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In my journey to hope for a better understanding of incomplete markets I have decided to focus on the Trinomial model (maybe some of you have seen my previous questions). I have decided to consider first the one period market with 3 assets : the risk free, the risky asset, and the third is an option on the previous risky asset, say a European call with a strike different of course. The idea was to complete the market in order to be able to find a replicating strategy as in the Binomial model by solving a system of three unknows with three equations. The problem is that when I tried to make an example, I have solved the system but... I have found an arbitrage strategy, that is a strategy where the initial value of the portfolio is $V_0(\theta)=0$ while $\mathbb{P}(\{V_T(\theta)>0\})>0$.

I don't understand why since I have imposed the arbitrage free condition $1+r\in]\min(d_1,d_2),\max(u_1,u_2)[$

If someone has an idea on what's going on it would be very kind to tell me please

Thanks !

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