Some interview questions
Hey all. Interview coming up in about 2 weeks and I came across some questions that I wasn't sure the answer to. Just hoping any more math-y type person could reason them out for me if possible. Note, these are all from someone's DRW interview review, if you're curious.
You roll 3 dice. What would you pay for a call option with a strike at 16, assuming a $1 payoff for each dot above the strike. (I say about 30 cents assuming mental math. Expected value is 3/216 * 17 + 1/216 *18 = 69/216 = .32.)
Give me quotes for call options with a 2-year duration at strikes of 10,12,14, and 20 assuming a current price of 8. (I don't even know where to start. Wouldn't you atleast need to know interest rates, then you can set lower bounds using max(S0 - K*e^-rT?, 0))...
Also, I keep seeing a lot of these make me a market type questions where the interviewer will then hit/lift you depending on the market you make. At the end they want you to determine your pnl. The only problem is that the interviewer knows the correct answer and you don't. That means you're stuck making arbitrary markets based off what he's doing. For example. If actual value is 500 and you make a market at 300-350. The interviewer will lift you. You adjust market to 350-400. He lifts again. Okay maybe widen your spread a little bit. 340-420. He lifts again. This could theoretically go on forever, and the only time you (the interviewer) is making money on a trade is if your spread contains 500, in which case the interviewer knows and then stops. Ultimately, you've spent so much time making bad markets cause you don't know the actual value, you've lost so much money on bad trades. Is there a good strategy to answering these types of questions? I only ask because I don't have trading experience, so I'm curious to here what the thought process was from actual traders.
I wrote a quick Python script for the first one and I keep getting 2.3 cents. Not sure where I am going wrong. Here is the code.
http://www.codesend.com/view/a7e59cf3cd0f7b0baefaa856d5eeb535/
Oh duh I see where you went wrong. You are multiplying by 17 and 18 respectively when you should only be multiplying by 1 and 2 ( since you will only make $1 and $2 dollars on the transaction not 17 and 18). If you do that math it comes out to be 0.02314 or 2.3 cents.
Oh wow... Didn't even realize that. Interpreted the question wrong. But simple mistake. Thanks for confirming my ideas about how to approach it. Any idea on the other questions?
No, sorry. Not involved with derivatives or trading so I wouldn't even want to attempt those.
I'll take a crack at it:
"You roll 3 dice. What would you pay for a call option with a strike at 16, assuming a $1 payoff for each dot above the strike."
What you're looking for is the total of dots. These can be anything between 3 (3x1) and 18 (3x6). If you roll any total below and including 16 your option will expire worthless. If you roll 17 or 18 your option will be in the money and you get $1 (if you get 17 dots) or $2 (if you get 18 dots).
There is one distinct way to roll a 18, that's three sixes. The chance for this is (1/6)^3 or 1/216.
There are more ways to roll a 17. You could roll 6-6-5 or 6-5-6 or 5-6-6. So getting a 17 is three times more likely than getting an 18. In other words the probability is 3*(1/6)^3 or 3/216 or 1/72.
There are no ways to roll anything above 18 and anything below 17 results in your call to expire worthless. So your payoff is:
$1 * 3*(1/6)^3 + $2 (1/6)^3 = 0.02314.... in other words a little more than 2 cents.
If you rise the strike to $17, then the only positive outcome for you is if you roll three sixes as this gives you $1. Chance for this to happen is (1/6)^3, so in that case your option is worth almost nothing.
If you drop the strike to $15 then you just adjust the calculation: Rolling a total of 16 dots gives you $1 now, rolling 17 gives you $2 and rolling 18 gives you $3. You multiply these with the probability that this happens and get your value.
"Give me quotes for call options with a 2-year duration at strikes of 10,12,14, and 20 assuming a current price of 8. (I don't even know where to start. Wouldn't you atleast need to know interest rates, then you can set lower bounds using max(S0 - K*e^-rT?, 0))..."
First when discussing options you don't talk about duration (that term refers to the sensitivity of BONDS to parallel changes in the yield curves, something totally different) but about expiry.
So you have some stock worth $8 now and you want to price the right to buy this stock for a price of $10, $12, $14 and $20. Intuitively you should realize that it's more likely for the stock to rise to say, $11, than to $25. So the lower the strike the more expensive your call option. In fact the strike 20 call options is so far out of the money that it's probably worth nothing.
Now they want an approximation of the price from you. That means you ignore a lot of stuff, e.g. interest rates. A good approximation for at the money calls, that is for a call with strike price 8 in a market trading at 8 is 1/sqrt(2*pi) * vol * sqrt(remaining time).
That can be even made easier by knowing that 1/sqrt(2*pi) is pretty much exactly 0.4
So this formula needs vol and you don't have vol. You need to pick a value here. A smart choice could be: Hey, I'll estimate at daily vol of about 1% and this translates into about 16% vol per year (vol moves by squareroot of time), as these are volatile times I'll up the ante and calculate with 25% which is about 1.5x 16%.
So you get Price_call_strike8 = 0.4 * sqrt(2) * 0.25 = 14% (of $8) or $1.13. That would be an approximation of the price of a call option with strike 8 on a stock with price 8 with 2 years til expiry. (It would also be the approximation of a put. yes, really.)
now i don't know of any good approximations for otm calls, but one way to get the other prices could be to make the reasonable assumption: "it's so rare for a stock to almost tripple in value in just 2 years that I can reasonably say that the right to buy this stock for $16 is almost worthless"
You would then have the two data points
Call with strike $8 => worth $1.12 Call with strike $16 => worth maybe $0.01
You should also know that simply drawing a line between these two data points is bullshit, because lower prices are more likely than higher prices. The smart thing to say here is "returns are log-normally distributed".
So you could simply say: Okay, i know this is oversimplifying it a lot, but we want to price the call options with strikes $8, $10, $12, $14 and $16. That's five data points. I have approximated the at the money call to be worth $1.12 and I am quite confident with this approximation given my assumed volatility. I also approximated the $16 call to be almost worthless. So that leaves three strikes, namely $10, $12 and $14 between a price of $1.12 and $0. I will assume that for $10 the price will have be only half, so I say the option with strike $10 has a price of $0.60. I also assume that for the strike of $12 the price is only half of that again, so I say it's worth $0.30. And for $14 it might be half of that was it was before, so $0.15"
anyways, until the atm approximation my answers is pretty good and correct. after that i kinda make a lot of wild assumptions however i think my guesses prices are only a few cents off
Wow Mueller27. Great answers. Thank you very much. For the first one you answered, I redid mine and ended up getting the same thing. As for your second answer, I knew the formula for atm call option, but I didn't think to use that in approximating the others. But your answer makes a lot of sense, and I'll definitely use the logic if I come across and interview question like that. Thanks for the help!
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