# How to derive no-arbitrage conditions w.r.t. the variance of a trinomial tree?

For a trinomial pricing tree, some notes say there are two no-arbitrage conditions:

(1) $$E[S(t_{i+1})|S(t_{i})]=e^{r{\Delta}t}S(t_{i})$$

(2) $$Var[S(t_{i+1})|S(t_{i})]=[S(t_{i})]^2\sigma^2\Delta{t}$$

where $$\sigma$$ is constant volatility of the underlying which follows the geometric Brownian motion.

Could anyone tell me how to get the condition (2)?

Condition (2) is not related to no-arbitrage. It only states that in the trinomial tree the local stock price return variance is set to $$\sigma^2 \delta t$$, so that when $$\delta t \to 0$$ the trinomial tree converges to the continuous time diffusion process $$dS_t/S_t = r dt + \sigma dW_t$$