulb went off in my head and I felt like an idiot!

1] Quick way:

S(T) is lognormally distributed as:

     ln(S(T)) = ln(S(t)) + (r - 0.5*v*v)*(T-t) + v*sqrt(T-t)*N(0,1)

Therefore so is S(T)^(-1). Taking logs:

    ln(S(T)^(-1)) = -ln(S(T) = ln(S(t)^(-1)) - (r - 0.5*v*v)*(T-t) - v*sqrt(T-t)*N(0,1)              (#)

Use the standard result that for a lognormal variable X:

    E[ X (T) | X (t) ] = exp(m + 0.5*s*s]

where m and s are the mean and standard deviation of the corresponding normal distribution. Discount the expectation at the risk free rate to get the price.

Long way: Use Ito to find the SDE for Y=1/S, note that it is lognormal, and then Ito again to find the SDE for ln(Y) and which gets you to point (#) above.

7) The payoff Y = ln(S(T))*ln(S(T)) is Chi-Squared as it is the product of two normals. Simply square the normal distribution for ln(S(T)), take a simple expectation and discount the result. 

 i used to just hate that completing-the-square stuff

Dude, if there is one thing I have memorized and forgotten more than anything else it's that damn completing-the-square formula. I always have to look it up.

And so you must also derive the formula for the roots of a quadratic everytime you need it, yes?