Analysis of a binary option trading website.

a) In a one-step model with risky asset prices \(S_{0}, S_{1}\) at times \(t=0\) and \(t=1\), compute the price at time \(t=0\) of the binary call option with payoff

in terms of the probability \(p^{*}=\mathbb{P}^{*}\left(S_{1} \geqslant Kight)\) and of the risk-free interest rate \(r\).

b) Compute the two potential net returns obtained by purchasing one binary call option.

c) Compute the corresponding expected (net) return.

d) A website proposes to pay a return of \(86 \%\) in case the binary call option matures "in the money", i.e. when \(S_{1} \geqslant K\). Compute the corresponding expected (net) return. What do you conclude?

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C = 1 [K,00) (S1) = $1 if S > K, if S < K, 0