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I'm not too sure what the answer is to this. You have a call option on a security worth 100 now that will either be worth 110 or 95 dollars at a future date. The strike of this option is 105. What is an estimated value of this call option?

Thanks

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  • $\begingroup$ Hint: the payoff is either 110-105 = 5 or 0. How much would you pay for such a gamble... $\endgroup$
    – nbbo2
    Commented Sep 11, 2017 at 21:10
  • $\begingroup$ @noob2 Is it 2.5 since the probability of either appears to be half? $\endgroup$
    – Jojo
    Commented Sep 11, 2017 at 21:17
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    $\begingroup$ That was my first guess, then I noticed that 100 (the starting price) is closer to 95 than to 110, so maybe we should assume 1/3 probability for up and 2/3 for down. So that would be 1.66 dollars. But in any case they are looking for a rough estimate. $\endgroup$
    – nbbo2
    Commented Sep 11, 2017 at 21:20
  • $\begingroup$ @noob2 if this is an interview question, then surely it's better to ask if you can make these assumptions... You should ask what the probability of being at each of those two prices is at expiry. $\endgroup$
    – will
    Commented Sep 11, 2017 at 22:42

1 Answer 1

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I would think you are supposed to assume that cash is worth 1 at all times. There is miniscule interest across a day in any case. They are testing if you can do a one-step binomial tree.

You can then either price by replication or risk-neutral valuation. The RN probability of an up-move is $q$ such that $$ 10 q -5(1-q) =0. $$ So $q=1/3$ so the price is $$ \frac{1}{3} \times (110-105) + \frac{2}{3} \times 0 = \frac{5}{3}. $$

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