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I am struggling with the following:

an investor is considering a call option on the shares of XYZ. the strike price is 510p and the option can only be exercised in exactly 2 months time. at the moment the share price of XYZ is 500p. in 2 months time the value of the option = Max[shareprice-510,0]

During the first month the investor believes that the probability the share increases by5% is 0.35, the probability that it increases by 2% is 0.5 and the probability the share falls by 4% is 0.15.

In The second month the investor believes that the probability the share increases by 4% is 0.3, the probability that it increases by 1% is 0.45 and the probability the share falls by 3% is 0.25

Calculate the price the investor is willing to pay for the option assuming they want to make 3% expected return over the period.

I calculate the expected return using a tree graph (in the picture below). The result is 11.177 (summing up all the value of the option by the probabilities) and that is a return of 2.2%. (Please keep in mind that if the share price is below the strike price the value option is 0)

enter image description here

My problem here is that I need to get the fair price, knowing the expected return... so I need to do exactly the opposite.

What is the formula to get the price the investor is willing to pay in order to get a 3% expected return?

Thanks

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Please improve the layout and typing of your question. –  Richard Jan 30 at 12:49
    
Hi Richard, what do you mean with improve the layout? My question is: what is the formula to get the price the ivestor is willing to buy to get a 3% expected return? –  user7130 Jan 30 at 14:59
    
Use latex for formulas. Do not itemize with different symbols and so forth. It is not too much fun to read your question in the way it is now. Furthermore sentences like "What I did is... I calculate the using a tree graph the expected return that" are not correct and I am not willing to take time answering your question if you don't take time to formulate it. –  Richard Jan 30 at 15:02
    
I hope that know it is a bit clear. Thanks :) –  user7130 Jan 30 at 15:23
    
You are aware of general option pricing theory with no-arbitrage arguments? How can we interpet the probabilities of the investor? The expected return that he wants to have? In reality the market does not care what the investor wants. Is this a problem of no-arbitrage option pricing or real option pricing? –  Richard Jan 30 at 16:36
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1 Answer

If I understand correctly you have calculated our investors expected payoff using his probabilities to 11.177USD. He wants a three percent return so the value he assigns is 11.177/1.03 = 10.85USD. Simple as that.

You can then have another argument a la Black and Scholes to show that you can replicate the payoff to another cost. If that cost is lower, your investor have another incentive to buy.

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