Dice Rolling Option Valuation Quant Interview Question
A type of quant question that gets asked very regularly by companies such as Jane Street, Citadel, and SIG (Susquehanna International Group)
A question that gets asked very regularly by companies such as Jane Street, Citadel, and SIG (Susquehanna International Group), are questions that ask you to price an option that changes the outcome of some game as a test to see how well you would do this in a trading setting. Here, I will show you a good example.
Question:
There are 2 people and you’re one of them. You can both roll a 4 sided die and get paid whatever shows up on your die. Now, you have an option that allows you to reroll. How much would you value this option? Please take into account the fact that both of you have to bid for this option, and only one of you can get it before the game starts.
Try to think about how you would solve this and how you would approach this without a calculator before looking into the solution. And if you got the answer, see if there is a faster way you can do this.
Solution:
Very quickly, you see that your expected value or mean value for the one single dice roll is:
We then see that, with the option, we have now two scenarios:
If we roll 1 or 2 in our first roll, we say no thank you and try again because rolling one or two returns less value than our reroll’s theoretical value. So 1/2 of the time we defer to another roll.
If we roll 3 or 4 we keep them. This gives us 1/2 of the time where we get 3 or 4.
Based on this we can now calculate our expected value given our option, which is:
We can now see that the price of our option is just:
Hence, we should be looking to pay anything at 0.5 or less, regardless of the other person bidding.
But is this question really that easy? Yes, but no. The right way to think about this question is that the theoretical value is really just at 0.5. But if the interviewer probes you further, you should be able to point out that:
The option also reduces the variance
Risk-averse you should be willing to pay to reduce the standard deviation of the play
Using quick calculations, you see first that:
You then see that, post the option, you get:
You see that in fact, your standard deviation goes from roughly 1.12 to 1, which is a 12% decrease in standard deviation (a quick way to estimate this standard deviation below in your head).
The wide-open question is how you would value this. Suffice it to say, that this question is extremely hard to answer and depends on the interviewer's (and their firm’s) risk tolerance as well as their competitive nature.
If they do not much care and only care purely about the expected value, then it could just purely be a quick phone interview type question (or a lack of knowledge of how tail risk can crush the firm). Otherwise, it should generally be a positive thing that you would want to factor in variance.
If the firm is competitive, they might be willing to earn less even at a lower profit.
If you are valuing the option with the amount of variance you are reducing in mind, you can consider taking some value k such that:
Essentially k values how much you value the risk reduction the option gives you, a question which is fairly up for debate. This should be good enough to get you a good discussion with the interviewer though.
If the firm is competitive, and the interviewer prompts you as such. You might want to consider maybe even paying $1 for this option, such that you break even when the roll is $1 but win every other time. This drastically improves your chances of winning the bid for the option, although reduces your potential profit (which begs the question of why you would do this for a single iteration play at least).
Some aside:
What is the dice roll distribution?
For the game without an option, this is simply a Uniform Discrete Distribution
For the game with an option, it is a Non-Uniform Discrete Distribution
That’s all for today! Hope you enjoyed this and learnt something out of this.
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