Suppose you have a portfolio of 100 options. Then I give you a subset of trades in which you can make. The trades consist of possible buys/sells of different options from different clients. Discuss how would you design a trading system/algorithm to determine which trades to accept and which to deny? Hint: Use portfolio theory.
The question is pretty vague. Don't let that get to to you and just take a stab at some sort of answer:
Assuming all the options are on the same underlying (say the S&P 500), the option portfolio will have overall stats such as Delta, Gamma, Vega (perhaps a few others) which can be easily computed. The managers will also have targets in mind for these stats (either as optimal values $\delta^*,\gamma^*,\nu^*$ or perhaps as ranges which they consider acceptable e.g. $\delta_L\le \delta\le \delta_H$ etc.).
The algorithm to review possible trades would work like this: You accept a trade if it brings the parameters of the portfolio closer to the desired values. For example if Delta of the portfolio is already too high, then you do not accept any trades with a positive delta, but you do accept negative delta trades, which help to bring the portfolio delta down. Some details would have to be filled in for the algorithm to be exactly specified, but this would be the basic idea.
If the options are on different underlyings it gets more complicated, we might have to keep track of multiple Deltas etc. with respect to multiple underlyings.
The question is vague on purpose. I think that the interviewer just wanted to hear what you know about options and how they work in combination with each other.
This way, he will get a feeling about how well you can handle these securities and portfolios.
The amount of things you could say about this type of scenario is sheer endles...