In a two-step trinomial market model \(\left(S_{t}ight)_{t=0,1,2}\) with interest rate \(r=0\) and three return rates \(R_{t}=-0.5,0,1\), we consider a down-an-out barrier call option with exercise date \(N=2\), strike price \(K\) and barrier level \(B\), whose payoff \(C\) is given by

a) Show that \(\mathbb{P}^{*}\) given by \(r^{*}=\mathbb{P}^{*}\left(R_{t}=-0.5ight):=1 / 2, q^{*}=\mathbb{P}^{*}\left(R_{t}=0ight):=1 / 4\), \(p^{*}=\mathbb{P}^{*}\left(R_{t}=1ight):=1 / 4\) is a risk-neutral probability measure.

b) Taking \(S_{0}=1\), compute the possible values of the down-an-out barrier call option payoff \(C\) with strike price \(K=1.5\) and barrier level \(B=1\), at maturity \(N=2\).

c) Price the down-an-out barrier call option with exercise date \(N=2\), strike price \(K=1.5\) and barrier level \(B=1\), at time \(t=0\) and \(t=1\).
Hint: Use the formula \[
\pi_{t}(C)=\frac{1}{(1+r)^{N-t}} \mathbb{E}^{*}\left[C \mid S_{t}ight], \quad t=0,1, \ldots, N \]
where \(N\) denotes maturity time and \(C\) is the option payoff.

d) Is this market complete? Is every contingent claim attainable?

Transcribed Image Text:

C = (SNK) +1 min St> B t=1,2,..., N (SN - K) if 0 if min t=1,2,...,N min t=1,2,...,N St> B, St B.