Separate No-Arbitrage Drift from Variance Matching in a Trinomial Tree

Summary

The discussion distinguishes the role of two conditions used when building a trinomial pricing tree. The conditional expected stock price is set to grow at the risk-free rate under the tree’s transition probabilities. This is the no-arbitrage condition: it makes the discounted stock price a martingale.

The variance condition has a different purpose. It sets the local variance of stock returns to match the volatility assumption, so that as the time step becomes small, the tree approaches a continuous-time diffusion with risk-free drift and the specified volatility. Thus the variance equation is a calibration or convergence choice, not an independent no-arbitrage requirement. The explanation is limited to the stated geometric Brownian motion setting and does not derive transition probabilities or address alternative tree constructions.

Key ideas

• The risk-neutral expected price condition enforces the martingale property of discounted stock prices.
• Matching local variance to volatility is distinct from enforcing no-arbitrage.
• The variance specification helps a trinomial tree converge to the assumed diffusion as its time step shrinks.
• Tree conditions should be interpreted according to whether they impose pricing consistency or process calibration.

Tags

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


# 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)?

## Answer by Antoine Conze (score 2, accepted)

https://quant.stackexchange.com/a/43623

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.