Assume that a non-dividend-paying security with price \(S_t\) at time \(t\) can move to either \(S_t u\) or \(S_t d\) at time \(t + 1\). The continuously compounded rate of interest is \(r\), and \(u > e^r > d\). A financial derivative pays \(\alpha\) if \(S_{t+1}\) = \(S_t u\) and \(\beta\) if \(S_{t+1} = S_t d\).

A portfolio of cash (amount \(x\)) and the underlying security (value \(y\)) at time \(t\) exactly replicates the payoff of the derivative at time \(t + 1\).

I have two comments before I answer:

Firstly, the title of your question doesn't really relate to the problem description. Let me clarify what I mean, an at-the-money call or put option is an option where the current underlying security price \( S_t \) is the same as the strike price \( K \) (i.e. \(S_t = K \) ). The question that you're beginning to ask looks like building a hedging portfolio (which is, I guess, a generalisation of pricing the puts and calls).

Secondly, you haven't asked a question. You've written the first half of a problem statement. Let us know what you want answered, what you have tried in order to solve the problem and then where you are getting stuck.

Help me help you.

PS good luck with your test!