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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)?

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Condition (1) is the no-arbitrage condition: it states that under the trinomial tree transition probabilities, the discounted stock price is a martingale.

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 $$ dS_t/S_t = r dt + \sigma dW_t $$

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