Lattice Option Pricing and Convergence to Continuous-Time Models

Summary

The document asks when prices from a Cox–Ross–Rubinstein binomial tree converge to Black–Scholes prices as the time partition becomes finer, and whether the same behavior applies to Black pricing for options on forwards or futures. The answer gives an intuition applicable to many lattice models: the tree approximates a contract’s strike or barrier with a terminal node, and finer partitions bring that node closer to the true level.

That moving approximation can create a sawtooth pattern in convergence. The response does not state formal conditions, prove convergence, or resolve the Black formula question specifically. It is therefore a useful qualitative observation about discretization error in lattice pricing, but not a complete convergence result or a guide to selecting tree parameters.

Key ideas

• A binomial tree approximates a contract’s strike or barrier using a node in the tree.
• As the partition is refined, the approximating node approaches the true strike or barrier.
• The changing node alignment can produce sawtooth behavior in lattice price convergence.
• The answer provides intuition rather than formal conditions or a proof.

Tags

# Convergence in the CRR model


# Convergence in the CRR model












Under certain conditions, the option price of the CRR (Cox-Ross-Rubinstein) Binomial model converges to the Black-Scholes price as the maximal step size of the partition converges to zero (i.e. a smoother partition of $\left[0,T\right]$ is taken).

What are these conditions? Does the same convergence can be observed in case of Black formula (the lognormal pricing model of options on forwards/futures), or is it only applicable to Black-Scholes pricing?

## Answer by Quasar (score 1)

https://quant.stackexchange.com/a/76123

In general, in lattice models, you are approximating the true strike $K$ or barrier $H$ of an option with $\hat{K}$ or $\hat{H}$, a terminal node in the tree. As you take finer partitions, $\hat{K} \to {K}$. So, intuitively, the sawtooth pattern of convergence must apply to most lattice models.

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