The payoff in binomial model is a martingale.

Let $V_N$ the payoff of a security at time $N$, recurssvely define \begin{equation} V_n=\frac{1}{r+1}(\tilde{p}V_{n+1}(H)+\tilde{q}V_{n+1}(T)) \end{equation} where $\tilde{q},\tilde{p}$ are the risk free probabilities. I am trying to prove that $V_n$ is a martingale, what I was doing is given a sequence of results $(\omega_1,...,\omega_n)$ of $H,T$, then \begin{equation} \begin{aligned} V_n(\omega_1,...,\omega_n)&=\frac{1}{r+1}(\tilde{p}V_{n+1}(\omega_1,...,\omega_n,H)+\tilde{q}V_{n+1}(\omega_1,...,\omega_n,T))\\ &=\frac{1}{1+r}\tilde{\mathbb{E}}(V_{n+1}|\mathcal{F}_n) \end{aligned} \end{equation} where $\{\mathcal{F}_n\}$ is the information given by $(\omega_1,...,\omega_n)$, but this means that is not a martingale, so I don't know what I am wrong, or what is my mistake?

Solution 1:

Your calculation is correct. However, it is not the payoff process that should be a martingale, but rather the discounted payoff process. This is what you have shown in your calculation.