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Question: The price of a stock is 100. With equal probabilities, it either goes up to 130 or down to 70. What is the price of a 1 year call option with exercise price 100. Risk free rate is 5%.

Attempt: I use black scholes.

  • Given: $Su=130. Sd=70.rf=5% S_0=100$
  • $Cu=max(0,130-100)=30. Cd=max(0,70-100)=30$

Now, $$HedgeRatio = (30-0)/(130-70)=1/2$$ $$B=(Cd-Sd \cdot HedgeRatio)/(1+rf)=(0-70/2)/(1.05)=-(33+1/3)$$ So the price of the call option is $$C_0=S_0*Hedgeratio+B= 100/2 -(33+1/3)=16+2/3$$ but it is wrong.

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  • $\begingroup$ Seems you are using the binomial option pricing method, for which I get the same answer (assuming discrete discounting of 1 year). $\endgroup$
    – MikeRand
    May 16, 2015 at 10:31

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I don't know the BS formula you are trying to use.

The price is the expected value of the discounted payoff under the risk neutral probability measure (I.e. Under which S is a martingale)

So the you need to compute the risk neutral probabilities for S to go up or down. The probabilities given in the problem have no impact. They are just there to trick the candidate.

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