# Super Hedging in incomplete Trinomial Tree

I have a question concerning the super-replication of a call in a trinomial tree which has the following characteristics:

Suppose we have one risky asset $S_t=2+\sum_{k=1}^tZ_i$, where $P(Z_i=0)=P(Z_i=-1)=P(Z_i=1)=\frac{1}{3}$, where $P$ denotes the objective (or real-world) probability measure and one bond $B_t=1$ for t=1,2 (so the risk-free rate is assumed to be 0 over time). The call that is supposed to be super replicated is denoted by $C_2=(S_2-1)^+$.

One can show that there exists an equivalent martingale measure in this market and therefore is is arbitrage-free but obviously not complete.

If $\Delta_t$ denotes the amount of shares and $\beta_t$ denoted the units of bonds we need to hold at time $t$ for our hedging strategy, I calculated the following for the superhedge portfolio $(\beta_t,\Delta_t)$:

On $\{Z_1=1 \}$ $$4\Delta_2 + \beta_2 \geq 3\\ 3\Delta_2 + \beta_2 \geq 2\\ 2\Delta_2 + \beta_2 \geq 1\\$$ which holds for $\Delta_2=1$ and $\beta_2=-1$.

On $\{Z_1=0 \}$ $$3\Delta_2 + \beta_2 \geq 2\\ 2\Delta_2 + \beta_2 \geq 1\\ 1\Delta_2 + \beta_2 \geq 0\\$$ which holds for $\Delta_2=1$ and $\beta_2=-1$.

On $\{Z_1= -1 \}$ $$2\Delta_2 + \beta_2 \geq 1\\ 1\Delta_2 + \beta_2 \geq 0\\ 0\Delta_2 + \beta_2 \geq 0\\$$ and here the last inequality implies the middle one, hence for $\Delta_2=\frac{1}{2}$ and $\beta_2=0$ this holds.

So far these are the values for the super-replicating portfolio at $t=2$. In order to calculate the amounts that need to be held at $t=1$ we need to keep in mind that a superhedge needs to be self-financing, i.e.

$$3\Delta_1 +\beta_1 = 3\Delta_2 + \beta_2 = 3*1-1 = 2 \text{ on \{Z_1=1 \} }\\ 2\Delta_1 +\beta_1 = 2\Delta_2 + \beta_2 = 2*1-1=1 \text{ on \{Z_1=0 \} }\\ 1\Delta_1 +\beta_1 = \frac{1}{2}\Delta_2 + \beta_2 = \frac{1}{2}*1-0=\frac{1}{2}\text{ on \{Z_1=-1 \} }\\$$ which has no solution, since $\Delta_1$ and $\beta_1$ need to be constant. So I can't find a self-financing portfolio that is superhedging the call $C$. I am really confused as I don't see where the flaw in my logic is. I have been trying to find a solution to this since a couple of days so I would really appreciate any help. Cheers.