We consider a two-step binomial market model \(\left(S_{t}ight)_{t=0,1,2}\) with \(S_{0}=1\) and return rates \(R_{t}=\left(S_{t}-S_{t-1}ight) / S_{t-1}, t=1,2\), taking the values \(a=0, b=1\), and assume that

\[ p:=\mathbb{P}\left(R_{t}=1ight)>0, \quad q:=\mathbb{P}\left(R_{t}=0ight)>0, \quad t=1,2 \]

The riskless account is \(A_{t}=\$ 1\) and the risk-free interest rate is \(r=0\). We consider the tunnel option whose payoff \(C\) at time \(t=2\) is given by

a) Build a hedging portfolio for the claim \(C\) at time \(t=1\) depending on the value of \(S_{1}\).

b) Price the claim \(C\) at time \(t=1\) depending on the value of \(S_{1}\).

c) Build a hedging portfolio for the claim \(C\) at time \(t=0\).

d) Price the claim \(C\) at time \(t=0\).

e) Does this model admit an equivalent risk-neutral measure in the sense of Definitions \(2.12-2.14\) ?

f) Is the model without arbitrage according to Theorem 2.15?

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So = 1 S = 2 S = 1 S = 4, C=0 S2 = 2, C = 1 S = 1, C = 0.