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I am fairly new to all this, merely read the first few chapters of "The Concepts and Practice of Mathematical Finance". I recently had a job interview and was asked a simple question about options pricing:

Given that a stock today is worth 100 and tomorrow has a 20% chance of being worth 70, and would be worth 107.5 otherwise, what is the fair price of an at-the-money call option?

I answered, using the risk-neutral approach, that the option was worth 3.75, however, the interviewer told me I am wrong, as I need to weigh the probabilities, i.e the option should be worth 0.8* 7.5 =6. This is precisely what Joshi argues is wrong due to the possibility of hedging all the risk away. Am I missing something?

Filipe Miguel
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    Risk neutral probability of up move is equal to 50% iff up move and down move is symmetric (and rfr =0%), which your case obviously isn't. For rfr = 0% and $S_u$ = 107.50 and $S_d$ = 70, risk neutral probability of up move is actually 0.8. Therefore the interviewer gave you risk neutral probability already. – emot Jan 11 '22 at 11:34
  • Out of curiosity, what calculations did you do that led you to the wrong answer of $3.75? –  Jan 22 '22 at 21:02
  • @Jack Bueller, I calculated the average lf the two because I had assumed they had equal chances in the risk neutral approach. I clearly did not understand this. – Filipe Miguel Jan 23 '22 at 22:10

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Call option gives the right to buy the stock. ATM call option struck at 100 would be worth zero in the lower state and would be worth 7.5 units of money in the upper state (you can buy at 100, when the stock is worth 107.5).

As @emot points out, the risk-neutral probability is given at 80% for the upper state, so the option is worth 0.8 * 7.5 = 6. You can check that the probabilities given are risk-neutral by focusing on the stock alone: the stock value today has to equal the risk-neutral expected stock price in the future states discounted to today. You can use high-school maths to compute the risk-neutral probabilities yourself using this technique; denoting risk-neutral probability of an up-move with $p$ and assuming rates are zero:

$$p*107.5 + (1-p)70=100 \rightarrow p = 0.8$$

Which then gives the option price of 6 as discussed above.

PS: if you argue with the interviewer, it'll not only screw up that one interview, but the feedback might also disqualify you from future opportunities at the firm. In all honesty, the example you gave is so basic that it's a waste of your own time (as well as the firm's time) for you to have applied in the first place. You should first spend time on the basics before applying.

Jan Stuller
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    I edited the question, I meant a call option. The reasoning is of course the same. Thanks for your answer, I now understand. I guess I was confused because all the examples in Joshi were price symmetric. Regarding why I am applying, I do not have a background in finance, and I am applying for an internship only, not a position. – Filipe Miguel Jan 11 '22 at 12:52
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    Ok. I didn't mean to be harsh: From my own experience, the Quant Finance industry can be quite unforgiving, so it's deff worth dedicating the time to the basics, before going to lots of interviews. The good news is that if you have the right background in mathematics, you'll be able to learn the basics quite fast. Joshi's book on Quant Finance interviews is a great start, but also this forum and focusing on the very basicc principles of risk-neutral pricing will help you a lot @FilipeMiguel – Jan Stuller Jan 11 '22 at 13:41
  • No worries, thank you very much for your reply. I will definitely dedicate more to getting to grips with the basics, I simply haven't yet spent more than a couple of days learning about these topics, it is a work in progress. – Filipe Miguel Jan 11 '22 at 13:48
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    @FilipeMiguel: this thread should help a lot, all the answers combined provide a good foundation for risk-neutral pricing. – Jan Stuller Jan 12 '22 at 10:53
  • Thank you very much @Jan Stuller – Filipe Miguel Jan 12 '22 at 13:09