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Suppose it is known that the price of a certain security after one period will be one of the $m$ values $s_1,\ldots,s_m$. What should be the cost of an option to purchase the security at time $1$ for the price $K$ when $K < \min s_i$? (This problem is Exercise 5.3 in Sheldon M. Ross, An Elementary Introduction to Mathematical Finance, 3/e.)

I know that the value of the call option at time $1$ will be $s_i-K$ if the price of the security will be $s_i$ at time $1$. Then one unit of the security at time $1$ will be equal in value to one unit of the call plus a sure amount of $K$. Then if the interest rate is $r$ compounded continuously, the security price $S$ and the option price $C$ at time $0$ should satisfy $$S=C+Ke^{-r}.$$ However I am not sure how to calculate the exact cost of this option, given that neither the price of the security nor the interest rate is known. This chapter is mainly about arbitrage so I think this problem should be solved via arbitrage.

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That is the answer, $S-e^{-r}K$. It depends on the values $S$ of the security at time $0$, on $K$ and on the interest rate $r$. All of these you can assume know at time $0$, i.e. now.

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You use no-arbitrage theorem:

There exist probabilities on each outcome such that expected gain on every wager is 0.

Or, risk neutral pricing:

The expected return on each wager grows at the compounded discount rate. In particular, consider the wager of buying the stock today and selling tomorrow.

By either argument the 'risk neutral' probabilities associated with each possible outcome are such that expected gain on above wager is 0. Now price the option as expected payoff under these probabilities. You arrive at the answer as posted above.

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