# fair price for a call option

I am struggling with the following problem:

An investor is considering a European call option, whose price $C_0$ is yet to be determined, on the shares of a company called XYZ. You know that :

• the share price at $t=0$ for company XYZ is denoted $P_0 = 500$.
• the strike price of the option is $K=510$
• expiration date is $T=2$.

In 2 months, the value of the option will be $C_2= Max[P_2-510,0]$

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)

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?

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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
Hi Richard, I am not aware of the difference between no-arbitrage option pricing and real option pricing. I am studying finance and this was a problem the teacher gave us. and the data I have is just what I wrote here. But I am keen to understand how to solve this problem. Thanks –  user7130 Jan 30 at 16:48
The first chapter of S. Shreve, "Stochastic Calculus for Fiance I: the binomial asset pricing model" explains this task in a step-by-step fashion. –  user1157 Jan 30 at 21:24
Also be aware that in a scenario where the investor expects 3% growth, the value at the end is not probability*payoff. See en.m.wikipedia.org/wiki/Time_value_of_money –  user1157 Jan 30 at 21:27
Thanks Anna very useful –  user7130 Jan 30 at 23:15

From the format of your question, I imagine it comes from some exercises set.

If so, I would be curious to see it, because it looks really weird to me.

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

That doesn't make sense. Indeed, what you are trying to price is a European call option. Technically, its value can be calculated using a hedging argument. The fact that the investor wants to make 3% is irrelevant. The price is fair according to the underlying model.

To compute the price, you cannot use real-world probabilities, because you would need to adjust the expectation of the future according to the investor's utility function which depends on his risk aversion. The way to go is to use risk-neutral probability measure, but I don't think you can do this with the information you provide us with.

So, please have a look at the source of your question and see if they do specify that the probabilities are risk-neutral. Please specify where that comes from.

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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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