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Suppose a competitive, frictionless market provides European call options on an asset with current price $S0$ for all strike prices $K$ at market price $C(K)$ and European put options for all strike prices $K$ at market price $P(K)$. The asset at the date of the option expiration can only take discrete values $S=1,2,\cdots$ .

What is the state price for state s?

The answer is: $C(s−1)+C(s+1)−2P(s)$

Question: I have been thinking about this question but I cannot see why that is the right answer. How would that expression replicate the pay-off? I do not think this would require the binomial tree approach.

Thanks in advance.

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I think the correct solution should be:

$$C(s-1) + C(s+1) - 2C(s)$$ or (yielding the same payoff function):

$$P(s+1) + P(s-1) - 2P(s)$$

The answer you were given does not replicate the correct payoff.

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  • $\begingroup$ Thanks for your answer. After drawing the strategy in a payoff stock price diagram, I obtained a triangle with the centre on s. How would this replicate the stock price at state s. Why start at s-1 and not at s-2? $\endgroup$ Dec 16, 2023 at 18:46
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    $\begingroup$ This thing has value only if the stock price at maturity is S, otherwise the value is zero. That's what a State Price is like, it is a bet that only pays off when you end up in the desired state and yields nothing otherwise. $\endgroup$
    – nbbo2
    Dec 16, 2023 at 19:45

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